Symmetry:Art and Science - Sint-Lucas

Symmetry:Art
and
Special Issue edited by the
Department for Architecture
Sint-Lucas Brussels BELGIUM.
The Quarterly of the
International Society for the
Interdisciplinary Study of Symmetry
(ISIS - Symmetry).
Science
Volume 2 (new series),
numbers 1 - 4, 2002.

Symmetry: Art and Science (formerly Symmetry: Culture and Science) is the journal of
the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry).
The views expressed are those of individual authors and not necessarily shared by the
Society or the Editors.
Regular Editors:
George Lugosi
R&D&I (Research-Develop-Invent)
2 Union Street
Kew, Victoria 3101
Australia
Fax: +61-3 9852 – 8344
Email: g.lugosi@hfi.unimelb.edu.au
Dénes Nagy
Institute for the Advancement of Research
Australian Catholic University
Locked Bag 4115, Fitzroy, Victoria 3065
Australia
E-mail: d.nagy@patrick.acu.edu.au
© ISIS-Symmetry. No part of this publication may be reproduced without written
permission from the Society.
ISSN 1447- 607X
Cover layout: Gunter Schmitz.
Image on the front cover: Atomium Anamorphosis by Phillip Kent.
Images on the back cover: An artistic design by means of an algebraic structure, by
F. Ruiz and M. Peñas; 3D model of Schindler’s Braxton House, by Jin-Ho Park.
Ambigram on the back cover by John Langdon (Wordplay, 1992).
Book Production: D. Gillis, Brussels, BELGIUM.
Tel +32 (0)2 522 39 69; fax +32 (0)2 520 03 78; dirk.gillis@gillis.be.

Symmetry:
Founding editors: G. Darvas and D. Nagy
The Quarterly of the
International Society for the
Interdisciplinary Study of Symmetry
(ISIS-Symmetry)
Published by the International Symmetry Foundation.
Volume 2 (New Series), Numbers 1 - 4, 2002.
SPECIAL ISSUE:
Papers presented at the
Mat mium Euro-workshop,
a Regional Congress of ISIS-Symmetry,
supported by the European Commission.
Guest Editing:
Department of Architecture Sint-Lucas Brussels, BELGIUM.
1

EDITORIAL
This issue contains papers presented at the Euro-workshop Mat mium in Brussels, held
9-13 April 2002 at the Department for Architecture Sint-Lucas in Brussels, Belgium.
The Congress was co-presided by Dénes Nagy with the collaboration of Slavik Jablan.
The papers are arranged in order of participation.
The cover followed the ISIS format, and thus it was printed on soft paper. A hard cover
would have been more suitable, but then the larger ISIS network would not have been
accessible for its distribution. Eric Blanckaert took most of the photos on the different
locations.
An apology is owed because of the delay in the publication of these proceedings.
However, there is a very good reason: after a heavy traveling schedule, Dénes Nagy fell
ill. Actually, while this issue was prepared, he was still recovering from treatment. We
express our thanks to all participants for their patience, and to all authors for their
assistance and understanding in preparing this issue.
The Euro-workshop Mat mium was supported by the European Commission, Research
DG, Human Potential Programme, High-Level Scientific Conferences, under contract
number HPCF-2001-00377.
The information provided is the sole responsibility of the authors and does not reflect
the Community's opinion. The Community is not responsible for any use that might be
made of data appearing in this publication.
2

CONTENTS
Event Page
Day 1, At the Atomium, 2002 April 9
Presentation of the speakers
Special Feature: TV-Brussels broadcast.
1.Metadesign, LAB[AU] Laboratory for Architecture and Urbanism. 19
2. Footprints Literacy: The Origins of Art and Prelude to Science, Tsion
Avital.
23
Day 2, At the Institute Architecture, 2002 April 10
25
Special Feature: Fractal art exposition.
26
1. Architecture, mathematics, and a “symmetric link” between them (From the 31
Atomium building to the Mat mium project), Dénes Nagy.
2. The Role of Mathematics in the Gothic Architecture Structural Analysis,
Javier Barallo.
65
3. From Itten’s Tower to Virtual Towers: a generative Algorithm, Elena
Marchetti and Luisa Rossi Costa.
75
4. Programmed Design: The Systematic Method and the Form of Pattern,
Karen Y. Li.
85
5. Defining Basic Design as a Discipline, William S. Huff. 91
6. Design and Cognition: Contribution to a Design Theory, Claudio Guerri.
7. Growth, Curvature and Computation, Chaim Goodman-Strauss.
99
111
8. Sliceform, Surfaces and a Serendipitous Discovery, John Sharp. 123
9. Art of Anamorphosis, Phillip Kent. 131
10. Basic Crystal Symmetries Generated by Molecular Dimers, Alajos
Kalman, Laszlo Fabian and Andrea B. Deak.
135
Day 3, At the Africa Museum of Tervuren, 2002 April 11
139
Special Feature: conference movie
142
1. Folded Structures, Tibor Tarnai. 147
2. Symmetry and Ornament, Slavik Jablan. 161
3. How Plato Designed Atlantis, Leslie Greenhill. 163
4. The Mnemonics of the Cretan Labyrinth, Tessa Morrison. 203
5. The Parthenon Design Measurements, Anne Bulckens. 219
6. Artistic Designs by Means of Algebraic Structures, F. Ruiz and M. Peñas. 231
7. Form World – Generated by Integer Permutation, George Lugosi. 251
8. Visualization vs. Verbalization, Insight into the Morphology of Polyhedra,
Irit Wertheim and Nitsa Movshovitz-Hadar.
255
9. Virus Model, Florian Kovacs. 265
3
5
6
16

Day 4, At the Horta van Eetvelde Hotel, 2002 April 12
269
Special Feature: conference cartoons
270
1. Pattern Design by Improper Use of Mathcad, Patrick Labarque. 275
2. Growing Visible, Nonexistent Trees and Building four-Dimensional
Polytopes, Virpi Kauko.
281
3. Virtual and Real States – The Structure of Things and Objects, Tomek
Michniowski.
287
4. Tessellations of Euclidean, Riemannian and Hyperbolic Plane, Radmila
Sazdanovic and Miodrag Sremcevic.
299
5. A Remarkable Horta Type Spiral, Annie Van Maldeghem. 305
6. Rudolph M. Schindler’s Braxton House: The Fibonacci and Lucas Series,
Jin-Ho Park.
313
7. An Unexpected Encounter with the Mathematician E. B. Christoffel, Helena
Alexandra and Robert Willem Van der Waall.
325
8. SPACE PRODUCTION, 51N4E. 333
9. Frustration: Source of Complexity, Tohru Ogawa. 351
10. Digital Shaping of Spatial Structures, Janusz Rebielak. 353
11. Proportions and Dissections in Polygons, Encarnacion Reyes Iglesias. 357
12. Gothic Town Halls in and Around Flanders, 1350-1550: A Geometrical
Analysis, Han Vandevyvere.
365
13. Nonperiodic Selfsimilar Tilings and Symmetry, Dirk Frettloeh. 379
14. Symmetry Groups in Mathematics, Architecture and Art, Vera Winitzky de
Spinadel.
385
15. Useful Mathematics: Advantages of Decentralized Electricity and Heat
Supply for Buildings, using Fuel Cells, Erico Spinadel.
403
Day 5, Bruges, 2002 April 13. 415
4

Location.
Day 1
At the Atomium, 2002 April 9.
The location for the first days’ talks was the
Atomium, the lasting symbol of the 1958
Brussels World's Fair and Belgium's answer
to the Eiffel Tower or the Statue of Liberty.
Architects André and Jean Polak realized the
project together with engineer André
Waterkeyn. The Atomium became
Belgium’s landmark, and one of the most
visited attractions in the country.
The first session at the Atomium started with the presentation of the participants, in
“order of appearance”, by the chairpersons of each day:
- Day 1: ATOMIUM: Dirk Huylebrouck, Brussels.
- Day 2: W&K ARCHITECTUUR: Patrick Labarque, Brussels.
- Day 3: AFRICAN MUSEUM TERVUREN: Dénes Nagy, Melbourne.
- Day 4: HOTEL VAN EETVELDE, VICTOR HORTA: Slavik Jablan, Belgrade.
5

PRESENTATION OF THE SPEAKERS
51N4E Space Producers
"SPACE PRODUCTION"
Arch. Peter Swinnen
Sint-Lucas Brussels, Architectural Association
London; editor for Financieel Economische Tijd, de
Architect; coordinator Stichting Stad & Architectuur
Leuven.
Arch. Freek Persyn
Sint-Lucas Brussels, collaborator Xaveer de Geyter
Architecten.
Arch. Johan Anrys
Sint-Lucas Brussels, tutor architecture design
Campus Faydherbe Mechelen.
Brussels
BELGIUM
51n4e@pi.be.
Tsion AVITAL
"FOOTPRINTS LITERACY: THE ORIGINS OF ART AND
PRELUDE TO SCIENCE"
Holon Academic Institute of Technology
Department of Design and Art, Holon Campus,
Holon
ISRAEL
avital@hait.ac.il
www.hait.ac.il/staff/Avital/Avital.htm
Javier BARRALLO
"THE ROLE OF MATHEMATICS IN THE GOTHIC
ARCHITECTURE STRUCTURAL ANALYSIS"
University of the Basque Country,
San Sebastian
SPAIN
mapbacaj@telepolis.com
www.sc.ehu.es/mathema1/BelVie.htm
www.sc.ehu.es/mathema1/Anglet.htm
www.sc.ehu.es/mathema1/fas.html
6

Anne BULCKENS
"THE PARTHENON DESIGN MEASUREMENTS"
Dr in Architecture
Jakarta
INDONESIA.
annepaul@cbn.net.id
Luisa Rossi COSTA
"WORKING WITH AFFINE TRANSFORMATION: THREE
VIRTUAL TOWERS"
Dipartimento di Matematica "F.Brioschi"
Politecnico di Milano
Milano
ITALY.
luiros@mate.polimi.it
Vera W. de Spinadel
"SYMMETRY GROUPS IN MATHEMATICS,
ARCHITECTURE AND ART"
Director of the Center of Mathematics & Design MAyDI
Faculty of Architecture, Design and Urban Studies
University of Buenos Aires
Buenos Aires
ARGENTINA.
vspinade@fibertel.com.ar
Erico de SPINADEL
"USEFUL MATHEMATICS: ADVANTAGES OF
DECENTRALIZED ELECTRICITY AND HEAT SUPPLY FOR
BUILDINGS, USING FUEL CELLS"
Argentine Wind Energy Association AAEE
University FASTA
Mar del Plata
ARGENTINA.
vspinade@fibertel.com
7

Laszlo FABIAN
"BASIC CRYSTAL SYMMETRIES GENERATED BY
MOLECULAR DIMERS"
Junior research assistant
Chemical Research Center
Hung. Acad. Sci.
Budapest
HUNGARY.
Dirk FRETTLOEH
"NONPERIODIC SELFSIMILAR TILINGS AND SYMMETRY"
Dipl. Math.
FB Mathematik
Universitaet Dortmund
Dortmund
GERMANY.
Dirk.Frettloeh@udo.edu
Chaim GOODMAN-STRAUSS
"NEW MODELS OF GROWTH AND FORM"
Mathematician, graphic designer, artist.
Dept. Mathematics
Univ. Arkansas
Fayetteville
USA.
cgstraus@comp.uark.edu
www.delojo.com/be.html
Leslie GREENHILL
"HOW PLATO DESIGNED ATLANTIS"
Dip. P.S.P.
Educated at the Royal Melbourne Institute of Technology
Background in public education and public affairs in State
government in Victoria, Australia. He presented early findings on
ancient design systems at the World Mathematical Year 2000
Conference at the University of Melbourne.
AUSTRALIA.
lesgreenhill@yahoo.com.au
8

Claudio GUERRI
"DESIGN AND COGNITION: CONTRIBUTION TO A DESIGN
THEORY"
Architect
Buenos Aires
ARGENTINA.
claudioguerri@fibertel.com.ar
William S. HUFF
"DEFINING BASIC DESIGN AS A DISCIPLINE"
Professor Emeritus
State University of New York at Buffalo
USA.
wshuff@earthlink.net
Slavik JABLAN
"SYMMETRY AND ORNAMENT"
Mathematical Institute
Belgrade
YUGOSLAVIA.
jablans@mi.sanu.ac.yu .
www.mi.sanu.ac.yu/~jablans/
Alajos KALMAN
"BASIC CRYSTAL SYMMETRIES GENERATED BY
MOLECULAR DIMERS"
Ph.D., D.Sc., FASc
President of the Hungarian Chemical Society
Chemical Research Center
Hung. Acad. Sci.
Budapest
HUNGARY.
akalman@chemres.hu
9

Virpi KAUKO
"GROWING VISIBLE, NONEXISTENT TREES AND
BUILDING FOUR-DIMENSIONAL POLYTOPES"
Department of Mathematics and Statistics
University of Jyväskylä
Jyväskylä
FINLAND.
virpik@maths.jyu.fi
Phillip KENT
"ART OF ANAMORPHOSIS"
School of Mathematics, Science and Technology
Institute of Education
London
UNITED KINGDOM.
p.kent@mail.com
www.anamorphosis.com
Patrick LABARQUE
"PATTERN DESIGN BY IMPROPER USE OF MATHCAD"
Architect
Hogeschool voor Wetenschap & Kunst
Departement "Sint-Lucas" Architectuur, campus Brussel
Brussels
BELGIUM
patrick.labarque@archb.sintlucas.wenk.be
LAB[au]
"://> project - sPACE, navigable music"
LAB[au] = laboratory for architecture and urbanism
M. Abendroth, J. Decock,
A. Plennevaux, G.Verhaegen.
BELGIUM – GERMANY.
lab-au@lab-au.com
www.lab-au.com
www.lab-au.com/space
10

Karen Yuqing LI
"PATTERNS OF SYMMETRY AND ANTISYMMETRY -
A PROGRAM DESIGN"
Designer Architect
Pittsburgh
USA
Karen_yli@yahoo.com
George LUGOSI
"FORM WORLD - GENERATED BY INTEGER
PERMUTATION"
Chemical Engineer, Electronic Engineer, Patent Attorney
Director of Forarc Pty. Ltd., R&D&I (Research, Development,
Inventions) trading. Former Computer Scientist at Howard Florey
Institute, University of Melbourne
Melbourne.
AUSTRALIA.
g.lugosi@hfi.unimelb.edu.au
Elena MARCHETTI
"WORKING WITH AFFINE TRANSFORMATION: THREE
VIRTUAL TOWERS"
Dipartimento di Matematica "F.Brioschi"
Politecnico di Milano
Milano
ITALY.
luiros@mate.polimi.it
Tomasz MICHNIOWSKI
"VIRTUAL AND REAL STATES - THE STRUCTURE OF
THINGS AND OBJECTS"
Surname: Tomek.
pool@kul.lublin.pl
11

Tessa MORRISON
"THE MNEMONICS OF THE CRETAN LABYRINTH"
School of Fine Arts
The University of Newcastle
Newcastle
AUSTRALIA.
c9520975@alinga.newcastle.edu.au
Nitsa MOVSHOVITZ-HADAR
"VISUALIZATION VS. VERBALIZATION, INSIGHT INTO
THE MORPHOLOGY OF POLYHEDRA"
Ph.D., Head of Kesher Cham - National Center for Mathematics
Education, Director of the Israel National Museum of Science,
Planning, and Technology
Technion
ISRAEL.
nitsa@techunix.technion.ac.il
Denes NAGY
"INFORM-A-TOMIUM"
Nagy is the president of the ISIS Society.
d.nagy@patrick.acu.edu.au
Tohru OGAWA
"FRUSTRATION: SOURCE OF COMPLEXITY"
Japan.
He renewed upon a paper written together with Y. Nakajima:
Frustration, Degeneracy, and Forms: A View of the
Antiferromagnetic Ising Model on a Triangular Lattice,
Progress of Theoretical Physics, Supplement No. 87 (1986), pp.
90-101.
JAPAN.
ogawa-t@koalanet.ne.jp
12

Jin-Ho PARK
"RUDOLPH M. SCHINDLER’S BRAXTON HOUSE: THE
FIBONACCI AND LUCAS SERIES"
School of Architecture
University of Hawaii at Manoa
Honolulu
USA.
jinhpark@hawaii.edu
Maria PENAS
"ARTISTIC DESIGNS BY MEANS OF ALGEBRAIC
STRUCTURES"
Department of Didactics of Mathematics
University of Granada
Faculty of Education
Campus de Cartuja
Granada
SPAIN.
mtroyano@ugr.es
Janusz REBIELAK
"DIGITAL SHAPING OF SPATIAL STRUCTURES"
Professor at Wroclaw University of Technology
Department of Architecture
Wroclaw
POLAND
j.rebielak@wp.pl
Encarnacion REYES IGLESIAS
"PROPORTIONS AND DISSECTIONS IN POLYGONS"
Mathematician.
Departamento de Matemática Aplicada Fundamental
E.T.S. Arquitectura
University of Valladolid
Volladolid
SPAIN.
ereyes@maf.uva.es
13

Francisco RUIZ
"ARTISTIC DESIGNS BY MEANS OF ALGEBRAIC
STRUCTURES"
Department of Didactics of Mathematics
University of Granada Faculty of Education
Campus de Cartuja
Granada
SPAIN.
fcoruiz@ugr.es
Radmila SAZDANOVIC
"TESSELATIONS OF EUCLIDEAN, RIEMANNIAN AND
HYPERBOLIC PLANE"
Faculty of Mathematics
University of Belgrade
Belgrade
Yugoslavia
seasmile@galeb.etf.bg.ac.yu
John SHARP
"SLICEFORM SURFACES AND A SERENDIPITOUS
DISCOVERY"
Watford Herts
ENGLAND.
Sliceforms@compuserve.com
www.counton.org/explorer/sliceforms ,
www.nmsi.ac.uk/visitors/surfaces/
Tibor TARNAI
"FOLDED STRUCTURES"
Member of the Hungarian Academy of Sciences
Budapest University of Technology and Economics
Budapest
HUNGARY.
tarnai@ep-mech.me.bme.hu
14

R.W. VAN DER WAALL
"AN UNEXPECTED ENCOUNTER WITH THE
MATHEMATICIAN E.B.CRISTOFFEL"
KdV-Instituut, Universiteit Amsterdam
Amsterdam
THE NETHERLANDS.
waallr@science.uva.nl
Han VANDEVYVERE
"GOTHIC TOWN HALLS IN AND AROUND FLANDERS,
1350-1550: A GEOMETRICAL ANALYSIS"
ir.-arch.
Research group caad and design methodology
Departement architectuur, stedenbouw en ruimtelijke ordening
K.U.Leuven
Leuven
BELGIUM.
asro.kuleuven.ac.be.
Annie VAN MALDEGHEM
"A REMARKABLE HORTA TYPE SPIRAL"
Hogeschool voor Wetenschap & Kunst
Departement "Sint-Lucas" Architectuur, campus Gent
Ghent
BELGIUM.
annie_van_maldeghem@yahoo.com
Irit WERTHEIM
"VISUALIZATION VS. VERBALIZATION, INSIGHT INTO
THE MORPHOLOGY OF POLYHEDRA"
Architect, Ph.D.
Technion – Israel Institute of Technology
Department of Education in Science and technology
Haifa
ISRAEL.
Main research: morphological approach to 3-D geometry.
weririt@techunix.technion.ac.il.
15

Special Feature: TV-Brussels broadcast.
TV Brussels is the local TV-station
in Brussels. Some sites are:
http://www.tv-brussel.irisnet.be/
http://tv-brussel.vgc.be/ .
This TV station broadcasted briefly on the Mat mium event, during its news broadcast.
Below are some images from that broadcast.
General view on the conference room,
inside one of the spheres of the Atomium.
D. Nagy during his introduction.
16
D. Nagy and T. Tarnai, attentively
listening to the introduction.
D. Nagy linked Horta to mathematics and
the Belgian environment.

Next was LABau’s turn, with a musical
performance inside the Atomium.
“What we demonstrate here tonight is
called: Space Navigable Music.
The project was initially meant for the
Internet …
17
It took a lot of preparation to get
everything installed in the 9 spheres
building with its many stairs and
escalators.
The project links architecture, music and
images.
... but it has grown to a live-act with
musicians.

The project is based on a very simple
principle.
By moving though that space, you make
music.
LABau’s presenter became part of his own
show.
18
One has a three-dimensional space where
movement is free.
You can record your music and let your
friends listen to it through the Internet.”
Spectators came in large numbers, and
even had to sit on the ground

METADESIGN.
LAB[AU] LABORATORY FOR ARCHITECTURE AND URBANISM
Name: LAB[au] is Manuel Abendroth, Jérôme Decock, Alexandre Plennevaux.
Address: 19, Quai au Foin, B-1000 Brussels.
E-mail: lab-au@lab-au.com
Fields of interest: Architecture, urbanism, interfaces, new technologies of information and communication.
Awards: 2002, ‘Culture2002’ first prize in category Art for the project ‘Worldebt, you can count on it’
http://www.encorebruxelles.org/jk 1999: 'Prix de la Brique Belge', 'Lightscape(s), displacement maps' - light
plan study for the Heizel plateau http://www.lab-au.com/lightsc 'Tech-art prize', Vlaamse Ingenieurs Kamer.
Publications and/or Exhibitions: A+ magazine, nr. 168 April-May 2001 'http://mind space', A+ magazine, nr.
172 Dec.2001/ Jan.2001 'http://e.motional space', A+ magazine, nr. 173 Feb./March 2002 'http://soundspace'
Keywords: Hypermedia – sonic space – soundscapes - inFORMation processes – connectivity - eSPACE
CONSTRUCTionS.
Abstract: The transposition of inFORMation processes, transmission and computation,
in textual, graphical bi-dimensional, three-dimensional and biomorphic (autogenerative;
n-dimensional) forms explores new constructs proper to the electronic
medium and outlines the spatial and semantic mutation provoked by technologies on the
perception and conception of our environment. ‘Metadesign’ thus can be understood as
a technology determinism that constitutes the main vector/thought in the concern of
networked, information-based societies.
1 METADESIGN
A technology is not an independent or alien object, it complements integrally our
sensorial and cognitive system; as a medium, it conditions not only communication
modes but also the way we perceive and conceive our environment.
The increasing implication of communication and information technologies in the
process of production and knowledge leads to the fundamental re-thinking of the
organization and definition of space. Technology based on the transmission and
computation of information influences organization models (modes of production, work
and knowledge) and affects the communication process (code, symbol) and the social
relations as well as their spatialization. The affectation of traditional articulations
19

etween information, space and time leads to the augmenting need to flatten the
electronic realm into the concrete space.
If, as all communication systems, new technologies induce a transmission channel
(signal-medium), a message (information) and a code, their property is to operate on any
kind of information, even space, a reduction in a sequence of elementary information
coded in a binary language, 0/1 or bit/second. However, contrary to its analogue
counterparts within which information was materially fixed on a medium, the digital
media celebrate the loss of inscription; it is the transposition of all stable “FORM” into
transmissible and editable “inFORMation”, processes.
As a consequence, the investigation in information space constructs shows the shift from
traditional architecture into a metadesign, exploring new spatio-temporal structures as
well as their representation practices such as architecture and urbanism. New
technologies therefore perform a transformation on semantic and spatial structures
(architecture) as much on the level of language (code, style) as on other levels such as
social/spatial/economical/political relations. “e-SPACE CONSTRUCTionS” display the
theme of new space constructs relative to information processes, as the formalizations of
communication and computation processes.
In relation to >INFORMation< processes, metadesign is Information architecture,
related to the structuring of information, its textual, graphical, spatial and biomorphic
transcription and interfacing grounded on the inherent logics of computation and
communication technology in networked societies.
Metadesign deals with the setting of new ‘senses’ as components of language, while
improving, increasing our cognitive capacities and influencing in a major way our
psychic state (consciousness), our emotional and social behavior and thus participate as
much in the individual project as to the collective. Consequently, in the field of new
medias, it is important to understand the relation, which is established between
perception (the use of senses), recognition, comprehension and the representation (the
extraction of sense/meaning), and the action that results from it (production of
sense/meaning).
In this manner, information architecture deals with intelligible electronic constructs not
only as modalities of perception and cognition, but as mental and psychic settings of
behavior, ontological concerns, as well as the production of active and functional space
settings, spaces of intervention within the constitution of e.SPACE CONSTRUCTionS.
Metadesign thus deals with information as programming and meta-inscription, versus as
an output of interpretation - and data as objective reality versus information as narrative
and simulation. ‘Metadesign’ displays the theme of new space constructs relative to
information processes, as the formalizations of communication and computation
processes according to social, semantic and spatial structures (architecture) as much on
20

the level of language (code, structure) in order to build up connectivity and
effectiveness.
1.1 “Space, navigable music” project
Figure 1: screen of the online project “space, navigable music” (http://www.labau.com/space).
‘sPACE, Navigable Music’ is an online project investigating the impact of IC
technologies and particularly, 3D Real Time modeling languages (such as VRML) in the
construct of space. According to the objectives of LAB[au] the project constitutes as
much a space for theoretical research as a space of experimentation on the forms of
interactions in networked systems exploring the possibilities of space settings in shared
processes in order to build up connectivity.
In sPACE, navigable music, the object or architecture is generated in real time
according to the position and movements of the user (mix color, mix image, mix sound).
Operating on structural parameters, the integration (recombination) of spatial (x,y,z),
temporal (t-movements) sonic (frequency, pitch) and generative image sequencing
functions, each interaction by the user, displacement, transforms this visual and sonic
environment. In addition, the recording of movements allows users to produce a
traveling according to camera movements, montage and image sequencing. The
21

established relation between the spatial, visual and sonic formalization processes and the
editable interactivity of users lead to an experience combining architecture, music and
cinematic techniques through movement patterns. The ‘Navigable Music’ thus
constitutes a space, in which the user experiments cyberspace by dropping sounds into
space, mixing music throughout space and navigation, record its movements to produce
an animation, a traveling in its sonic space architecture, a kinetic music clip.
As such, inFORMation processes, computation and communication through codes /
language, VRML, thus describe programmatic relations between these different media
fusing them into a hypermedia, which can be experimented through networks, extending
the construct of space to the digital matrix (mixed reality), where the multi-user space
even more enlarges this experience to shared and collaborative processes based on
sound and e.space.
Year the Work was created: 2001
Project URL: http://www.lab-au.com/space
Technical requirements: VRML, Blaxxun 5 plug-in, Flash player.
22

FOOTPRINTS LITERACY: THE ORIGINS OF ART
AND PRELUDE TO SCIENCE.
Tsion AVITAL
Abstract: Art is a far too complex cultural phenomenon to have been invented ex-nihilo.
However, no adequate explanation has so far been given regarding the graphic and
cognitive skills, which preceded prehistoric art, and made its actual emergence
possible. This essay proposes that prehistoric art was preceded by a more primitive
kind of pictorial literacy, namely footprints literacy. The obvious attribute common to
many early prehistoric paintings and footprints is that both represent their subjects by
contour and negative. A deeper analysis of these two kinds of visual literacy reveals
many other common attributes: connectivity-differentiation, classification, abstraction,
generalization, signification, visual class-names, symmetry-asymmetry, schematization,
complementarity, induction, deduction, hypothetical thinking and others. Thus, it is
probable that footprints are the proto-symbols from which figurative art evolved. It is
striking that the same attributes which appear in footprints literacy about 4 millions of
years ago and in figurative art since its beginning about 40.000 years ago, appear
much later as basic attributes of modern science, but at a much higher level of
sophistication. Possibly, these three domains represent successive stages of noetic
evolution. Probably, this finding points to fundamental cognitive attributes or
"mindprints" that are basic not only to these areas, but also to human intelligence itself
and probably to all other phases of Being. Pointing out the origins of art might be a
substantial contribution to the lifting of the veil from the most fundamental attributes of
art since its very beginnings. This may provide a new key to the delineation of the
demarcation lines between art and non-art, which seems to be the most haunting
problem of modern art. It can be shown that works of nonrepresentational art do not
share those unique fundamental attributes or mind prints shared by footprints literacy,
figurative art, science and other branches of culture. Hence, there is room for doubts if
works of nonrepresentational art are culturally relevant and if they are works of art at
all.
23

This contribution was related to the publication Footprints Literacy: The Origins of Art
and Prelude to Science, Symmetry: Culture and Science. Vol. 9. No. 1. pp. 3-46 (1998),
and to a special volume of the ISIS journal (Vol. 1, No, 2, 1999). An electronic
reference is Visual Mathematics, at the Internet address members.tripod.com/vismath/ .
The talk served as "opening act" to mark the aimed level of abstraction, scope,
interconnectedness and directness. It gave the participants a feeling they were allowed to
really speak their mind, and to the best of their ability.
Avital worked all his life to develop new ideas, rather than repeating old stuff. He was
really involved in his own research, intellectually and emotionally. Therefore, he states,
simple-minded conformists get angry because they have never read about it before and
they think it is outrageous that one dares to speak his mind. The truly intelligent and
open-minded people get excited because they are exposed to new ideas. For sure, one
way or the other his audience is never indifferent: those who hate his ideas and those
who love them, feel that he talks about something that many feel but do not dare to
express or do not have the words and understanding to express.
24

Location.
Day 2
At the Institute for Architecture, 2002 April 10.
The location for the first full day of the
Mat mium conference, and the base location of
the organization, was the W&K Institute for
Architecture, W&K being short notation for the
Dutch words Wetenschap (science) and Kunst
(art). The contributions of this chapter were
presented there.
The participants were embedded in the Brussels’
environment by an additional evening visit to the
Grand-Place of Brussels, a mandatory tourist spot
for every dedicated artist or architect.
25

Special Feature: Fractal art exposition.
At this location, during the time of the Mat mium workshop, a mathematical art
exposition was simultaneously held at the W&K Institute for Architecture. The initiative
for this exposition was due to Prof. Javier Barallo, of the University of the Basque
Country (Spain), or, in the Bask language "Euskal Herriko Unibertsitatea Donostia” or
else, in Spanish, "Universidad del Pais Vasco San Sebastian”.
Barallo’s group The border between Art and Science made its first exposition in
September 1997. Since then, it realized numerous exhibitions in Germany, Argentina,
Austria, Brazil, Spain, France, Japan and Serbia-Montenegro (Yugoslavia). The
paintings are grouped under the denominator “The Frontier between Art and Science”.
After the Mat mium workshop, the exposition moved to another Belgian city, Ghent,
where it was shown in a major high school for a broad audience.
This survey exposition consists of works by different authors. They all share the use of
mathematics in their work. All canvasses were made with computer programs, in which
mathematical formulas, based on fractals, were especially designed for the graphical
representation. The mathematical form and the used parameters give to each image a
unique and singular structure, form and color.
Each author works meticulously on his oeuvre by trying different values for the
equations, formulas and parameters. In this way, the final image was obtained. Such a
canvas should not be seen as a synthetic, cold and mechanically generated image set up
by a computer, but as an artistic expression capable of transferring emotion and
sensitivity.
For more information, see: www.sc.ehu.es/mathema1/Anglet.htm,
www.sc.ehu.es/mathema1/BelVie.htm, or www.sc.ehu.es/mathema1/fas.html. If you
would like to have the exposition at your university or hometown, please do contact
Prof. Javier Barallo (mapbacaj@telepolis.com).
Below, the 12 images shown in Belgium are presented, but of course infinitely more
fractals are available. This artwork is copy right protected, and thus shown here in very
reduced format, just to give the reader a taste.
26

27
The announcement of the exposition.
Javier Barallo in front of one of the fractal exposition
images.
Barallo’s explanations about how the fractal artworks were
obtained were artworks too. Below right shows Kedaja or
triptych by Kerry Mitchell (left), Damien Jones (center) and
Janet Preslar (right).

"Faeries" by Domenick Annuzzi.
"Volcano" by Javier Barrallo.
"Alien Blood" by Earl Hinrichs.
28
"The Coral Reef" by Linda Allison.
"Avalanche" by Sylvie Gallet.
"Spade" by Damien Jones.

"Weathered" by Daniel Kuzmenka.
"Big Bang" by Kerry Mitchell.
"Jewelry" by Frederik Slijkerman.
29
"Polygon 1" by Luke Plant
"Taupensky" by Janet Preslar.
"The Mysterious Conjunction" by Mark
Townsend.

Dedicated to
ARCHITECTURE, MATHEMATICS, AND A
“SYMMETRIC LINK” BETWEEN THEM.
(From the Atomium building to the Mat mium project)
DÉNES NAGY
- János Bolyai, the co-discoverer of non-Euclidean geometry, who was born 200 years
ago in 1802, and
- Victor Horta, the pioneer of Art Nouveau, who designed the Tassel House, his landmark
building, 110 years ago in 1892.
1 Architecture and mathematics? (from the dome of the Eskimos to
the St. Paul Cathedral)
There is an intricate relationship between architecture and mathematics since prehistoric
times. Making an artificial dwelling obviously required some sort of planning and
inspired geometry-related ideas. We may call this knowledge as proto-mathematics or
ethnomathematics. Gradually, some people developed special skills in such questions,
and they were able to initiate more complex designs. Such an interaction between protoarchitecture
and proto-mathematics led to the development of both sides and to various
ingenious ideas. For example, the Eskimos invented the structure of the dome in the
form of igloo, their typical dwelling, and widely used it perhaps earlier than Roman
architects. Turning to African culture, the studies of Paulus Gerdes demonstrated the
importance of various mathematics-related ideas in design, while Ron Eglash pointed
out the appearance of self-similarity and even fractal geometry in traditional
architecture. We may believe that some of these geometrical structures are very old and
kept “invariant” by generations of craftsmen. We also should pay a special attention to
wooden buildings that were developed in many regions in different forms. Although the
ancient versions of these buildings do not survive, the lashing technique used in Oceania
and the art of joinery mastered in China, Korea, and Japan give an insight into the
possible methods that were used for fixing these structures. Both of these are associated
with mathematical problems, including knot theory and solid geometry. Turning to the
31

architecture with stone blocks, the art of cutting stones and fitting them together, often
without using binding materials, required a high-level of geometrical accuracy. This
craft is called stereotomy, but in mathematics we may characterize it as constructing 3dimensional
tilings. “Geometry” (geo + metron, i.e., land-measuring) was obviously
necessary for marking out the foundation of a new pyramid, palace, or temple, and,
during the completion of the buildings, further mathematical ideas were used. The Rhind
Papyrus, for example, includes a mathematical problem that is associated with the angle
of a pyramid. Around the 4th century B.C., the birth of deductive mathematics with
axioms, theorems, and proofs marked a split between mathematics and architecture. In
Euclid’s Elements (Stoicheia, perhaps around 300 B.C.) there is no reference to
architectural problems; this is a work of “pure” mathematics. In Vitruvius’s Ten Books
on Architecture (De architectura libri decem, 1st century A.D.) we see some
mathematics, but there is no reference to Euclid. I even suggested speaking about
Euclidean (or deductive) mathematics vs. Vitruvian (or craftsman’s) mathematics (c. f.
Mathematics and Design 98, San Sebastian, 1998, pp. 17-25). However, the two types
of mathematics were not totally separate. For example:
- a part of the so-called 15th book of Euclid was written, with great probability, by a
pupil of Isidorus of Miletus, the architect of the Church of St. Sophia in Constantinople
(now Hagia Sophia or Aya Sofya in Istanbul), in the 6th century A.D.,
- the builders of the Milan cathedral consulted with the mathematician Gabriele
Stornaloco, whose sketch and calculations survive form the late 14th century,
- the mathematician Al-Kashi included problems related to architecture in his
mathematical treatise in the 15th century, and
- major figures of the Renaissance dealt with both architecture and mathematics.
In most cases, however the master masons or the architects constructed their buildings
without mathematical assistance and used their own skills. Interestingly, they “solved”
higher-degree equations and “simulated” complicated curves and surfaces without the
utilization of advanced mathematics and long before the age of computers. Their usual
secret was a set of simple geometric rules and the method of trial-and-error. Indeed,
many Gothic churches collapsed several times before the master masons determined the
suitable shape of the structural elements. In some cases, geometric scale models helped
the work, including the problems of constructions and stability. A famous example was
the brick and plaster model of the Church of San Petronio in Bologna in the 14th
century. Note, however, that the questions of strength cannot be studied in scale models,
as Galileo pointed it out in 1638. He illustrated the problem by the different shapes of
leg-bones of small and big animals. Instead of geometric similarity, we see much thicker
bones in the case of large animals. Specifically, the mechanical strength of a structural
element varies with the cross-sectional area, hence with the square of its linear
32

dimensions, while the volume varies with the cube of its linear dimensions (square-cube
law). Perhaps masons also used an original method of “simulating” the optimal curves
and surfaces by mechanical analogies. For example, by hanging a cord between two
points we get a curve that can be used, after inverting it up-side down, as the shape of an
arch. Although we do not know too much about such methods in the Middle Ages,
perhaps these were kept secret by the free masons, we have an insight into the
importance of hanging experiments from later periods. Here we may mention Sir
Christopher Wren’s design of the St. Paul’s Cathedral in London, and more recently
Antonio Gaudi’s workshop at the Sagrada Familia in Barcelona. With the names of
Galileo Galilei and Sir Christopher Wren, we also reached a point where applied science
started to replace the Vitruvian mathematics and the medieval traditions in architecture.
Galileo’s referred to work contributed to the birth of structural design, and Wren was
originally a professor of astronomy and made important contributions to mathematics,
including the rectification of the cycloid (“On finding a straight line equal to that of a
cycloid and the parts of thereof”, Philosophical Transactions of the Royal Society,
November 1673). Characteristically, the second edition of Newton’s Principia (1713)
mentioned Wren, together with Wallis and Huyghens, as the leading geometers of his
age, while the general public recognized him as the person who rebuilt many churches
after the Great Fire of London in 1666. However, the specialization of mathematics
made less probable the “personal union” of the work of the architect and of the
mathematician. The rapid developments led to separate fields, which are manifested in
Departments of Mathematics and Schools of Architecture at the universities. These
institutions rarely cooperated, although the “Euclidean distance” between them on the
campuses became smaller than that one in the Hellenistic age.
2 From the “handcuff” of historic styles to the “freedom” of form in
Art Nouveau (from a gardener’s contribution to an architectural
revolution in Belgium)
Before turning to the modern relationships between architecture and mathematics, we
should make an outlook to some interesting developments in the history of design and
architecture. In the 18th and 19th centuries, most architects still focused on the
revitalization of some historic styles and the decorative art also followed this tendency.
However, the political changes that were associated with the French Revolution in 1789
and later with Napoleon’s empire opened the door for new ideas in art and architecture
and made Paris the center of influence for a long period. Interestingly, a revolutionary
movement in French architecture did not follow the events, but became a forerunner.
Shortly before the revolution, Étienne-Louis Boullée’s and Claude-Nicolas Ledoux’s
presented their utopian designs, and later their student Jean-Nicolas-Louis Durand
introduced his composition principle based on the combination of basic geometric units.
Although most of their designs, including Boullée’s enormous spherical building, the
Cenotaph for Sir Isaac Newton (1784), were never executed, their works and
33

geometrical ideas became influential for later generations via Durand’s textbooks. Their
fresh approach, however, did not prevent the majority of architects from continuing the
application of historic styles. It is true, that this historicism was not always a way of
simply looking backwards. Eugène-Emmanuel Viollet-le-Duc studied the Gothic
construction principles as an example for “rational architecture” and his goal was to
create a modern parallel of it by using iron. As a prolific writer, he devoted a ten-volume
dictionary to this topic (Paris, 1854-1868). He was also a leading, although
controversial, figure of restoration of ancient monuments and wrote a monograph on the
geodesic and geologic “construction” of the Mont-Blanc and its glaciers. In the mid-19th
century, another and more radical inspiration came from “outsiders” who introduced a
new structural rationality into architecture and design. The rapid development of
industrial production provided new materials and new possibilities. The gardener Sir
Joseph Paxton’s iron-and-glass Crystal Palace for the Great Exhibition in London in
1851, which was also marked by the 1851-foot length of the building, contributed to the
birth of a new aesthetics. Later the engineer Alexandre-Gustave Eiffel continued this
process. He designed a 300-meter-high wrought iron Tower for the World’s Fair in Paris
in 1889. (The earlier invention of the elevator was also essential for the Eiffel Tower,
while the utilization of steel was, surprisingly, missed.) The changing focus of these
“expos” is also characteristic. While the central part of the Crystal Palace was a gigantic
hothouse for a horticultural exhibition with a hidden drainage system that utilized the
hollow cast-iron pillars, the later Paris World’s Fairs gave more emphasis to machines
and industrial products. The “machine age”, however, gained not only support, but also
a strong opposition with social criticism: the Arts and Crafts Movement, which was
initiated by John Ruskin and William Morris in England, promoted the value of
handicrafts.
34

Figure 1.Victor Horta’s Tassel House, Brussels (1892-93).
This tradition was also continued in the “New Art” or Art Nouveau, which suddenly
appeared in architecture, first in Belgium in the 1890s. Its innovator, the architect Victor
Horta, ignored the traditional styles and introduced a new form-language for
architecture, which is dominated by curves, organic motifs, and the interplay between
symmetry and asymmetry. His Tassel House in Brussels (1892-93) was a
Gesamtkunstwerk, the synthesis of the arts, where all details, from architecture to
internal decorations, from metalwork to furniture, were carefully executed according to
his plans. Horta’s building is a celebration of the freedom of form, the imaginative use
of geometry, and the harmony among various fields of art, as well as the promotion of
the use of iron as structural material. Remember that Viollet-le-Duc’s also suggested the
latter. Horta made this move in a period when even the machines were often decorated
with Greek columns or Gothic arches. This “stylization”, however, became immediately
nonsense and the designers of Art Nouveau were able to concentrate on the optimal form
of objects, which is required by their function. Horta was an architect who became
involved in the design of almost everything inside the building, while another Belgian
35

artist, Henry van de Velde took the opposite direction from paintings to design and
architecture. This was the time when the role of Paris as the “global center” of new art
movements was challenged: an interesting “anti-globalization” and “de-centralization”
was marked by pioneering achievements in architecture and design in Belgium and
many other countries. In Germany, Velde moved there in 1897, the new style was first
called as “Veldean” (Veldische) or “Belgian” (Belgische) and later simply Youth-Style,
Jugendstil, which reflected not only the young age of the artists, but also the related
interest of the Munich-based magazine Jugend. In Austria and Hungary, however,
Secession-style or simply Secession (Sezessionstil in German, Szecesszió in Hungarian)
became the conventional expression. There were about a dozen different names for very
similar movements. Some of these new centers were directly related, while others
worked independently, from Catalonia (Antonio Gaudi) to Hungary (Ödön Lechner). In
Art Nouveau, the main inspiration of a new form-language came for organic nature,
especially from botany. Samuel Bing, who opened his gallery and shop L’Art nouveau in
Paris, claimed that nature is “the infallible code of all laws of beauty” (in the journal The
Craftsman, Vol. 5, October 1903). Some artists published botanical and horticultural
articles (Emile Gallé, Eugène Grasset), while Ernst Haeckel, a leading biologist with a
special interest in evolution and morphology, released his beautifully illustrated album
Art-Forms of Nature (Kunstformen der Nature, Leipzig, 1899-1904). Note that in the
late 18th century, Johann Wolfgang von Goethe, beyond his literary and political
activities, was also involved in morphological studies and wrote an essay on the spiral
tendency in vegetation. In the 19th century, there were important mathematicalbiological
works on spirals in nature, including the discovery of the spiral phyllotaxis by
German and French scholars in the early 1830s. It is a tendency that the number of left
and right spirals on the surface pinecones, sunflowers, and cacti are neighboring
Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, ...; each term is the sum of the previous two
and the ratios of the neighboring numbers approximate the value of the golden section
(a/b = b/(a + b) = 0.618…). Sir Theodore Andrea Cook’s books Spiral in Nature and
Art (London, 1903) and The Curves of Life (New York, 1914), and, especially, Sir
D’Arcy W. Thompson’s monograph On Growth and Form (Cambridge, 1917) attracted
a special attention. The latter was written originally for biologists, but influenced rather
designers and geometers of the forthcoming generations. Last, but not least, Art
Nouveau was also inspired by the “discovery” of Japanese art, following the end of
seclusion of that country in the late 19th century, and by a growing interest in objects of
art from Africa, Asia, and Oceania that were collected by enthusiastic expeditions and
displayed in museums. Japanese architecture offered many interesting features for
designers: the use of organic motifs, the striking geometric simplicity, the delicate
interplay of the inside and the outside of the building, and the modular system of units.
The half-size model of the Katsura Imperial Villa was exhibited at the World Exhibition
in Chicago in 1893 and attracted a great attention among visitors, including the
American architect Frank Lloyd Wright. The influence of Japanese art had two levels:
partly reinforced an interest in floral motifs, partly contributed to a new level of “organic
art” where the nature of the materials and geometric simplicity dominates without
36

depicting flowers and other motifs. Later we shall return to Wright and the importance
of his “organic simplicity”, which was also inspired by his reading of Herbert Spencer’s
biological-philosophical works. In Europe, Rudolf Steiner, originally a Goethe-scholar
and a supporter of Theosophy (ancient “divine wisdom” in East and West), later the
founder of Anthroposophy, initiated a form of organic architecture and an
interdisciplinary educational movement. (From the 1960s, the Hungarian Organic
Architecture, the schools of György Csete and of Imre Makovecz, gave a new emphasis
to organic forms and materials, as well as to Hungarian folk art.)
3 Industrial design and geometric abstractions with “style” in the
period of war and piece (from the Werkbund in Germany to
verhouding in the Netherlands, as well as beelding instead of
bleeding)
The popularity of new objects d’art, from furniture to wallpaper, from doors to lamps,
was growing, but their prizes remained high. The obvious question was the following:
do you need the expensive object made by handicraft or the cheap industrial product that
lacks the artistic quality? Since many people preferred the advantages of both, some
artists initiated a compromise, first by making multiple copies of the same object, and
later by cooperating with the industry to mass-produce it. Van de Velde, in his 1897
paper “A chapter on the design and construction of modern furniture”, explained his
principle of “systematically avoiding designing anything that cannot be mass-produced”
(in the Berlin journal Pan, Vol. 3, pp. 260-264, 1897; English trans. in Tim Benton at
al., Form and Function, London, 1975, pp. 17-19). Indeed, a new field was born, which
is called, by a later term, industrial design. One of its early representatives was Peter
Behrens, a German painter turned designer and architect, who become the artistic
advisor of the large electric company AEG in 1907. Interestingly, Behrens’ office
employed Mies van der Rohe, Walter Gropius, and Le Corbusier, three architects who
later played crucial role in the 20th century developments. The cooperation between
artists and engineers also required some exact mathematical methods in order to
describe the parameters of the artist’s original design and to organize its industrial
multiplication. The keyword was “types” and even the mass-produced furniture was
named as “type-furniture” (Typenmöbel). The central figure of integrating art and
technique, as well as modernizing the system of art education in Germany, was Hermann
Muthesius. Earlier he studied the arts and crafts movement in England as a diplomat,
wrote a book about The English House (Das englische Haus, Berlin, 1904-05), and
returned to work in the Prussian ministry. In 1907, he founded the German Werkbund
(Work-Federation). This organization, which institutionalized the cooperation among
architects, artists, engineers, and industrialists, contributed to the extensive
standardization and mechanization. Sadly, this was utilized not only by the civil sphere,
but also by the military industry.
37

The new freedom that Art Nouveau gave to designers, together with the influence of
Japanese and African art, paved the way to a more aggressive geometric interest in art in
the early 20th century. This was first manifested in Cubist paintings and Futurist
sculptures, then in abstract compositions where the figurative motifs totally disappeared
and simple geometric figures dominated (Kandinsky, Malevich). This period became the
age of manifestoes and the foundation of new artistic centers in Italy (Futurism), Russia
(Rayonism, Suprematism, and the early form of Constructivism), Hungary (Activism),
and the Netherlands (Neo-Plasticism, Elementarism). The number of –isms rapidly grew
and started to approximate the number of significant artistic groups. In architecture, the
preference for more simple geometric form replaced the Art Nouveau’s extensive usage
of floral motifs. The Austrian architect Adolf Loos, who had a great influence on the
next generation of architects (including André Lurçat, Eric Mendelsohn, Richard Neutra,
and Raymond Schindler), did not hesitate to speak about Ornament and Crime, which is
actually the title of his book (Ornament und Verbrechen, 1908). We should not forget,
however, the debt of modern architecture to Art Nouveau and its revolution against
historic styles. The tensions before and during World Word I (1914-1918) led to various
contradictory feelings with illusions and disappointments, enthusiasm and nihilism, but
also to the search for new ideas in small groups. Shortly before the war, the artists of the
Futurist group had a great hope in technology. They also liked to refer to mathematics,
at least metaphorically, as Filippo Tommaso Marinetti’s Manifesto of Geometrical and
Mechanical Splendour, and the Sensibility of Numbers (1914) demonstrates, while
Antonio Sant’Ellia predicted in his designs many aspect of the “new city”. As a
numerical obsession, they published the majority of their manifestoes on the 11th day of
the actual month. Their interest in military technology, however, led to a tragic
direction: the leaders of the movement volunteered to serve in a motorcycle-battalion
during the war and Boccioni and Sant’Ellia were killed. The disillusionment with the
war and the military machine also marked the end of the movement (although Marinetti
remained active as a writer and later joined the fascist leader Mussolini).
The case of the Netherlands needs a special attention. Since the country remained
neutral in the war and the only military goal was to defend its borders, as well as to help
the occupied Belgium economically, the developments in Dutch architecture and design
were not interrupted. In the late 19th century, the architect Jan Hessel de Groot and
some of his colleagues generated a special interest in mathematical aesthetics, especially
in proportional systems. Perhaps this was associated with the idea of “rational
architecture” that Viollet-le-Duc initiated earlier. De Groot summarized his views in
many papers and books; here we refer just to one of his later work, a monograph on
Form-Harmony in Dutch (Vormharmonie, Amsterdam, 1912). For him the importance
of proportion was to create a whole, “to make from, say, twenty forms, one form”. J. L.
M. Lauweriks suggested combining the cosmic significance of mathematical proportions
and the ethics of socialism. In Dutch both terms, the international proportie and the
native verhouding, can be used in the sense of proportion, although the latter also means
“relationship”, including a love affair and the relationship with God. Perhaps this was
38

suitable for Lauweriks, who had a “love affair” with proportions and also headed a lodge
of the Theosophical Society. (Indeed the title of his paper on proportion is
“Verhouding” in the journal Architectura, Vol. 5, pp. 175-176 and 178-180, 1987.) The
case of Lauweriks is interesting to us, because he linked many ideas that we discussed
earlier: started with Art Nouveau (Nieuwe kunst) decorations, in 1904 moved to
Germany, following the invitation of Peter Behrens, and became an important figure of
applied art and also designed various houses, in 1916 returned to the Netherlands and
worked as a professor. With great probability, he was the designer of the villa at
Bremen, which inspired Le Corbusier’s interest in mathematical proportions. (Corbu’s
data on the villa are fuzzy in his book Modulor, but Reyner Banham later pointed out
that it must have been Lauweriks’s building, cf., Theory and Design in the First
Machine Age, London, 1960, p. 142). Hendrikus Peter Berlage, another influential
figure of Dutch architecture, was also associated with the 19th century “rational”
movement, as well as the Dutch school of proportions. He visited Frank Lloyd Wright in
1897 and again in 1911, and, following his reports, the American architect became an
influential figure in Europe, too. The young generation of architects in the Netherlands,
Belgium, and some other countries were confused with the sharp criticism against Art
Nouveau and against the use organic decorations, even Horta switched to a more
academic style. The helping hand came from Berlage who distributed information on
Wright’s works that offered “organic simplicity”. It was new, but also gave a feeling that
the earlier “organic” approach had a meaning. Berlage, in his textbook in German,
pointed out that the role of the architect is “the creation of space, not the sketching of
façades” (Grundlagen und Entwicklung der Architektur, Berlin, 1908).
The importance of geometry and creation of space, while requiring simplicity, provided
the intellectual basis for the painter Theo van Doesburg and some young artists and
architects who founded the journal De Stijl (The Style, 1917-1928 and 1932). Even the
title was associated with the views of de Groot and Berlage who often referred to
Gottfried Semper’s book The Style in Applied Arts (Der Stil in den technischen Künsten,
2 vols., 1860-63) and his approach to practical aesthetics. The “Dutch phase” and the
“international phase” of De Stijl, with a special emphasis of architecture, are discussed
by Reyner Banham’s referred to book on theory of design in detail (pp. 138-200). I
would say, however, that the first phase was also international in some sense. Their
group included, beyond Theo van Doesburg who edited the journal, not only the Dutch
painters Piet Mondrian, Bart van der Leck and the Dutch architects J. J. P. Oud, Gerrit
Rietveld, Rob van t’Hoff, but also the Hungarian painter Vilmos Huszár, the Belgian
sculptor and painter Georges Vantongerloo (van Tongerloo), and the Italian ex-futurist
Gino Severini. Around 1922, at the beginning of the “international phase” the group
included just two non-Dutch members, the German Hans Richter and the Russian El
Lissitzy, while Theo van Doesburg and two architects, Gerrit Rietveld and Cornelis van
Eesteren, were the Dutch participants. Mondrian and some other members of the
original group advocated total abstraction, without any subjectivity, and called their
approach Neo-Plasticism (Nieuwe beelding). Mondrian was influenced not only by
39

cubism, but also by the spiritual-mathematical works of the Thesophist M. H. J.
Schoenmaekers, who wrote a book about the Principles of Plastic Mathematics in Dutch
(Beginselen der beeldende wiskunde, 1915). Although Mondrian’s interest in
Theosophy is often mentioned, Robert P. Welsh presented some important new details
more recently (see, e.g., his paper “Mondrian and Theosophy” in the book The Spiritual
Image in Modern Art, Wheaton, Illinois, 1987). While a large part of Europe was
“bleeding” during World War I, the peace in the Netherlands led to remarkable studies
in beelding, plasticism. The main meaning of the noun beeld is “image”, but it is also
associated with “beauty”; its derivative is used in beeldende kunsten “plastic arts”. (In
German, both van Doesburg and Mondrian rendered the Dutch expression as
Gestaltung.) What is Neo-Plasticism? Mondrian reduced everything to horizontal and
vertical lines and three primary colors, red, blue, and yellow. Later van Doesburg also
allowed diagonal lines in his “counter-compositions” and called his new approach as
Elementarism, which move led to a split with Mondrian. As a counterpoint to the
extensive decorations and the use of curved lines in Art Nouveau, strong geometrical
restrictions took command. Note that modern music made a similar step in Arnold
Schonberg’s serial music. The scientific interest of van Doesburg and his colleagues is
well represented by the fact that the subtitle of De Stijl was changed from “Monthly
Magazine for Plastic Arts and Crafts” (Maandblad voor de beeldende vakken), the last
word means not only arts and crafts, but also professions, trades, subjects of study, and
even, as a nice play on words, squares and panels, to “Monthly Magazine for New Art,
Science and Culture” (Maandblad voor nieuwe kunst, wetenschap en kultuur). This
reference to science was not an empty word: the journal published, for example,
excerpts from a paper by Henri Poincaré, a leading figure of mathematics (“Why does
the space have three dimensions?”, De Stijl, Vol. 6, No. 5, pp. 66-70, 1923). The
movement associated with the journal De Stijl had an important impact on architecture
where geometric simplicity had an advantage. Although modern technology does not
exclude curved surfaces, but economical reasons may limit the structure. The architects
demonstrated how to enrich the world of form, how to make new harmonies in the
restricted “universe” of right angles, straight lines, and interlocking planes.
These new ideas from Cubism to Neo-Plasticism, together with the needs of rebuilding
many parts of Europe after World Word I (1914-1918), contributed to the
popularization of modern design and the establishment of two movements with
educational institutions:
- the Constructivists (1918-1922 and, in a restricted form, 1922-1934) in Russia, soon
after the socialist revolution in 1917, and
- the Bauhaus (1919-1933) in Germany, soon after the establishment of the Weimar
Republic in 1919.
40

In both countries, the dramatic political changes contributed first to the birth of these
movements and later to their suppression. However, their influence remained essential
via their achievements and via the activities of some of their members who continued
their works in other countries, as well as via their failure. (It is a very important that we
should also consider the failures and learn from these; the method of trial-and-error has
two sides and both are important.). Let us see these two brothers (or sisters), the
Constructivists and the Bauhaus in the context of design and mathematics.
4 Constructivism: art industrial design architecture ( art)
(an architect turned professor of geometry)
The Russian Constructivists interest in scientific and technical questions were
immediately emphasized in the names of some of their institutions that were quickly
established after the Socialist Revolution in October 1917. In Moscow, they united two
schools of art and architecture as the Free State Art Workshops or shortly Svomas in
1918, with the leadership of Vladimir Tatlin, but soon renamed as Higher State Art-
Technical Workshops or Vkhutemas (the abbreviation of Vysshie gosudarstvennye
khudozhestvenno-tekhnicheskie masterskie, 1920-1926), which became the Higher State
Art-Technical Institute or Vkhutein (1926-1930). In Petrograd, later Leningrad, now St.
Petersburg, they organized similar institutions and Tatlin himself moved there for a
longer period. Some constructivists and avant-garde artists also participated at the
foundation of the Russian Academy of Artistic Sciences (Rosiiskaya akademiya
khudozhestvennykh nauk or shortly RAKhN) in 1921. Since the expression “artistic
sciences” is puzzling, we may illustrate this diverse field with two examples. The painter
Wassily Kandinsky (Kandinskii) became the head of the Physico-Mathematical and
Physico-Psychological Department of this new Academy, and he initiated various
research topics from the basic elements of art to the psychology of aesthetics. At the
same department, Leonid Sabaneyev, who earlier studied both music and mathematics,
conducted his research on the possible importance of the golden section, the proportion
a/b = b/(a + b), in the temporal organization of musical works and analyzed 1,770 works
by 42 composers. Characteristically, some of the constructivists had training at
engineering schools. Thus, El Lissitzky (Lazar Lisitskii) was educated at the Technische
Hochschule Darmstadt (1909-14), while Naum Gabo, one of the Pevsner brothers,
studied medicine (1910), science (1911), and engineering (1912) in Munich.
Constructivism originally benefited from the various geometrical and abstract tendencies
that were popular among Russian artists (Larionov, Goncharova, Malevich, Gabo, and
others) and even from the combination of abstract art with a psychological and spiritual
interest (Kandinsky). However, these tendencies of pure art were soon overshadowed by
the emphasis on industrial design and production, including such diverse fields as textile
design and typography. “Down with guarding the traditions of art. Long live the
Constructivist technician”, claimed Aleksandr Rodchenko and Varvara Stepanova in
their manifesto of the “Productivist Group” in 1920. These “constructivists” or “artist-
41

engineers” believed that they are able to build a new industrial culture and participating
in the social changes. (Perhaps we may find some similarities between the views of the
Russian “Artistic Left” in the 1920s and the British “Scientific Left” in Cambridge in the
1930s; indeed, John Desmond (“Des”) Bernal’s representative book The Social
Function of Science, London, 1939, frequently refers to science and industry in the
Soviet Union, although not to the artists.) Constructivism was started as a new form of
art (or an “anti-art”), became associated with industrial production, tried to serve a
socio-political system, but its main stream was politically destroyed from 1922 by the
“socialist-realist” artists. The “Declaration of the Association of Artist of the
Revolutionary Russia” claimed that their real goal is to depict “the life of the Red Army,
the workers, the peasants, the revolutionaries, and the heroes of labor”. (Obviously,
there were enough “volunteers” to pose for such pictures and sculptures!) On the other
hand, the constructivists also contributed to this attack against them by their mistakes.
They did not try to defend the autonomy of art, but became involved in “agitating
propaganda” with poster design and street decorations, then in “production”, without
even checking that the politicians like or dislike their works. In fact, the leading political
figures, with the exception of Lunacharskii, the People’s Commissar for Enlightenment
(1917-1929), had little sympathy with their art or even opposed it. They also provoked
other artists by claming that they represent the only form of artistic expression. The first
sign of troubles were well demonstrated by the fact that even Kandinsky and Gabo left
Russia in 1921 and 1922.
However, constructivism did not die immediately, but survived in various fields of
applied art and even advanced in architecture where the goals of depicting the
revolutionaries had no meaning. The architects partly responded for the great demand
for building new industrial complexes, houses, and other objects, partly tried to gain
support for some advanced building. For example, the Vesnin Borthers proposed a
transparent glass-iron-concrete building for the Leningrad Office of the newspaper
Pravda with the elevators outside (1923), El Lissitzky initiated skyscrapers (“Cloud
Hengers”) where the offices or the dwelling units were on the top of elevator-towers
(1924), and I. I. Leonidov designed a spherical auditorium with 4,000 seats for the
Lenin Library (1927-28). Most of these futuristic ideas had the same fate as the earlier
French plans: they were not built, but influenced later generations. This period also
produced many important theoretical works where architecture and mathematics were
linked, including
- Nikolai Karsil’nikov’s quantitative method for testing all the possible arrangements of
given elements in architectural design (in the journal Sovremannaya Arkhitektura, i.e.,
Modern Architecture, 1927),
- Yakov Chernikov’s book Construction of Architectural and Machine Forms
(Konstruktsiya arkhitekturnykh i mashinnykh form, Leningrad, 1931), which was written
in Russian, but also advertised in a brochure in English.
42

Perhaps the swansong of constructivism was another book by Chernikov: Architectural
Fantasies (Arkhitekturnye fantazii, Leningrad, 1933). Ironically, this title immediately
hints that the goal was not anymore “real” architecture, but play with geometrical forms.
John E. Bowlt, in his book Russian Art of the Avant-Garde (London, 1988, p. 156),
concluded that constructivism both began and ended as art. We may add that this final
form of constructivism is such an art that has a special openness toward mathematics.
Using compass, ruler, and even mathematical formulas was not unusual, as it is
demonstrated by El Lissitzky’s picture Tatlin at Work on the Third International (pencil,
gouache, photomontage, ca. 1920). In Tatlin’ s hand there is a ruler and at his eye a
compass. It is true that the mathematical formulas written next to Tatlin’s body are
meaningless, but these were obviously “workable” for the general public as a
metaphorical reference to higher mathematics. However, Chernikov went far beyond
basic geometry and even anticipated the field of shape grammar that we should discuss
later. Characteristically, Chernikov earned a degree in Architectural Sciences in 1935
and left Leningrad to take up the position of the head of the Department of Descriptive
Geometry and Graphics at the Institute of Engineering Economy. Indeed, the
constructivists lost their last hopes to continue their work as artists. In 1930, the
movement had a second death by “renaming” its main educational institution, the
Vukhtemas/Vukhtein (1920-1930), which pioneered a strong cooperation between art
and crafts and also initiated the links to industry. The new name, Moscow Institute of
Art, had no reference to technique (-te- in the earlier names) anymore. Finally, in 1934
the third and final death of constructivism came with the political declaration that all
forms of art and literature should be based on Socialist Realism, which ideology soon
occupied the field of architecture and design by rejecting the luxury “formalist” ideas.
However, constructivism retained an influence via some of its members who moved to
other countries and via the help of its “younger brother” in Germany.
5 Bauhaus: art industrial design architecture ( basic design)
(Emaille by email order, or else the roots of computer-aided
manufacture)
The Bauhaus (1919-1933), “building” (Bau) a “house” (Haus) where architects,
sculptors, and painters work together and there will be no barrier between artists and
craftsmen, is perhaps the best known movement and educational institution in the field
of modern architecture and design. However, its history was not so straightforward as it
is widely believed and there were many struggles in order to find its direction. The
Bauhaus was established, similar to the Vukhtemas in Moscow, by uniting two existing
art colleges in the German city Weimar. It is interesting to note that the director of these
institutions was the same Henry van de Velde who contributed to the birth of Art
Nouveau in Belgium and had a strong interest in industrial design. The founding director
43

of the Bauhaus was the architect Walter Gropius, who was nominated by van de Welde
some years earlier, but no move was made until the end of World War I and the birth of
the republic system in Germany. Gropius, in his manifesto of 1919, immediately
announced the establishment of a new department of architecture, the reunification of
sculpture, painting, handicrafts, and the crafts as “inseparable component of a new
architecture”, and promised “constant contact with the leaders of the crafts and industry”
(Programme of the Staatliches Bauhaus in Weimar, April 1919). However, no real
emphasis was given to industrial design and functionalism until the arrival of the
Hungarian self-taught artist László Moholy-Nagy in 1923 and no department of
architecture was established until 1927, which was also the last year of Gropius at the
school. These steps show a striking similarity with the Russian constructivists’ initial
interest in pure art, then a shift to industrial design and later to architecture and applied
arts. Incidentally, some authors call the very first period of the Bauhaus, until 1923, as
expressionist, which was followed by a constructivist phase. What factors did delay an
immediate interest in industrial design at the Bauhaus? What was the reason of
interrupting those links between art and industry that were available in Germany via van
de Velde and, more intensively, via the German Werkbund? Obviously, the horrors of
Word War I and the importance of standardization and mechanization for the military
industry gave some negative feelings and did not encourage the artists to rush to the
industry. The second reason could have been the personal attitude of the first instructors.
Gropius hired creative artist, painters and sculptors, of great ability (Lyonel Feininger,
Johannes Itten, Gerhard Marcks, later Paul Klee, Oskar Schlemmer, Vasily Kandinsky,
and others), but they had less interest to move towards industry. Still, the early period
was very important to work out a progressive structure of education with a half-year,
later one-year, Preliminary Course (Vorkurs) and then a three-year dual system of
instruction in form and in crafts problems (Formlehre and Werklehre) in various
workshops organized according to the used materials, specifically stone, wood, metal,
clay, glass, colors for wall-painting, and textiles. The idea was that two instructors
should lead each workshop: an artist and a craftsman whose title is “form-master”
(Formmeister) and “craft-master” (Werkmeister), respectively. The Swiss painter
Johannes Itten was the most influential instructor of the first period. He organized the
Preliminary Course, with an intention “to liberate the student’s creative power”, and also
headed some of the workshops. The students created individual pieces of art-works;
these were carefully crafted, but without too much consideration of the possibility of
industrial reproduction. Since Itten also favored a kind of Oriental mysticism and
introduced breathing exercises, diets, and prayers, while Gropius had a growing interest
in social questions and the reestablishment of the links to industry, a growing tension
appeared between them. The move towards the constructivist phase of the Bauhaus was
also encouraged by the private seminars of Theo van Doesburg, the founder of the
journal De Stijl, who spent a longer period in Weimar in 1921-22. Finally, Itten left the
school in 1923 and Gropius’ view became the dominating force: “Art and technology: a
new unity”.
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The appointment of László Moholy-Nagy, who was associated with the Hungarian
Activism, a Constructivism-related group, became a very successful step into this
direction. In 1922, shortly before his arrival at the Bauhaus, Moholy-Nagy introduced
the idea of “telephone-pictures” (Telefonbilder), which were made using vitreous enamel
(or, in German Emaille or Email) on steal, and the description of the figures were such
exact that, theoretically, a factory was able to produce it via a telephone order, without
seeing its sketch. This method included, in preliminary form, those ideas that later led to
the algorithmic description of figures, their presentation by computer graphics, and
producing by computer-aided manufacture. (Nomen est omen: we are able to order
Emaille pictures not only by telephone, but also by email.) Moholy-Nagy was in charge
of the new one-year Preliminary Course, with the participation of Albers, Klee, and
Kandinsky, and he also headed the metal workshop where many prototypes were
developed for industrial production. Perhaps the best-known “objects” of the Bauhaus
were those chairs that Marcel Breuer, a Hungarian student and later a leading architect,
designed in the furniture workshop. On the other hand, Moholy-Nagy did not follow
those Russian constructivists who favored “production art” and rejected “pure art”, but
suggested a balanced view. He intensively dealt with the use of new media, from photo
and film experiments to works with kinetic light and mobiles. Moholy-Nagy also had a
special interest in graphic design, which lead to his “asymmetric” typography for the
Bauhaus books. In this series, edited by Gropius and Moholy-Nagy, 14 books were
published between 1925 and 1929, including not only the works of the Bauhaus
members (Kandisnky, Klee, Schlemmer, and the editors own books), but also
representative works of the French cubism (Gleizes), the Russian suprematism
(Malevich), and the Dutch De Stijl (van Doesburg, Mondrian, J. J. P. Oud).
The Bauhaus “brotherhood” with the Russian constructivists and other avant-garde
movement was also reinforced with the appointment of Kandinsky in 1922 and the short
visits by El Lissitzky and Malevich. Although the Bauhaus did not have such a member
as the Russian Chernikov, who continued his career as a professor of geometry, the
importance of mathematics was directly emphasized by Gropius in his second manifesto
“The theory and organization of the Bauhaus” in 1923. Specifically, he pointed out that
the dual system of instruction in form and crafts problem needs a general coordination
(Harmonisierungslehre), and he included here, among others, mathematics, physics, and
mechanics, as well as the synthetic study of space. The program of the instruction in
form (Formlehre) also had a section of descriptive geometry. In 1926, the Bauhaus
moved to Dessau into a new building designed by Gropius. This glass and reinforced
concrete building, with an asymmetric shape, symbolized the Bauhaus interest in
functionalism and demonstrated the relationship between design and modern
technology. Finally, in 1927, the Bauhaus established its Department of Architecture
with the chairmanship of the Swiss architect Hannes Meyer, who also become the new
Director in 1928 when Gropius, together with Moholy Nagy and some other masters,
left the school. Meyer represented a Marxist ideology and was strongly productionoriented,
while reduced the teaching commitments. In 1930, he was removed from office
45

and replaced by the German architect Mies van der Rohe. The Bauhaus, similar to the
Russian constructivists’ workshops Svomas/Vukhtemas/Vukhtein (1918-1930), had
three different full names: Staatliches Bauhaus in Weimar (1919-1925), Bauhaus
Dessau: Hochschule für Gestaltung (1925-1932), and Bauhaus Berlin (1932-1933). The
lack of founding in Weimar did not cause too much problem and the Bauhaus moved to
another place. However, the closure in Dessau and later in Berlin were politically
motivated by the Nazis.
6 Beyond the Bauhaus (Briefly about the International Style, two
schools, and two personalities)
The Bauhaus-idea survived in various forms, including some remarkable works of the
“classical period” (the Bauhaus chairs are still available commercially) and the later
teaching activity of its former members (for example, Gropius at Harvard University,
Mies van der Rohe at the Illinois Institute of Technology in Chicago, Albers at the Black
Mountain College, North Carolina). The Bauhaus’ impact on education of design is
significant, although not without controversy. Peter Collins remarked, with some irony,
that at Harvard University, where Gropius was teaching from 1937, virtually all
elements of the Bauhaus curriculum, with the exception of the Preliminary Course, were
abandoned. Moreover, the Bauhaus’s only graduate who furthered the ideal of
“architectonic art” in the case of buildings was Marcel Brauer, who studied only
furniture design there (Changing Ideals in Modern Architecture, London, 1965, p. 269).
Perhaps the best legacy of the Bauhaus in the field of education is those Basic Design
Courses or Foundation Courses that became popular almost elsewhere at schools of
architecture and design in order to develop the students’ creative skills. Here we should
pay tribute to Johannes Itten who “designed” the first Basic Design courses before his
“forced” departure. The fact that such type of courses may also contribute to
mathematics is demonstrated by the Rubik’s Cube, which was originally a simple tool
for design students in Hungary, later became an international obsession, and ended up as
a research topic for mathematicians. Interestingly, the Bauhaus had a very strong impact
on Japanese design education: many schools introduced similar foundation courses and
all Bauhaus books were translated into Japanese. In the history of architecture, the
expression International Style was introduced for those achievements that the European
architects represented from Gropius to Mies van der Rohe, from Oud to Le Corbusier,
although they never formed a coherent group in such a broad scale. The name was
obviously inspired by Gropius’s book entitled International Architecture (Internationale
Architektur, 1925), the first volume in the series “Bauhaus Books”, but he himself never
referred to a new “style”. In 1932, the title of the first architecture-related exhibition in
the Museum of Modern Art in New York, just three years after its opening, and the book
published for this occasion, “canonized” the name “International Style”.
46

There were two institutions that emphasized the Bauhaus-traditions in their names,
although none of them tried simply to “copy” it. The first institution had three names
(this number became “magic” in this field): the New Bauhaus in Chicago (1937-1938),
which became School of Design (1939-1944), then Institute of Design (1944-), which
was later incorporated by the Illinois Institute of Technology in Chicago.
Moholy-Nagy was the first head of this institution and his commercial works was
marked, among others, the designed of the fountain pen Parker 51 in 1941. (Note the
conflicting interest of Hungarians in the world: Moholy-Nagy designed the best-known
fountain pen, but Biro invented, with a little help of some friends in the Budapest cafés,
the Biro pen, the original form of the ball pen.) Sadly, Moholy-Nagy’s wide-ranging
activities were stopped by his early death of leukemia in 1946. Between 1946 and 1951,
Serge Chermayeff was the new director. His name is known as both a successful
industrial designer, including various chairs, and the author of theoretical works. Gyorgy
Kepes, a close coworker and compatriot of Moholy-Nagy, became a leading figure of
building bridges between art and the newest results of modern science and technology,
first in Chicago and later, as the founding director of the Center for Advanced Visual
Studies at MIT, in Cambridge, Massachusetts. Kepes’s books, with contributions by
very many artists and scientists, attracted a special attention (Language of Vision,
Chicago, 1944; The New Landscape in Art and Science, Chicago, 1956; Seven volumes
of the series “Vision + Value”, New York, 1965-72). The other institution, the
Hochschule für Gestaltung in Ulm (1953-1968), was initiated soon after World War II,
by a foundation for the memory of Hans and Sophie Scholl, who were members of the
“White Rose” resistance group and executed by the Nazis. Similar to the Bauhaus and
the Constructivist Movement, the founders were eager to contribute to the solution of
social and intellectual problems after the war. Note that the German name of the
institution is identical with the secondary name of the Bauhaus in Dessau, while it is
usually referred to in English as Ulm School of Design. As an emphasis on the
traditions, Henry van de Velde, Walter Gropius, and Mies van der Rohe were among the
early visitors. We should discuss this institution in more details because of an interesting
controversy in connection with mathematics. The Ulm School of Design was first
headed by the Swiss artist Max Bill, a former student of the Bauhaus, and later by a
collective leadership including the Argentine designer Tomás Maldonado, the German
artist Otl Aicher, and others. During the first years Johannes Itten and Josef Albers,
former “masters” of the Bauhaus’ Preliminary Course, participated in the teaching of the
new school’s Foundation Course (Grundkurs), and later, among others, the
mathematician Hermann von Baravalle and the architect William Huff (who is a
Honorary Member of ISIS-Symmetry) continued this work. In the meantime, the
curriculum was changed from a Bauhaus-type teaching to a new one with two types of
courses: (1) design practice, and (2) complementary sciences, which included both
natural and social sciences. The school had four departments: visual communications,
information, industrial design, and building departments. While the Bauhaus attracted
some professors with great ability in intuitive mathematics, the Ulm School of Design
47

made a further step and regularly invited mathematicians, crystallographers, and
engineers to give courses or lectures (Z. S. Makowski, Frei Otto, Paul Schatz, K. L.
Wolf, and others). The school was eager to introduce the newest results of
communication and information theory, cybernetics, and semiotics into the curriculum
and attracted a remarkable group of people from these fields as instructors (Max Bense,
R. Gunzenhäuser, Abraham A. Moles, H. Stachowiak, and even a visit by Norbert
Wiener). This scientific component of the curriculum was never challenged, however,
the question how much “mathematization” led to a conflict. Horst Rittel, originally a
mathematician who was in charge of design methods and also a member of a threeperson
team that headed the school between 1959 and 1961, preferred the use of more
mathematics in all fields of industrial design, for example modular systems and
complicated grids, while the pragmatists rejected the “worshipping” of some
mathematical methods. They suggested concentrating on the links to industry. The battle
was “won” by the latter group.
We should emphasize that not just the Russian Constructivists and the Bauhaus made
contributions to linking architecture and mathematics in the “machine age”, the first
decades of the 20th century. For example, Le Corbusier and Ozenfant’s aesthetic
program Purism (ca. 1918-1926) in Paris and later Buckminster Fuller’s Dymaxion
house in America (from the 1920s) also initiated such links. In the case of both “Corbu”
and “Bucky”, this interest remained long-lasting, which is well-illustrated by the goldensection-related
proportional system Modulor and the theoretical works on Synergetics,
the cooperative forces (“energies”) in tensile structures, respectively. (We should not
confuse, however, Bucky’s Synergetics with the later usage of the same expression by
the physicist Hermann Haken and his interdisciplinary group.) We should stop at Bucky
for a while. He was trained as an engineer, not as an architect. Thus, his Dymaxion
house and geodesic dome, which became popular first at expos and later at many places,
reiterate the 19th century story with Eiffel’s contribution to the development of
architectural design. Playing on words, Bucky used not the analogy of the handcrafts,
but immediately the aircraft. (Much later Frank O. Gehry used a computer program for
shaping the surfaces of the Guggenheim Museum in Bilbao, which was originally
developed for designing the French Mirage fighters; indeed, the shape immediately hints
some relationship with aircrafts and the design encouraged some “fight” around the
building, fortunately without fighter jets, just verbally.) Bucky himself was a “fighter”
for his own ideas, which were often not fully his, but “adopted” from others and
popularized with a remarkable efficiency. One may say that the hexagonal shape of the
Dymaxion house was also an adoption, but it was legal because the bees did not patent
it. However, Bucky went far beyond the economy of nature and introduced prefabricated
high-tech components made by aircraft construction methods. The structural
components were prewired, preplumbed, and equipped with various devices. Thus, the
house became fully operational immediately after assembling. Fuller’s design took
advantage of the tensile strengths of materials and later used the same idea in the case of
his geodesic domes, which were shaped with grids based on the great circles (geodesic
48

lines) on spherical surfaces. In the same time, Bucky made a sharp criticism against the
Bauhaus and the architects of the International Style, claiming that they “never went
back of the wall-surface to look at the plumbing”. He accused the Bauhaus with “designblindness”
and the ignorance towards modern technology. Reyner Banham, in the
conclusion of his book Theory and Design in the First Machine Age (London, 1960),
agreed with Bucky’s view and declared that the architects of the First Machine Age did
not understand the contemporary science and technology and, in this sense, they were
wrong. Fortunately, we have here just a smaller problem: architecture and modern
mathematics. Do architect utilize modern mathematics? Do mathematician care those
problems that are important for architecture and urban design? The answer is: usually
not, but there are exceptions...
7 Mathematical architecture (?): the systematic usage of modern
mathematics in architecture and urban design (a Cambridge center
with some encouragement from constructivism)
I am not quite sure that the expression “mathematical architecture” is the best one, but,
using the analogy of mathematical crystallography, mathematical physics, mathematical
biology, and even mathematical aesthetics, why not to try this combination. Of course,
this is not about the “architecture of mathematics” or the “foundations of mathematics”,
which usually refers to the system of axioms, definitions, theorems in mathematics and
to the related questions of logic, but a field of architecture and design or, using the idea
of the Ulm School, a “complementary science” to architecture. From the 1950s
structural engineering made important advances. Beyond the geodesic domes
(Buckminster Fuller reinvented in the U.S.A.), the thin shell structures (Félix Candela in
Mexico and Pier Luigi Nervi in Italy), and its “inverse”, the suspended roofs (Frei Otto
in Germany) became very popular. Of course, these structures provided further
mathematical questions. We may consider “mathematical architecture” as a topic
associated with “building science” or “architectural science”, which became a wellestablished
field with strong links to applied physics and engineering. Let us see the
partnership of mathematics and architecture from the mid 20th century. As we have
seen, various groups of architects and designers turned to mathematical methods. Most
of these people, however, used elementary mathematics, for example proportional
systems, modules, basic geometric transformations. A few others applied higher
mathematics, but just occasionally in a few concrete cases. The complex problems in
architecture and urban design, however, led to the reappearance of “mathematiciansarchitects”.
(We should not forget that Sir Christopher, as we have seen in Chapter 1,
was both a mathematician and an architect.) Some architects decided to learn the
necessary mathematics or, in some fortunate cases, they were trained in both fields.
Since we “spent” a longer period in continental Europe, it is time to visit the United
Kingdom and then the U.S.A. again. Earlier we have seen the birth of Art Nouveau,
Jugendstil, Sezessionstil in continental Europe, but we should add that the Glasgow style
49

in Scotland and the Tiffany style in the U.S.A. represented similar developments and
their representatives had regular interactions with other movements. Remember that the
Belgian Horta was interested in the Arts and Crafts Movement in England (Chapter 2)
and the German Muthesius published a book about the English house (Berlin, 1904-05),
before he founded the Werkbund (Chapter 3). About a decade later, in 1915, the British
architect W. R. Lethaby wrote a paper entitled “Modern German architecture and what
we may learn from it”, which is available in his book Form in Civilization (London,
1922, pp. 96-105). In another paper, he wrote about the need of systematic research on
the geometry of architectural structures, as Lord Kelvin investigated the geometry of
crystalline structures. The first help came, however, from the side of Constructive art.
The architect Leslie Martin, later Sir Leslie, met the Constructive artist Naum Gabo,
later Sir Naum, who left Russia in 1922, and these two and the English painter Ben
Nicholson, edited a collection of papers with almost 200 plates:
- Circle: International Survey on Constructive Art (London, 1937).
Among the contributors, we may find, among others, Gropius, Le Corbusier, Moholy-
Nagy, Mondrian, and, as Lethaby “dreamed” earlier, a leading British crystallographer,
J. D. Bernal (earlier we mentioned his name as a representative of the “Scientific left” in
Britain). Fortunately, J. D. Bernal remained active in “building” bridges between
crystallography and architecture and he wrote two more papers, with almost symmetric
titles, “Architecture and science” and “Science in architecture” for the Journal of the
Royal Institute of British Architects (Vol. 44, No. 16, 1937 and Vol. 53, No. 5, 1946).
The latter was based on a lecture given at an informal meeting of the same organization.
All of the mentioned three papers were reprinted in Bernal’s book The Freedom of
Necessity (London, 1949, pp. 185-213). However, these contributions were brief survey
papers, without going into technical details. Their importance was to identify two
mathematical topics that are useful for architects: symmetry and topology.
If we are looking for the systematic usage of mathematical methods in architecture and
urban design and the establishment of an educational institution where such problems
became prominent, we should remain with the intellectual circles of Sir Leslie Martin.
After World Word II, he faced a situation that was similar to Sir Christopher Wren’s
circumstances after the Great Fire of London in 1666: he helped the post-war
reconstruction of London as the Chief Architect. From 1956, he taught at Cambridge
University, where he became the first professor of architecture, then, in 1967, the
Director of the new Centre for Land Use and Built Form Studies at the same university,
which was later renamed after him as the Martin Centre for Architectural and Urban
Studies. Martin declared at the very beginning of his professorship that it is necessary to
establish a bridge between faculties, between the arts and the sciences. His institution
became, beyond dealing with social and economical problems of architecture, a center
for the application of mathematical methods to architecture. Later some of the members
of his circles founded the Centre for Configurational Studies at the Open University in
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England and started a cooperation with the School of Architecture and Urban Planning
at the University of California Los Angeles (UCLA).
The desire to synthesize new ideas in Europe and America, architecture and modern
mathematics led the Austrian-born Christopher Alexander to graduate in both
architecture and mathematics at University of the English Cambridge, then earning a
doctorate in architecture at Harvard of the American Cambridge. His thesis was later
published as a book, while he moved to University of California, Berkeley:
- Christopher Alexander Notes on the Synthesis of Form (Cambridge, Mass., 1964).
From graphs theory to statistics, he applied various fields of modern mathematics,
together with some results from social sciences. His next work attracted a special
attention: “A city is not a tree”, claimed Alexander in the title of his paper, which refers
not a green tree with leaves, but to a type of semi-lattice with mathematical inspirations
(Architectural Forum, April and May 1965, pp. 58-62 and 68-71). He concluded that if
we develop our cities like trees, they will “cut” our life into pieces. This paper became a
widely discussed document in the field of city planning and was reprinted several times
(Design, February 1966, pp. 46-55; Jeffrey Cook, editor, Architectural Anthology,
Tempe, Arizona, 1969, pp. 580-590; etc.). Alexander also worked with Serge
Chermayeff (Community and Privacy: Towards a New Architecture of Humanism, New
York, 1963) and contributed a paper to the series of books edited by Gyorgy Kepes
(“From a set of forces to a form”, in: The Man-Made Object, New York, 1966, pp. 96-
107). Note that earlier we referred to both Chermayeff and Kepes in the case of the
history of Moholy-Nagy’s New Bauhaus in Chicago (Chapter 5). In 1967, Alexander
became the president of the Center for Environmental Structure in Berkeley, California,
and worked on a pattern language that considers verbal criteria for generating
architectural designs. Interestingly, Alexander decided not to use mathematical
weighting or valuation of the criteria, the often conflicting requirements by the client,
the local council, and so on, but to consider these as verbal statements and the achieve
the best possible solution with a concrete pattern. (Some problems of this methodology,
and a sympathy with Bruce Archer’s alternative approach based on operations research
with weighted criteria, were discussed in Lionel March’s paper “The logic of design and
the question of value” in the book The Architecture of Form, Cambridge, 1976, pp. 1-
40). Alexander and his coworkers continued refining their methodology and published a
large number of works on this topic, partly in the book series of the center, partly at
other publishers, for example:
- Christopher Alexander at al. A Pattern Language: Towns, Buildings, Construction
(New York, 1977).
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The interested reader may find the list of more recent publications of Alexander and his
colleagues at the home page: patternlanguage.com. Note that Alexander united the
theoretical works and the practice of the architect: he designed more than 200 buildings.
Returning to the circles of Leslie Martin and the activities in the triangle of Cambridge
University, Open University, and UCLA, a group of scholars published many works on
mathematical architecture (note again that it is not their term, but my trial the name an
important movement), where they applied a variety of mathematical methods, from
symmetry groups to matrices, from graphs to networks. Their activity is marked with
monographs and collections of papers, for example:
- Lionel March and Philip Steadman The Geometry of Environment: An Introduction to
Spatial Organization in Design (London, 1971),
- Leslie Martin and Lionel March, editors, Urban Space and Structures (Cambridge,
1972),
- George Stiny Pictoral and Formal Aspects of Shape and Shape Grammars: On
Computer Generation of Aesthetic Objects (Basel, 1975),
- Lionel March, editor, The Architecture of Form (Cambridge, 1976),
- Philip Steadman Architectural Morphology: An Introduction to the Geometry of
Building Plans (London, 1983),
- as well as, by papers in the journal Environment and Planning, B [series] (1974-).
Note that the first item in this list, March and Steadman’s book, also has an American
edition (Cambridge, Massachusetts, 1974), while its Hungarian translation has a
modified title Geometry in Architecture (Geometria az épitészetben, Budapest, 1975).
Independently of this, William Blackwell used the same English title for his book (New
York, 1984). The second item of the list, which was edited by Martin and March, also
inaugurated a new series of books “Cambridge Urban and Architectural Studies”, which
is, of course, not limited to mathematical architecture. Perhaps it is necessary to discuss
the idea of “shape grammar”, which features in the title of George Stiny’s book. The
word “grammar” was frequently used and even overused in architecture and design,
starting with Owen Jones’s monumental work The Grammar of Ornament. (London,
1856) and Charles Blanc’s Grammar of the Arts of Drawing in French (Grammaire des
arts de dessin, Paris, 1867); both books were frequently used by architects. In most
cases, including these two examples, “grammar” is just a metaphor: we have a set of
geometrical patterns or designs, these are somehow arranged, and we should “feel” that
the arrangement is based on a grammar. In Stiny’s case, however, “grammar” is not a
metaphor, but it actually manipulates shapes, which are defined by sets of lines. Shape
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grammars are very useful to generate all the possible designs in the framework of a
given grammar. We may compare it with a corner stone of mathematical
crystallography: finding all the possible types of periodic patterns (crystallographic
symmetry groups). Incidentally, the 17 two-dimensional or wallpaper patterns are useful
for analyzing ornamental art and urban patterns: March and Steadman’s book devotes a
chapter to this topic. Although some of the mathematical methods in the listed books
were worked out prior to the micro-electronic revolution, the research in the field of
mathematical architecture, old and new, contributed to the “computerization” and also
befitted from it. Even more methodologies were worked out for using computers in the
design process. Here we mention just one monograph:
- W. J. Mitchell Computer-Aided Architectural Design (New York, 1977).
We separated this book from the others since it represented an emerging new field.
Incidentally, Mitchell is a frequent collaborator of Stiny, who initiated the application of
shape grammar in the field of computer-aided design. On the other hand, the same
methodology can be used not only for generation of shapes, but also for formal analysis
of objects of art, as Terry W. Knight demonstrated in her book Transformations in
Design: A Formal Approach to Stylistic Change and Innovation in the Visual Arts
(Cambridge, 1994). It is a special benefit that modern methods can be applied for
dealing with historic topics. For example, Stiny and Mitchell coauthored papers on such
diverse topics as “The Palladian grammar”, which is continued by “Counting Palladian
plans”, and “The grammar of paradise: on the generation of Moghul gardens”
(Environment and Planning, B, Vol. 5, pp. 5-18 and pp. 189-198, 1978; Vol. 7, pp. 209-
226, 1980). In the first two papers, they dealt with the layouts of the villas of Andrea
Palladio, the 16th century Italian architect, and enumerated all the possible plans of
fixed sizes. Returning to the early inspiration from the Russian constructivism, the
names of Gabo and Martin appeared together again in 1957, exactly twenty years after
the publication of the book Circle: International Survey on Constructive Art (London,
1937). Specifically, Gabo’s album was published by Martin’s preface (London, 1957).
Moreover, the publisher of Circle came out with another interdisciplinary and
international collection Data: Directions in Art, Theory and Aesthetics (London, 1968),
which was edited by the artist Anthony Hill. Although “data” here is an acronym of the
subtitle (it is better to write it in all caps), we may find a nice metaphor here for the
changing time: from the to utopian discussions in a circle to modern data processing,
from the perfect geometric figure to the information age. The theoretical works of
Russian architects (Chapter 4) were also rediscovered, March started his 1976 book with
a reference to Karisl’nikov, and Stiny applied Chernikhov’s drawings for illustrating a
shape grammar. The growing in interest in Chernikhov and Krasil’nikov is represented
by many publications, including Arthur Sprague’s paper “Chernikov and
constructivism” (Survey, No. 39, pp. 69-77, 1961), Catherine Cooke’s book
Chernikhov: Phantasy and Construction (London, 1984), and her paper “Nikolai
53

Krasil’nikov’s quantitative approach to architectural design” (Environment and
Planning, B, Vol. 2, pp. 3-20, 1975).
We do not refer to more recent works in the field of mathematical architecture, because
the home pages of the referred to institutions and some related groups provide detailed
information. We also confess that our survey is very far from being comprehensive, we
missed many important developments. Our main goal was to demonstrate the beginnings
of a new field and its early inspirations.
8 A symmetric link (and two notes on the golden section)
In modern times, no architect and designer can graduate without completing courses in
mathematics. It is true, that this statement is not ‘‘symmetric’’: most mathematicians
graduate without taking any course in architecture and design, but it would be not a bad
idea to offer such courses in the framework of general studies programs. More recently,
computer aided architectural design (CAAD) is a field where mathematicians,
computer scientists, and architects should cooperate. Obviously, there are many
problems that require discussions in broader circles of architects and mathematicians.
For example, modern architecture may need new transformations and new algorithms
that are available in mathematics. In the same time, some questions in architecture and
design may inspire mathematicians. The historic reconstruction of building is another
problem where we may have fruitful cooperation among representatives from various
fields. Even history of mathematics may provide some help in special cases (I will give
an example later).
In this paper, we mentioned the topic of symmetry as a link between mathematics and
architecture several times. Thus, the crystallographer J. D. Bernal recommended it for
architects, March and Steadman’s referred to book The Geometry of Environment:
(London, 1971) has a chapter on the basic symmetry operations and another one
symmetry groups in the plane, Steadman’s book Architectural Morphology (London,
1983) deals, among others, with the symmetries of rectangular plans. We may give
further examples, including a mathematical contribution
- J. A. Baglivo and J. E. Graver Incidence and Symmetry in Design and Architecture
(Cambridge, 1983),
and a video program made by the Open University in Great Britain,
- P. Steadman Symmetry for the course ‘‘Design: Principles and Practice’’ (1992).
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If we are interested how the topic of symmetry became a part of the curriculum in
design education in a systematic form, we should go back to the complementary science
courses at the Ulm School of Design (Chapter 6). The leaders of the school invited the
chemist K. L. Wolf to give lectures on symmetry. He coauthored two interdisciplinary
books on symmetry in German:
- K. L. Wolf and D. Kuhn, D. (1952) Gestalt and Symmetry: A Systematic Presentation
of Symmetric Bodies (Gestalt und Symmetrie: Eine Systematik der symmetrischen
Körper, Tübingen, 1952),
- K. L. Wolf and R. Wolff Symmetry: An Attempt towards an Instruction in Seeing
Gestalt and Meaningfully Creating Gestalt, Systematically Described and with
Numerous Examples Explained (Symmetrie: Versuch einer Anweisung zu gestalthaftem
Sehen und sinnvollem Gestalten, systematisch dargestellt und an zahlreichen Beispielen
erläutert, Münster, 1956).
Note the presence of the term Gestalt in both tiles, which also featured in the name of
the Ulm School of Design (Hochschule für Gestaltung). The second book has not only
an extraordinary long subtitle, but also an interesting structure: the first volume gives the
main text, while the second one is an exciting selection of illustrations. The American
architect William S. Huff, who was the guest instructor of the same school’s Foundation
Course between 1963 and 1968, made the next step, a “symmetric” response. While the
chemists wrote a book on symmetry that is useful for designers, too, the architect
published a series of booklets as a visual introduction into the subject, which is also
interesting for people with scientific background. The series includes not only many
rarely seen illustrations, but also presents interesting documents from the history of the
related scientific fields:
- William S. Huff Symmetry: An Appreciation of its Presence in Man's Consciousness,
Parts 2-6, Designed by Tomás Gonda (Pittsburgh, 1967-1977).
This publication was distributed in Northern America for those universities that had
design programs in that time, while the Oppositios: A Journal for Ideas and Criticism in
Architecture reprinted three parts of the series (Nos. 3, 6, 10, 1974-1977). Another step
was due to a duo, a designer and a mathematician, who wrote a textbook jointly:
- Sadao Kumagai and Yasuaki Sawada Ornamental Patterns and Symmetry (Moyou to
shinmetorii, Kanazawa, 1983).
This book presents a visual approach to symmetry groups in the plane (wallpaper
groups), including black-and-white and colored ones. Although the book is in Japanese,
it is dominated by illustrations and tables that are given in English. A similar
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interdisciplinary cooperation between an anthropologist and a mathematician led to a
comprehensive book with the symmetry-analysis of very many ornaments:
- Dorothy K. Washburn and Donald W. Crowe Symmetries of Culture: Theory and
Practice of Plane Pattern Analysis (Seattle, Washington, 1988.
Polyhedra with various symmetry properties are frequently discussed in both
mathematics and design; these are very important tools for the development of 3dimensional
thinking and also can be applied in practice. Beyond a large number of
specialized mathematical books on polyhedra, many authors wrote non-technical, or
less technical, books that favor visual approach and consider the needs of designers.
The modern pioneers of this approach, beyond Leonardo, Dürer, and some Renaissance
artists, were H. S. M. Coxeter, who added a chapter on polyhedra to W. W. Rouse
Ball’s book Mathematical Recreations and Essays (London, 1939), and H. M. Cundy
and A. P. Rollett, who presented many useful illustrations and data on polyhedra in
their book Mathematical Models (Oxford, 1951). Later, both designers and
mathematicians contributed to the growing number of such works, including Ugo
Adriano Graziotti’s Polyhedra: The Realm of Geometric Beauty (San Francisco, 1962),
Magnus Wenninger’s Polyhedron Models (New York, 1971), Anthony Pugh’s
Polyhedra: A Visual Approach (Berkeley, 1976), Peter and Susan Pearce’s Polyhedra
Primer (New York, 1978). We should also refer to three more complex books where the
study of polyhedra play a central role: Keith Critchlow’s Order in Space: A Design
Source Book (London, 1969), Alan Holden’s Space, Shapes, and Symmetry (New York,
1971), and Robert Williams’s The Geometrical Foundation of Natural Structures: A
Source Book of Design (New York, 1979). As a clearly interesting development, some
scholars with backgrounds in architecture and design achieved not only interesting
structures, but also produced new mathematical results:
- A. Wachman, M. Burt, and M. Kleinman Infinite Polyhedra (Haifa, 1974),
- Haresh Lalvani Transpolyhedra: Dual Transformations by Explosion-Implosion (New
York, 1977),
- Koji Miyazaki An Adventure in Multidimensional Space: The Art and Geometry of
Polygons, Polyhedra, and Polytopes (New York, 1986),
- Michael Burt The Periodic Table of the Polyhedral Universe (Haifa, 1996),
as well as, some papers in the Canadian journal Structural Topology (1979-).
Of course, the more recent works are available at home pages. We suggest starting with
the mathematician-artists George Hart’s web site.
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Last, but not least the concept symmetry is also associated with proportions, which
attracted, from Vitruvius to Le Corbusier, many architects, including serious theoretical
works and non-sense proportion “hunts”. Here I would like to discuss just one “special”
proportion, the golden section, a/b = b/(a + b), which attracted, unfortunately, very few
serious works and very many non-sense trials for “gold digging”, although these people
always claimed that they found some gold. We referred to some of the more serious type
of works: the discovery of the spiral phyllotaxis in the 1830s and the survey in D’Arcy
Thompson’s book in 1917 (Chapter 2), the Russian musicologist Sabayev’s study in the
1920s (Chapter 3), and Le Corbusier’s modulor (Chapter 6). We may add that Klee and
Moholy-Nagy, two masters of the Bauhaus (Chapter 5), also used the golden section in
their teaching practice. It is widely believed that the golden section (or, in Latin, sectio
aurea) was often used in ancient and medieval art and architecture. Some scholars even
used the argument that the preference of the golden section was associated with an
interest in gold, which was used in decorations or occupied the minds of medieval
alchemists, and so on. However, there is a serious problem with this argument: the
golden section was not related to “gold” until the 19th century, but was called the
extreme and mean ratio (since Euclid’s time), the divine proportion (Luca Pacioli), the
continuous proportion (German mathematicians), and the medial section (British
mathematicians). Incidentally, the mathematical-historical works on the subject give
credit to Max Ohm, the brother of the physicist Georg Simon Ohm, for the first known
printed usage of the expression “golden section” (der goldene Schnitt.) in 1835. This
information is repeated in many etymological dictionaries in English, French, German,
and Italian. Some years ago, I reported an earlier example of 1833 and now, for the first
time, I present a further one of 1830, which is available in the first edition of Ferdinand
Wolff’s textbook of geometry (Lehrbuch der Geometrie, Berlin, 1830). Interestingly,
Wolff was a professor of the Gewerbe-Institut (Trade Institute), which played an
important role of popularizing industrial design about 70 years later (Chapter 3). His
textbook, however, deals with pure mathematics and there is no evidence that artists and
craftsmen contributed to the new terminology. Interestingly, the sectio aurea is not an
ancient or medieval expression, but a 19th century Latinization of the German term; it
demonstrated that the mathematical concept, not the expression, is ancient. I still
maintain my earlier view that the expression “golden section” was coined by educators
of mathematics. Although we moved “backwards” just by five years, from 1835 to 1833,
then to 1830, this result is still significant from the point of view of an interesting
question. Did the discovery of the botanical importance of the golden section in the case
of the spiral phyllotaxis inspire the new term “golden section”? The answer is negative:
the recently discovered example of 1830 shows that it was available shortly before the
discovery of the phyllotaxis by German and French scholars, Braun, Schimper, and the
Bravais brothers, in the early 1830s.
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I should refer to a new finding in connection with Leonardo, too. Many authors claim
that Leonardo was interested in the golden section: he illustrated the polyhedra in Luca
Pacioli’s book Divina proportione (Venice, 1509), which title refers to the golden
section, and introduced it into his proportional system of the human body. The latter is
best represented in Leonardo’s “Vitruvian man”, which was associated with a
proportional system that Vitruvius (1st c. A.D.) described in an unclear paragraph. The
first part of this statement that Leonardo worked with Pacioli is correct, but the referred
to book is not about the application of the golden section in art. This proportion is
discussed in the mathematical parts of the book, while Pacioli’s separate essay on
proportion in art suggests using simple ratios of integers. Even more problematic is the
statement that the “Vitruvian man” is based on the golden section. The usual view is that
the navel divides the height of the man according to the golden section. Let us disregard
the possible chronological problem that the “Vitruvian man” was drawn around 1490,
many years before Leonardo met Pacioli. Indeed, one may argue that the dating of
Leonardo’s manuscripts is also problematic or Leonardo’s interest in the golden section
could have started earlier. The “Vitruvian man” with outstretched arms and legs is
inscribed into both a square (homo ad quadratum) and a circle (homo at circulum),
which are not concentric. This was Leonardo’s important contribution, all earlier trials
to illustrate Vitruvius’s text considered concentric figures. In Leonardo’s case, the center
of the square is the point where the penis begins, while the center of the circle is the
navel. On the other hand, the distance between these two points is not given, and this
missing particular prompted various studies to reconstruct Leonardo’s canon. These
trials usually introduce new regulating lines and other figures, and many of these are
associated with the golden section directly or indirectly. Here I offer an alternative
approach, which is based on the accompanying text and does not require the extensive
use of new regulating lines. Let us start with the square. It is drawn on the basis of the
following observation: “The span of a man’s outstretched arms is equal to his height” (I
quote Edward MacCurdy’s translation with some corrections). Of course, a side of the
square is equal to the man’s height. Leonardo described various lengths of the body by
simple ratios of integers: 1/2, 1/4, 1/6, 1/7, 1/8, 1/10 lengths of the man’s height. I
projected the marked “maximum width of the shoulders”, which is ”a fourth part of the
man”, to the top horizontal side of the square, and calculated the distance between an
end-point of this projected line-segment to the point where the circle intersects this
horizontal line. The point of intersection is marked by the tip of the middle finger of an
outstretched arm. I calculated the described horizontal distance by considering the righttriangle
where the other two sides are: (1) the vertical distance “from the top of the
breast to the crown of the head”, which is “the sixth of the man”, and (2) the length of
the arm “from the elbow to the tip of the middle finger is the fourth part; from this elbow
to the end of the shoulder is the eighth part”, that is, all together, 1/4 + 1/8 = 3/8 height
of the man. Using the Pythagorean theorem, the horizontal distance in question is
√65/24 = 0.335...times the man’s height. If we know the exact position of this point, it is
easy to calculate the radius of the circle as the function of the height: (325 + 3√65)/576
= 0.606... times the height. Since the center of the circle is at the navel, this number also
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gives the ratio of the height of the navel to the full height of the body. It is significantly
less than the golden number, (√5 – 1)/2 = 0.618... (Incidentally, the 19th century
German scholar Adolf Zeising was the “pioneer” of considering the navel as the goldensection-point
of the height; he tried to explain almost everything with his new theory of
human proportions.) I also noticed that Leonardo’s ratios included a redundant one: the
distance from the top of the breast to the crown of the head can be calculated from
earlier data and it is 47/280 (1/8 + 1/7 – 1/10 = 47/280), but later he gave it as 1/6,
which is a good approximation of the other one. This means that Leonardo did not
hesitate to round the complicated ratios in order to have simple ones. Consequently, we
may believe that our “key ratio” of 0.335... should be 1/3. I am less enthusiastic to round
0.606... as 3/5 = 0.6. Moreover, by fixing 1/3, we determine the entire system. Thus, the
ratio 1/3 is either a well kept “secret” of Leonardo or, at least, the key to a method to
approximate his system. If we mark this point on the top horizontal side of a given
square, it is very easy to construct the center of the circle on the vertical midline of this
square, which also marks the man’s height at the middle. First, we connect our point
with the midpoint on the bottom horizontal side of the square (where the circle and the
square should have a tangential point), and then the perpendicular bisector of this new
line-segment will intersect the midline of the square at the necessary point. I do not go
into further details, but will publish this approach elsewhere. This result is a further
piece of evidence that the books on basic design should not rush to explain Leonardo’ s
“Vitruvian man” with the golden section or other sophisticated methods. The importance
of the golden section in the history of art is strongly overemphasized in the literature, as
we discussed in the paper “Golden section(ism): From mathematics to the theory of art
and musicology” (Symmetry: Culture and Science, Vol. 7, 413-441 and Vol. 8, 74-112,
1996-97). Although computer-aided design programs are able the handle proportional
systems based on the irrational value of the golden section (earlier it was not so easy), I
do not recommend “introducing” this proportion for the analysis or reconstruction of
historic monuments in an uncritical way. Since the golden section was much less popular
among artists and architects of the past than it is believed, we should use this proportion
only in those cases where we have some evidence that it was really used. We may end
up with less gold, but a few relevant cases may shine better...
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9 The Symmetry Society and Brussels
Although the cooperation between architects and mathematicians may have various
advantages, the meetings between them are relatively rare. Most symposia deal with
specialized questions and there are very few broadly interdisciplinary meetings where
representatives of science and of art may actually meet. The International Society for the
Interdisciplinary Study of Symmetry (ISIS-Symmetry), or shortly Symmetry Society,
pioneered such a forum with a triennial congress and exhibition: (1) Budapest, 1989; (2)
Hiroshima, 1992; (3) Washington, D.C., 1995; (4) Haifa, 1998; (5) Sydney, 2001; (6)
planned in Brussels, 2004. The concept “symmetry” is very useful to symbolize the
interdisciplinary co-operations since it is widely known that it has both scientific and
aesthetical meanings since ancient times. Interestingly, the very term “symmetry” was
strongly associated in particular with mathematics and architecture in a complex way
(cf., D. Nagy “The 2500-year old term symmetry in science and art and its ‘missing link’
between the antiquity and the modern age”, Symmetry: Culture and Science, 6, No. 1,
18-28, 1995):
- the original Greek expression was first a mathematical term in the sense of
commensurability, but later it was also used in aesthetics as good proportion (the term
was not associated with “mirror symmetry”!)
- it was adopted into Latin as “symmetria”, but was rarely used since the mathematical
meaning of the Greek term was rendered in Latin as commensura or commensuratio,
while the aesthetical one as proportio; the expression still survived because Vitruvius
frequently used it by making a small distinction between the theoretical vs. practical
aspects of proportions as symmetria vs. proportio,
- the expression “symmetry” became an international word in architectural texts and was
used in many languages following the Renaissance period, when many artists-scholars
translated the Vitruvian text and they needed not only the equivalent of proportio, but
also symmetria (the term was not yet associated with balance in architecture)
- it was readopted by mathematicians as “mirror symmetry” since the original Greek
meaning can be associated with the “common measure” of two equal parts; this modern
understanding of symmetry became the everyday meaning of the expressions and it was
also utilized by architects;
- architects utilized the new meaning of symmetry and used it to describe arrangement
with a mirror-reflection; however, many of them emphasized that we need a more
sophisticated balance, not just geometric symmetry
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- mathematicians, crystallographers, and biologists generalized the concept by
considering not only mirror reflection, but also other operations that transform a
geometric figure or figure-system into itself (isomorphism); thus we may consider,
among others, rotational symmetry, translational symmetry, crystallographic symmetries,
color symmetries, similarity-symmetry, biosymmetries, etc. (later physicists continued
this process and considered symmetry as invariance of a property, not definitely a
geometric one)
- it became clear that the idea of crystallographic symmetries and biosymmetries is also
useful in the theory of architecture, partly as a composition principles in the case of
individual buildings, partly as an organizing principle in town planning.
Although the main interest of the Society is symmetry and the related concepts
(proportion, balance, equilibrium, invariance, etc.), as well as the relationship between
symmetry and asymmetry, it is not limited to these. We may understand the concept also
metaphorically as the search for “symmetry” or “bridges” between concrete fields of art
and science. Last, but not least, we should remark that the some of the major figures of
the historic developments discussed earlier are associated with the Symmetry Society
directly or indirectly. The mathematician Heinrich Heesch, an Honorary Member of the
Society, discovered various tilings in the 1930s and 1940s, which were later
commercially produced. He established connections with both Peter Behrens, a pioneer
of industrial design (Chapter 3), and Johannes Itten, the initiator of the first Preliminary
Course of the Bauhaus (Chapter 5). Earlier we referred to the architect William S. Huff
(Chapters 6 and 8), another Honorary Member of the Society, who gave the Foundation
Course of the Ulm School of Design and published a series of booklets on symmetry. He
is a regular contributor of the Society and attended all of the five congresses and
exhibitions in four continents. Indeed, we are very happy to have him and his fresh ideas
at our events, which links architecture and mathematics.
There is a symbolic meaning that Brussels became a European center of the Symmetry
Society in the symmetric year of 2002 (we should wait until 2112 for the next one).
Almost fifty years ago, Brussels hosted the World Exhibition of 1958. Interestingly, but
not surprisingly, some people who featured in our survey were also there. The architect
Le Corbusier and the composer Edgar Varése presented their joint multimedia show
entitled Electronic Poem in the Philips Pavilion, which was designed by the composer
Iannis Xenakis who worked as an architect in the office of Le Corbusier in that time.
The designer Tomás Maldonado presented the new curriculum of the Ulm School of
Design during a lecture at the Brussels Exhibition. The text of this lecture was printed in
the journal of the school; it was the first declaration of some distance from the Bauhaus.
The Exhibition was also special for the engineer Frei Otto. At the beginning of this
paper (Chapter 2), we discussed how the “expo architecture”, the Crystal Palace and the
Eiffel Tower, contributed to the development of modern architecture and design. Now
the suspension structures, which were suggested by Frei Otto in his book of 1954,
61

ecame dominant. The main symbol of the exhibition, however, was the Atomium, a
building representing an iron molecule enlarged 150 billion times (designed by Polak
and Waterkeyn). Eight spherical rooms (“atoms”) were arranged, according to the bodycentered
cubic crystal-system, around a central one. The cubic structure stands on one of
its vertex (the bottom spherical room) in a position where one of the diagonals is vertical
(the elevator shaft is there). Earlier we discussed various utopian buildings that were
never built, including Chernikov’s designs and Leonidov’s “spherical library” (Chapter
4). As a further coincidence, Ulrich Conrads and Hans G. Sperlich’s book The
Architecture of Fantasy (New York, 1962, pp. 90-91) put the Atomium, the sketch of
Chernikov’s Observation Station, and Leonidov’s referred to design after each other
after. The Atomium, however, is not only fantasy, but became reality...
The first Mat mium conference of 2002 was opened in the Atomium building, in the
presence of the Brussels Television, and concluded in a Horta Building...
62

Appendix: MM = Manifesto on Mat mium (in the third
millennium)
The Atomium Building symbolizes modern architecture, the beauty of mathematical
structures, the importance of chemistry and crystallography, and, directly or indirectly,
many fields of art, science, and technology. Since it was built for the Brussels Expo and
preserved some of the original exhibitions, it also gives an insight into history and its
meaning for the future. Last, but not least the Atomium is a symbol of Brussels, the
capital of the united Europe, which is just before a historic expansion by reuniting
Western and Eastern, Northern and Southern Europe, including such regions that were
divided by political walls. This historic time is useful to express our wishes to reunite
some efforts in different disciplines, symmetrically from all directions. Of course, the
goal is not the unification of these disciplines, but to provide an informal forum where
some representatives of different fields of art, science, and technology may come
together and discuss mutually interesting topics. Another important goal is to contribute
to the modernization of education.
The discussions in design about a century ago focused on the question: “standardization
or individuality”. The U-turns in the history of design show that we need both. The too
much standardization, for example, contributed to the military machine before World
War I, while the too much individualism led to very many –isms and movements in art
and design, which disappeared very soon. The Constructivists and the Bauhaus tried to
unite the positive aspects of all of these, with a strong social commitment and, indeed,
produced some remarkable results. They tried to unite various forms of arts and crafts,
as well as to educate artists-craftsman in a unified system, but some of their idea
remained naïve utopia, while they failed to make a real unity between modern art and
modern science and technology. There are many examples in our history that too much
unification leads to failures. Thus, our goal is not to find some artists-craftsmen, as the
Bauhaus tried, or the few people who can be considered as an architect-mathematician, a
painter-biologist, a sculpture-chemist, a musician-physicist, and so on. Our actual goal is
to find artists and scientists, including architects and mathematicians, painters and
biologists, and others, who are interested in the dialogues and have a social commitment.
Instead of a unified global view, we should consider the different opinions and
contributions from all regions. We need unity in plurality, using a phrase that was used
from ancient philosophy to modern architecture by various groups of people.
In the age of globalization, we should make a special effort to listen to the views in
different countries and regions, including the smaller ones. Both design and mathematics
provide good examples where something totally new came from small countries. For
example, the initiator Art Nouveau was Victor Horta in Belgium, while one of the codiscoverers
of the new non-Euclidean geometry was János Bolyai in Hungary. We are
63

facing complex problems and the search for solutions should be also a complex
interdisciplinary and international effort. For the preparation of this, however, we need
regular meetings for dialogues, trialogues, quadrologues, ... and a “parliament” of the
disciplines of art and science...
D & D in B & B (and A & A):
Dénes in Budapest (and Australia)
Dirk in Brussels (and Africa)
64

THE ROLE OF MATHEMATICS IN THE GOTHIC
ARCHITECTURE STRUCTURAL ANALYSIS
Javier BARRALLO
Name: Javier Barrallo,
Address: University of Basque Country, San Sebastian, Spain.
E-mail: mapbacaj@telepolis.com
Abstract: For centuries the restoration works made on gothic heritage buildings were
based on experience and intuition, as there was no methodologies or mathematical
models to simulate accurately the complexity of Gothic structures.
Nowadays, the development of new technologies like the Finite Element Method, the
computer monitoring of structural elements or the stress measurements techniques
provide the necessary data to establish the intervention criteria based on scientific
results.
The geometry of the Gothic structure must be carefully measured using techniques like
topography or photogrametry. The data obtained from the measurement process, the
experimental techniques and the mechanical analysis is used to create a computer
model that represents the structural properties of the structure.
This work will show the mathematical processes involved in all these processes. We will
discuss the mathematical idea philosophically, with an emphasis on presenting a wide
range of graphical applications.
Below is the entire Power Point presentation of Prof. Barallo’s contribution.
65

FROM ITTEN’S TOWER TO VIRTUAL TOWERS:
A GENERATIVE ALGORITHM
ELENA MARCHETTI – LUISA ROSSI COSTA
Name: Elena MARCHETTI, Mathematician, (b. Milan, Italy, 1948) - Luisa ROSSI COSTA, Mathematician,
(b. Casatisma, Pavia, Italy, 1947).
Address: Department of Mathematics F. Brioschi, Milan Polytechnic, Piazza Leonardo da Vinci, 32, 20133
Milano, Italy.
E-mail: elemar@mate.polimi.it, luiros@mate.polimi.it.
Fields of interest: Functional and Numerical Analysis, Geometry, Architecture and Arts.
Publications: Numerous papers about Functional or Numerical Analysis in specialized journals. Several
papers about didactics of Mathematics in Engineering and Architecture Faculties. Same papers in Nexus
Journal referred to Mathematics and Architecture or Art.
Abstract: This paper deals with the mathematical reconstruction of different models of
Itten’s Fire Tower, which appear in his Tagebücher. A strong geometrical spirit
characterizes the configuration of the towers: consequently, we decided to describe
them using affine transformations. Vectors and matrices are the basic elements of linear
algebra, which allow us an appropriate and straightforward description of the
mathematical interpretation of the towers. Using generative algorithm to build up
virtual different towers enables us to compare them and decide which is more practical
and more pleasing to the eye. Speaking about the towers planned from other artists
around the beginning of the twentieth century, we try to remember some examples,
underlining that the spirals are involved as crucial lines.
1 INTRODUCTION
We were inspired by the numerous towers or monuments planned around the beginning
of the twentieth century to create a connection between Mathematics and Arts. With this
aim, from the point of view of Geometry, we described and built virtually The Fire
Tower of Johannes Itten (Marchetti, Rossi Costa, 2002), a monument realized only as a
prototype, known by only few photographs dated around 1919-20 (Fig.1a). Carrying out
our investigations, we found in Itten’s Tagebücher drawings and sketches related with
other configurations of towers, never realized (Fig.1b, c). We were intrigued by the idea
to reconstruct virtually, not only the known prototype, but also other different models, to
75

compare them and discover analogies and differences, in order to establish how far
mathematics unconsciously or consciously influenced Itten’s aesthetic choices.
Figure 1: a) Itten’s tower prototype, b) and c) Sketches in the Tagebücher.
In this paper in Section 2 we recall some information about Itten’s production and we
briefly describe other typical towers planned at the beginning of the twentieth century.
In Section 3 we underline the different configurations of the Fire Tower presenting their
mathematical description and a generative algorithm. In Section 4 we mention some
points of discussion, which came out during the Mat mium talk, and we give some
final conclusions.
2 THE ARTISTIC CONTEXT
2.1 Johannes Itten and the early Bauhaus
The Bauhaus School, founded by Walter Gropius in 1919 in Weimar, involved many
important artists of the beginning of the twentieth century, like Feininger, Itten,
Kandinsky, Klee, Marks, Moholy-Nagy, Muche, and subsequently students who were
particularly clever became teachers, like Albers, Bayer, and Breuer.
The School had as a purpose the mass-production of common-use things but aiming at a
fine design, as in the English Arts and Crafts movement. Therefore teachers and students
were engaged not only in the artistic field but also in mathematical or geometrical
studies, in analyzing forms and colors, in preparing models and respecting proportions.
Among the teachers, Johannes Itten as teacher of form was one of the most important,
bringing in that context his artistic and mathematical formation.
76

The members of the Bauhaus School were also involved in the philosophical and
metaphysical field and made choices characterizing their way of life. In this sense
Johannes Itten, very close to Gropius at the beginning of his stay in Weimar, had a big
clash with the founder and he left the school in 1923.
During the Weimar period Itten concluded an interesting study on towers, already begun
in Vienna before 1919. He had probably different motivations: he planned a bell-tower
or a beacon for the Weimar airport as it is evident reading his Diary (Tagebücher)
(Badura-Triska, 1990). He realized also a prototype now lost, but well known by few
photographs.
The promoters of two important Bauhaus Exhibitions in Nuremberg (1971) and in Milan
(1995) gave two different reconstructions of the prototype (De Michelis-Kohlmeyer,
1996). In our paper The Fire Tower (NNJ- Spring 2002) we examined in detail the
photograph of the prototype realized by Itten and were led to accept the Milan
reconstruction as being the more faithful to the original idea (Fig.2b). Some of Itten’s
drawings in the Tagebücher (Fig.1c) may well have been used for the Nuremberg model
reconstruction (Fig.2a), which is however marked differently from the one presented in
Itten’s photographs. Other sketches, abandoned by Itten, suggest different forms of the
tower, which he never developed beyond this primary stage (Fig.1b).
Figure 2: a) Nuremberg reconstruction, b) Milan reconstruction.
77

We decided to build virtually three different models - Model A: based on Itten’s
prototype and the Milan reconstruction; Model B: the Nuremberg; Model C: an
unrealized plan using the same method, contained in the Tagebücher.
2.2 Towers around the twentieth century
As we mentioned before we discover numerous towers planned as monuments by artists
around 1900. Some of these contain characteristic cubic or prismatic parts, anticipating
the artistic movement called Cubism; others present the spiral, an important line full of
significance. We think it is interesting to present some examples in order to underline
the two different leit-motif, frequently connected in the same monument, as in Itten’s
towers.
Probably most of these new monuments were inspired by very ancient or less ancient
examples, because all the History of Art of any age is rich in buildings having analogous
form. We only give examples dated around the turn of the twentieth century.
Figure 3: a) Rodin’s tower, b) Obrist’s tower, c) Tatlin’s monument.
The most famous monument is certainly The Tour Eiffel (1889), but we can recall
another work of the French artist Rodin, who planned in neoclassical style, at the end of
the nineteenth century, the tower named Tour du travail (1893-94), in which the spiral
appears evident (Fig.3a). It is also important to mention the prototype of Obrist for a
monument (1898-1900), a strange cone with oblique axis and spiral motif (Fig.3b).
Quite unusual is Tatlin’s Monument for The Third International (1919-20): structure
conceived in steel bars, forming a spiral developed around an oblique axis, evoking the
one of the earth (Fig.3c). The steel bars are connected in a pattern remembering a cage,
78

like the Tour Eiffel, but in Tatlin’s tower it is not possible to recognize any of the
symmetries, which characterize the Paris symbol.
Figure 4: a) Gropius monument, b) Hablik’s Exhibition Building, c) Klint’s spiral tower.
Gropius and Itten themselves used polyhedra in planning, respectively, the March Dead
Memorial (1922) (Fig.4a) and the Composition with dice (1919) (Marchetti, Rossi
Costa, 2002). Among the students of the Bauhaus School we recall two artists involving
polyhedra or spirals: Wenzel Hablik for an Exhibition Building (1919) (Fig.4b) and
Hilma af Klint for a Spiral Tower (1920) (Fig.4c).
3 MATHEMATICAL GENERATION OF ITTEN’S TOWERS
Itten studied basic Mathematics and Geometry but certainly he never applied linear
algebra techniques. The three projects for the towers presented in the previous section
have in common a strong geometrical formation. The virtual prototypes we constructed
on the base of different drawings and descriptions follow the same mathematical
procedure. Our method was to generate towers by adopting affine transformations of
only a few geometric elements. In fact basic elements can be seen to appear in the tower
many times - always reduced, rotated and translated.
We can synthesize our mathematical process by matrix operators. The points of the
structure, intended as points of the three-dimensional space 3
R , are represented in a
suitable homogeneous Cartesian system Oxyzu by four real component vectors
T
v = [ x , y,
z,
1]
.
For − l 2 ≤ x,
y ≤ l 2,
0 ≤ z ≤ l the vectors v describe a cube of side l as the basic
element of the tower’s interior. We apply to vectors v a ( 4,
4)
matrix M , which
synthesizes the three fundamental transformations recognizable in the construction of the
tower: rotation, scaling and translation.
79

The matrix M is the product of the three matrices related respectively to each single
transformation.
The product of the matrices R, S and T , related to these transformations, gives the
( 4,
4)
matrix used in the process:
⎡k
cosϑ
− k sin ϑ 0 0⎤
⎢
⎥
⎢
k sin ϑ k cos ϑ 0 0
M = TSR =
⎥
⎢ 0 0 k l ⎥
⎢
⎥
⎣ 0 0 0 1⎦
where 0 < ϑ < π 2 is the rotation angle around the z − axis, 2 2 ≤ k < 1is
the scaling
factor, l > 0 the translation factor and
⎡cosϑ − sinϑ
R =
⎢sinϑ
cosϑ
⎢ 0 0
⎢⎣
0 0
0
0
1
0
are the corresponding matrices.
0⎤
0⎥
,
0⎥
1⎥⎦
⎡k
S =
⎢0
⎢0
⎢⎣
0
The vectors w = Mv describe the cube transformed from the basic element. By
applying this process eleven times, we obtained the twelve cubes of the tower.
A similar procedure can be followed to generate the two supports and the conic surfaces
decorating the sides of the tower.
In Fig.5 you can see the projections on the Oxy plan of the cubic system and of the two
supports (like in Fig.6 and Fig.7, from left to right, the drawings are referred to the
Itten’s prototype, the Nuremberg reconstruction and the virtual model with ϑ = π 4 ,
respectively).
Figure 5: Square projections.
80
0
k
0
0
0
0
k
0
0⎤
0⎥
,
0⎥
1⎥⎦
⎡1
T =
⎢0
⎢0
⎢⎣
0
0
1
0
0
0
0
1
0
0⎤
0⎥
l ⎥
1⎥⎦

From the detailed mathematical study of the bi- and tri-dimensional figures obtained, we
have deduced that the vertices of the squares or cubes belong to arcs of logarithmic
spirals, whose equations can be easily evaluated. In Fig.6 the 3D spirals and their
corresponding plan projections are shown. The detailed mathematical description, the
particular vector-equations and the matrices involved are in (Marchetti, Rossi Costa,
Internal report, 2002).
Figure 6: Spirals in 3D.
From the mathematical point of view the different towers are characterized by a different
choice of the parameter ϑ , being the value k depending on ϑ . A generative algorithm
can give all the infinite models; in all of them the initial idea is recognizable at every
step. In this way we have generated (Fig.6) the three towers proposed by Itten,
corresponding to the value θ = arctan(1/3), θ = arctan(1/2), θ = π/4, respectively.
81

Figure 7: Elements of the virtual reconstructions.
We can observe that the shortest tower is for ϑ = π / 4 , corresponding to k = 2 2 ;
increasing ϑ between 0 and π 2 the height h ~ of the tower can be represented as a
~ ~
function h = h ( ϑ)
symmetric with respect to ϑ = π / 4 , having its minimum in
ϑ = π / 4 .
In analogy with the ancient question known as cube duplication, connected with Delo’s
altar, during the discussion in the Mat mium we were asked about the value of ϑ
giving at every step a cube of half- volume. We verified that it is possible to solve this
problem with two angles ϑ symmetric with respect to ϑ = π / 4 , corresponding to
* 3 k = 1 2 . It must be noted that in this case the limitation 2 2 ≤ k < 1 is satisfied; on
the contrary if you would like to have a sequence of cubes with a volume a third of the
previous, you can’t do it applying this process, as the corresponding k is out of the
range.
4 DISCUSSION AND CONCLUSIONS
Our talk gave us the opportunity to examine different aspects of the subject, at the end of
the presentation of the work and during informal talks. We were encouraged to give
more historical information about the production of artists around the change of the
century and about the Bauhaus period.
In the previous section 3 we answer to an interesting question related to the
mathematical study of the tower, that it is possible to halve the volume of the initial
82

cube. We appreciate the friendly atmosphere and the easy approach to Mat mium
participants.
Going back to the towers, we underline once again that all of them have the same leitmotif.
Thinking to the different versions, we built up virtual examples, which tend to be
squat (as model B and C), so we can conclude that Itten’s prototype (model A) is
graceful, soaring towards the sky. Not only is it more pleasing to the eye, but also it is
more practical, if as it is thought, it was designed to function as an airport beacon.
The mathematics is always the same; the change of angle gives the aesthetically more
pleasing model. The choice of the angle makes the difference!
References
Altamira, A. (1997) Il secolo sconosciuto, Rossellabigi Ed., Milano.
De Michelis, M. and Kohlmeyer, A. (edited by) (1996) 1919-1933 Bauhaus, Fondazione Mazzotta, Milano
Badura-Triska, E. (edited by) (1990) Johannes Itten- Tagebücher, 2 vols, Vienna.
Bogner, D. (edited by) (1994-95) Das früheBauhaus und Johannes Itten, Weimar-Berlin-Bern.
Marchetti, E. and Rossi Costa, L. (Spring 2002) The Fire Tower, Nexus Network Journal,
[http://www.nexusjournal.com/MarRos-it.html], 4, no. 2.
Marchetti, E. and Rossi Costa, L. Un algoritmo generativo per particolari trasformazioni affini, (Internal
report) Quad. Dip. Mat. Politecnico di Milano, n. 516/R, (2002).
Ray, S. (1987) L’Architettura dell’Occidente, La Nuova Scientifica, Roma.
83

PROGRAMMED DESIGN:
THE SYSTEMATIC METHOD AND THE FORM OF
PATTERN
Name: Karen Y. Li
Address: Pittsburgh, Pennsylvania, U.S.A.
Email: Karen_yli@yahoo.com
KAREN Y. LI
Abstract: A programmed design project I did in school after studying in Professor
William Huff’s Basic Design Studio has resulted in my interests in pattern study and the
systematic method. What I found particularly fascinating is, how complex form can be
generated by a simple system, and how the rigorousness of logic and the variability of
form, or simplicity and complexity can work so perfectly together, and further, how this
can be applied to architectural design and urban study, or other artistic creation.
A Programmed Design: A group of four elementary squares, each subdivided by
triangles in an asymmetrical manner, are arranged in a 9-square square – each dictated
by three regulating subsquares. This square is continuously transformed as it translates
four times across and six times down while the three regulating subsquares shift in the
numerical patterns of 123/412/341/234 across and 123/432/134/243/142/324 down –
this closed group forms one panel. By rearranging the position of triangle in the
elementary squares or changing their sequences, a different panel is generated. Through
combinatorics, a total of 24 panels are generated – each displays areas of mirrored
symmetry and mirrored antisymmetry.
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Programming “cells” and a closed panel.
The above programmed design is presented here both as an example of how to generate
a complex pattern by a mathematic system, and a case study of pattern. But the
application of systematic or structural methods is not limited in the design field. There
have been many works from various disciplines that are created in this approach, such as
the music of Arnold Schoenberg and Steve Reich; or the literary writings of Jorge Luis
Borges and Italo Calvino; or the “serial” constructions and wall drawings by Sol LeWitt.
For example, Sol LeWitt’s Variations of Incomplete Cube is a juxtaposition of all the
open cubes of 3-edge (minimal sides of a cube) to 11-edge (12-edge will be a complete
cube). His wall drawing All Combinations of Arcs From Corners & Sides; Straight, Not
Straight, And Broken Lines is another such kind of construct through combinatorics. It is
a sequence of all the 190 combinations of any two basic elements listed on the title – an
exhaustion of all the possibilities of a preset system. For Sol LeWitt, the appeal of this
approach lies on being “the way of creating art that did not rely on the whim of the
moment, but on a consistently thought out process that was interesting and exciting”,
and from which, “the realizations of straight-forward ideas often turn out to be
unexpectedly beautiful”. (Sol LeWitt and Andrea Miller-Keller, from Sol LeWitt:
Twenty-Five Years of Wall Drawings, 1968-1993, p39.)
Sol LeWitt: Variations of Incomplete Cube (left) and All Combinations of Arcs
From Corners & Sides; Straight, Not Straight, and Broken Lines (right).
86

Part I of the final pattern.
Combinatorics, or “ars combinatorial”, also takes a prominent role in the formalism of
Italio Calvino’s writings. Take his Invisible Cities as an example. The book consists of a
series of short stories of eleven subjects (“memory”, “desire”, etc.), and is structured in
the order of number “5” in a music-like sequence. If we letter the 11 subjects as A to K,
then Chapter One – the introduction – will be in the order of (A1/A2, B1/A3, B2,
C1/A4, B3, C2, D1). As soon as the number goes up to “5”, the book enters into Chapter
Two of (A5, B4, C3, D2, E1) and then Chapter Three of (B5, C4, D3, E2, F1). As such,
the same pattern is repeated from Chapter Two through Chapter Eight, while the
subject/letter continuously fading in and out. Then in Chapter Nine, the order becomes:
H5, I4, J3, K2/I5, J4, K3/J5, K4/K5 – as an ending, formally if not thematically, appears
both as an echo and an inversion of Chapter One. And by this, all combinations of
numbers and letters are complete. Through this compositional method, what is presented
in the book is not a conventional storyline, but the interweaving of multiple layers and
the complex texture of a city.
Compare these examples with the programmed design at the beginning, there are some
common characteristics of the projected patterns, either visual (3D & 2D) or textual or
tonal, that need to be further noted and studied.
First is the contrast and interplay between the rigorous order and the apparent
randomness of the pattern, and the simple repetition of basic elements and the complex
variation of the results. What this observation tells us is: on one hand, the phenomena
may seem random or even chaotic, but a logic can be always assumed hidden behind
even though it may not be easy to discern; on the other hand, the sum of the set of orders
may be simple, but its working can be highly complicated.
Further, what results from the generative system is not a composed figure, but a pattern
by the autonomous operation of the system. In other words, the orders of the system may
be conceptually or contextually referential, but the form they generate is completely selfreferential.
Thus, the usual figure-field relation is irrelevant here. A pattern itself is a
field without singular focus or favours to any privileged point. Contrasting with the
classical tradition that parts are governed by the overall formal relations (proportion,
axis, etc.), in a pattern, parts are simply parts and all parts are equal. Each part is
indifferent to or unaware of the form of the whole.
87

Part II
Yet, is defined only by its relations with its neighbouring parts. For example, in the
programmed design at the beginning, a rotational relation defines each unit against its
preceding and following unit without knowing the consequential symmetric or
antisymmetric relations of the whole.
We should also note that the form of each basic element (e.g. the elementary squares
with triangle in the programmed design) is independent of the system: while the under-
lying structure may be the same, the form of the element is changeable, and each change
will cause substantial changes to the pattern, and vice versa. To use architecture and
urbanism as an example: “absolute liberty is granted to the single architectural fragment,
but this fragment is situated in a context that it does not condition formally: the
secondary elements of the city are given maximum articulation; while the laws
governing the whole are rigidly maintained.” (Manefredo Tafuri: Architecture and
Utopia, 1973, p38.)
Versailles (left, painted by Pierre Patel, ca. 1668) is an example of
classical composition that emphasizes focal points and axes. Right:
Bucuresti 2000 master plan by O.C.E.A.N. is an urban design example
that uses systematic method emphasizing field conditions and patterns.
Finally, similar to algorithms, the set of rules that comprise the system usually address
the conditions and procedures rather than an intended form. Thus, we have seen more
uses of diagram for representation in the practice of such an approach than perspective
and illustration. Also, since there are various factors, individuals, forces, and operational
methods that are addressed by these rules, each system in fact comprises a number of
smaller systems.
88

Part III
As such, the proceeding of the system that results in the unfolding of the pattern is a
complex act of superimposition and interaction between each individual system.
Therefore, each system is in fact an open system, and is capable of variation and
adjustment by accepting any changes or interference from any additional elements.
It is my belief that the understanding of pattern can help us to understand the complexity
of nature and culture. What have been discussed above are just some basic observations,
and I believe that the current cross-disciplinary studies on Complexity and System will
provide a much boarder and stronger theoretical base. In a world that is both complex
and dynamic, the static and homogenous nature of absolute control and unity has
rendered the notion itself obsolete and inappropriate. However, a world without any
order would prove to be too chaotic. It seems to be the balance and union between
system and individuality, orders and chaos, logic and chance, structure and randomness,
and repetition and variation that we would like to pursue. What is suggested here is, on
one hand, we need to continue to find out the systems that are hidden behind the
complex phenomena, on the other hand, we can apply these systems that we have
understood to guide us, and help us to set up a minimal structure that can both maintain
an order and allow the maximum possibilities and freedom.
References
Allen, Stan: From Object to Field, Architectural Design 1997, May-June, P27.
Hurtt, Steven: The American Continental Grid: Form and Meaning, Threshold, Journal of the School of
Architecture, University of Illinois at Chicago.
Calvino, Italio (1972-74): Invisible Cities, translated by William Weaver, Harcourt Brace & Company.
89

DEFINING BASIC DESIGN AS A DISCIPLINE
WILLIAM S. HUFF
Name: William S. Huff, Professor Emeritus (b. Pittsburgh, Penna, 1927).
Department of Architecture, School of Architecture and Planning, State University of New York at Buffalo,
Buffalo, N.Y., U.S.A.
Home address: 1326 Murray Avenue, Pittsburgh, PA, U.S.A.
E-mail: wshuff@earthlink.net
Fields of interest: Basic Design (including symmetry, theory of structure, visual topology, Gestalt principles,
color perception, function of color), Architectural Design
Award: Honorary Member of the Board, ISIS-Symmetry, 2001
Publications: “Ordering Disorder after K. L. Wolf,” in: Forma, 15, Tokyo (2000), 41-47; “The Landscape
Handscroll and the Parquet Deformation,” in: Katachi U Symmetry, Tokyo: Springer-Verlag (1996), 307-314;
“The Programmed Design: Probing the Discernibility of Properties of Symmetry,” [Abstracts] Symmetry:
Culture and Science, 6 (1995), 254-257; “That Unordinary Mirror-Rotation Symmetry,” [Abstracts]
Symmetry of Structure, 1, Budapest, Hungarian Academy of Sciences (1989), 228-232
Abstract: Johannes Itten’s original Bauhaus Vorkurs took from the great movements of
Cubism, Constructivism, and Expressionism, all developed by the end of the second
decade of the 20 th century, creating thereby an introductory medley of Modern Art.
Josef Albers revolutionized Basic Design by purging it of unrestrained expressionism
and established it as a full-fledged discipline that deals only with universal and
persistent, non-objective formal elements of the visual world. At the Hochschule für
Gestaltung, Tomás Maldonado retained the Albers model of strict non-objectivity and
linked it interdisciplinarily to abstract fields of knowledge, among which are symmetry
theory, topology, and Gestalt principles. In the discipline of basic design, the
opportunities for self-determined, non-applied research are copious.
1 BACKGROUND AND BASIS
Johannes Itten’s original Bauhaus Vorkurs borrowed from the great movements of
Cubism, Constructivism, and Expressionism, all of which had been fully developed by
the end of the second decade of the 20 th Century—that is, by the time the Bauhaus
opened in 1919. In his Vorkurs, Itten created, consequently, a sort of introductory
mélange of Modern Art. Josef Albers revolutionized the Vorkurs at the Bauhaus by
purging it of unrestrained expressionism and established it as a full-fledged discipline
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that deals only with persistent and universal, non-objective, formal elements that inhabit
the visual world.
With the closing of the Bauhaus, Albers took his strict regimen of non-objective design
studies to Black Mountain, N.C. The teaching of color in any manner had been reserved
at the Bauhaus for Albers’s elders, particularly Kandinsky and Klee; so it was at Black
Mountain that Albers developed the most notable of his several basic design topics,
based not on the theory of color (the teaching of which he depreciated), but on the
perception of color. That study culminated at Yale with his ambitious book, The
Interaction of Color. At the Hochschule für Gestaltung, where Albers had twice served
as a guest teacher, his model of strict non-objectivity was retained, first, by Max Bill, a
Bauhaus student of Albers, and subsequently by one of a group of Argentinean
vanguards of art concret, Tomás Maldonado. Maldonado went one step further; he
linked non-objective design interdisciplinarily to abstract fields of knowledge—among
which were symmetry theory, topology, and Gestalt principles.
Laying this down as a background, I claim that basic design is a discipline unto itself—
so long as it deals only with non-objectivity. As a complete domain—perhaps, a strange
domain—it is one that is all-empowered, self-contained, and possessed of all the
prerogatives of sovereign sway. Another way of putting it, non-objective basic design
does not involve issues that are specific to any of the disciplines of applied design:
architecture, graphic design, or product design, and allied fields. I will not take the time
to make that argument here. Permit me, at the same time, to introduce a consummate
statement, regarding this point. It comes neither from Albers nor Maldonado, but from
Iakov Chernikov of the Vhuetmus school of design of Moscow, a design school that
predated the Bauhaus:
Specific functions or subject matter as such, do not play any part in this course of
teaching. Not once do we use real briefs and problems. The whole methodology is based
upon the development of “combinations” and “assemblages” of lines, planes and
volumes, independent of what the given elements may represent. Just as an appropriate
assembly of sounds gives us musical products, so too we construct and assemble a
representation in which lines, planes and volumes can be musically turned. Thus we
create a skilled composer of new forms.
The interesting result of this non-objective approach is that we produce an executant
who is freely capable of handling tasks that are based on real subject matter, for the nonobjective
and real are erected upon identical principles of form. 1
1
Catharine Cooke, Iakov Chernikov: AD Profile, vol. 55, IC, 22.
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Taking another cue from the writings of Albers (“the discrepancy between physical fact
and psychic effect” 2 ), my basic design studio (which I had recently preferred to call a
“formative design studio”—though I now think it should be called a “non-objective
design studio”) dealt principally with two issues: how structure actually is and how
structure is perceived. By structure, I mean elements (or parts) of a design and how they
are related or (arranged) in a design. In regard to the visual manifestations of design, the
elements are basically shape and color; and to arrange such elements is to design. When
concentrating on the actual nature of structure, its perception was always brought into
consideration; and it was the other way around when concentrating on the perception of
structure.
2.1 ACADEMIC RESEARCH
Working on non-applied research has the disadvantage that there is not much internal
institutional funding and, often, even less external funding; and deans do not regard
unfunded research to be of much importance. The flipside is that researching formal
issues very often requires little or no funding; and it affords, therefore, relative freedom
for the researcher to do exactly what he chooses to do—that includes not having to meet
arbitrary deadlines. In my case, at least, it seems that research found me, perhaps
haphazardly, but welcomely. My deep-rooted curiosity, companioned with a certain
facility of observation and a decided visual orientation, had something to do with that, to
be sure.
When I studied at Yale, just about every one of my design critics practiced: Eliot Noyes,
Philip Johnson, Louis Kahn, Paul Schweikher, Edward Stone, and Richard Neutra,
among others. Frederick Kiessler practiced very little; rather, he sculpted and exhibited
eccentric architectural projects. Two of my critics were in-house faculty, who did not
have practices, even if they received an occasional commission; and they handled most
of the perfunctory activities around the school to make up for that deficiency. At the
same time, the curriculum in those days was put out by the department’s chairman, who
took counsel with his faculty rather sparely.
When I began teaching in a school of architecture, research was still not a requirement
for a sideline career of critiquing design students; the mode still was to have a practice.
Then, non-practicing graduates of schools of architecture filtered into the academy; and
they began to justify their fulltime faculty positions through research. Thus, I entered
teaching at the cusp between design critics who were practitioners and design critics
who were research academician.
2
Josef Albers, Search Versus Re-Search, (Hartford, Conn.: Trinity College Press, 1969), 10.
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I was, in fact, doing architectural work on my own at the beginning of my teaching
career; and I was one of the first (July 1965) to be awarded an Arts and Humanities
Program (then NEH) grant, which led to the series of booklets, Symmetry: an
appreciation of its presence in man’s consciousness. The project’s justification,
however, did not hang on its own merits as an artifact of abstract knowledge; instead, it
was framed as “a tool for design education,” as indicated by the title of my proposal,
“Issues of Symmetry in Design Education.” I had been introduced to the topic of
symmetry by Tomás Maldonado at the Hochschule für Gestaltung (HfG), mainly
through the text of K. L. Wolf and Robert Wolff; and it is my recollection that
Maldonado had encouraged me to develop such a study—with one intention being the
inclusion of other members of the HfG family.
Excepting for a small supplemental grant on the same project, this was the only research
funding that I ever received. I might add, this was the only research project on which I
ever had to meet deadlines; and an absence of deadlines has allowed projects to
mellow—to improve with age. Since, then, a large part of my research, but not all by far,
has come out of the classroom.
2.2 RESEARCH IN THE BASIC DESIGN STUDIO
To conduct research deliberately in the undergraduate classroom can dilute and abuse
pedagogic purposes. Students should not be shortchanged for the sake of research. The
fact is, however, research topics can just turn up in the design studio (the design studio
being a rather different environment from the typical academic classroom) in many an
unpredictable instance and does so, almost inevitably, as a byproduct—when the
curriculum itself is solid and forward-looking. A large part of the studio way is the
facilitating of the student’s inventiveness. But the student is often not equipped to judge
what is truly inventive, in respect, for instance, to solutions that other students had
already hit on. It takes the discerning eye and seasoned experience of the instructor to
catch what is really new and significant. Furthermore, the grist for research can occur
when the experiences of previous studios are built into subsequent ones. Unforeseen
solutions often create unforeseen questions.
Without a doubt, I can make the case that my students were never deprived—even while
material that was destined for research was ensuing from my assignments—unawares to
myself, often, at the time of its occurrence. My objective with my students was mainly to
train them to become proficient and self-sufficient in aesthetic judgment; their
inventions—innovations, which would not have happened without the context of their
instructor’s studio task—were more of a dividend than a demand. (I should make clear
that the identities and contributions of individual students have been scrupulously
acknowledged in all publications that have included their work—so long as editors
honored my watch over that.)
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In my design studio I assigned two types of tasks: the exercise and the project. Exercises
tested the students’ acquisition of new information, attuned them to the basic formal
issues of design, trained them experientially in perceptual acuity, and implemented them
with the called for skills of presentation. Exercises are precisely prescribed. At the same
time, their drill tends to have a timelessness that works across the ages.
Projects challenged students to the extents of their inherent and newly acquired formal
abilities, having been awakened, it was hoped, by the exercises, and educed their
ingenuities. Projects allow great latitude within the framework of a challenging
geometrical proposition or perceptual ambiguity. Though their topical particulars
become dated, they make openings for creativity.
Exercises, it is submitted, are to design what scales and etudes are to music; projects are
preludes to composing.
3 FOUR THREE-DIMENSIONAL PROJECTS
Periodically, I offered basic designs studios that were conducted in a three-dimensional
format, rather than in a more usual two-dimensional one. The topics are ones that have
only three-dimensional physicalities. Both types of studios covered the same general
formal issues of design. Four of the three-dimensional projects that I developed are
presented here; many remarkable results have come out of them.
1. The Twofold Mirror-Rotation Project [Fig. 1] came out of my one funded project,
“Issues of Symmetry in Design Education.” A grasp of the concept of mirror-rotation,
even in its simplest twofold form, was made difficult by K. L. Wolf’s German text—a
text that Germans found difficult to penetrate. This intriguing type of symmetry, initially
baffling me, led to its exploration and exploitation in the design studio. If its properties
took inordinate time for our research team to learn, how many others in our midst still
do not know what it is? The basic design studio seemed to be a good place to make
demonstrative, didactic models of twofold mirror-rotation, as well as to try to bring out
some of its aesthetic potential.
Of a sudden, while trying to find a good way to explain mirror-rotation to others, I
realized that I had a simple didactic example of this symmetry operation in my own two
hands—literally: I mirror my hands (like Dürer’s “Praying Hands”)—then, I rotate one
hand in respect to the other by 180°—now, I have 2-fold mirror-rotation. This is what
Wolf called a “coupled coverage operation”—simultaneously mirroring while rotating,
or simultaneously rotating while mirroring.
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See:
2. The Fourfold Mirror-Rotation Project [Fig. 2] came to light while I was familiarizing
myself with crystallography. At the beginning of the funded project, no texts, which I
had referenced, made clear that symmetry theory—that is, a whole system of
symmetry—came from crystallography, or that molecular chemistry was decades behind
crystallography—mainly because the nature of the molecule was not well understood at
the time that the 32 finite Crystal Classes had been established.
It was from papers of Federov, translated into English through David Harker, that I had
learned about famed Auguste Bravais’s missing a thirty-second Crystal Class: the
symmetry of fourfold mirror-rotation. For well over a century, the discovery of that class
had been attributed to Hessel—and, in fact, he is the one to whom all the other early
crystallographers referred. It was relatively recently, in 1986 (only three years before the
organizing meeting of ISIS-Symmetry in Budapest), that it became widely known about
one Moritz Ludwig Frankenheim who, in 1826, had preceded Hessel’s publishing all 32
Classes in 1830—and Frankenheim had done it in a brilliantly simpler way than Hessel,
through combinatorics. 3
While materializations of this symmetry type exist in some numbers at the molecular
level, fourfold mirror-rotation is rare at human scale. It is known in crystals; but no manmade
artifact is known to employ it, at least not known to me. Though some students in
earlier studios produced a few objects with this property of symmetry, in 1986, when I
taught some sixty beginning design students at the University of Hong Kong, the
majority of them made fourfold mirror-rotation objects, though some made sixfold ones.
May I claim that never in the history of the world were there so many differently
designed and produced fourfold mirror-rotation artifacts in one place at one time?
3. The Project of the Trisection of the Cube into Congruent Parts [Fig. 3] has its own
odd history. Already exploring for a decade the sectioning of the cube into any number
of congruent parts—and the congruent sectioning of other regular and semi-regular
solids, I concentrated on the trisectioning of the cube with my students, after Martin
Gardner had faltered in his presentation of this particular topic in his September 1980
column, “Mathematical Games,” for Scientific American.
Since then, I have found a number of noticeably different types of trisections. Though I
can surmise, not being a topologist, I cannot be certain which trisection types are unique
in respect to others (there are a minimum of three, I believe) and which types are only
variations of others.
3
J. J. Burckhardt, Die Symmetrie der Kristalle, (Basel: Birkhäuser Verlag, 1988), 34-47.
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4. The Project of the Non-orientable Surface [Fig. 4]. Max Bill had come to the HfG
with a history of sculpting the Möbius band, more generically called the non-orientable
surface. Bill produced such a band for the Breuer-Roth Doldertal Apartments in Zurich,
but he did not know about the nature of its peculiar topological properties until a
mathematician informed him. It was Maldonado, however, who introduced the topic to
his Grundlehre classes at the HfG. When I continued studies of non-orientable surfaces
in my own studio, our first question of the Möbius band was whether any properties of
symmetry could be introduced into such a type; for upon the casual viewing of its usual
depiction, it appears to be irregular. Being a topological entity, the Möbius has no fixed
cross-ratio and is, therefore, subject to infinite deformation (as Bill’s original sculpture
serves to demonstrate); however, in its usual presentation, a special case to be sure, it
conforms to twofold rotational symmetry; thus, this property of symmetry can be
imposed at will on such an artifact. One student demonstrated that, by starting with a flat
pattern of a fourfold rotational nature, she could easily attain fourfold mirror-rotation in
her configuring that pattern into a duplex non-orientable object.
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Fig. 1. Twofold Mirror-Rotation Artifact; Fig. 2. Fourfold Mirror-Rotation Artifact;
Basic Design Studio of William S. Huff; Basic Design Studio of William S. Huff;
Student: Lisa Lu, SUNY at Buffalo, 1980. Student: Sally Kilmer: SUNY at Buffalo, 1990.
Fig. 3. Array of Three Trisected Cubes; Fig. 4. Non-orientable Surface (with surgical clamp);
Basic Design Studio of William S. Huff; Basic Design Studio of William S. Huff;
rows from right to left: Student: Habib Khalafi, SUNY at Buffalo, 1980.
Student: Vincent Marlowe, SUNY at Buffalo, 1978;
Student: Fernando Polletta, SUNY at Buffalo, 1983;
Student: Michael Cooke, SUNY at Buffalo, 1987.
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DESIGN AND COGNITION:
CONTRIBUTION TO A DESIGN THEORY
CLAUDIO GUERRI
Name: Claudio F. Guerri, Architect (b. Rome, Italy, 1947).
Department of Morphology and Communication, Faculty of Architecture, Design and Urbanism, University
of Buenos Aires, Buenos Aires, Argentina.
Home-Address: Olleros 2532 2ºA, (1426) Buenos Aires, Argentina.
E-mail: claudioguerri@fibertel.com.ar
Fields of interest: Theory of Design, Graphic Languages, Space and Visual Semiotic, Architecture, Graphic
and Industrial Design
Awards: Article of the Year (Best Art Critic Article) AICA Award, 1986;
Cientific and Technological Production Award, SICyT-University of Buenos Aires, 1995.
Publications:
(1988) Article: “Architectural, Design, and Space Semiotics in Argentina” in The Semiotic Web 1987. A
yearbook of Semiotics by T. A. Sebeok and J. Umiker-Sebeok (eds.), pp. 389-419. Berlín: Mouton de
Gruyter. ISBN 3-11-011711-8
(1997) Symposium paper: “Deep structure and design configurations in paintings” in Semiotics of the Media:
state of the Art, Projects and Perspectives by Winfried Nöth (ed.), pp. 675-688. Berlín: Ed. Mouton de
Gruyter. ISBN 3-11-015537-0
Abstract: The traditional graphic languages used for designing, orthogonal projective
geometry and perspective, account for only quantifying or qualifying space. They do not
allow for conceptualizing or specifically working with pure design, the morphological
and aesthetic aspects in the practicing of design. TSD is a new graphic language that
systemizes necessary and sufficient morphic and tactic dimensions of planar and
volumetric forms, allowing all possible relationships of selection and combination, basic
operations of pure design. Symmetry is a property of high isotopic synthesis and,
therefore, affords an effective cognitive strategy in the practice of design that is widely
employed in it, whether obvious or hidden in a design’s final outcome.
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1. INTRODUCTION
Although there is no certainty, sedentarism, durability of constructions and geometric
thinking emerged at the same time. In his famous History of Architecture, Fletcher
(1896 [1956]: 1) claimed that Architecture, with all its varying phases and complex
developments, must have had a simple origin in the primitive efforts of mankind to
provide protection against inclement weather, wild beasts, and human enemies. This
explanation about the origin and also most writings about architecture stress function,
highlighting construction as an inhabiting-protective device. The time has come to
change the discourse and analyze the pure form related aspects, “art”, “the artistic side”,
“the esthetic concern”, etc. from the angle of the prefiguring possibilities that enables
the design operations through the available graphic languages in the framework of an
incipient “Grammar” or Theory of Design.
Architecture was born at a very complex crossroad. Architecture results from the
application of design to the formal elaboration of constructions mankind builds to
inhabit the world. Architecture is both the use of built space and the delimitation of
inhabited space. On the other hand, Design is creation, delimitation, and articulation of
form, articulation of form to take under control the construction and dwelling
necessities. As autonomous knowledge—from construction and dwelling—Design
enables, —through a project—modifying the first anthropoid’s monotonous and
utilitarian constructions. (Jannello 1983: unpublished).
Geometry was probably the first development in human thinking about the world that
had not emerged directly from sensory experience. Geometry belongs to the world of
forms, and its application to inhabitable constructions allowed for its exact formal
determination. The history of architecture, as one kind of applied design, has produced
in five thousand millenniums a vast typological repertoire of combined shapes with their
own syntax. Not much has been said about this syntax or combination.
One of the first problems is to understand that Geometry was not conceived for purposes
of Design, although since ancient times some of its intuitive and non-systematic
properties and techniques have been used, such as geometrical tracings (Ghyka
1927: 221-322; Naumann 1930; Lawlor 1982 [1992]: 23-64). These still are and have
always been the quintessence for appreciating the value of symmetry operations,
generally hidden in constructions but present in tracings. Even though Geometry deals
with forms only from an entitative viewpoint, geometrical tracings were generally used
to explain the structure of symmetry. Designers, however, have conceptual and practical
needs: to compare and establish relationships between forms and to select and combine
them.
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Graphic languages, such as Monge System (orthogonal projections) and Perspective
(conic projections), traditionally used for designing, account ‘only’ for quantifying or
qualifying space, respectively space of construction or dwelling space. They do not
allow for conceptualising or working with pure design (Jannello 1980; Guerri
1988: 394) understood as selection and combination of forms without pertaining to an
object. Because of Jannello’s original research context —Argentina 1950-1985, pure
design is some how related to the Bauhaus’s Formlehre and basic design (Huff
1990: 76-85).
In turn, going to the geometry-symmetry relationship, we must acknowledge that a
geometric figure as a whole, as an entity, as a problem of simple selection of
differentiated recognition, lacks symmetry. Symmetry implies parts, which are in
symmetry. It could be the parts of a figure or several figures. When there is symmetry,
there must be a combination of parts. According to Nicolle (1950: 16), ce qui nous
frappe, c’est donc une certaine répétition, soit d’un objet, soit d’éléments qui le
constituent. A geometric figure considered as a whole has no parts. Its entitative
consideration would not allow for a symmetrical approach. The only way of finding
symmetry in a figure is to consider partial, syntactic, or combining aspects based on
subdivision criteria. But Geometry does not refer to comparisons among figures, nor to
combination aspects between figures—either flat or volumetric—in relation to constant
and comparable dimensions (Jannello 1984; Guerri 1988: 399). Geometry was not
“created” specifically for designers.
2. GRAPHIC LANGUAGES
The traditional graphic systems, such as orthogonal and conic projections, were
developed as methods of execution. They were never conceived as ideological systems
implicit in any language, either graphic or verbal.
The first graphic system explicitly defined as such is the graphic language TSD. TSD is
the acronym for Theory of Spatial Delimitation used in honour of its first proponent
César Jannello (1918-1985). This graphic language systematizes all possibilities of
selection and combination of flat and volumetric figures. It establishes necessary and
sufficient morphic and tactic dimensions to account for all possible relationships of
selection and combination, the pure design operations.
The morphic dimensions are Formatrix, Size and Saturation (Jannello 1984; Guerri
1988: 406-408). These three dimensions can be justified to be necessary and sufficient
conditions from a geometrical point of view: they determine a point in space, in the
infinitely high semi-cone of morphic selection possibilities (Guerri 1988: 398), and from
a logical-semiotic (Peircean) point of view, because they are firstness, secondness, and
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thirdness respectively (Peirce CP 2.235-2.241). The tactic dimensions are Tactrix,
Separation—in two or three directions, and Attitude (Guerri 1988: 409).
TSD systematizes the execution of tracings and subsequent description in hierarchical
tree structures of corresponding complex configurations of pure design. (Guerri
1988: 389-419). Thus, insofar as each graphic language supposes different types of
analogies, each of them has its specificity: Perspective to speak about quality of space,
Monge System to state the quantity of matter present in that space, and TSD to account
for the relationships in which matter and space abide.
Regarding Symmetry, a variety of conclusions may be drawn from the TSD proposal
and its use through CA-TSD—Computer Aided TSD—as a specialized-graphic
software:
1. Although Geometry is taken as its foundation, given morphic and tactic dimensions,
considered necessary and sufficient for both flat and volumetric figures; what is
considered a figure and a simple configuration for this graphic system is thus redefined.
For example: a rhomboid and a trapezium are no longer figures for the system and will
be considered as design configurations. A simple configuration is the combination of at
least two figures. The tactrix of a simple configuration is defined by a description of the
morphic relation of the intervening figures.
2. The morphic paradigm generates the figures as a continuous group of substitution
possibilities. As to symmetry, figures should be considered on account of their lineal
components and of the dimensions proposed by the graphic language TSD itself. For
example: Saturation, insofar as it is a qualitative evaluation of figures, implies a problem
of constance-variation, which is usually known as proportion—but that, is a concept that
cannot be used to compare two irregular figures. Saturation also posits a problem of
presence-absence, there will always be some kind of saturation-proportion in any figure,
but there may or may not be saturation-symmetry.
3. The tactic paradigm generates simple configuration, as a continuous set of integration
possibilities. The set of morphic and tactic paradigms constitutes theoretically
exhaustive systems of all possible—flat and volumetric—figures and configurations. It
demonstrates that symmetry operations are a specific convergence of morphic and tactic
dimensions; and it demonstrates that given any two forms, any simple flat or volumetric
configuration, all derived symmetry variables can be determined a priori.
4. For Design, Symmetry is an operation of a higher isotopic synthesis and, as a result,
it is one of the most widely used design operations. Since its origin, it has been known as
a concept linked to regularity that has been used to keep order, a way to skirt chaos,
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monstrosity, and shapelessness. History has it that symmetry was patently used or
hidden, denied or reconsidered, but has always been present in design processes.
3. SYMMETRY
In Western architecture, order is construed in an agitated dialectics between historical
periods –taken as concrete units- and the general structure of architectural design since
its origin, up to current times. Thus, the concept of order in design (style) is not a
constant concept but the manifestation of structured fluctuations. In this history of
fluctuations, of permanence and change, the concept of Symmetry follows its own
development, and will be considered by Western thought as a concept of regularity.
In architecture, symmetry has been one of the pillars for design—to the extent that all
epochs have been ordered around it. Thus, symmetry anchors an abstract practice of
design in a concrete design operation. As stated, it becomes the first design algorithm.
In Greece, symmetry is ostensible, generally, mirror and twofold rotation. The
Renaissance continued making it ostensible, but in a more complex way: there was a
tendency toward higher degrees of rotation; Villa Rotonda, with fourfold rotation;
Bramante’s San Pietro in Montorio with multi-fold rotation. At the break of the XXth.
Century, Modernism meant a significant change in the application of symmetry, insofar
as it was not concretely built into the constructed object. This rupture in ostensible
symmetry lead to the development of hidden symmetry in subjacent design operations,
never shown on plans or in buildings. Post-modern architecture attempts to establish a
new order completely different from the modern order: it counters functional and
constructive principles that, in their discourse, backed the basic principles of
Modernism, and once again welcomes symmetry; but symmetry becomes a reference
and a parody of other symmetries. In post-modern architecture, there is a transposition,
and designed or built symmetry is a reflex or reference to some historical symmetry.
The history of architecture reveals that, as in any human aspect, there are no laws, but
rather a series of different rules that have been diachronically occurring over time.
CONCLUSIONS
TSD is presented as a temporary improvement regarding control over pure design
operations. Within pure design operations, the possibilities of symmetry, in any
configuration, are predetermined a priori by the tactic paradigm.
As shown in the computerized and transparency examples, the use of TSD for analyzing
architectural design—Palladio, Le Corbusier, graphic design—El Lissitzky—or pictorial
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design—Velázquez, Malevich—allows for systematizing one’s approach to aesthetic
values, thus contributing to the understanding of both historical practices as well as new
design possibilities.
Putting aside the problems of perception, TSD implies a new code of graphically
representing the problem of Symmetry; and thus, it is also a new cognitive approach.
References
FLETCHER, Banister (1896) A history of Architecture on the comparative method. London: Batsford
[1956].
GHYKA, Matila (1927) Esthétique des proportions dans la nature et dans les arts. Paris: Gallimard.
GUERRI, Claudio F. (1984) "Semiotic characteristics of the architectural design based on the model by
Charles S. Peirce" in Semiotic Theory and Practice, M. Herzfeld and L. Melazzo (eds.), 347-356.
Berlin: Mouton de Gruyter, 1988.
——— (1988) "Architectural design, and space semiotic in Argentina" in The Semiotic Web 1987, T. A.
Sebeok and J. Umiker-Sebeok, eds., 389-419. Berlin: Mouton de Gruyter.
HUFF, William S. (1990) “What is Basic Design” and “Basic Design Studios of William S. Huff” in
Intersight 1, 76-86, New York. ISSN 1049-6564
JANNELLO, César (1980) Diseño, lenguaje y arquitectura. Buenos Aires: Course lectures, Facultad de
Arquitectura y Urbanismo-UBA (mimeo).
——— (1984) "Fondements pour une semiotique de la conformation delimitante des objets du monde
naturel" in M. Herzfeld & L. Melazzo, eds., Semiotic Theory and Practice, M. Herzfeld and L.
Melazzo (eds.), 483-496. Berlin: Mouton de Gruyter, 1988.
LAWLOR, Robert (1982) Sacred geometry. London: Thames and Hudson [1992].
NAUMANN, Hans Heinrich (1930) Das Grünewald-Problem und das neuentdeckte Selbstbildnis des 20
jährigen Matis Nithart aus dem Jahre 1475. Jena: Eugen Diederichs.
NICOLLE, Jacques (1950) La symmétrie et ses applications. Paris: Albin Michel.
PEIRCE, Charles S. (1931-58) Collected Papers of Charles Sanders Peirce, vols. 1-6 by C. Hartshorne, P.
Weiss, vols. 7-8 by A. W. Burks. Cambridge: Harvard University Press.
PRIGOGINE, Ilya (1983) ¿Tan sólo una ilusión? Una exploración del caos al orden. Barcelona: Tusquets.
On the next pages are parts of Prof. Guerri’s Power Point presentation.
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GROWTH, CURVATURE AND COMPUTATION
CHAIM GOODMAN-STRAUSS
Name: Chaim Goodman-Strauss, Mathematician, (b. Austin, Texas., U.S.A., 1967).
Address: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701, U.S.A. E-mail:
cgstraus@uark.edu
Fields of interest: Geometry, topology, symmetry and ornament.
Publications:
Goodman-Strauss, C. (1998) Matching rules and substitution tilings, Annals of Mathematics. 147, 181-223.
Goodman-Strauss, C. (2000) Open questions in tilings, preprint.
Goodman-Strauss, C. (2002) Regular Production Systems and Triangle Tilings, preprint.
Abstract: We discuss some new mathematical techniques for describing growth and
form across a range of mathematical, biological and artistic applications. The essential
idea is in the air these days: to describe emergent complex behaviour through simple
local interactions. These interactions are shaped by local combinatorial restrictions,
and in turn give rise to global geometric properties and tremendously rich behaviour is
possible. New tools are proving to be quite powerful, producing a body of interesting
mathematical, but too they seem to shed light on real phenomena, such as the way that
tissue grows to produce complex geometric forms such as the human ear. We will
discuss the mathematical idea philosophically, with an emphasis on presenting a wide
range of graphical applications.
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In this talk, I’d like to present a model of growth and form that arises from local
interactions among little combinatorial agents. These agents can be thought of as puzzle
pieces which may fit together in certain ways; two pieces can be neighbors only if they
are compatible. To visualize this, imagine a sea filled with copies of a few kinds of the
organisms in Fig 1: these organisms have hands of certain shapes and sizes that can fit
together in only certain ways with each other. Even though it is very simple to tell what
is happening locally, the global behaviour of the system may be quite subtle. The study
of such agents immediately touches on fundamental issues in the theory of computation,
and rich, beautiful behavior arises.
Figure 1: A cartoon model of local combinatorial structure.
We will focus on how curvature can arise through these interactions; in particular, we
will see that there will be no method, a priori, of determining just what the global
curvature of a given system will be. Conversely, all kinds of behaviors will be possible.
Curvature
Before getting underway, I’d like to take a moment to discuss the curvature of surfaces.
These ideas are well known and quite old, but may be unfamiliar to the general
audience. A flat surface is a good place to start (Fig 2): a flat surface has the same local
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metric properties as the Euclidean plane. In particular, as a circle grows on a flat
surface, the circumference grows linearly with as the radius of the circle increases. The
cylinder, for example, is in fact a flat surface. (An easy way to see this is simply to roll
up a sheet of paper). This rasies an important point: “flatness” and curvature are
intrinsic properties of the surface. It doesn’t matter one bit that a cylinder is “curvy”: it’s
flatness depends on the way we measure things when we are confined to the surface.
Figure 2: Two flat surfaces.
This isn’t so strange– the earlier image of the plane was in perspective, and distance was
certainly not accurately rendered in that image either. In Figure 3 we see two images of a
checkerboard. The images appear quite different but they are intrinsically the same,
measuring distances so that all the squares are to be of uniform size. An inhabitant
would not be able to tell which of the two spaces he lived in.
Figure 3: Geometry is intrinsic; two more flat surfaces.
We can contrast this with positively curved surfaces, such as the sphere, and negatively
curved surfaces such as a crinkly piece of lettuce.
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A sphere really is curved; there is no way to make a sphere out of flat pieces. On a
sphere, the circumference of a circle grows much more slowly than the radius, and the
amount of curvature can be measured by this “deficit”. As before, the particular
rendering is unimportant; the intrinsic notions of distance on the surface define
curvature. We are quite used to seeing this: the Earth is not flat despite the image at right
in Figure 4.
Figure 4: Positive Curvature.
On a negatively curved surface, the circumference of a circle grows exponentially as the
radius increases. A physical object with negative curvature will typically be quite
crinkly, or have numerous “tubes”. In Figure 5, at middle at right, we see two forms,
drawn by Ernst Haeckel, with high negative curvature.
Figure 5: Negative Curvature.
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Combinatorics and Curvature
Consider fitting together puzzle pieces to make a surface. For the moment, we will
suppose we that the surface is “growing” outwards along some boundary. (Fig 6, again
due to Haeckel)
Figure 6: Growth of a surface along a boundary.
Locally the curvature of the resulting surface depends on how fast this boundary is
expanding. If the boundary is expanding rapidly with each step, then the curvature will
be negative. If the boundary is contracting, or growing very slowly, the curvature of the
surface will be positive. This is precisely the effect we see here: as the shell accretes,
the boundary is growing extremely rapidly and we have high negative curvature. This is
precisely the effect we see in Figure 7: as the shell accretes, the boundary is growing
extremely rapidly and we have high negative curvature.
Figure 7: Highly negatively curved surface.
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One may enjoy performing a small experiment to see how local conditions give rise to
curvature. Consider gluing many equilateral triangles together to make a tiling or
polyhedron. If exactly six triangles meet at each vertex, the resulting surface will be flat.
If fewer than six meet at each vertex, the result will be a surface with positive curvature
(for example, if five meet at each vertex, we will obtain an icosahedron– a spherical
surface). And if more than six meet at each vertex, we will obtain a surface with
negative curvature. In fact, this is all true regardless of the specific geometry of the
triangles used– what is important is how they meet, i.e. the nature of the local
combinatorics.
Now suppose the local curvatures are uniformly mixed: in some spots the curvature is
positive, and elsewhere negative. What will the overall curvature be?
If we can calculate the overall proportion of each local behavior we can easily calculate
the overall curvature of the surface. For example, consider triangles that can meet in
two kinds of ways: 113 and 331 (that is, in 5’s and in 7’s) There is in fact only one
possible solution: the resulting surface must be flat: the two kinds of local curvature
must balance out perfectly.
Combinatoria
I’d like to turn now to the kinds of local combinatorial objects I’m considering. Such
gadgets can be thought of as little puzzle pieces that can fit with one another. One
example of such gadgets are tiles in the Euclidean plane. Here, of course, we know a
priori what the local curvature will be.
But consider the following question:
Given a set of tiles, can you tell whether you can cover the entire Euclidean plane with
copies of these tiles? For example, can you cover the plane with copies of the tile at left
in Figure 8? How about with copies of that at right? (These examples are due to Jarkko
Kari).
Figure 8: Which of these two tiles can admit a tiling?
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This question is known as the “Domino Problem”. The real question is:
Is there a way to decide the answer in general? That is,
Is the Domino Problem “decidable”?
The Domino Problem is exceedingly subtle, even when we have only one kind of tile.
(As a hint of this, even the two tiles in Figure 8 are tricky to analyze. The tile at left in
Figure 8 can tile a large region but no more (Figure 9). The tile at right in Figure 8 can
tile, but the simplest way this can be achieved is fairly complicated (Figure 10).
Figure 9.
Figure 10.
In 1966, R. Berger showed the Domino Problem is undecidable in the Euclidean plane.
That is:
There is no general method to decide, for a given set of tiles, whether they can form a
tiling
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Any computation can be modeled by some set of tiles.
A very nice corollary to this is that there exist aperiodic sets of tiles– tiles that can tile
but can never tile periodically. The Penrose tiles are the most famous example (Figure
11).
Figure 11: The Penrose rhombs.
Now for our game here we are laying down layer after layer of these combinatorial
objects; can the overall rate of growth be determined?
Symbolic Subsitution Systems
A first model is symbolic substitution systems. One begins with an alphabet, say 0, 1,
and a set of replacement rules, say 0→1, 1→10
This defines a map on the set of all words in this alphabet. For example:
0110 →1 10 10 1 and in turn 110101 → 1010110110
We can define superwords; these arise from applying this map repeatedly to our
alphabet. So for example, here our superwords are
0→1→10→101→10110→10110101→1011010110110 etc
Now note:
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a) Such symbolic substitutions can model the kind of growth we’ve been discussing. The
“letters” describe our combinatorial objects. The “words” describe our boundaries.
b) Classical theorems describe precisely the rate at which word-lengths grow under the
substitution, and the overall distribution of letters in the superwords.
In our example 0→1, 1→10 words tend to grow by the golden ratio and in the large,
the ratio of the number of 1’s to the number of 0’s will also tend to the golden ratio.
In particular, word length grows exponentially.
The tiling of the hyperbolic plane shown in Figure 11 is precisely described by the
system 0→1, 1→10. And indeed all symbolic substitution systems can be modeled in
similar fashion.
1
1
0
1
0
1
Figure 12: A tiling of (the Klein model of) the hyperbolic plane based on the system
0→1, 1→10.
Regular Production Systems
However, the general situation is much more subtle. In brief, one considers an alphabet
as before. However we restrict ourselves to a “regular language” of allowed words. For
example, let our alphabet be 0, 1, 2 and take the language of words where: 0 can only be
followed by 1, 1 can only be followed by 2, and 2 can only be followed by 0 or 1. So
for example, the words 1201 and 012120 are in the language, but 1120 is not.
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For each letter we take one or more replacement rules. In this example we take
0→12 1→12 2→20 1→21 2→01
Here the replacement is not deterministic. For two words W, V in our language, we
write W→V if there is some choice of replacements on the letters that takes W to V.
But Note: a given word may be mapped to one, no or many words. For example the
word 012120 can only be mapped to 12 12 01 21 20 12; the word 0120 can’t be mapped
to any word at all. And the word 1212 can be mapped to either 12012120 or 21201201.
The point of these systems is that they can model any arrangement, with any curvature,
of any set of combinatorial objects. The letters describe the pieces, the language how
these pieces fit together in layers, and the rules how each layer may fit with the next. A
cartoon of this is shown in Figure 13.
Figure 13: A cartoon illustration of how regular production systems can be used to
model growth by the accretion of combinatorial elements along a boundary.
However, quite unlike the classical substitution systems, it is likely that it is undecidable,
for example, whether one may repeatedly substitute ad infinitum; it is quite likely that it
is undecidable whether a given rule will be needed; it is likely that it is undecidable how
frequently a given rule will be applied.
In particular, it is certainly undecidable, given a particular regular substitution system,
what the curvature of the corresponding surface will be.
Or to put it another way, any desired behaviour can be attained.
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Post Tag systems and growth
I’d like to demonstrate this with an adaptation of an universal computer due to Emil
Post, called “Post Tag Productions”
One has an alphabet, a set of rules and a starting letter:
a→abc
b→a abbca
c→ba
At each step, one has a word. You cross off the first two letters, and depending on the
original first letter, add a word to the back. So for example:
abbca→ bca abc (crossing off ab and adding abc)
bcaabc→ aabca→ bcaabc etc. This particular system repeats quite quickly. However, a
little experimentation shows it is difficult to predict whether the words in a given system
will eventually repeat, grow forever, or shrink away to nothing. I encourage the reader
to try a few more examples– simply select different rules, a different alphabet, or a
different starting word. In fact the long term behaviour is in general undecidable and
indeed any computation can be modeled as a Post-tag system.
In a manner that we cannot explain here, this implies that the curvature induced by a
regular production system (that is, how fast the words in aregular production system
grow) is undecidable; correspondingly, anything can be achieved. This is precisely, in
fact, why nature has access to such a staggering variety of forms through simple growth
along a boundary, using elementary rules.
A complete bibliography and proper treatment of this material can be found in the
papers “Open Problems in Tiling” and “Regular Production Systems and Triangle
Tilings” both available at http://comp.uark.edu/~cgstraus/papers.
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References
Berger, R. (1966) The undecidability of the domino problem, Memoirs Am. Math. Soc. 66 (1966).
Gardner, M. (1977) Extraordinary nonperiodic tiling that enriches the theory of tilings, Scientific American
236, 110-121.
Goodman-Strauss, C. (1998) Matching rules and substitution tilings, Annals of Mathematics. 147, 181-223.
Goodman-Strauss, C. (2000) Open questions in tilings, preprint.
Goodman-Strauss, C. (2002) Regular Production Systems and Triangle Tilings, preprint.
Grünbaum, B and Shepherd, G.C. (1987) Tilings and patterns, W.H. Freeman and Co.
Minsky, M.L. (1967) Computation and infinite machines, Prentice-Hall, Englewood Cliffs, N.J.
Queffelec, M (1987) Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics 1294,
Springer-Verlag, New York.
Robinson, R.M, (1971) Undecidability and nonperiodicity of tilings in the plane, Inv. Math. 12, 177-209.
Thurston, W. (1989( Groups, tilings and finite state automata: Summer 1989 AMS Colloquim Lectures,
unpublished notes.
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SLICEFORM SURFACES
AND A SERENDIPITOUS DISCOVERY
JOHN SHARP
Name: John Sharp, Chemist, Freelance Technical Author, mathematician and artist, (b. Kingston-upon-Hull,
England, 1945).
Address: 20 The Glebe, Watford, Herts, WD25 0LR England E-mail: sliceforms@compuserve.com.
Fields of interest: Geometry and art
Publications /Exhibitions: 1) Illustrator of David Wells' "The Penguin book of Curious and Interesting
Geometry" Penguin1991. 2) Counton, the UK government website to promote mathematics at
www.counton.org, sets of pages on Sliceforms, Morphing Tilings, GridWarps, and Anamorphic art. 3)
Sliceform sculptures: part of the mathematical models collection at the Science Museum in London featured
in their exhibition on “Strange Surfaces” 4) Contributor to “Art and Mathematics 2000” exhibition Cooper
Union, New York 5) Numerous articles and papers on geometry and Art and Mathematics in Nexus Journal,
Times Education Supplement, Art Review, Mathematical Gazette and other educational mathematics
journals.
Abstract: At the end of the nineteenth century, mathematicians created many models of
geometrical surfaces which were beautifully crafted. They are as artistic as the many
mathematical computer graphics created at the end of the last century. Among these
was a group of models of quartic surfaces constructed as sets of circular slices. They
were slotted together in such a way as to continuously deform. The artist Naum Gabo
saw these models in Munich around 1910 and they influenced his construction of
sculptural heads. My own work in constructing such surfaces I call Sliceforms has
created a number of new surfaces. The technique has a wide range of possibilities, but
one of the most interesting, resulting from a mistaken assembly of a model, is a new
relationship between surfaces with an unusual change of symmetry. The work described
here is work in progress.
1. SLICEFORMS
I have developed, but not invented, a method to create models of mathematical surfaces
and to create collapsible paper sculptures which I have called Sliceforms [4].
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At the end of the nineteenth century, mathematicians created many models of
geometrical surfaces which were beautifully crafted. They are as artistic as the many
mathematical computer graphics created at the end of the last century. Among these was
a group of models of quartic surfaces constructed as sets of circular slices. They were
slotted together in such a way as to continuously deform. They were discovered by a
mathematician called Olaus Henrici and are often seen described in books on
mathematical models [1] and three dimensional geometry [2] and were produced as a set
for use in teaching German mathematician Alexander Brill.
It is thought that the Russian constructivist artist Naum Gabo saw these models in
Munich around 1910. He then used them as inspiration for his sculptured heads which
are now in the Tate Gallery, London’s museum of modern art. [3]
Mathematically, the idea does not generally appear other than as illustrations of quartic
surfaces. I have taken the idea further in my Sliceform sculptures [4]. Some of these are
shown alongside the Henrici/Brill models in the exhibition “Strange Surfaces” at the
Science Museum in London.
1.1 Construction of Sliceforms
Pictures and displays of Sliceform models do not show their full beauty. Only by making
and physically handling the models can their true dynamic qualities be fully appreciated.
Their three-dimensional forms and surfaces are defined or suggested by two intersecting
sets of parallel slices (see Figure 1). These intersections act as a multitude of hinges and
as a consequence each model can be made to collapse flat in two different ways.
Between these two extreme positions the surface passes through a host of different but
related shapes. By using different colours for the slices in each direction, the patterns
generated as the model is manipulated can be very attractive and unexpected. The play
of light on them and the shadows they create as they are moved also offer more insights
into the interplay of art and mathematics.
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Figure 1, Construction method

Designing and constructing the sculptures offers insights for both artists and
mathematicians. They can also be used as a starting point for other types of sculpture
such as plaster casting since the gaps between the slices form hollow tubes.
1.2 Symmetry of slices
Surfaces can have many different types of symmetry and this affects whether the shape
of the slices in each direction. With a surface like a sphere, the slices (like the left slice
in Figure 2) are identical in both directions. In the case of surfaces of rotation which do
not have spherical symmetry, the slices are identical in shape, but the slots are placed
differently as in the two slices of an egg surface (like the right two of Figure2).
Figure 2, shape of slices
2. SLICEFORM REVERSALS
My own work in constructing Sliceforms has created a number of new surfaces. The
technique has a wide range of possibilities, but one of the most interesting, resulted from
a mistaken assembly of a model.
By accident, in attempting to create a model where the slices were identical in shape, but
had slots in different directions, I used two sets of slices for the same direction.
Surprisingly, the slices fitted perfectly, although I was later to find that this is not always
the case. Moreover, depending on the properties of the original surface, the resulting
surface can have different symmetry properties. The resulting transformation is not a
standard one since it involves a series of rotations and translations for each slice and not
a transformation of the surface as a whole.
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I am currently exploring this serendipitous discovery to find out the way different
surfaces of revolution behave. Since I originally read chemistry, my scientific intuition is
to experiment before starting to understand what is happening mathematically, in
particular the constraints on the resulting surface. Since it is by no means a trivial
transformation, the variety of results have vindicated this approach. This is still work in
progress, and the results are more intriguing and surprising than I can understand at the
present.
2.1 The hemisphere reversal
Figure 3 shows the result for the simple case of the hemisphere, which at first appears to
be the developable surface formed by the union of two cylinders, but in fact shows
subtle differences.
Figure 3, reversal of a hemisphere
2.1.1 Why the hemisphere reversal does not work
However, one must not take the evidence of one’s eyes in attempting to determine the
mathematics of such an object. Close inspection reveals that the slices may not match
exactly. Normally, when flattened, the slices fit together so that there is no motion along
an axis perpendicular to the slots. In this case, there is in fact some movement possible
which distorts the Sliceform. That the two sets of slices do not match is shown as
follows.
By taking the central slice of one direction and then fitting all the slices in the opposite
direction I calculated the expected shape of the other slices in the original direction. The
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shape of the slices which are needed are no longer part circular sections. Figure 4 shows
the result. The left diagram shows the shape of a slice from a hemisphere. The right slice
shows how the true shape of the slices needed to match have a rounded shape.
2.2 The paraboloid reversal
Figure 4, shape of the slice
In the case of the surface of revolution of a parabola, the reversal of the Sliceform does
give a set of slices that fit. Figure 5 shows the original surface and the reversal version.
Figure 5, paraboloid and reversal
The slices when slotted together in the reversal, fit perfectly as can be seen in figure 6.
Moreover, as with the apparent reversal of the hemisphere in figure 3, the resultant
surface is one that has been created by the intersection of a pair of parabolic cylinders,
where the axis of each cylinder is tangent to the “peak” of the other cylinder.
The question then comes as to why the slices of the fit together perfectly. The answer, as
yet unproved mathematically, is shown in figure 7. The slices of the surface of
revolution of a parabola are all the same parabola. The diagram at the left of figure 7
shows the central slice which is rotated to form the surface. The one at the right is
another slice. They are congruent parabolae.
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Figure 6, views of the paraboloid reversal, showing cylindrical cross section
2.2 Other reversals
Figure 7, slices of the surface of revolution of a parabola
Another reversal of a surface of that appears to work is of an oval. This gives rise to an
egg shaped surface. The central slice giving rise to the surface of revolution, together
with the Sliceform is shown in figure 8.
The result of reversing the slices is shown in figure 9. The surface has a “pointed” end
which points up when viewed orthogonally to one set of slices as shown in the right
view, but down when the surface is rotated. The half way view at the left of figure 9
shows the difference from the original egg of figure 8, although it is not possible to
appreciate the full effect of the reversal in two photographs like these. You need to
manipulate the object to appreciate its properties.
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Figure 8, egg as surface of revolution of an oval
Figure 9, reversal of surface of revolution of an oval
The accuracy of the reversal here has not been checked mathematically.
When a surface will obviously not fit when reversed, the method is still a powerful one
for the artist. Figure 10 shows a curve with five fold symmetry which has been used to
create a surface of revolution. The reversal of the slices shown in figure 11 do not match
at the outermost slots. However, this gives a visually interesting reversal where some of
the slices can be moved to give a range of surface possibilities.
Discussion and conclusions
Many of the audience were intrigued by the beauty of the display of models shown. In
answer to a request for help for working on the mathematics, no one was forthcoming.
However, informal discussion afterwards showed that this type of symmetry change is
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out of the normal stream and delegates were unsure how it fits in with conventional
theories.
References
Figure 10, curve with 5-fold symmetry and its surface of revolution
Figure 11, curve with 5-fold symmetry and its surface of revolution
[1] H M Cundy and A P Rollet, Mathematical Models, Oxford University Press 1960
[2] W H McCrea, Analytical geometry of three dimensions, Oliver and Boyd, Edinburgh 1953
[3] Steven Nash and Jörn Merket, Naum Gabo, Sixty Years of Constructivism, 1985. A catalogue of a
touring exhibition which began at the Dallas Museum of Art and ended in the Tate Museum in
London,
[4] John Sharp, Sliceforms, Tarquin Publications, Stradbroke,Norfolk, England, 1995
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ART OF ANAMORPHOSIS
PHILLIP KENT.
Name: Phillip Kent.
Address: School of Mathematics, Science and Technology, Institute of Education, London, WC1H 0AL,
United Kingdom.
E-mail: p.kent@mail.com.
This presentation will be based on text, artworks and software produced for an
exhibition held in the United Kingdom in the spring of 2001.
Anamorphosis is concerned with the creation of distorted images according to the
mathematical rules of perspective and mirror reflection. The distinctive thing about
these distortions is that the undistorted form of the image can be recovered by looking at
the image from a particular location, or using an unusual shape of mirror (cylindrical,
conical).
The relationship between anamorphosis and mathematics is interesting in several ways.
Anamorphosis is an application of mathematical transformations — several of which
have been implemented in a piece of free software, Anamorph Me!, that can be
downloaded from the web site. This software can be used to turn any image into an
anamorphic distortion, often with beautiful and fascinating results. Furthermore, the idea
of anamorphosis developed in the European Renaissance, at a time when art and
mathematics were deeply connected: it was artists who first developed the geometrical
understanding of perspective on which anamorphosis is based.
Reference
See www.anamorphosis.com.
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134

BASIC CRYSTAL SYMMETRIES GENERATED BY
MOLECULAR DIMERS
ALAJOS KÁLMÁN AND LÁSZLÓ FÁBIÁN
Professor Alajos Kálmán X-ray Crystallographer, (b. Budapest 1935) and Dr. László Fábián X-ray
Crystallographer (b. Kecskemét, 1973),
Institute of Chemistry, Chemical Research Center, Hungarian Academy of Sciences, Budapest, II. Pusztaszeri
ut 59-67, P.O.B. 17, H-1525, Hungary. E-mail: akalman@chemres.hu
(Chemical crystallography , isostructurality )
Academic Award, 1975; Golden Medal of the Labour Order of Merit, 1986; The Hungarian State Prize
bearing the Name of Count István Széchenyi, 1994.
375 original papers in the field of crystal structure analysis and chemical crystallography.
Abstract: In the last years numerous attempts have been made to recognize
deterministic elements of crystallization among the stochastic factors. E.g. a series of
crystals, formed by equal amount of mirror related cyclic molecules (CH2)n n = 5→8,
bearing two vicinal hydrogen-bond donor/acceptor functions (e.g. OH, COOH) were
found to represent seven of eight patterns of close packing. In each structure there are
H-bonded homochiral (➾➾➾➾) or heterochiral (➾➞➾➞) chains assembled either in
antiparallel or in parallel mode. We also revealed that these patterns of basic crystal
symmetries can be deduced from the linear or lateral association of molecular dimers
with either Ci, C2 or Cs symmetry.
1. BASIC PATTERNS OF CRYSTAL ARCHITECTURE
The crystal structure determinations of numerous 1,2-disubstituted cycloaliphatic
molecules such as cyclopentanes, cyclohexanes, cycloheptanes and cyclooctanes
resulted in the recognition of basic (almost canonical) patterns of supramolecular close
packing (Kálmán et al., 2001; 2002a). In accordance with the files of the Cambridge
Structural Database (CSD) the patterns represent the most popular space groups
assumed by organic molecules in the solid state: P21/c (35%), P⎺1 (19%), C2/c (7%).
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1.1 Early fact gathering
The simplicity of the compounds offered an opportunity to study intermolecular
hydrogen bond interactions generated by two vicinal (cis- or trans) functions located on
an alicyclic ring: (CH2)n n = 5, 6, 7 or 8. Each 2-hydroxy-1-cycloaliphatic-carboxylic
acid (functions: OH and COOH) forms two intermolecular hydrogen bonds, while each
2-hydroxy-1-cycloaliphatic-carboxamide (functions: OH and CONH2) forms three. The
hydrogen bonds HB1 (OH…OC) and HB2 (XH…OH, X = O or N) are independent of
the stereoisomerism exhibited by these compounds. Every crystal is racemic, which
means that each crystal has the same amount (1:1) of mirror related molecules in the unit
cell. Overall, the crystal structures investigated (Kálmán et al., 2001) reflected six of
eight possible basic patterns of supramolecular self-assembly organised by the parallel
or antiparallel arrangement of hydrogen bonded homo- (➾➾➾➾) or heterochiral
(➾➞➾➞) chains of the molecules.
1.2 Deduction of the basic forms of close packing
The ninth pattern, observed first in trans-2-hydroxy-1-cyclooctanecarboxylic acid
(hereinafter 8T), could not be inferred from the parallel/antiparallel arrays of the
molecular chains. It was revealed only by the combination of the HB1 and HB2 bonds.
They, depending on symmetry relationships between the molecules, may generate four
motifs: translation forms homochiral tapes (T), screw axis generates helices (H), glide
plane forms heterochiral meanders (M), while inversion center joins heterochiral rings
(R). One of their independent combinations (R:R) corresponds to the linear association
of heterochiral (Ci) dimers. Studying this pattern, it became apparent that the linear
association of Ci–dimers, joined by twelve membered rings, generates automatically new
dimers which are the links between them. These dimers differ in the acceptor group(s) of
the hydrogen bonds. The “OC” dimers are formed by the OH...OC hydrogen bonds,
whereas the “OH” dimers are formed by the XH…OH hydrogen bonds. Along the
parallel ribbons of 8T molecules these OC and OH dimers alternate. In other words, the
heterochiral connection between two dimers of either type generates the other dimer,
and this linear array is therefore unique. It can be regarded as the corner stone of close
packing patterns recognized in our works.
This recognition prompted us to generate all of the lateral and linear associations of the
dimers maintained by twelve-membered rings with Ci, or C2 symmetry and assembled
either in antiparallel or in parallel mode. They can be classified into six basic patterns of
close packing as depicted in Fig. 1 (1-3) and Fig. 2 (4-6).
1. Linear assoc. of Ci–dimers (OC and OH dimers together): space group P⎺1,
2. Lateral assoc. of Ci–dimers in parallel array: space group P⎺1,
3. Lateral assoc. of Ci–dimers in antiparallel array: space group C2/c,
4. Linear assoc. of C2–dimers polymerized into parallel helices: space group Pca21,
5. Linear assoc. of C2–dimers polymerized into antiparallel helices: sp. group. P21/c,
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6. Lateral assoc. of C2–dimers polymerized into antiparallel helices: sp. group. P21/c.
In addition, glide planes (Cg) may also form heterochiral dimers without ring closure:
7. Lateral assoc. of Cg–dimers in parallel array: basic space group Pc,
8. Lateral assoc. of Cg–dimers in antiparallel array: basic space group Pn,
the alternative crystal packings: space group either P21/n or Pna21.
Both glide plane-generated patterns 7 and 8 are hallmarked by 18-membered antidromic
rings (Jeffrey & Saenger, 1991) which generate significant total dipole moment. Since
dipoles must cancel out over the whole crystal, antiparallel stacking of the layers is
compulsory. This can be done by screw axes (21) either perpendicular to (space group
P21/n), or parallel (space group Pna21) with the best planes of the rings.
The synthesis of novel compounds enabled us to demonstrate experimentally all dimer
forms (except 7) by X-ray diffraction.
1.3 Future: Crystal engineering
From these works it was recognised (Kálmán et al., 2002a) that (a) Pattern 1 could
easily be deduced from pattern 2 if all HB1 bonds turn simultaneously from the
respective homochiral chains to their neighbouring enantiomers and vice versa. (b) The
carboxamide analogue (COX, X = NH2) of 8T (COX, X = OH) also crystallises with
pattern 1, therefore they are isostructural (Kálmán et al., 2002b). This prompted us to
extend our “symmetry versus structure” investigation, termed combinatorial crystal
chemistry, to novel chemical structures. In accordance with expectation, a superposition
of patterns 1 and 2 was obtained by the replacement of –OH function with –NH2 moiety.
The series of cis-2-amino-1-cyclopentane- to -cyclooctanecarboxylic acids is
isostructural in space group P⎺1. This is a unique case in which the common form of
crystallisation is controlled by the deterministic formation of the expected hydrogen
bond pattern, rather than by the stochastic effects of anisotropic forces.
Support from OTKA (Grant No T034985) is thanked.
References
Kálmán, A., Argay, Gy., Fábián, L., Bernáth, G. and Fülöp, F. (2001). Basic forms of supramolecular selfassembly
organized by parallel and antiparallel hydrogen bonds in the racemic crystal structures
of six disubstituted and trisubstituted cyclopentane derivatives. Acta Crystallographica Section B,
57, 539-550.
Kálmán, A., Fábián, L., Argay, Gy., Bernáth, G. and Gyarmati, Zs. (2002a). Novel, predicted patterns of
supramolecular self-assembly, afforded by tetrameric R4 4 (12) rings of C2 symmetry in the crystal
structures of 2-hydroxy-1-cyclopentanecarboxylic acid, 2-hydroxy-1-cyclohexanecarboxylic acid
and 2-hydroxy-1-cycloheptanecarboxylic acid, Acta Crystallographica, Section B, 58, in the press.
Kálmán, A., Fábián, L., Argay, Gy., Bernáth, G and Gyarmati, Zs. (2002b). Combinatorial crystal
chemistry: Novel, predictable close packing similarities between cis- and trans-2-hydroxy-1cyclooctanecarboxylic
acids and trans-2-hydroxy-1-cyclooctanecarboxamide. Acta
Crystallographica, Section B, submitted for publication.
Jeffrey, G. A. and Saenger, W. (1991). Hydrogen Bonding in Biological Structures, Springer Verlag, Berlin,
Heidelberg.
137

c
a
2 P1 C2/ c 3
_
b
P2 1/
c
a c
c
Pca2 1
1
Fig. 1. Linear ( 1) and lateral ( 2 and 3)
associations of OH and OC
dimers of C i symmetry.
5a
_
P1
Pca2 1
a P2 1/ c
P2 1/
c b
6
4
b
5b
P2 1/
c
Fig. 2. Linear and lateral associations of homochiral dimers polymerized
into parallel ( 4) and antiparallel ( 5, 6)
helices.
138
C2/ c
a
b
d

Day 3
At the Africa Museum of Tervuren, 2002 April 11.
Location.
The location for the Thursday talks was the
Africa Museum of Tervuren, nearby
Brussels. Brussels counts several Leopold
II style buildings, but this Institute for non-
European Art and Sciences was an
inspiring choice.
Built in 1897, there is a permanent
exposition of animals, plants,
ethnographical objects and sculptures
linked to Central-Africa. After 1908, it
became the Belgian Congo Museum,
changing its name again in 1960 to become
the Royal Institute for Central-Africa and
one of the most important centers for
scientific research about Africa.
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Special Feature
At the Africa Museum of Tervuren Eric
Blanckaert, a professional moviemaker shot
a video and images reproducing the mood
of the math-art conference in the Africa
museum. Below are some of these images.

Impatient participants.
View on the conference room.
Prof. Tarnai in action.
140
The participants, waiting to get started.
View from another angle.
The projection system worked perfectly in
the African environment.

Co-organisor Slavik Jablan explaining the
mathematics of the surrounding African
ornaments (and many other objects).
Les during his talk.
The traditional coffee break was really
traditional.
141
Chairman Denes Nagy, presenting Leslie
Greenhill.
Anna Bulckens during her talk.
Conference organization member Ann
Ratinckx prevented elephants to enter the
conference room.

Prof. Ruiz with junior researcher Maria Peñas during their joint lecture.
It combined expert research with refreshing ideas through a contagious energy.
The quality of the movie images is lower, of course, and they do not give the correct
impression of the video. Here are some images, in reduced format, but they hopefully
will allow the reader to get an impression about the video.
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146

FOLDED STRUCTURES
TIBOR TARNAI
Name: Tibor Tarnai, Structural Engineer, Appl. Mathematician, (b. Hatvan, Hungary, 1943).
Address: Department of Structural Mechanics, Budapest University of Technology and Economics,
Müegyetem rkp. 3, Budapest, H-1521 Hungary.
E-mail: tarnai@ep-mech.me.bme.hu.
Fields of interest: Kinematically indeterminate structures, discrete geometry (packing and covering
problems).
Publications: Tarnai, T. (1996) Symmetry of golf balls, In: Ogawa, T., Miura, K., Masunari, T. and Nagy, D.,
eds. Katachi U Symmetry. Tokyo: Springer-Verlag, 207-214.
Tarnai, T. (1996) Geodesic domes: natural and man-made. International Journal of Space Structures 11, 13-
25.
Fowler, P.W. and Tarnai, T. (1999) Transition from circle packing to covering on a sphere: the odd case of 13
circles. Proceedings of the Royal Society of London A 455, 4131-4143.
Tarnai, T. and Gáspár, Zs. (2001) Packing of equal regular pentagons on a sphere. Proceedings of the Royal
Society of London A 457, 1043-1058.
Abstract: The paper gives an overview of folding and its applications in some fields of
natural and man-made structures, such as symmetric latticed cylinders, braced domes,
deployable space structures, insect wings and swollen form of viruses. Application of
folding to buckling of thin-walled box columns and expandable polyhedra are also
investigated.
1. ORIGAMI
In traditional Japanese origami, folding of straight line is used, and in this way a great
variety of shapes is produced. Origami with much more sophisticated curved lines is
also available. Many lampshades are made with this type of folding. Origami occurs also
in mathematics, however, the most striking constructions were produced not in Japan but
in the US. Kaleidocycles and flexagons were invented in the 1930s. Nowadays a new
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architectural style is called "folding" with a slight reference to paper folding, though,
architectural forms based on origami have been used for a very long time.
2. SYMMETRIC LATTICED CYLINDER
The inextensional buckling form of a thin-walled circular cylinder is the Yoshimura
pattern. Probably, it is not an accident that this buckling form was discovered in Japan,
because this form of many boxes, lampshades and vases was known before. The
Yoshimura pattern, in fact, is a column composed of equal antiprisms. The edge network
of the Yoshimura pattern can be considered as a cylindrical bar-and-joint structure. If
the lower joints are attached to a foundation and the upper joints are free, then for this
latticed cylinder, the necessary condition of rigidity is fulfilled. However, it is rigid only
if the number of nodes along the circular base is odd. For even numbers, the structure is
not rigid (Tarnai, 1980). There are many examples of non-rigid latticed cylinders:
braced cooling towers, interstage structures in multi-stage rockets, the timber octagon in
Ely cathedral, children climbers, the Gemini telescope, etc.
3. BUCKLING OF THIN-WALLED BOX COLUMNS
Hollow columns with polygonal cross-section usually are made of thin metal plates by
folding and welding. Their inextensional local buckling form is like a curved origami. A
deeper insight into this buckling pattern is available by studying folding of uniform
plane tessellations. Tessellation of Schläfli symbol (3,3,4,3,4) is used as a base of the
investigation. The polyhedron obtained by folding leads to a curved origami that
approximates the buckling pattern (Tarnai, 1997). Here, neither the polyhedron, nor the
curved origami has complete straight-line folds. To maintain zero Gaussian curvature we
need some theorems of differential geometry.
4. HIGHER-ORDER MECHANISMS
When folding a plane tessellation into a cylinder the whole structure behaves like a onedegree-of-freedom
mechanism with finite displacements. There are bar-and-joint
assemblies called higher-order infinitesimal mechanisms, which produce the feeling that
they move with finite displacements, but in fact they have only infinitesimal free
motions. This feeling is caused by imperfections. In the Madrid codex, there is a sketch
of Leonardo about a chain-like bar-and-joint structure composed of T shaped units. The
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horizontal and vertical segments of the T shape are composed of two parts connected to
each other by a joint at an angle. This structure shows an exponential increase in forces
in bars. On the other hand, if the member pairs are straightened, then the assembly
becomes a higher-order infinitesimal mechanism, which shows an exponential decay of
motion.
5. DEPLOYABLE STRUCTURES IN LIVING NATURE
Folded structures are common in nature. At plants, for instance, leaves of trees and
petals of flowers are in folded position in buds. When they deploy, usually a growing
process is coupled to deployment. At insects, the situation is different. For butterflies,
wings deploy at the beginning of their adult period, and their shape is kept until the end
of their life. For beetles, however, transparent wings under hard wings are deployed and
retracted at every flight. Different species apply different folding patterns.
6. APPLICATIONS IN SPACE RESEARCH
Space structures are usually compactly packaged on earth, and deployed in orbit in
space. Packaging of deploying masts, solar panels, membranes and antennas is a basic
problem. There exist several different solutions to it. An inextensional folding system of
rectangular membranes is the "Miura-ori", where the folded membrane deploys with one
degree of freedom. The principle of this folding system came from the analogy with the
large displacement buckling pattern of thin elastic plates that was experimented recently
by metal-coated shrinking jelly layers. Miura-ori is a real origami that is used also in
map folding. If a circular membrane should be folded, then folding about a central hub
can be used by rotation.
7. BRACED DOMES
Nowadays many large domes are built for sport arenas. These structures are mainly
constructed as space frames with large height. In order to reduce scaffolding, M.
Kawaguchi has developed the "Pantadome" construction system, in which the main part
of the structure is erected on ground, but some bars are deliberately left out the
construction. The assembly in this phase is a mechanism that can be pushed up to the
final position, then by adding the missing bars to it the structure becomes complete. This
large-scale origami technique is frequently used in Japan and elsewhere. For temporary
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use, transportable deployable latticed domes (tents) have also been developed. It is a
requirement that a stadium or a sport dome is adapted to weather conditions. Therefore,
retractable roofs are built. One group of these structures is the "Iris Dome". The latticed
domes work as iris of an eye: increasing or decreasing the pupil, that is, the opening at
the top of the dome. Hoberman has realized such a structure, but his structure requires
very difficult nodes. Recently, F. Kovács suggested a much simpler solution.
8. EXPANDABLE POLYHEDRA
There are several toys in which a polyhedron is transformed to another one by
continuous motion. An early construction, the "jitterbug", is due to Fuller. It transforms
an octahedron to an icosahedron then a cuboctahedron. Such a structure was constructed
in the Heureka exhibition in Zurich about ten years ago. Not long ago, it was
experimentally observed that certain icosahedral viruses have a swollen form if pH of
the environment changes (Speir et al., 1995). The known expandable polyhedra having
icosahedral symmetry, which were developed by engineers and mathematicians
(Hoberman's (1991) expanding globe, Verheyen's dipolygonids, Yananose's Juno's
spinner), do not provide proper model of the expanding virus. We have started a
research to describe the motion of the expandable virus. Until now we could develop a
simplified model, an expandable dodecahedron, which however has six additional
degrees of freedom those are not required for the investigated motion (Kovács and
Tarnai, 2000).
9. CONCLUSIONS
At the beginning, origami itself, and folded configurations were considered only as toys.
Later, it turned out that folded structures have serious applications in space research,
medical research and architecture. Importance of folded structures will increase in the
future.
Acknowledgement. Research reported here was supported in parts by OTKA Grant No.
T031931, and by OMFB and the British Council Grant No. GB-15/99.
References.
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Hoberman, C. (1991) Radial expansion/retraction truss structures. United States Patent, Patent Number
5,024,031.
Kovács, F. and Tarnai, T. (2000) An expandable dodecahedron. In: Gerrits, J.M., ed., Proceedings of the 4 th
International Colloquium on Structural Morphology. Delft: University of Technology, 227-234.
Speir, J.A., Munshi, S., Wang, G., Baker, T.S. and Johnson, J.E. (1995) Structures of the native and swollen
forms of cowpea chlorotic mottle virus determined by X-ray crystallography and cryo-electron
microscopy, Structure 3, 63-78.
Tarnai, T. (1980) Simultaneous static and cinematic indeterminacy of space trusses with cyclic symmetry.
Int. Journal of Solids and Structures 16, 347-359.
Tarnai, T. (1997) Folding of uniform plane tessellations, In: Miura, K. ed., Origami Science and Art.
Proceedings of the Second International Meeting of Origami Science and Scientific Origami. Otsu:
Seian University of Art and Design, 83-91.
Below is the Power Point presentation of the contribution.
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160

SYMMETRY AND ORNAMENT
SLAVIK JABLAN
Name: Slavik Jablan.
Address: Mathematical Institute, Knez Mihailova 35, P.O.Box 367, 11001 Belgrade, Yugoslavia.
E-mail: jablans@mi.sanu.ac.yu
Web site: http://www.mi.sanu.ac.yu/~jablans/
Abstract: Throughout history there were always links between geometry and the art of
painting. These links become especially evident when to the study of ornamental art we
apply the theory of symmetry. Therefore, ornamental art is called by A. Spaiser "the
oldest aspect of higher mathematics given in an implicit form" and the "prehistory of
group theory".
The idea to study ornaments of different cultures from the point of view of the theory of
symmetry, given by G. Pólya and A. Speiser, was supported by the intensive
development of the theory of symmetry in the 20th century. This caused the appearance
of a whole series of works dedicated mostly to the ornamental art of ancient
civilizations, to the cultures which contributed the most to the development of
ornamental art (Egyptian, Arab, Moorish, etc.), and to the ethnical ornamental art.
Only in some recent works, research has turned to the very roots, the origins of
ornamental art- to the ornamental art of the Palaeolithic and Neolithic. The extensions
of the classical theory of symmetry- antisymmetry and coloured symmetry, made
possible the more profound analysis of the "black-white" and coloured ornamental
motifs in the ornamental art of the Neolithic and ancient civilizations.
This work gives the results of the symmetry analysis of Palaeolithic and Neolithic
ornamental art. It is dedicated to the search for "ornamental archetypes"- the universal
basis of the complete ornamental art. The development of ornamental art started
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together with the beginnings of mankind. It represents one of the oldest records of
human attempts to note, understand and express regularity- the underlying basis of any
scientific knowledge. The final conclusion is that the most of ornamental motifs, which
have been discussed from the standpoint of the theory of symmetry, are of a much
earlier date than we can expect. This places the beginning of ornamental art, the oldest
aspect of geometric cognition, back to several thousands years before the ancient
civilizations, i.e. in the Palaeolithic and Neolithic.
The contents of his talk were published in Jablan’s recent book with the same title.
162

How Plato designed Atlantis
Leslie Greenhill © 2002
Address: P.O. Box 314, Mentone, Victoria 3194 Australia
E-mail: lesgreenhill@yahoo.com.au
Fields of interest: geometry and architecture in antiquity; Neoplatonism; metrological
systems of Ancient Egypt, Greece and the Roman Empire
Some recent achievements: presenter at Mathematics 2000 Festival, University of
Melbourne; divisional winner in Australian short story competition (2000)
163

Remarks on references and abbreviations
The works of classical writers often have ciphers printed in the margins of the
translations to indicate a standard means of precise reference to passages. For example,
in the case of Plato they indicate the pages in the edition of the philosopher’s works by
Stephanus (Henri Estienne), Geneva, 1578. To accord with tradition, references to
Plato’s writings are presented in this manner along with references to more accessible
popular versions of his books. Hence, in the text of this exposition, references to
classical works are presented in the following way:
Plato.
Lee, D. Timaeus and Critias, p. 137/S114
S114 indicates the page in the Stephanus edition.
Herodotus. In the case of Herodotus (The Histories), the numbering system is that used
by J. Marincola from the de Sélincourt translation, e.g.,
Marincola, p. 88/H2.9
H means Herodotus and 2.9 indicates Book Two, passage nine.
Vitruvius. For Vitruvius (The Ten Books on Architecture):
Morgan, p. 27/V1.6.9
V means Vitruvius and 1.6.9 indicates Book One, chapter 6, paragraph 9.
Plutarch. Lastly, for Plutarch (Moralia V),
Babbitt, pp. 175–7/S381
S381 indicates page 381 in the Books of the Moralia, the edition of Stephanus, 1572.
Sometimes, for clarity, the letters BI are used to distinguish British imperial measures
from ancient Greek and Roman measures: for example, 607.5 BI feet.
164

SPECIAL NOTES
(1) Historians of metrology have noted the use of a range of “feet” of varying values in
ancient Greece. In all important cases the present writer is able to demonstrate they are
proportionally related in a specific geometric configuration to the so-called
Greek/Parthenon foot, the ancient Egyptian remen and the ancient Egyptian cubits. The
matter is comprehensively dealt with in a new paper tentatively entitled The Creation of
Measure in Antiquity now in preparation (February 2003). The only Greek measures
discussed at length in this account are the so-called Parthenon foot of about 12.15 inches
or 308.6 mm, the Parthenon cubit (equal to 1½ Parthenon feet), and the stade of 600
Parthenon feet. As an example of a proportional relationship, the so-called Doric foot is
the Parthenon foot multiplied by 36 /35 twice. This measure, about 326.5 mm, had a
special design function, as did the 36 /35 proportion; they are explicated in the new paper.
(2) Since the Matomium Conference in Brussels in April 2002, when this paper was first
presented, the measures in the design system described herein have been discovered,
inventively encrypted, in a famous classical Roman text. Ancient Egyptian, Greek and
Roman measures have been expressed in the text in terms of the so-called British
imperial inch, with the values exactly the same as shown in this exposition because the
same geometry was used. This far-reaching discovery (February 2003) is presently
being incorporated into The Creation of Measure in Antiquity.
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1. Preamble
1.1. The material and the sources.
How Plato designed Atlantis
This exposition is extracted from a larger work entitled How Plato designed Atlantis
and where Vitruvius obtained his model for Man. The two subjects are combined in the
larger exposition because the designs are linked. In fact, in the larger work it is
demonstrated that Vitruvius (fl. 1 st century BC), author of The Ten Books on
Architecture and an admirer of Plato (Morgan, p. 195/Introduction to Book Seven), has
reformulated the Greek philosopher’s design for Atlantis in several ingenious ways, one
of which is the mathematical design for a “well shaped man”. The larger exposition,
though, contains more evidence to prove the case on Atlantis than is presented here.
That said, there is sufficient evidence in this version to justify the title’s contention.
1.2. Plato.
Figure 1: Vitruvius’s “well shaped man” ,
as realized by Cesar Cesariano in his edition of Vitruvius (1521).
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The philosopher Plato, 427–347 BC, a disciple of Socrates, is a seminal figure in the
Western intellectual tradition. Two of his late works Timaeus and the continuation
Critias contain the earliest accounts of the mythical lost continent of Atlantis. Plato says
the information about Atlantis was given to Solon, an Athenian and “the wisest of the
wise men”, by an Egyptian priest (Lee, Timaeus and Critias, p. 33–7/S20–5). The
principal physical description of the country is contained in Critias, which has been
described by Desmond Lee, a translator of the books, as the “first work of science
fiction”. The most detailed description of the place given by the philosopher, though, is
of the capital city, in particular the inner citadel, a circular island surrounded by a
concentric arrangement of rings of water and land. Amongst other things, the inspiring
source of the design concept of the overall layout of the land is unveiled here along with
its encrypted mathematical nature.
1.3. Plato and time measurement.
In the Introduction to Timaeus and Critias Lee writes:
Plato was aware of the close connection between time and time measurement.
Can we speak of one without the other? And if not, are we not bound to say
that ‘time came into being with the heavens’, that is, that time in the sense we
use it cannot be conceived without the instruments, processes, and movements
by which we measure it? (Lee, Timaeus and Critias, p. 11)
The quote is from the Timaeus. The following is the passage from which it was
extracted.
So time came into being with the heavens in order that, having come into being
together, they should also be dissolved together if ever they are dissolved; and
it was made as like as possible to eternity, which was its model. For the model
exists eternally and the copy correspondingly has been and is and will be
throughout the whole extent of time. (Lee, Timaeus and Critias, p. 52/S38)
It will be demonstrated that the nature of time, arising from unexpected and surprising
geometric sources, is inseparable from the nature of Atlantis.
1.4. Plato and The Laws.
That twelve is a key time number is beyond dispute. In The Laws, believed to be Plato’s
last major work, the philosopher sets out his ideas for the ideal state. This is his
description of the ideal city:
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After this, the legislator’s first job is to locate the city as precisely as possible
in the center of the country, provided that the site he chooses is a convenient
one for a city in all other respects too (these are details which can be
understood and specified easily enough). Next, he must divide the country into
twelve sections. But first he ought to reserve a sacred area for Hestia, Zeus and
Athena (calling it the ‘acropolis’), and enclose its boundaries; he will then
divide the city itself and the whole country into twelve sections by lines
radiating from this central point. The twelve sections should be made equal in
the sense that a section should be smaller if the soil is good, bigger if it is poor.
The legislator must then mark out five thousand and forty holdings, and further
divide each into two parts; he should then make an individual holding consist
of two such parts coupled so that each has a partner near the center or the
boundary of the state as the case may be. … He should also divide the
population into twelve sections, and arrange to distribute among them as
equally as possible all wealth over and above the actual holdings (a
comprehensive list will be compiled). Finally, they must allocate the sections
as twelve ‘holdings’ for the twelve [Olympian] gods, consecrate each section to
the particular god which it has drawn by lot, name it after him and call it a
‘tribe’. Again, they must divide the city into twelve sections in the same way as
they divided the rest of the country; and each man should be allotted two
houses, one near the center of the state, one near the boundary. That will finish
off the job of getting the state founded. (Saunders, pp. 215–6/S745)
Plato follows this up by stating:
Now that we have decided to divide the citizens into twelve sections, we should
try to realize (after all, it’s clear enough) the enormous number of divisors the
subdivisions of each section have, and reflect how these in turn can be further
subdivided and subdivided again until you get to 5040. This is the
mathematical framework, which will yield you your brotherhoods, local
administrative units, villages, your military companies and marching-columns,
as well as units of coinage, liquid and dry measures, and weights. The law
must regulate all these details so that the proper proportions and
correspondences are observed. (Saunders, pp. 217–8/S746)
1.4.1. In a commentary on Plato’s most famous work the Republic, James Adam, author
of The Republic of Plato writes:
We know from the Laws that Plato counted 360 ‘days’ in the year. (Adam, p.
301)
Adam’s footnote to this passage states:
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The number of Senators in the Laws is 360: these are to be divided into 12
sections of 30 each, and each section is to administer the State for one month.
The number 60 with its multiples and divisors is the dominant number
throughout the Laws. 360 ‘days’ is of course only an ideal division of the year:
see § 6. Plato elsewhere recognizes (with Philolaus) 364 ½ days (Rep. IX 587
E …). (Adam, p. 301)
The number 364 ½ is half 729 which is 27 squared. The number 27 makes several
notable appearances in this exposition.
1.5. Plato and the Pythagoreans.
In the Translator’s Introduction to Plato’s most famous work, the Republic, Lee
remarks:
In 388–387 BC Plato visited South Italy, perhaps in order to make the
acquaintance of some of the Pythagorean philosophers living there. The
Orphic-Pythagorean belief in the after-life and the Pythagorean emphasis on
mathematics as a philosophic discipline certainly influenced him strongly, as
can be seen in the Republic. (Lee, Republic, p. xvii)
Given Plato’s familiarity with Pythagorean ideas there can be no doubt that he was
familiar with the well-known symbol of the Pythagoreans: the pentagram.
2. Central Atlantis
2.1. Plato’s description.
Figure 2: the pentagram, the symbol of the Pythagoreans.
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The philosopher’s description of the citadel and ring arrangement in Atlantis is as
follows:
The largest of the rings, to which there was access from the sea, was three
stades in breadth and the ring of land within it the same. Of the second pair,
the ring of water was two stades in breadth, and the ring of land again equal to
it, while the ring of water running immediately around the central island was a
stade across. The diameter of the island on which the palace was situated was
five stades. (Lee, Timaeus and Critias, p. 139/S115)
The full ring arrangement measures 27 stades in diameter: see figure 3 below. An
ancient Greek stade measured around 607.5 BI feet (based on the so-called “Parthenon”
foot of about 12.15 inches). The stade and the Greek/Parthenon foot are discussed later.
Lee makes the following points in his commentary on Atlantis (Lee, Timaeus and
Critias, p. 152):
a) “But what Plato is most interested in and spends most time describing is the capital
city itself, and more particularly its inner citadel.”
b) “The inner citadel is shown as a series of small rings at the center, … Its basic form
was determined by Poseidon who ringed the small hill where Cleito lived with two rings
of land and three of water; but its equipment and buildings are the work of the
inhabitants.”
c) “The breadth of the rings of land and water is 3 + 3, 2 + 2, and one stade, and the
central island is 5 stades across. (Is it significant that this gives a total of 27 = 3 3 ?)”
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Figure 3: the rings of water and land around the central island of Atlantis five stades in
diameter (3 + 3 + 2 + 2 + 1 + 5 + 1 + 2 + 2 + 3 + 3 = 27 stades,
the diameter of the full ring arrangement).
2.1.1. Plato provides clues on the original design source for the illustration in figure
3.
He [Poseidon] begot five pairs of male twins, brought them up, and divided the island of
Atlantis into ten parts, which he distributed between them. … The eldest, the King, he
gave a name from which the whole island and surrounding ocean took their designation
of ‘Atlantic’, deriving it from Atlas the first King. His twin, to whom was allocated the
furthest part of the island towards the Pillars of Heracles and facing the district now
called Gadira, was called in Greek Eumelus … . (Lee, Timaeus and Critias, p.
137/S113–4)
Attention is drawn to four matters: the numbers five (notably the five-stade diameter of
the central island) and ten, how the name Atlantis is derived from Atlas, and the mention
of Heracles (Hercules). Atlas is frequently depicted in art bearing the globe of the
cosmos or the world on his shoulders.
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Figure 4: statue of Atlas in Collins Street, Melbourne.
3. Dissecting the Pythagorean symbol
3.1. Five and ten.
In the Pythagorean symbol in figure 5 below—a five-pointed “star”—references to five
and ten can clearly be seen.
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Figure 5: a five-pointed star, the pentagram.
ABC above, one of five isosceles triangles, is a one-tenth segment of a decagon (see
figure 6 below). Angle A is 36°, angles B and C each 72°.
3.2. Angles.
Figure 6: a decagon inside a circle.
In the pentagram illustrated below the following angles can be discerned:
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Figure 7: angles in a pentagram.
Note that 36 is 6 squared, 72 + 72 (144) is 12 squared, and 108 + 108 (216) is 6 cubed.
3.2.1. If 36° is taken as base one then figure 7 above can be illustrated thus:
Figure 8.
Instantly the source of Plato’s 3 + 3 + 2 + 2 + 1 ring arrangement becomes apparent:
see figure 3 and the preceding description by Plato of the ring arrangement in 2.1.
3.3. Time and the world.
What made Plato, the Pythagoreans and others so interested in the pentagram? An
examination of the nature of time shows why.
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A day contains 24 hours. Each hour contains 60 minutes and a minute contains 60
seconds. An hour contains 3600 seconds, twelve hours 43,200 seconds and a full 24hour
period, 86,400 seconds.
In antiquity, the 24-hour period was divided into twelve daylight hours and twelve
nighttime hours. The latter arrangement has an ancient Egyptian origin.
The Egyptians were the first to divide the day into 24 hours; there were twelve hours of
the day and twelve hours of the night. (Gardiner, p. 206)
3.3.1. In figure 9 below it can be seen that from point A to B the sum of the angles is
432°. A to C also adds up to 432°. A to B plus A to C sum to 864°. As stated, twelve
hours contains 43,200 seconds and 24 hours contains 86,400 seconds.
Figure 9: (A to B or A to C) 36° + 72° + 108° + 108° + 72° + 36° = 432°.
Furthermore, the sum of the angles in the pentagon component of the pentagram is 540°
(108° x 5) and the sum of the angles in the five isosceles triangles is 900° (180° x 5):
540° + 900° = 1440°. A day contains 1440 minutes.
3.4. Connecting to ancient Egypt.
In ancient Egypt, right back in the pyramid age, ancient Egyptians used the five-pointed
hieroglyph illustrated below to represent, amongst other things, time.
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Figure 10: the five-pointed “star” hieroglyph.
The following information is from Sir Alan Gardiner’s opus, Egyptian Grammar. It has
been extracted from a description of the “star” hieroglyph pictured above, which is N14
in his Sign-list.
ideo or det. [ideogram or determinative] in ‘star’, ‘teach’, time as indicated by stars,
‘month’, ‘hour’. (Gardiner, p. 487)
An important and far-reaching deduction can be made. As no star in the sky has ever
looked anything like the five-pointed hieroglyph above, this geometric figure, abstracted
from a pentagram (see figure 11 below) must have been chosen in antiquity as a symbol
for time because it had an appropriate mathematical constitution. It is relevant to point
out that an ancient Egyptian month of thirty days contained 720 hours, that is, 360
daytime and 360 night-time hours: cf. the angles in figure 5. Drawing lines from the
apexes of the triangles to the center of the pentagon can also create the “star”
hieroglyph.
Figure 11: the “star” in the pentagram.
3.5. Eratosthenes and his measure of the world.
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Eratosthenes, who became head of the Alexandrian Library around 240 BC, is credited
with producing the earliest sensible measure of the world: 252,000 Greek stades.
Vitruvius tells us of the matter in The Ten Books on Architecture (Morgan, p.
27/V1.6.9). Curiously, a few paragraphs after discussing the measure of the world the
architect remarks:
Some people do indeed say that Eratosthenes could not have inferred the true
measure of the earth. (Morgan, p. 28/V1.6.11)
There is a reason for this comment. Vitruvius, Eratosthenes, Plato, Plutarch (ca. 45–120
AD), Herodotus (ca. 490–420 BC) and other notable historical personages knew the real
measure of the world was 216,000 (60 x 60 x 60) stades. It was a designed measure,
just like the French decided a few hundred years ago that the world would have a
circumference of 40,000 kilometers, that is, 40 million meters. In antiquity, the real
figure was disguised from enemies and the uninitiated. Proof of the matter follows.
3.5.1. The pentagram is further examined. The lettering in the diagram below is based
on that in figure 5 with the addition of letters X, D, and E. X, like B, signifies an angle
of 72°. D and E are each 108°.
Figure 12.
The sum of B 72° + D 108° + X 72° is 252°. The sum of A 36° + B 72° + D 108° is
216°. (So is the sum of D 108° + E 108°.) The fake and the real measures, in reduced
form, are therefore located in one geometric figure—and they are part of each other. A
remarkable analogue of this is presented later. In the larger version of this exposition, it
is demonstrated that both the fake and the real measures of the world are encrypted in
the mathematical constitution of Vitruvian Man.
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The Greek (sometimes called the Periclean or Parthenon) foot is estimated to have
measured around 12.15 inches (Zupko, p. 6 and Petrie, MW, p. 5). As a stade contained
600 Greek feet, the stade was estimated to be around 7290 inches or 607.5 feet (Zupko,
p. 6). 1 Multiply the latter number by 216,000 and the product is 131,220,000 feet which
is 24,852 3 /11 miles. The modern polar circumference measure of the world is
24,859.82 miles. Most contemporary encyclopedias provide this information. The
difference is about 7.5 miles or about 12.1 km.
The Parthenon foot is readily found. It is generally agreed that the top step of the
Parthenon’s base was designed to have a length of 225 Greek feet (about 2735 inches)
and a breadth of 100 Greek feet (about 1215 inches). Contemporary measurements
(Encylopaedia Britannica, Vol. 9, p. 173) show the length of the top step to be 2737.7
inches (228.14 BI feet/69.54 m) and the breadth to be 1216 inches (101.34 BI feet/30.89
m). The rate of accuracy for the layout is better than 99.9%.
Some remarks about the Great Pyramid are pertinent. The base of the structure was
originally 440 royal cubits square and the height 280 royal cubits (Petrie, TPTG, p. 183).
In Number and Divinity in Antiquity, another exposition by the present writer, it is
demonstrated that the chief design particulars for every major ancient Egyptian pyramid
stem from one ingeniously contrived geometric configuration ― even the layout of the
three main pyramids at Giza. The designs for the pyramids and the Giza layout are a
spectacular example (perhaps the greatest) of Vitruvius’s maxim on symmetry and
proportion:
The design of a temple depends on symmetry, the principles of which must be
carefully observed by the architect. They are due to proportion, in Greek
àναλογία. Proportion is a correspondence among the measures of the members
of an entire work, and of the whole to a certain part selected as a standard.
From this result the principles of symmetry. Without symmetry and proportion,
there can be no principles in the design of any temple, that is, if there is no
precise relation between its members, as in the case of those of a well-shaped
man. (Morgan, pp. 72/V3.1.1)
The royal cubit is estimated to have measured around 20.62 inches, about 52.375 cm
(Petrie, TPTG, p. 139 and Klein, p. 59). If the height of the Great Pyramid, 280 royal
cubits, were divided by the time-related number 8.64 (refer 3.3. and 3.3.1.) the quotient
would be 32 11 /27 royal cubits, which is about 668.24 inches. W. M. F. Petrie recorded
the height of the entrance to the structure as being about 668.2 inches above the level of
pavement of the pyramid (Petrie, TPTG, p. 55). Note that 55 Greek feet of 12.15 inches
is equal to 668.25 inches. There is a 297:175 proportional relationship between the
royal cubit and the Greek/Parthenon foot (proved later). The two measures are
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generated in a remarkable way in the geometric configuration mentioned above, along
with the well-known 22:7 proportion.
The number 8.64 is also a prominent mathematical feature in the Atlantis design and
other designs that are discussed later.
3.5.2. The stade measure can be readily detected in ancient Egypt. In an article A
Ground Plan at Giza, British researcher John Legon, using the measurements of the
renowned archaeologist W. M. F. Petrie, observed the appearance of a design strategy in
the rectangular layout of the three main pyramids at Giza. Legon demonstrated that the
axial distance from the west side of the pyramid of Menkaure (Mycerinus) to the west
side of the pyramid of Khafre (Khephren), the second largest of the pyramids, was
7289.5 inches (Legon, AGPG, p. 40). Compare this with the estimated measure of the
stade above, 7290 inches; there is a near perfect fit. Note that a stade is equivalent to
500 ancient Egyptian remen, which can be expressed as 10,000 remen digits. Zupko and
others have recorded the remen digit as measuring 0.729 inch and equal to the Roman
digit (Zupko, p. 6 and Klein, p. 71). Petrie also noted the link between ancient
Egyptian, Greek and Roman measures (Petrie, MW, p. 5). An examination is now made
of the way Vitruvius, Plutarch and Herodotus referred in cryptic form to the real
measure of the world.
3.6. Vitruvius.
The following information is from The Ten Books on Architecture:
a) In the Introduction to Book Five Vitruvius says:
Then again, Pythagoras and those who came after him in his school thought it
proper to employ the principles of the cube in composing books on their
doctrines, and, having determined that the cube consisted of 216 lines, held that
there should be no more than three cubes in any one treatise. … The
Pythagoreans appear to have drawn their analogy from the cube, because the
number of lines mentioned will be fixed firmly and steadily in the memory
when they have settled down, like a cube, upon a man’s understanding.
(Morgan, p. 130)
It can be readily guessed where the Pythagoreans obtained the number of lines from—
nowhere else but from the measure of the world. (That said, there are astonishing
geometric reasons as well. They are explicated in the larger version of this exposition
and in Number and Divinity in Antiquity where more detailed accounts of the geometric
knowledge of Plato and the Pythagoreans are given.)
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) The 600 Greek-foot stade was equivalent to 625 (25 squared) Roman feet. A Roman
mile contained 5000 Roman feet, which was equivalent to 8 (2 cubed) stades (Zupko, p.
6). Consequently, the ancient world measure can be expressed as 27,000 (30 cubed)
Roman miles. Vitruvius tells us in a discussion of aqueducts, wells and cisterns that:
It is not ineffectual to build reservoirs at intervals of 24,000 [Roman] feet, …
(Morgan, p. 245/V8.6.7)
Twenty-four thousand Roman feet was equal to 4.8 Roman miles. This was a
mathematical formulation for those in the know, and possibly a conundrum for initiates.
The quotient of the world measure 27,000 Roman miles (equal to 135,000,000 Roman
feet) divided by 4.8 Roman miles is 5625, which is 75 squared. Other instances of such
mathematical formulations can be found in The Ten Books on Architecture. Here is
another; again related to the measure of time.
c) In a discussion on hoisting machines, Vitruvius reports:
In our own times, however, when the pedestal of the colossal Apollo in his
temple had cracked with age, they were afraid that the statue would fall and be
broken, and so they contracted for the cutting of a pedestal from the same
quarries. The contract was taken by one Paconius. This pedestal was twelve
feet long, eight feet wide and six feet high. (Morgan, p. 289/V10.2.13)
The volume of the pedestal is 576 cubic feet: 576 is the time-number 24 squared.
Apollo returns to the story later.
3.7. Plutarch.
E. A. Wallis Budge, the renowned Keeper of the Egyptian and Assyrian Antiquities in
the British Museum, wrote that the ancient Egyptian god Thoth, usually depicted as a
man with the head of an ibis, was, amongst other things,
“… he who reckons in heaven, the counter of the stars, the enumerator of the
earth and of what is therein, and the measurer of the earth;” (Budge, Vol. 1, p.
400)
Plutarch, apart from being a teacher and a philosopher, was also a Delphic priest.
(There was a famous Oracle at Delphi that was located in the temple of Apollo. The
Oracle, seated on a tripod, went into a trance before she spoke.) Plutarch writes of the
ibis in Isis and Osiris:
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The strictest of the [Egyptian] priests take their lustral water for purification
from a place where the ibis has drunk: for she does not drink water if it is
unwholesome or tainted, nor will she approach it. By the spreading of her feet,
in their relation to each other and to her bill, she makes an equilateral triangle.
(Babbitt, p. 175–7/S381)
Here is an inventive reference to the ancient measure of the world. An equilateral
triangle, which the feet of a tripod also form, has a particularly suitable mathematical
property that connects to the ancient measure of the world: each internal angle is 60°
and there are three of them.
Figure 13: an equilateral triangle
And, as discussed many times already, 60 x 60 x 60 = 216,000. Why Thoth was the
counter of stars and the enumerator and measurer of the earth is no longer a mystery,
thanks to Plutarch.
3.8. Herodotus.
Three striking examples from the writings of Herodotus provide more evidence. Book
Two in The Histories, which deals with ancient Egypt, is the information source.
3.8.1. Herodotus tells us that the coastline of Egypt is 60 shoeni long, and that a
schoenus (an Egyptian measure, he says) is equal to 60 stades. The coastline is thus
3600 stades long (Marincola, p. 88/H2.6–7). Clearly, the historian has designed the
measure to be exactly one-sixtieth of the ancient measure of the world, 216,000 stades
which is equivalent to 3600 (60 squared) schoeni. Everything is in proportion.
3.8.2. Herodotus also reports:
From Heliopolis to Thebes is a nine days’ voyage up the Nile, a distance of
eight-one schoeni or 4860 stades. (Marincola, p. 89/H2.9)
The number nine is 3 squared and eighty-one is 9 squared. If the measure of the world
is divided by 4860 stades the quotient is 44 4 /9 (44.444…), which is 6 2 /3 (6.666…)
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squared. Later, the geometric source of this formulation is unveiled.
Note that nine days contains 216 (6 cubed) hours. This and the material in 3.8.1.
provide compelling evidence, indeed, about the reality of the ancient world measure of
216,000 stades.
3.8.3. The famous historian tells us about the city of Buto “at the Sebennytic mouth of
the Nile”:
The city also contains two other temples, one of Apollo, the other of Artemis.
The shrine of Leto [mother of the two gods], where the oracle is, is a building
of great size with a gateway sixty feet high, but the most remarkable sight it has
to offer is not the temple itself, but a small shrine within the enclosure made out
of a single block of stone; it is cubical in shape, each side sixty feet long and
sixty high. (Marincola, p. 144/H2.155)
The purpose of the shrine is clear: the cube’s volume is 216,000 cubic feet and thus it
memorializes the number associated with the ancient measure of the world.
Leto, as a matter of interest, suffered terrible birth pangs for nine days and
nights, that is, 216 hours, before giving birth to Apollo (Grant, p. 118).
3.8.4. One further mention of Apollo.
Robert Graves tells us that the name has two possible meanings: “destroyer” or “appleman”
(Graves, Vol. 1, p. 57 also Vol. 2, p. 381). Apples are discussed shortly.
3.8.5. Ancient Egypt provides more proof.
E. A. Wallis Budge notes in a translation from an ancient Egyptian text that:
Soon after his marriage with Thi, Amen-hetep [Amenhotep] III. dug, in his
wife’s city of Tcharu, a lake, … “its length 3600 cubits, its breadth 600 cubits.”
(Budge, Vol. 2, p. 70)
The surface area of the lake dug by Amenhotep III (c. 1391–1353 BC) is consequently
2,160,000 square cubits. The significance of the number is obvious and it provides
further evidence that the measure of the world has been around for a very long time
indeed. In this regard, recall the discussion in 3.5.2. of the manifestation of the stade
measure in the Giza, pyramid layout.
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4. In and around Atlantis
4.1. Time in Atlantis.
A further examination of Plato’s design for Atlantis proves fruitful.
A Greek cubit was one-and-a-half Greek feet and had 24 digit divisions (Dilke, p. 26).
A stade therefore contained 400 Greek cubits making the measure of the world
equivalent to 86,400,000 Greek cubits: cf. 86,400 seconds in a day.
In the following calculations, “real” pi or 22 /7 can be used. Plato used 22 /7. This is
proved in the larger version of this exposition and in Number and Divinity in Antiquity.
For ease and meaningful outcomes, the fraction 22 /7 is used here.
4.1.1. The concentric ring arrangement of Atlantis was detailed in 2.1. The central
island, five stades in diameter, has an area of 19 9 /14 ( 275 /14) square stades. The area
of the largest ring island is 169 10 /14 ( 2376 /14) square stades.
The quotient of 169 10 /14 divided by 19 9 /14 is 8.64. A day contains 86,400 seconds
and the ancient measure of the world was 86,400,000 Greek cubits. Review the
pentagram material in 3.3.1.
4.1.2. The area of the smaller ring island is exactly one-third that of the larger ring
island, 56 8 /14 ( 792 /14) square stades. It is exactly equal to the area of a course for
horseracing on the larger ring island. Plato writes:
On the middle of the larger island, in particular, there was a special course for
horseracing; its width was a stade and its length that of a complete circuit of the
island, which was reserved for it. (Lee, Timaeus and Critias, pp. 140–1/S117)
Note how the philosopher has drawn attention to this feature and has provided mensural
details. This setup is for one of the most unique mathematical feats ever to have come
from antiquity. It is dealt with later.
4.1.3. The area of the largest water ring, the outer ring, is 226 8 /28 ( 6336 /28) square
stades. It is equal to the sum of the areas of the two ring islands. This area is,
consequently, 4 /3 times the area of the largest ring island. The 4:3 ratio is also found in
a 3:4:5-proportion triangle. The significance of this triangle to the nature of Atlantis
becomes apparent soon.
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4.1.4. The Atlantis ring arrangement, as already mentioned in 2.1. is 27 stades in
diameter. The quotient of 216,000 stades, the ancient measure of the world, divided by
27 is 8000, which is 20 cubed.
4.2. Atlantis and the pentagon.
Figure 14: angles in a pentagon.
As can be seen in the above illustration two angles are prominent in the pentagon
component: 72° and 54°.
a) In the Atlantis concentric ring arrangement, the area of the largest water ring is 226
8 /28 square stades. This is equivalent to the area of a circle that has a radius of √72
stades.
b) The area of the largest ring island is 169 10 /14 square stades. This is equal to the
area of a circle that has a radius of √54 stades.
All the components of the Atlantis concentric ring arrangement are dealt with later.
4.3. Dodecahedron.
In the Timaeus where Atlantis is first mentioned by Plato, there is a discussion of the
five Platonic solids. Plato equates the dodecahedron with its twelve faces, each face a
regular pentagon, with “the whole heaven” (Lee, Timaeus and Critias, p. 78/S55). In
Phaedo, however, the philosopher tells us, through Socrates, that:
… ‘the earth’s true surface, viewed from above, is supposed to look like one of
those balls made of twelve pieces of skin, …’ (Tredennick and Tarrant, p.
177/S110)
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Figure 15: a dodecahedron.
Here is Plato exercising his remarkable imaginative faculties. He chooses the
dodecahedron to represent “the earth’s true surface” because information on the size of
the planet and the nature of time is bound up in the geometric properties of this unique
solid.
4.4. Plain facts.
The other main physical feature of Atlantis, we are told by Plato, is of a great
rectangular plain 3000 stades by 2000 stades (Lee, Timaeus and Critias, p. 141/S118).
It was mentioned in 3.5.1. that a stade contained 600 Greek feet. Hence, the plain can
be said to measure 1,800,000 by 1,200,000 Greek feet, an area of 2,160,000,000,000
square Greek feet. There are no prizes for guessing the number alludes to the ancient
measure of the world. Recall the mention by Herodotus of the cubical block of stone
with a volume of 216,000 cubic feet in the shrine of Leto at Buto (3.8.3.) and the area of
the lake created by Amenhotep III in 3.8.5.
4.5. Finding time, 22 /7 and much else in 3:4:5 proportions.
The pentagram was not the only geometry involved in Plato’s design for Atlantis. The
3:4:5-proportion triangle had a key role too. (Much more is made of this triangle in
How Plato designed Atlantis and where Vitruvius obtained his model for Man and in
Number and Divinity in Antiquity.)
The proportion clearly attracted interest in the ancient world:
(i) At least eight ancient Egyptian pyramids, including the second largest, the
pyramid of Khephren (Khafre) which is next to the Great Pyramid on the Giza
plateau, exhibit such proportions. The angle of slope (tangent 4 /3) is about 53°
7’ 48” (Baines and Malek, pp. 140–1 and Petrie, TPTG, p. 202).
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(ii) Plato discusses the 3:4:5 proportions in the Republic (S546) in relation to
the “divine creature’s cycle”; Vitruvius mentions them several times,
particularly in the Introduction to Book Nine of The Ten Books on
Architecture; in Isis and Osiris (Moralia V), Plutarch associates the “three”
side with Osiris, the “four” side with Isis and the hypotenuse, the “five” side,
with their son, Horus (Babbitt, 135/S373–4).
4.5.1. The geometry in figure 16 below is of great historic interest. In this one diagram,
solutions to important ancient mysteries can be found: Atlantis; the pyramids; key
metrologies of antiquity. The following is a description of what has taken place in the
illustrated geometry. Everything has been realized by the use of straight edge and a pair
of compasses ― mostly straight edge. It is instructive to study this geometry carefully;
there is a kind of rhythm to the process:
i) ABC is a 3:4:5–proportion triangle. Squares are drawn on the three sides.
ii) K is the midpoint of BC. K is connected to D on the “four” square by a line. The
line crosses AC at R.
iii) A dotted line continues from K to (lower case) “b” on the base of the “five” square
GF. From “b” a dotted line parallel to AC is drawn to (lower case) “h” on BG. The
small triangle hGb has 3:4:5 proportions.
iv) From K a perpendicular line is drawn to L on AC.
v) From B a line is drawn through L to O on DE.
vi) A perpendicular line is drawn from A to I on GF. AI crosses BC at M. AM is 2.4
units. MY is constructed to be also 2.4 units. YI is 2.6 units.
vii) BO crosses AM at N. Through N, a line parallel to AC is drawn from P on AB to Q
on BC. PQ crosses DK at U.
viii) From U a line parallel to BC is drawn to V on AB. VU crosses AM at Æ.
ix) From M a line is drawn to S on AB; SM is parallel to AC. From S a perpendicular
line is drawn to W on BC. SW crosses BO at X.
x) From K, the middle of BC, a line parallel to AC is drawn to T, which is the midpoint
of AB. From T a perpendicular line is drawn to Z on BC.
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4.5.2. Features and outcomes
Figure 16.
a) The perimeter of ABC measures 12 units (3 + 4 + 5): cf. the twelve greater Olympian
gods and Plato’s many references to the number twelve in his writings: The Laws, for
instance. A British imperial foot has twelve inches. Most analogue clocks express time
in terms of hours numbered from one to twelve and this can be directly linked to ancient
Egyptian water and shadow clock designs. A year contains 12 months, just like the
ancient Egyptian year.
The perimeter of the entire configuration measures 36 units (3 + 3 + 3 + 4 + 4 + 4 + 5 +
5 + 5). A British imperial yard contains 36 inches; a circle contains 360 degrees; an
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ancient Egyptian sacred year contained 360 days (Gardiner, p. 203). Also see Adam’s
remarks in 1.4.1.
The product of 3 multiplied by 4 multiplied by 5 is 60: cf. the sixty-second minute and
the sixty-minute hour.
In 3.3.1. it is shown that the sum of the angles in a pentagram is 1440° and that a day
contains 1440 minutes; SM measures 1.44 (1.2 squared) units. (The pentagon
component of the pentagram contains five by 108°; SB measures 1.08 units.)
b) AM measures 2.4 units and SV 0.24 unit. A day contains 24 hours. The Greek cubit
was divided into 24 parts called daktyloi (Dilke, p. 26). The Roman cubit and the
ancient Egyptian common (short) cubit also had 24 digit divisions. (In Number and
Divinity in Antiquity it is shown how the Greek cubit was created in the geometry
presently under examination and how the geometry, in an astonishing natural process,
automatically divides the measure into 24 digit divisions.
c) MK is 0.7 unit. BD is ten times larger and is 7 units. A week contains seven days.
QC is 1.4 units: a fortnight contains 14 days. The body of the ancient Egyptian deity
Osiris is said, according to Plutarch, to have been cut into 14 parts by his brother Seth
and those parts scattered throughout Egypt.
d) Triangle BPQ has three measures: BP 2.16 units, PQ 2.88 units and BQ 3.6 units, a
total of 8.64 units. A 24-hour day contains 86,400 seconds. It was shown in 4.1.1. that
the area of the larger ring island in Atlantis was 8.64 times the area of the central
circular island. In 3.5.1. the entrance to the Great Pyramid was described and it was
shown that the original height of the structure, 280 royal cubits, divided by 8.64
accurately produced the height above pavement level of the entrance. SW measures
0.864 unit. All the main traditional numbers associated with the way time is measured
have thus been located in full or miniature form: 12; 24; 60; 7; 14; and 360.
e) Triangle ABC has an area of 6 square units. RKC, a scalene triangle, has an area of 1
10 /11 square units. The quotient of 6 divided by 1 10 /11 is 22 /7, the value often used in
mathematical calculations to represent pi, a transcendental number. Its appearance in
the design of the Great Pyramid has been much discussed: for example, see Petrie,
TPTG, p. 183.
KC is 2.5 units and RC is 2 6 /11 units. Although a perpendicular line from R to BC has
not been drawn to minimize clutter, RK slopes at tangent 56 /17. This leads to a
momentous finding.
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f) This material is of major historic interest. RC is 2 6 /11 units and AR is 1 5 /11 units, a
total of 4 units. The tangent of right-angled triangle DAR is 2.75. This is readily
established. T is the middle of AB 3 units, as K is the middle of BC. AT is, therefore,
1.5 units and DT is 5.5 units (AD 4 + AT 1.5); TK is 2 units; DT 5.5 ÷ TK 2 = 2.75.
Triangle DAR clearly has the same proportions as triangle DTK.
RC can be expressed as 28 /11 units and EC as 44 /11 units. Instantly the numbers
associated with the Great Pyramid arise: the height of the structure was originally 280
royal cubits and the base was 440 royal cubits square (Petrie, TPTG, p. 183).
The first true pyramid constructed was at Meydum, about 40 miles south of the Great
Pyramid. It originally had a base 275 royal cubits square and a height of 175 royal
cubits (Legon, Meydum, pp. 18–9):
AD 4 units ÷ AR 1 5 /11 units = 2.75. RC 2 6 /11 units ÷ AR 1 5 /11 units = 1.75.
Two pyramids in one design. It is possible to understand now why Khephren’s pyramid,
the largest 3:4:5-proportion structure ever built, has been located next to the Great
Pyramid. There are other remarkable mathematical reasons as well and they are
explicated in Number and Divinity in Antiquity.
Something else of interest: AP is 0.84 unit and DP is 4.84 units; 4.84 is 2.2 squared.
Since the tangent of right-angled triangle, DPU, like DTK, is 2.75 then PU must be 1.76
units. VU is 2.2 units and PV is 1.32 units. VPU is yet another 3:4:5-proportion
triangle. The sum of its sides is 5.28 units. A mile contains 5280 feet or 1760 yards.
Furthermore, the perimeter of the base of the Great Pyramid originally measured 1760
royal cubits, its half base 220 royal cubits. TP is 0.66 unit, so is TV: a chain in 66 feet
and a furlong is 660 feet. The sum of VU 2.2 units and PU 1.76 units is 3.96 units
which is half 7.92 units; a British imperial link measures 7.92 inches and there are 100
links in a 22-yard chain. The yard, as stated earlier, is 36 inches or 3 feet. It is possible
to see quite clearly the birthplace of the so-called British imperial system of linear
measures in this geometry. Even the number associated with the number of inches in a
mile, 63,360, can be found; it is explicated in further developments that are at the far
edge of the human imagination in Number and Divinity in Antiquity.
The royal cubit is one of the most famous and controversial measures of history. The
controversy usually concerns its origin. The matter can now be settled. It derives from
the geometry presently under examination.
W. M. F. Petrie thought the best value was one ascertained from the dimensions of the
King’s Chamber in the Great Pyramid. He estimated the measure, which has 28 digit
189

divisions, to be 20.620±0.005 inches (Petrie TPTG, p. 179). The acceptable range is,
therefore, 20.615 to 20.625 inches.
The construction of the small 3:4:5-proportion triangle hGb was described in 4.5.1.
(point iii). Gh measures 165 /224 unit. Twenty-eight times 165 /224 produces 20.625.
As inches, this is the intended value for the royal cubit. The measure has a clear
geometric link to the design for the Great Pyramid as well as the 22 /7 proportion.
The quotient of AB 3 units divided by AR 1 5 /11 units is 2.0625.
LK is 1.875 units. If LK were extended to GF, the base of the “five” square, the line
would measure 6.875 units; 6.875 is one-third of 20.625. In Number and Divinity in
Antiquity, the measure as a whole, along with its digit division (not the one shown here),
is created in this geometry in a more startling fashion.
g) BP is 2.16 units. The real measure of the world in antiquity was 216,000 stades. The
3:4:5-proportion triangle PAN has sides that measure PN 0.63 unit, AP 0.84 unit, and
AN 1.05 units, a total of 2.52 units. Eratosthenes’ fake measure of the world was
252,000 stades. Here we see an analogue of the 216/252 connection in the pentagram
described in 3.5.1. WZ measures 0.252 unit.
The ratio 216:252 can be expressed as 6:7. Compare this with the opening chapters of
the Old Testament in the Bible, Genesis I and II: God created the world in six days and
rested on the seventh.
h) Right-angled triangle ODB has the proportions 7:24:25. OD measures 2 1 /24 units,
BD 7 units and BO 7 7 /24 units (7 7 /24 appears on a calculator display screen as
7.291666…).
BO, as inches, is exactly one-thousandth the measure of the stade (i.e. 7291 2 /3 inches),
as it was originally understood to be. Two times 7 7 /24 inches is 14 7 /12 inches
(14.58333…). The ancient Egyptian remen has long been estimated to measure around
14.58 inches (Zupko, p. 6 and Petrie, MW, p. 5). Recall the appearance of the stade in
the Giza pyramid layout described in 3.5.2. The axial distance from the west side of the
pyramid of Menkaure to the west side of the pyramid of Khephren is 7289.5 inches.
The rate of accuracy in layout intention is around 99.97 percent.
The ratio between the 20.625-inch royal cubit and the 14 7 /12–inch remen is 99:70, a
ratio that closely approximates the irrational number √2. (A4-size paper, which
measures 297 mm by 210 mm, expresses this ratio.)
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i) The stade of 7291 2 /3 inches when divided by 600 yields the Greek/Parthenon foot of
12 11 /72 (12.152777…) inches: cf. the historic estimates of 12.15 inches (Zupko, p. 6
and Petrie, MW, p. 5). In Number and Divinity in Antiquity, it is shown how the
measure of 12 11 /72 inches is generated naturally in the geometry illustrated in figure
16, along with the royal and common cubits, the remen, the Greek cubit, and the Roman
foot and cubit. The geometry even generates their digit divisions naturally. The socalled
British imperial foot and yard are readily seen: refer to point (a).
j) The ancient measure of the world can now be accurately fixed: 216,000 stades
multiplied by 7291 2 /3 inches produces 1,575,000,000 inches or 24,857 21 /22
(24,857.95454…) miles: cf. the modern polar circumference measure of 24,859.82
miles.
k) The Romans divided the stade into 625 (25 squared) Roman feet. Accordingly, the
Roman foot measured 11 2 /3 inches (296.333… mm). The measure of the world was
therefore equal to 135,000,000 Roman feet. In figure 16, NM measures 1.35 units.
l) BN measures 2.25 (1.5 squared) units: cf. the length of the Parthenon, 225 Greek
feet: see 3.5.1.
m) Triangle AMB has sides that measure BM 1.8 units, AM 2.4 units and AB 3 units, a
total of 7.2 units. Inside AMB is triangle BMN whose sides measure NM 1.35 units,
BM 1.8 units and BN 2.25 units, a total of 5.4 units. Compare this with the material on
72° and 54° in 4.2.
n) The areas of the three water rings, two ring islands and the central circular island of
Atlantis are as follows:
Largest water ring: 226 8 /28 square stades, which is equal to the area of a circle that
has a radius of √72 stades; the perimeter measure of triangle AMB is 7.2 units (also see
4.2.).
Larger ring island: 169 10 /14 square stades, which is equal to the area of a circle that
has a radius of √54 stades. The perimeter of triangle BMN measures 5.4 units (also see
4.2.).
Middle water ring: 81 5 /7 square stades, which is equal to the area of a circle that has a
radius of √26 stades; YI is 2.6 units (MY is 2.4 units, the same as AM; MI is 5 units.).
Smaller ring island: 56 4 /7 square stades, which is equal to the area of a circle that has
a radius of √18 stades; BM is 1.8 units.
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Smallest water ring: 18 6 /7 square stades, which is equal to the area of a circle that has
a radius √6 stades; the perimeter of triangle BTK is 6 units (BT 1.5 + TK 2 + BK 2.5).
Central island: 19 9 /14 square stades, which is equal to the area of a circle that has a
radius of 2.5 stades: BK is 2.5 units.
o) The ring arrangement of Atlantis is 27 stades in diameter. The circumference of
Atlantis is therefore 84 6 /7 stades, which is 618,750 inches or 9.765625 miles; 9.765625
is 3.125 squared. In figure 16, LB and LC each measure 3.125 units.
p) In 3.8.1. Herodotus’ description of the length of the coastline of ancient Egypt, 3600
stades, was examined. In figure 16 BQ is 3.6 units and the perimeter of the entire
configuration measures 36 units.
In 3.8.2. the historian’s report on the distance from Heliopolis to Thebes was discussed.
From Heliopolis to Thebes is a nine days’ voyage up the Nile, a distance of
eight-one schoeni or 4860 stades. (Marincola, p. 89/H2.9)
9 day’s voyage (216 hours): BZ measures 0.9 unit;
81 shoeni: BX measures 0.81 unit;
4860 stades: XW measures 0.486 unit;
216,000 stades, the measure of the world: BP measures 2.16, so does AV;
252,000 stades, Eratosthenes’ fake measure of the world: WZ measures 0.252 unit and
the perimeter of triangle APN measures 2.52 units.
The geometrical and numerical evidence that this is the source of Herodotus’
mathematical formulation for the voyage up the Nile is compelling indeed.
q) Eratosthenes’ measure of the world, 252,000 stades, was not chosen for use because
of numerical and geometric motives alone: there was a didactic reason as well.
In antiquity, the world at the equator turned 216,000 stades every 86,400 seconds (24
hours). Every second, at the equator, the world moved 2 11 /12 (2.91666…) stades; 2
11 /12 is one-quarter of 11 2 /3, the number of inches in a Roman foot; 2 11 /12 is onefifth
of 14 7 /12 ( 175 /12), the number of inches in an ancient Egyptian remen: see point
(h) above.
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Now 2 11 /12 stades is equal to (multiply by 7291 2 /3 inches) 21,267 13 /36 inches. The
latter number is 1750 /12 squared: cf. the number of inches in a remen in the preceding
paragraph. Furthermore, 1750 /12 inches is equal to 12 11 /72 BI feet; a Greek/Parthenon
foot measured 12 11 /72 inches: see point (i) above.
r) AV measures 2.16 (6 x 0.6 x 0.6) units. It is the hypotenuse of 3:4:5-proportion
triangle AÆV. AÆ measures 1.728 units; 1.728 is 1.2 cubed. VÆ measures 1.296 units
(3.6 x 0.36); the measure of 216,000 stades is equal to 129,600,000 Greek/Parthenon
feet. The number 3600 squared (12,960,000) is discussed in Lee’s footnote to Plato’s
reference to 3:4:5 proportions, the “divine creature’s cycle” and the number of days in a
“Great Year” (Lee, Republic, p. 299–300). The perimeter of triangle AÆV measures
5.184 units; 5.184 is 7.2 multiplied by 0.72.
4.6. A racecourse to remember.
The ancient measure of the world was 216,000 stades. It is equivalent to 24,857 21 /22
miles. The latter number can be written as 70,000,000 /2816. If 22 /7 is inverted and
dimidiated seven times, the outcome is 7 /2816. All this is worthy of contemplation.
Plato has recorded the number 24,857 21 /22 in a memorable way.
In 4.1.2. the following passage from Critias was cited:
On the middle of the larger island, in particular, there was a special course for
horseracing; its width was a stade and its length that of a complete circuit of the
island, which was reserved for it. (Lee, Timaeus and Critias, pp. 140–1/S117)
Plato has provided an explicit mathematical description of the racecourse. It is one
stade wide and it is in the middle of a ring island three stades wide. There is a reason
for the formulation: he has created a formidable mathematical conundrum that only
advanced students at his Academy or Pythagoreans, or privileged others could
understand.
(The calculations here use 22 /7 to represent pi. It is easier to do all this on a calculator
that has memory facilities. It is also instructive to do it without the calculator. The 22 /7
proportion is encrypted in the Timaeus; the location and method of encryption are
explicated in the larger version of this exposition.)
a) The area of the racecourse works out to be 56 4 /7 ( 396 /7) square stades which is
equal to (multiply by 7291 2 /3 inches twice) 3,007,812,500 square inches.
193

) Convert the area to square miles by dividing by 63,360 inches twice (63,360 inches is
the number of inches in a mile). The area works out to be 24,609,375 /32,845,824 square
mile. On a calculator display screen this will appear as 0.74923908… mile.
c) Divide the measure of the world 24,857 21 /22 miles by 24,609,375 /32,845,824 and the
quotient is 33,177.6, which can be expressed as 24 x 24 x 24 x 2.4.
That Plato or someone else in his Academy conceptualized all this is also worthy of
contemplation.
4.6. The Pantheon and a Revelation.
Other significant manifestations of the number 216 and thereby the measure of the world
can be found in famous architecture and literature.
(i) The first example is the Pantheon in Rome, one of the few great buildings of the
Roman Empire to survive to modern times basically intact. It is the work of Hadrian,
about 126 AD.
Figure 17: The Pantheon in Rome (illustration courtesy M. W. Jones)
Sir Mortimer Wheeler writes:
194

It was dedicated to the seven planetary deities and was in effect an architectural
simulacrum of the all-containing cosmos. (Wheeler, pp. 104–5)
Amongst the notable architectural features of this structure, normally the first to be seen
by the visitor is the portico: see the illustration above. The width of the portico is 108
Roman feet, that is, about 31.99 metres that is about 1260 inches (Davies, Hemsoll,
Wilson Jones, pp. 141 and 150). The number 108 is half 216 which is instantly
recognizable for its reference to the measure of the world. Also, compare the measure
with the angles of 108° in a pentagon as in figure 7.
Furthermore, 108 Roman feet is equal to 86.4 ancient Egyptian remen: the ancient
measure of the world was equivalent to 86,400,000 Greek cubits; there are 86,400
seconds in a day. One hundred and eight Roman feet is also equal to 1728 (12 cubed)
Roman digits. (A Roman foot had sixteen digit divisions: see Dilke, p. 26.) In figure
16, AÆ measures 1.728 units.
(ii) In the Bible, the last book of the New Testament is titled The Revelation of St. John
the Divine. It comes down to us in Greek and it is said it may have been composed in
Alexandria, in Egypt. The work contains material from a range of sources, some of it
ancient Egyptian. Chapter 21, the penultimate chapter, contains numerous mentions of
the number twelve. The two most interesting mentions are (1) the visionary New
Jerusalem, a giant cube whose length, width and breadth are 12,000 furlongs and (2) a
wall 144 cubits high surrounding the city.
Figure 18: a cube.
English translators long ago translated stade as furlong because of a vague similarity in
length. It is correct to say that the New Jerusalem cube has sides that measure 12,000
stades. This super solid has a surrounding wall 144 (twelve squared) cubits high. The
original text was Greek so it may properly be assumed the cubit is the Greek cubit.
195

A cube contains 24 right angles and 24 x 90° = 2160°. According to the Pythagoreans,
the cube consists of “216 lines”: see 3.6.
Each face of the New Jerusalem cube contains 144,000,000 square stades and since the
object has six faces, the total surface area of the solid is 864,000,000 square stades. A
day contains 86,400 seconds; the ancient measure of the world was equivalent to
86,400,000 Greek cubits. The number 864 has been shown to occur in the pentagram
(3.3.1.) and the number 8.64 in the designs for the Great Pyramid (3.5.1.) and Atlantis
(4.1.1.).
A cube has twelve edges. An edge of the New Jerusalem cube measures 12,000 stades.
The sum of the twelve edges is 144,000 stades, which is equal to 86,400,000 Greek feet.
The wall around the city is 144 Greek cubits high. A cubit (both Greek and Roman) is
equivalent to one and a half feet. Accordingly, the wall can be said to be 216 Greek feet
high. There can be no doubt as to what the height refers to.
More can be found in antiquity in relation to a square with sides that measure 12,000
stades.
E. A. Wallis Budge compared ancient Hebrew notions of the Underworld with ancient
Egyptian ideas of the Duat (or Tuat), the realm of the dead.
The commonest of the names, which the Hebrews gave to the abode of the
damned, is GÊ HINNOM, or Gehenna, … . According to the Rabbis
“Gehenna” was created on the second day of creation, with the firmament and
the angels, and just as there were an Upper and a Lower Paradise so there were
also two Gehennas, one in the heavens and one on the earth. As to the size of
Gehenna we read that Egypt was 400 parassangs long and 400 parassangs wide,
…; that Nubia was sixty times as large as Egypt; that the world was sixty times
as large as Nubia, and that it would require 500 years to travel across either its
length or its breadth; that Gehenna was sixty times as large as the world; and
that it would take a man 2,100 years to reach it.
In Gehenna, as in Paradise, there were seven “palaces” … , and the
punishments, which were meted out to their inhabitants varied both in kind and
in intensity. In each palace there are 6,000 houses, or chambers, and in each
house are 6,000 boxes, and in each box are 6,000 vessels fitted with gall.
(Budge, Vol. 1, pp. 273–4)
Gehenna is thus 216,000 times the size of Egypt. The number does not require
comment; enough has been said about the measure of the world and 6 x 6 x 6 variants
196

already.
The mythical area of Egypt, 400 parasangs square, is worthy of note. Herodotus informs
us that a parasang (Budge spells it parassang) is equal to 30 stades (Marincola, p.
88/H2.6). Therefore, Egypt was 12,000 stades square, exactly the same size as each
face of the New Jerusalem cube. Budge cites Eisenmenger, Entdecktes Judenthum, part
ii, p. 328 as the source of the information on the size of Egypt.
4.7. Closing remarks.
Atlantis is found. It rises from the depths of myth and speculation to be seen for what it
really is: a powerful mnemonic and didactic device for those initiated into certain
influential Mystery Schools in antiquity, notably those with Pythagorean and
Neoplatonist interests. A large body of data drawn from authoritative ancient and
modern sources, from the layout of the Giza pyramids, and from the properties of certain
basic geometric shapes known to Plato, Vitruvius and others in antiquity, has revealed a
distinctive approach to mathematical design. Unique and inventive mathematical feats
previously unknown from the past have been unveiled.
The design source for the Great Pyramid, too, has been located. So has the source for
important metrologies of the ancient world, including the so-called British imperial
system. Herodotus’ marvelous mathematical voyage up the Nile has not only been
explicated, it has been placed in the geometry set out in figure 16.
As has been repeatedly demonstrated, the ancient measure of the world was 216,000
stades. It has also been demonstrated that the measure of the world was linked to the
nature of time. In addition, it was shown that the measure of the world was established
at least as far back as the time of the construction of the pyramids at Giza. No
information has come down from the past, though, as to who it was that realized the
world was spherical—probably some astute astronomer who witnessed a number of
lunar eclipses. There is also no record of the survey method or of the survey work that
established the necessary distances for the stade and other measures to be created. The
survey work was likely to have been a religious duty as the world was considered a
deity, just like the sun or the moon. How many times the task was undertaken until
consistent results had been achieved cannot be estimated.
One last surprise remains.
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5. Golden apples
5.1. Storing knowledge.
From Biblical times, particularly through Christian imagery, the apple has been seen as a
symbol of knowledge (cf. the story of Adam and Eve in the garden of Eden). This fruit,
a member of the rose family of plants, 2 has also played an important role in Greek
mythology. One myth is especially relevant to the matter of Atlantis and beyond. It
concerns Heracles (Hercules) and one of his Twelve Labours.
Using data obtained from ancient sources, Robert Graves, author of The Greek Myths, in
the opening paragraph of chapter 133 entitled “The Eleventh Labour: The Apples of the
Hesperides”, writes (bolded words are the present writer’s emphasis):
Heracles had performed these Ten Labours in the space of eight years and one
month; but Eurystheus, discounting the Second and the Fifth, set him two more.
The Eleventh Labour was to fetch fruit from the golden apple-tree, Mother
Earth’s wedding gift to Hera, with which she had been so delighted that she
planted it in her own divine garden. This garden lay on the slopes of Mount
Atlas, where the panting chariot-horses of the Sun complete their journey, and
where Atlas’s sheep and cattle, one thousand herds of each, wander over their
undisputed pastures. (Graves, Vol. 2, p. 145)
What was so special about this fruit in ancient times? The photograph below says it all.
When the apple is sliced crosswise rather than downwards from where the stem
connects, the core is revealed as Nature’s pentagram, a geometric shape that encrypts the
ancient measure of the world and the way we measure time.
198

Figure 19: Nature’s pentagram in the core of an apple.
Thus, the key elements for the philosopher’s influential story can be found in an ancient
legend that predates Plato’s lifetime. Plato has already told us that Atlantis is named
after Atlas: refer 2.1.1. And we see that the apple-tree is a wedding gift from Mother
Earth (Gaea). Who more appropriate? And the apples grow on Mount Atlas: a fitting
location. Note, too, the mention of horses and recall Plato’s course for horse racing
(4.1.2. and 4.6.). Recall also one of the possible meanings of Apollo’s name: appleman
(3.8.4.). The apple truly is a symbol of knowledge.
Notes
1. Zupko incorrectly calculates the Roman foot to be 11.65 inches and the stade as 625
times this value, that is, 606.9 BI feet. Zupko (correctly) rates the 625 Roman-foot stade
as equal to 600 Greek feet like other historians of metrology. But he starts out with a
slightly incorrect value for the Roman foot: 296 mm. The links between ancient
Egyptian, Greek and Roman measures are well known and much discussed in works on
the history of metrology (see Petrie, MW, p. 5, Klein, p. 71 and Zupko, p. 6). The
Roman digit was the same as the ancient Egyptian remen digit, around 0.729 inch: see
the references just mentioned. The Roman foot of 16 digits was therefore around
11.664 inches or about 296.27 mm. Zupko has simply rounded this number to 296 mm
and obtained his value of 11.65 inches. Furthermore, when multiplied by 625 it led to
his value for the stade measure being out by some seven inches. The well-known ratio
(noted by Zupko) of 25:24 between the Greek foot and the Roman foot further
199

substantiates the corrected measure: the Greek (Parthenon) foot measured around 12.15
inches (Zupko, p. 6 and Petrie, MW, p. 5); twenty-four twenty-fifths of 12.15 inches is
11.664 inches.
The so-called Roman foot has been observed in Greece under various guises, including
the Nemean “foot”. In a critique of the book Athletics and Mathematics in Archaic
Corinth: the Origins of the Greek Stadion by David Gilman Romano, British academic
David W. J. Gill (see References) writes:
This provides the information that the Nemea stadion must have been around
178 m long, judging by the location of the 100-foot marker at a distance of
29.63 m from the starting line; this gives a foot of 0.296 m. (Gill, Internet
address in References)
Note how an error in the measure of the Nemean/Roman foot arises: Gill has rounded
29.63 divided by 100 to 0.296. It is more accurate to say the Roman foot was about
0.2963 m (about 11.665 inches).
2. This should interest members of the Rosicrucian Society.
3. Historians of metrology have noted the use of a range of “feet” of varying values in
ancient Greece. In all important cases, the present writer is able to demonstrate they are
proportionally related to the Parthenon foot, the remen and the ancient Egyptian cubits.
The matter is dealt with extensively in “Number and Divinity in Antiquity”, another
paper by the present writer. The only Greek foot and cubit discussed in this paper are
the so-called Parthenon foot and cubit.”
4. A few remarks on several notable Greek feet measures:
a) The so-called Doric foot of around 326 mm is simply the Parthenon foot of 12 11/72
(875/72) inches multiplied by 36/35 twice, that is, 36/35 squared. The Doric foot’s true
value is 12 6/7 (90/7) inches (326.57 mm). It has an interesting mathematical function
in relation to the measure of the world. As stated in the paper, 216,000 stades is equal
to 1,575,000,000 inches (see 4.5.2. point (j). The quotient of the measure of the world,
1,575,000,000 inches, divided by 12 6/7 inches is 122,500,000, which can be expressed
as 35 x 3,500,000.
b) 1,575,000,000 inches divided by the Parthenon foot of 12 11/72 inches (about 308.6
mm) is 129,600,000, which can be expressed as 36 x 3,600,000.
200

c) The so-called common Greek foot of about 316 mm is more accurately expressed as
12 4/9 (112/9) inches or 316.0888… mm. The quotient of 1,575,000,000 inches divided
by 12 4/9 inches is 126,562,500, which is 11,250 squared.
d) One last example: the Halieis stade of 166.5 metres produces a Greek foot of around
278 mm. The correct value is 10.9375 inches (277.8125… mm) and is 9/10 of the
Parthenon foot or three-quarters of an ancient Egyptian remen. The quotient of
1,575,000,000 inches divided by 10.9375 inches is 144,000,000, which is 12,000
squared.
e) The following website may help with these stade issues:
http://ccat.sas.upenn.edu/bmcr/1995/95.09.19.html
References
Adam, J. (with an introduction by D. A. Rees) The Republic of Plato. Vol. 2. Cambridge University Press,
London, 1965.
Babbitt, Frank Cole. Plutarch: Moralia. Volume V (Isis and Osiris). Harvard University Press, 1999 edition.
Baines, J. & Malek, J. Atlas of Ancient Egypt. Time-Life Books, Amsterdam, 1996 edition.
Budge, E. A. Wallis. The Gods of the Egyptians. (two volumes) Dover Publications, N.Y., 1969.
Davies, P., Hemsoll, D., Wilson Jones, M. “The Pantheon: Triumph of Rome or Triumph of Compromise?”
Art History. Vol. 10. No. 2, June 1987, ISSN 0141-6790
Dilke, O. A. W. Reading the Past: Mathematics and Measurement. British Museum Publications, London,
1991.
Encyclopaedia Britannica (The New) Volume 9, Micropaedia, 15
201
th edition. Encyclopaedia Britannica Inc,
Chicago, 1992.
Gardiner, A. Egyptian Grammar. Griffith Institute, Ashmolean Museum, Oxford. 1994 edition.
Gill, D. W. J. (University of Wales, Swansea) Internet address: http://www.swan.ac.uk/classics/dghp.html .
Grant, M. Myths of the Greeks and Romans. Mentor Books, New York, 1962.
Graves, R. The Greek Myths. (Vols. 1 & 2) Penguin, England, 1978 edition.
Klein, H. A. The World of Measurements. George Allen and Unwin Ltd., London, 1975.
Lee, D. Plato: The Republic. Penguin, England, 1987 edition.
Lee, D. Plato: Timaeus and Critias. Penguin, England, 1986 edition.
Legon, J. A. R. “A Ground Plan at Giza.” Discussions in Egyptology 30. Oxford, 1988. ISSN 0268-3083. .
(title abbreviation AGPG.) Also see the Internet: http://www.legon.demon.co.uk/index.htm .
Legon, J. A. R. “The 14/11 proportion at Meydum” Discussions in Egyptology 17, Oxford, 1990. ISSN
0268-3083. (title abbreviation Meydum)
Marincola, J. (revision of translation by Aubrey de Sélincourt) Herodotus: The Histories. Penguin Books,
1996 edition.
Morgan, M. H. Vitruvius: The Ten Books on Architecture. Harvard University Press, Cambridge, Mass.,
1914. (Republished by Dover Publications in 1960.)
Petrie, W. M. F. The Pyramids and Temples of Gizeh. Field and Tuer (Ye Leadenhalle Presse), London, 1883
(title abbreviation TPTG).
Petrie, W. M. F. Measures and Weights. Methuen & Co. Ltd., London, 1934. (title abbreviation MW)

Saunders, T. J. Plato: The Laws. Penguin Books, London, 1975 edition.
Tredennick, H. & Tarrant, H. The Last Days of Socrates. Penguin, England, 1993.
Wheeler, Sir Mortimer. Roman Art and Architecture. Thames and Hudson, London, 1994 edition.
Zupko, R. E. British Weights and Measures: A History from Antiquity to the Seventeenth Century. Madison
University of Wisconsin Press, 1977.
202

THE MNEMONICS OF THE CRETAN LABYRINTH
TESSA MORRISON
Name: Tessa Morrison, Ph.D. candidate
Address: The School of Fine Arts, The University of Newcastle, University Drive, Callaghan. NSW 2308.
Australia.
E-mail: c9520975@alinga.newcastle.edu.au
Fields of interest: geometric and algebraic topology, group theory and history of ideas
Awards: Certificate of Merit, Uiversitet im. Zhaksygarina, Aktobe, Zazakhstan, 1999
Diploma of Honour, Homage to the Poet, Ovidiu Petca. Cluj-Napoca, Romania, 2000
Best Graphic Print Bookplate, Australian Bookplate Design Award Exhibition, 2001
Publications and Exhibitions:
The Geometry of History; 032147658. Vismath, http://www.mi.sanu.ac.yu/vismath Volume 3. No. 4. 2001,
Het Stedelijk Museum, Sint-Niklaas, Belgium, 2001.
Printmakers Perceptions of the Past One Hundred Years of Federation, Maitland City Council Art Gallery,
Australia, 2001.
Stadtmuseum Bruneck, Brunico, Italy, 2001.
Roman Identity and the Roman Labyrinth, Journal of Inter-Cultural Studies, The University of Newcastle,
Australia 2002 (forthcoming issue).
Abstract: The aim of this paper is to examine the various possible symbolic mnemonic
devices that the ‘Cretan labyrinth’ can be constructed from. The ‘Cretan labyrinth’ has
been used throughout history, sometimes as graffiti. Yet it has a complex and difficult
structure, not a structure that you would expect to find in graffiti that would have been
constructed in a hurry. The structure does suggest that it was drawn from the center
expanded out form a very simple symbol. This paper concentrates on all the possible
different ways of constructing it. These simple mnemonic symbols may explain the
longevity and popularity of the complex structure of the ‘Cretan labyrinth’.
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1. HISTORY OF THE CRETAN LABYRINTH.
Symbols that consist of concentric levels appeared very early in history. Carved into a
Neolithic passage grave in Ireland are symbols that comprise of concentric circles and
are cut to the center by a line. These symbols are known as ‘Cups and Rings marks’.
Despite the simplicity of the design these symbol would have been laboriously carved
with no more sophisticated tool than a flint axe. This time-consuming process and
precision of the carving does indicate that they have a ritual significant. The design of
‘Cups and Rings’ continued to be carved into rocks well into the Late Bronze Age.
Regardless of the painstaking carving of these symbols they can be easily remembered,
once seen never forgotten. A ritual of significance would explain this symbols longevity
while the simplicity of the symbol would guarantee its exact repetition.
The next development of concentric level symbols was labyrinthine symbols. Although
the Cups and Ring are not labyrinths, they are possibly an embryonic form of these later
labyrinthine symbols. There are many examples of labyrinthine symbols that are dated to
the Bronze and Iron Ages. They have been found carved onto rocks in Italy, Spain,
England, Ireland and Sardinia (Kern 2000). A consistent structure began to emerge,
which has become known as the ‘Cretan labyrinth’. The Cretan labyrinth has one and
one only path from the outside to the center and it consists of eight concentric levels.
Each level runs in an opposite direction to the one before and the level sequence is from
the outside to the 3,1,2,4,7,6,5, levels and then into the center (Morrison 2001).
Although there are rock carvings of the Cretan labyrinth have been dated to the early
Bronze Age, the earliest dated Cretan labyrinth on the basis of archaeological and
historical criterion is on a clay tablet from the palace of Nestor, Pylos (see Figure 1).
Figure 1. The Pylos ‘Cretan Labyrinth’ thirteenth century BC.
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This tablet was originally unbaked and only survived because of the fire that destroyed
the palace. The writing on the other side of the tablet has no relevance to the labyrinth;
this has lead to the assumption that this symbol was a doodle by an idle scribe (Heller
1961). However, the other tablets that have been found in the same archive indicate that
the scribes had no time for doodles. The tablets show a city at war, fear and confusion
reigned, inventories of weapons, troop movements and sacrifices, possibly human, to the
gods were recorded (Chadwick 1976). Yet in this confusion a scribe draws this exact
and difficult geometrical structure, perhaps a private prayer. This structure does suggest
that it was drawn from the center, indicating that a mnemonic device was used. The
palace of Pylos did not survive the war and it was leveled to the ground. Its very
existence was forgotten until the excavations of Carl Blegen that begun in 1938.
The Cretan labyrinth remerges again and again throughout history. In a tomb near the
ancient city of Caera north of Rome, a seventh century BC Etruscan wine pitcher was
found. On this wine pitcher is a Cretan labyrinth. The Etruscan labyrinth is round, and
not square like the Pylos tablet, but they both have the same level sequence (see Fig. 2).
The center of the labyrinth has a distinct cross and the placement of the turning points of
rows 3 – 2, 2 – 1, 5 – 6, and 6 – 7 are equally placed in relation to the central cross. The
cross and the equal placement of these points are in distinct contract to the rough
execution of the rest of the labyrinth. This strongly suggests that here too a mnemonic
device was used to draw this labyrinth.
Figure 2. The Etruscan ‘Cretan Labyrinth’ seventh century BC
For over six hundred years from the fifth century BC the labyrinth was used on coins
from Crete. The symbol is used on the coins as a symbol of Crete (see Figure 3).
Although the earliest literary reference to the term ‘labyrinth’ is on a small clay tablet
found at Knossos, dated 1400BC, no other labyrinth survives on Crete prior to these
coins. The tablet has been translated to be “One jar or honey to all the gods, one jar of
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honey to the Mistress of the Labyrinth”(Chadwick 1976). However, the tablet gives no
clue as to what the labyrinth is or its meaning.
Figure 3. Cretan Coins (a) c190-100BC (b) c200-267BC
Graffiti found on the surface of a crimson-painted pillar, in the peristyle of a villa at
Pompeii, depicts a Cretan labyrinth (see Figure 4). It was preserved under the lava from
Vesuvius in 79AD. The labyrinth is accompanied by an inscription ‘Labyrinth. Here
lives the Minotaur’. The graffiti was scratched with a nail or stylus; one assumes that it
was executed with some speed given the insulting nature of the inscription.
Figure 4. Graffiti for Pompeii 79AD
Moreover, the Cretan labyrinth was used in Biblical manuscripts, as a symbol of the fall
of the walls of Jericho. Figure 5 shows a page of a ninth century manuscript. Alongside
the labyrinth is the inscription Uruem Gericho, a misspelling of Urbem Jercho, City of
Jericho. The Cretan labyrinth remains in Biblical manuscripts for at least a thousand
years (Kern 2000).
This symbolic representation of the Cretan labyrinth has been used unchanged in its
structure for thousands of years. The Cretan labyrinth is a complex structure, and it is
difficult to draw freehand, yet it is used in graffiti that would have been executed with
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some speed. The structure does suggest that it was drawn from the center with the
assistance of a simple symbol that was expanded out to the more complex labyrinthine
symbol. This simple symbol or symbols acted as a mnemonic and kept the Cretan
labyrinth alive throughout the millennium.
Figure 5. Cretan labyrinth representing the walls
of Jericho in a nine century Biblical manuscript.
2.1 MNEMONIC DEVICES.
Using mnemonic devices in training the memory was common in classical times. In a
world devoid of printing and notepaper a highly trained memory was of paramount
importance to an orator and rhetoric was an importance part of the classical education.
Texts on training the memory for oratory have survived. The earliest treatise, was
written c.86-5BC, was by an anonymous writer who came to be known as Ad
Herennium, this name came from the dedication. Ad Herennium refers to Greek writings
on the art of memory but these accounts have not survived. It is impossible to say how
far back these mnemonic systems for an orator go. However, the texts such as Ad
Herennium were written as if the knowledge of these mnemonic systems was common
place (Yates 1966).
In general a mnemonic is a device used to remember something that is otherwise hard to
recall in detail. The classical orators used devices than involved memorizing a
background and images. The background was like a wax tablet or papyrus and the
arrangement of the images on the background was like the script. This type of mnemonic
was short term and private. The orator would follow rules given in the rhetoric texts.
However, each speech would comprise of different backgrounds and images according
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to the orators own personal experience. A visual mnemonic for a symbolic depiction
would be far more suitable than training the memory. The longevity of the Cretan
labyrinth would depend upon its simplicity of the mnemonic and the clarity of the
algorithm that explained that mnemonic.
2.2. POSSIBLE MNEMOMIC SYMBOLS FOR THE
CONSTRUCTION OF THE CRETAN LABYRINTH.
One symbol and method of construct is continually pointed to as being the easiest way to
draw a Cretan labyrinth (Attali 1999; Kern 2000) (see Figure 6). Denote symbol Figure
6.a as M(a). To expand M(a) beginning at the top vertical of the cross then inserting a
right angle or arc between the vertical of the cross and vertical of the L-shape on the
right hand side (see Figure 6b). Second, begin with the vertical on the left-hand side and
follow the path made in the last step and terminate at the dot in the right-hand quadrant
(see Figure 6c). The proceeding steps shown in Figure 6 continue to build up the
labyrinth by beginning with the dot or line on the left-hand side, leaving the lines that
have terminated at the dots. Then traversing the symbol in the same direction and
terminating at the first dot or line on the right-hand side, again leaving the lines that have
terminated at the dots. Denote this algorithm A(1).
The nucleus M(a) to the expansion into the complete labyrinth has been reported to have
been a game called “Walls of Troy” that was well known at the beginning of the
twentieth century (Heller 1946) p.133. It is also claimed that this exact game continues
to be played in India (Phillips 1992) p.322. However, is M(a) the mnemonic that the
labyrinth on the Pylos tablet, the Etruscan wine pitcher or the Pompeii graffiti were
drawn from?
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Figure 6. (a) Is the mnemonic Symbol denote M(a),
(b-i) are the expansion A(1) for M(a)
The symbol M(a) does date back into history but this does not make it a mnemonic for
the Cretan labyrinth. M(a) has been found on Babylonian seals that date back to
2000BC. A pottery shard, dated 604BC, found in Ashkelon in Israel has M(a) as a part
of its design (see Figure 7a). The exact dating of the shard, can be established because
of the destruction of the city by Nebuchadnezzar, King of Babylon was recorded. The
pottery shard is of wild goat style and the wild goat style has lots of box type designs.
Figure 7b-d show other wild goat shards found at the same archaeological excavation at
Ashkelon. The pattern of M(a) may not have been used as a mnemonic in ancient times
since it is not complicated and it is an attractive symmetrical symbol that is easily
replicated accidentally. Moreover, M(a) gives the appearance that it was developed as a
mnemonic well after the Cretan labyrinth was established.
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Figure 7(a) the Ashkelon Shard containing M(a).
7(b-d) Examples of Wild Goat Pottery
The first noticeable thing about M(a) is that the algorithm to expand it is very simple.
The continuous joining of end points in the same direction is very straightforward.
However, the L-shapes do appear to be redundant, by removing the L-shapes the steps
of the algorithm are reduced. The Cretan labyrinth can be expanded from a simple cross
and four dots in the quadrant of the cross, (see Figure 8a), denote this symbol M(b).
First, begin with the top on the vertical of the cross moving right traverse the dot,
following a path between the central point and the dot. Then return to the left side and
terminate at the dot in that quadrant (see Figure 8b). Second, begin at the dot in the
upper right quadrant, follow the path made by the last arc then traverse the last
terminating point and proceed around to the horizontal of the cross on the right-hand
side (see Figure 8c). Third, begin at the horizontal of the cross on the left-hand side
follow the outside existing shape, traverse the dot following a path between the central
point and the dot on the lower right quadrant. Then return and terminate at the dot on
the left-hand quadrant (see Figure 8d). Finally, repeat the second step but begin at the
dot on the lower right-hand side (see Figure 8d). M(b) and this simple algorithm
complete the Cretan labyrinth in only a few simple steps. Denote the algorithm A(2).
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Figure 8. (a) Is the mnemonic Symbol denote M(b), (b-e) are the expansion A(3) of
M(b).
A third possible mnemonic consists of the four L-shapes (see Figure 9a); denote this
symbol M(c). The algorithm begins from the vertical line from the upper left-hand side
and then arcs over to the right quadrant and terminates in the center of this quadrant (see
Figure 9b). Second, begin at the horizontal line from the upper right-hand side return to
the left-side traversing the path created in the last step and terminate in upper left
quadrant level with the right terminating point (see Figure 9c). Third, begin at the
horizontal line from the upper left side; follow the path created in the last step until the
horizontal path between the quadrants is reached. Cross that path then continue around
the outside of the upper half and terminating at the horizontal line of the lower righthand
quadrant (see Figure 9d). Fourth, begin at the horizontal line on the lower left-hand
side, traverse the path created in the last step and terminate in the center of the lower
right-hand quadrant (see Figure 9e). Fifth, begin at the vertical line on the lower righthand
side traverse the path created in the last step and terminate in the center of the
lower left-hand quadrant (see Figure 9f). Finally, begin with the vertical of the lower
left-hand side traverse the entire symbol until the vertical path between the quadrant is
reached proceed through the center and terminate at the vertical of the right-hand side
(see Figure 9g). This completes the symbol of the Cretan labyrinth. Denote this
algorithm A(3).
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Figure 9. (a) Is the mnemonic Symbol denote M(c), (b-e) are the expansion A(3) of
M(c)
The mnemonic symbols M(a), M(b) and M(c) are expanded by different types of
algorithm. However, these symbols have variations and use algorithms A(1), A(2) and
A(3). The mnemonic Figure 10a, is constructed with nine dots, denote this mnemonic
M(d). M(d) is expanded by using A(1). The dots can be reduced to five dots, (see Figure
11a), denote this mnemonic M(e) and expand using A(2). Five dots and a vertical line
through the middle also form a mnemonic, (see Figure 11b), denote this mnemonic M(f)
and expand using A(3). A cross and eight dots, two in each quadrant (see Figure 11c),
denote this mnemonic M(g) and use A(1) to expand it. Also M(c) can include four dots,
denote M(h) and using the same algorithm, (see Figure 11d). Finally a vertical line a dot
or broken at the center and eight dots four on each side (see Figure 11e), denote this
M(i) and this is expanded using A(1). These nine suggested mnemonic devices are
variations on the same central region of the Cretan labyrinth. However, the prime
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purpose of this investigation is to consider what could be the mnemonic that the ancient
labyrinths were drawn from and if there were any developments of the mnemonic itself.
Figure 10. (a) Is the mnemonic Symbol denote M(d),
(b-e) are the expansion A(1) of M(d)
Figure 11. Further possible mnemonics
3. HOW WERE THE ANCIENT LABYRINTHS
DRAWN?
It would be very satisfying to discover an explanation that would clearly link a
mnemonic to the Minoan culture of ancient Crete. Assuming that the Cretan labyrinth
was originally of ritual significant, the mnemonic would be more in keeping with this
significance if it was based on a Minoan religious object or symbol. The swastika was an
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ancient symbol well known to the Minoans, it has been found on early Minoan seals and
sealings (Evans 1964 Vol, 4. p.570). A swastika can easily be constructed from the
mnemonic symbol M(b), (see Figure 12).
Figure 12. Left M(b) and right M(b) expanded into a swastika
On close examination of the seventh century Etruscan labyrinth, (see Figure 2) the
center of the labyrinth has a distinct cross and the placement of the turning points of
rows 3 – 2, 2 – 1, 5 – 6, and 6 – 7 these are equally placed in relation to the central
cross. Looking at the way this labyrinth was drawn the cross and turning point were
precisely placed then the walls of the labyrinth were drawn. The execution of the
labyrinth is in stark contrast with the center where it was very roughly drawn. The cross
and the equally placed turning point are M(b). The curve that traverses the turning point
excludes the mnemonics that contains the L-shape and the distinct pre-drawn cross in the
central exclude all those mnemonics without a cross in the center. The only other
possibility is M(g), however it would be more suited to a square labyrinth, rather than
the inaccurate curves that traverse the terminating points of the Etruscan labyrinth.
Figure 13. (a) The labyrinth in the Cathedral (b) the center of the labyrinth with the
swastika highlighted.
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The swastika does have a connection with a labyrinth in Algiers. A mosaic floor
labyrinth now in the Cathedral of Algiers is the oldest surviving floor labyrinth in a
church. It was originally at the Basilica of Reparata, which was founded in 324AD in
Al-Asnam Orleansville and the labyrinth is dated approximately the same age. The
labyrinth is a standard Roman layout of four quadrants, and in the center is a box that
contains a matrix of letters, thirteen across and thirteen down (see Figure 13a). The
center letter is ‘S’ on close inspection there is an inscription. From the center letter ‘S’
reads Sancta Eclesia, Holy Church, in all four directions forming a swastika, (see Figure
13b). The letters in the matrix are not well lined up so that the swastika is not visibly
clear immediately. Moreover, the swastika was relatively unknown in North Africa and
perhaps this is indicative of a strong Roman influence in the design and also in its
meaning. The connection between the labyrinth and the swastika is obscure.
Nevertheless, the inscription in the shape of the swastika in the center of the labyrinth in
Algiers does have a sacred meaning.
It is generally accepted that the origins of the word labyrinth is labyrinthos - laburifos
and this has a direct association with an ancient cult symbol the ‘double axe’ – labrys -
labrud (Evans 1964 Vol. 3. p.283). The symbol of the double axe is carved into the
walls of Knossos, it is on Minoan pottery and seals, in frescos, painted on sarcophaguses
and it is a character in the writing of the Minoans and Mycenaeans. There were sacred
knots in the shape of the double axe (Evans 1964 Vol. 1. p.430). Caves were used as
religious sanctuaries; here votive objects were left including small minute double axes
(Hogarth 1900). The double axe dates back to the borders of the Neolithic Age (Evans
1964 Vol. 1. p.57) and some of the mnemonics are in the shape of a double axe.
Figure 14. Mnemonics that are in the shape of the double axe
In Figure 14 there are four variations of mnemonics that have the shape of the double
axe, Figure 14a and 14b have the shape of the double blades while 14c and 14d have the
added vertical, the shaft of the axe. While the vertical line is the only difference between
the mnemonics 14a – 14c and 14b – 14d it does represent a visual difference in the flow
of the finished drawing of the labyrinth.
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Figures 14b and 14d are more suitable for a circular labyrinth, because the inner dots in
the quadrants restrict the flow of the curve around the turning points. Nevertheless, it
does not exclude them. Any of these four mnemonics in Figure 14 could have been the
bases of the square Pylos labyrinth, which was drawn in an era when the symbol of the
double axe appears to have been the most revered religious symbol. However, there is
something more satisfying about the mnemonic 14c as it appears more complete as an
image of the double axe.
4. CONCLUSION
The ancient rock carvings of the Cretan labyrinth are all circular. They are very
precisely carved and this precision leaves no hint as to how they were laid out before
carving. Furthermore, in most cases the dating has been inadequate. Many have been
dated to the early Bronze Age, purely on the evidence of the symbol itself. An example
of this inadequate dating was the Hollywood Stone. The Hollywood Stone was found in
Ireland it is inscribed with the Cretan labyrinth. It was originally dated to the early
Bronze Age, 2500-2000BC; however, it is now believed to be connected with the early
Celtic Church. The Cretan coins with the Cretan labyrinth are also too accurately
executed to reveal their pattern layout. In fact, the overall shape of the labyrinths, on the
coins are a square, rectangle or a circle while the graffiti and the drawing of the Etruscan
pitcher has a very defined raised entrance on one side. It is in the graffiti, of the Pylos
tablet and on the column of Pompeii and the rough drawing on the Etruscan wine pitcher
that hint at clues to their construction.
The graffiti from Pompeii has a distinct cross in the center, the turning points are equally
placed around this cross and vertical of the L-shapes are level with the opposite side.
The center has an extra heel to it, which make the labyrinth more squat. Nevertheless the
equally placed turning points and vertical of the L-shape around a precise cross suggest
that M(a) was used in the center. This is in complete contrast with the Etruscan labyrinth
where the orderly cross and the equally placed turning point are well defined but the
rough drawing and placement of the curve that traverses the turning points indicates that
the mnemonic M(b) was used and it was expanded by A(2). The swastika can also easily
be constructed from M(b) and the swastika is a symbol that is found on Etruscan funeral
urns dating to the same time that the labyrinth (Figure 2) and this wine pitcher was found
in a tomb.
Although the swastika was a religious symbol of the Minoans and Mycenaeans, the
symbol of the double axe permeates both cultures. The Pylos labyrinth does have a
distinct cross, the turning points are equally place around the central point yet the
verticals that traverse the turning points are not at all level, this is particularly evident at
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the verticals closest to the entrance. However, central endpoints of these verticals are
equally placed around the central point or cross. This indicates three possible mnemonic
M(d), M(g) and M(i) and using A(1) to expand. The cross would be formed by the
application of the algorithm and on a square labyrinth it would be impossible to say
which was used by visual evidence. Nevertheless, the deep religious significance of the
double axe would indicate M(d) and M(i) are the stronger contenders as being the
mnemonic device for the Pylos labyrinth. Also the general accepted etymology of the
word labyrinth is directly related the cult symbol the double axe – labrys and it would be
difficult to believe that the words layrinthos and labrys are not cognate.
Over thousands of years the Cretan labyrinth has remain unchanged in it structure,
sometimes round, sometime square yet the level sequence remains the same. This static
symbol has crossed time and cultural diversity. However, by examining the possible
different mnemonics and algorithms it become evident that what appears to be static on
the surface was undergoing continuous change. The continuous evolution of the
mnemonics of the labyrinth emphasizes the constant change in its meaning and its
religious significance as compared to the meaningless and rather pointless game it has
now become.
References
Attali, J. (1999). The Labyrinth in Culture and Society. Berkeley, North Atlantic Books.
Chadwick, J. (1976). The Mycenaean World. Cambridge, Cambridge University.
Evans, A. (1964). The Palace of Minos at Knossos, Volume 3. New York, Biblo and Tannen.
Evans, A. (1964). The Palace of Minos at Knossos, Volume 4. New York, Biblo and Tannen.
Evans, A. (1964). The Palace of Minos at Knossos, Volume I. New York, Biblo and Tannen.
Heller, J. L. (1946). “Labyrinth or Troy Town?” The Classical Journal Vol. 42: p.175-91.
Heller, J. L. (1961). “A Labyrinth from Pylos.” American Journal of Archaeology Vol. 65: p.57-65.
Hogarth, D. G. (1900). “The Dictaean Cave.” The Annual of the British School at Athens Vol. VI: p. 94-116.
Kern, H. (2000). Through the Labyrinth. Munich, Prestel.
Morrison, T. (2001). “The Geometry of History: 03217658.” Vismath Vol 3(4):
http://www.mi.sanu.ac.yu/vismath/
Phillips, A. (1992). “The Topology of Roman Mosaic Mazes.” Leonardo Vol. 25(3/4): p.321-329.
Yates, F. A. (1966). The Art of Memory. London, Routledge and Kegan Paul.
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218

THE PARTHENON HEIGHT MEASUREMENTS: THE
PARTHENON SCALE WITH ROOTS OF 2
A. M. BULCKENS
Name: Anne M. Bulckens, Ph.D. (Architecture), (b. Herentals, Belgium).
Address: Bulckens, Jalan Intan Ujung 75, Jakarta Selatan, 12430, Indonesia
E-mail: annepaul@cbn.net.id
Fields of interest: Classical, Renaissance and Baroque Art and Architecture
Publications:
Bulckens, A.M. (1998) Did Plato Ever Refer to a Section in Extreme and Mean Ratio in his Writings?,
Journal of Transfigural Mathematics, 3, no. 2, 27-31 and no. 3, 23-31.
Bulckens, A.M. and Shakunle, L. (1998) Logic Numbers and the music of Greek Architecture, Journal of
Transfigural Mathematics, 4, no.1, 21-47, 53-57.
Bulckens, A.M. (1999) The Parthenon’s Main Design Proportion and its Meaning. [Ph.D. Dissertation],
Geelong: Deakin University, 269 pp.
Bulckens, A.M. (2001) The Parthenon’s Symmetry, in Symmetry: Art and Science, (Fifth Interdisciplinary
Symmetry Congress and Exhibition of the ISIS-Symmetry, Sydney, July 8-14, 2001), no. 1-2, 38-41.
Abstract: The Parthenon (447 - 438 BC) epitomises the glory of Athens at the height of
Classicism. Although enticingly forming the subject of numerous works of research, the
understanding of several aspects of this temple to Athena still remains tentative. Many
scholars adhere to the view that the Parthenon was designed in accordance with a
simple 4:9 ratio. The dissertation, The Parthenon’s Main Design Proportion and its
Meaning, offers new insights. Through an analysis of the specific dimensions of the
temple’s building parts, this thesis establishes the measurement of a Parthenon Module
and Parthenon foot (with 16 Dactyls). It then explains the main design proportion of
the temple, expanding the idea that the Parthenon was built with a 4:9 ratio The
measurements and design of the Parthenon’s building elements exalt a 4:6 = 6:9
geometric proportion. This proportion involves the ratio 2/3, which is the perfect fifth
in the Pythagorean tetrachord. It is proposed here that the design of the Parthenon, in
regard to its height measurements, was based on a Dorian musical scale, which the
architects altered in order to incorporate simple ratios representing the first roots of 2.
In the next part of the article, not presented here, it will be shown how the Parthenon
design exemplifies these ratios (that closely approximate roots of 2 values) two-
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dimensionally, as well as three-dimensionally. Since it will then be revealed how the
subtle design of the Parthenon houses two cubes, with the large cube having double the
volume of the small cube, it then becomes clear that the Parthenon offers an answer to
'the Delian riddle of the cube’, which at the time was a hot mathematical issue
concerning 3√ 2.
1. THE HEIGHT MEASUREMENTS OF THE
PARTHENON
In the façade and flank, a total of twelve different height measurements are identified as
having importance. These measurements are listed below, and they are shown in figures
1 and 2.
All measurements are based on the dimensions provided by Penrose [7], since it is well
known that he was the leading archaeologist who measured the Parthenon in the most
accurate and thorough manner. Penrose’s measurements were then translated into a
system of Parthenon Feet consisting of 16 Dactyls (D), whereby each Dactyl measures
21.44 mm and each Module measures 40 Dactyls [See Bulckens, dissertation, 1 and
article, 2].
320 D = at the corner of the façade, acroterium + entire pediment (i.e. pediment face +
raking cornice + rise in curvature of the horizontal cornice)
440 D = corner column minus the stylobate rise along the flank, minus the capital
480 D = column proper = central column height minus the stylobate rise along the flank
512 D = in the centre of the façade, upper step + column
560 D = in the center of the façade, 3 steps + column
623 1/3 D = at the corner of the façade, 3 steps + column + architrave
640 D = at the corner and also in the center of the façade, column + entablature
666 2/3 D = at the corner of the façade, 2 steps + column + architrave + frieze + its
rise in curvature
720 D = in the center of the flank, 80 Dactyls of steps + column + entablature
800 D = in the center of the façade, column + entablature + pediment face
840 D = from the corner of the stylobate, column + entablature + pediment
960 D = from the corner of the stylobate, column + entablature + pediment + acroterium
Only two measurements are no integers: measurements 623 1/3 D and 666 2/3 D. The
fractions of these measurements indicate that the Dactyls are divided further in thirds of
Dactyls. As will be explained in this article, these two measurements were incorporated
to bring out root of 2 values.
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Clearly, all other Dactyl measurements are integers, yet all these Dactyl measurements
are very accurate conversions of Penrose’s dimensions into these Dactyl measurements,
working well within a margin of 0.15 %. (It is noted that the 320 D measurement
comprises the 200 Dactyl pediment (0.03 % off Penrose’s dimension) plus the height of
the acroterium that does not survive. Likewise, the 960 D measurement constitutes the
840 D measurement (0.03 % off) plus the same acroterium. An acroterium height of
120 Dactyl is a valid supposition; it is based on drawings by Orlandos.) Surmising,
then, that these accurate integers indicate that the Parthenon designers did their utmost
to obtain ideal measurements, it will be explored which design intentions underlie these
ideal measurements.
It might well be that it is the Athenian conviction that they were especially blessed by
the gods, hence superior of all others, which led the Athenians to demonstrate that this
was indeed so. The astonishing level of perfection achieved by the architects and
builders, as is evident in the Parthenon’s design and construction, puts the ‘self-image’
of Periclean Athens on show – for the whole world to admire.
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Figure 1: Parthenon diagram of the north-east corner and center of the façade, with curvature and
steps exaggerated; drawing not to scale; dimensions indicated to the right of their dimension line.
One of the key measurements is the 480 Dactyl height of the ‘column proper’, which is
the height of the central column of the façade minus the stylobate rise in curvature along
the flank. (The stylobate is the temple platform upon which the columns stand. As with
all ‘horozontal’ elements of the Parthenon, it is curved.) This measurement cannot be
seen, but it will be explained how it plays a crucial role. Among the more tangible
measurements at the center of the façade (shown in figure 1) are the 512 Dactyl
measurement comprising the upper step plus the central column (25 2/3 D + 486 1/3 D),
and the 560 Dactyl measurement consisting of the three steps and the central column (73
2/3 D + 486 1/3 D). Figure 1 shows also the 640 Dactyl measurement running from the
stylobate center to the top of the horizontal cornice (central column + entablature, 486 1/3
222

D + 153 2/3 D), and the 800 Dactyl measurement running from the stylobate center to the
top of the pediment face, excluding the perpendicular height of the raking cornice
(central column + entablature + pediment face, 486 1/3 D + 153 2/3 D + 160 D).
Figure 2: Parthenon diagram of the north-east corner and center of the north flank, with curvature
and steps exaggerated; drawing not to scale; measurements of the ‘Parthenon scale’ in bold color.
In the center of the flank, shown in figure 2, is the 720 Dactyl measurement, which goes
from the very bottom of the steps to the top of the horizontal cornice, including its rise
in curvature along the flank (steps + stylobate curvature at the flank + central column +
entablature, 73 2/3 D + 6 1/3 D + 486 1/3 D + 153 2/3 D). The 640 Dactyl measurement of
the column and the entablature can also be taken at the corner (corner column +
entablature, 486 2/3 D + 153 1/3 D). At this corner, the 623 1/3 Dactyl measurement
(shown in figure 1), runs from the bottom of the steps to the top of the architrave (three
223

steps + corner column + architrave, 73 2/3 D + 486 2/3 D + 63 D). And the 666 2/3 Dactyl
measurement (shown in figure 2) starts at the top of the lowest step and reaches the top
of the frieze, including the frieze’s rise in curvature at the flank (second step + upper
step + corner column + architrave + frieze + the frieze’s rise in curvature at the flank,
24 D + 25 2/3 D + 486 2/3 D + 63 D + 63 D + 4 1/3 D). Lastly, the height from the corner
of the stylobate to the top of the pediment is 840 Dactyls (corner column + entablature +
pediment, 486 2/3 D + 153 1/3 D + 200 D), and to crown the temple, the acroterium
height is added to this measurement, resulting in an overall temple height of 960 D.
2. TWELVE LARGE MEASUREMENTS
DETERMINING THE HEIGHT OF ALL THE
BUILDING ELEMENTS.
In the extended article, it will be explained in detail how these twelve measurements
determine the height of all the building elements of the façade and flank: the three steps,
the column and its capital, the entablature with its frieze, architrave and horizontal
cornice, the pediment with its pediment face and raking cornice, and the acroterium.
It is proposed that these measurements determine the heights of the building elements in
a peculiar way: when a large measurement is subtracted from a related larger
measurement, the height of the building element in between these two measurements is
obtained. And it is precisely the dimensions of the rise in curvature of the ‘horizontal’
elements, such as the stylobate and the entablature, which facilitate ideal dimensions and
simple ratios between large measurements. The reason for this method will become
clear in later paragraphs, but the method itself is easy to understand when giving a few
examples. Consider measurement 800 and measurement 640, which are both taken at
the center of the façade, starting from the top of the stylobate (the curved platform upon
which the columns stand). Take the largest measurement of 800 Dactyls, and notice that
it goes to the top of the pediment face. Subtract from this the 640 Dactyl measurement,
which only reaches the top of the entablature, including the rise in curvature of its
highest element, the horizontal cornice. What is left is a 160 Dactyl height. This is the
height that results from the subtraction (800 – 640), and it determines the height of the
pediment face because this is the building element between the two large measurements.
Equally important, the equation 800 x 4/5 = 640, establishes the relationship, i.e. the
ratio, between the two large measurements. Indeed, with this method of subtraction
between large measurements, the height of all the building elements will eventually be
determined, while simple ratios underlie the relationship between the large
measurements.
Continue the same sequence and procedure with the 640 Dactyl height running from the
center of the stylobate to the top of the entablature. Subtract from this the 486 1/3
Dactyl measurement comprising the central column. The result is the height of the
224

entablature: it measures 153 2/3 Dactyls. Interestingly, as explained on previous
occasions [1], it is the temple refinements, such as the rise in curvature of building
elements, which facilitate a system of simple ratios between measurements. This is also
the case with the height measurements, of which 640 cannot be readily multiplied by a
simple ratio in order to obtain 486 1/3. Instead, it is necessary to incorporate the
stylobate rise, in this case along the flank, to understand the measurement. The rise in
curvature along the flank is 6 1/3 Dactyls. The column proper, i.e. the height of the
central column minus the rise in curvature along the flank, measures 480 Dactyls. When
the 486 1/3 Dactyl height of the column is thus split up in the 6 1/3 Dactyl height of the
stylobate rise along the flank and the 480 Dactyl height of the column proper, the result
is that: (640 x 3/4 ) = 480. Thus, when 640 is multiplied by a simple 3/4 ratio, the
column proper is the result. And to obtain this column proper, a stylobate rise along the
flank of 6 1/3 Dactyls was established and then subtracted from the column height.
In this fashion, the height of all the building elements will eventually be determined in
the extended article. It will also be shown that there is a main sequence of
measurements, which are related to each other by a continuous series of the ratios 1/2,
2/3, 3/4, 4/5, 5/6.
Consider the next sequence of measurements, which starts with the 480 that originates
from the first sequence, and compare it to measurement 512. The 512 Dactyl
measurement, taken in the center of the façade, is the height of the upper step plus the
central column. Take the 512 Dactyl measurement and subtract from this the 480 Dactyl
column proper, which is the height of the central column minus the 6 1/3 Dactyl stylobate
rise in curvature along the flank. The result is 32 Dactyls. Hence, the height of the
upper step is (32 – 6 1/3) Dactyls, which is 25 2/3 Dactyls. Again, a simple ratio forms
the basis, since 512 x 15/16 = 480. And by subtracting the stylobate rise along the
flank, I was able to determine the height of the upper step.
I propose that the temple was not only built with tremendous care, but that the architects
already designed the temple adhering to this method, which allows working with a very
small margin of error, always below 0.2 %. I propose that this is one of the reasons why
the architects determined heights of building elements by taking simple ratios of larger
measurements, while equating the ‘left over’, i.e. the subtraction result, with the height
of the building element. This way, the architects could decide that, for example, 480 x
16/15 = 512, so that with this method employing 480, and a simple ratio, and 512, all
works out correctly. On the other hand, if they would equate the height of the upper
step, which in reality measures 25 3/4 Dactyls, with 25 2/3 Dactyls, the height of the step
is off by a margin of error of about 0.3 %. This example shows the typical order of error
of the building heights of the smaller elements, and what it makes clear is that: if it
would solely be the height of each building element from which design ratios would be
issued, the architects would not have enough leeway in order to form a coherent and
logic design with simple ratios. When trying to find accurate simple ratios for the height
of the upper step, the architrave, the frieze and the horizontal cornice, it becomes clear
that there are no such accurate relationships within the design of the façade or flank.
225

Moreover, small measurements easily produce larger margins of error. On the other
hand, if the architects worked with the large measurements as the ratio generators, and
if they employed the increments of the rises in curvature in various ways, they achieved
the necessary leeway to make an accurate, coherent and complete design from simple
ratios.
In the extended article, the heights of all the building elements of the façade are
established by working with twelve large measurements, simple ratios and the rises in
curvature.
The only temple height not yet determined by these twelve large measurements is the
height of the 2 steps of the cella. Beyond the façade, in the center of the cella, a 600
Dactyl measurement goes from the cella floor to the top of the entablature, excluding the
entablature’s rises in curvature. So, here is one more measurement that will be
considered. Comparing the 640 Dactyl measurement taken at the corner, and the 600
Dactyl measurement taken in the center of the cella, the equation is 640 x 15/16 = 600,
(This equation employs the same ratio as equation 480 x 16/15 = 512.) The difference
is 1 Module, which comprises the height of the 2 cella steps and the rise of the upper
floor in the 2 directions of the temple. (In fact, there is also a 600 Dactyl measurement,
with 599 1/3 being 0.11 % off 600, at the corner of the façade going from the second
step to the top of the architrave.)
3. THE PARTHENON SCALE WITH ROOTS OF 2
When the 600 Dactyl measurement is added to the height measurements, one gets a list
of all the large measurements needed to obtain the heights of the main building elements
of the temple. Measurement 512 can even be omitted, since it is possible, however
cumbersome, to arrive at all building heights without this measurement. Thus
measurement 512 is replaced by measurement 600, which also involves a ratio of 15/16
(the ratio of an enlarged half tone in music). (Concerning the ratios, the simplest route
was taken to arrive at the heights of all building elements. Once many of the heights of
building elements were determined, there were alternative routes on several occasions,
which were not explained. These alternative routes provide the remaining ratios found
between the heights in the ‘Parthenon scale’ shown in figure 3.)
When these twelve large height measurements are arranged in a linear scale as in figure
3, the temple heights reveal an elegant sequence of the ratios of the first ten integers,
10/9, 9/8, 8/7, 7/6, 6/5, 5/4, 4/3, 3/2, 2/1. Furthermore when 666 2/3 D is set as 100, it
becomes clear that most of the heights are the same numbers as the tones of the basic
Hindu–Greek musical scale [McClain, 5], later known as the ‘reverse Dorian scale’ and
currently the common scale: do re mi fa sol la ti do [Kappraff, 3]. This Dorian scale is:
72 80 90 96 108 120 135 144
C D E F G A B C
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Basically, to arrive from the ‘reverse Dorian musical scale’ to the ‘Parthenon scale’, 80
is enlarged to 84 and 135 is reduced to 126. The result is that the intervals from C to D
and B to C are enlarged whole tones, 7/6 and 8/7. When next 93 1/2 and 100 are
inserted, the ‘Parthenon scale’ is obtained. Interestingly, according to Kappraff, a
musician-mathematician, 100 falls on Fsharp [Kappraff, 3].
72 84 90 93 ½ 96 (100) 108 120 126 144
C D E F Fsharp G A B C
In figure 3, when 666 2/3 Dactyls are normalized to hundred, all Dactyl measurements
are divided by 6 2/3, so that all the numbers that are the same as the ones from the
Dorian scale result in integers, while their ratios become readily readable. As in the
Pythagorean tetrachord, 72–108 and 96–144 form perfect fifths, while 72–96 and
108–144 form perfect fourths, and 640–720 forms a whole tone. Surprisingly, 84–126
is a 3/2 ratio also. When this ratio is split up to reach the 100 in between 84 and 126,
the fourth root of 2 (which here is 100/84) and the cube root of two are obtained. In this
case, the cube root of 2 is the old Babylonian 126/100. (According to McClain [6], the
old Babylonian cube root of 2 was regarded as 125/100 + 1/100.)
In all, the only numbers that differ from the Dorian scale, are 126 and 84, while 93 ½
and 100 are inserted. These are the tones, which create ratios that closely approximate
the roots of 2. So, by altering the Dorian musical scale in a unique manner, the
‘Parthenon scale’ is created to provide the first roots of 2.
The column height is crucial in the ‘Parthenon scale’. The 480 D height of the column
proper forms the beginning of a musical octave. Its double is the overall temple height,
and this is exactly where an octave ends. Also, the column height assists in providing
the measurements 623 1/3 and 440. The ratio between these two measurements is 17/12,
which is a Pythagorean approximation of √2. [Lasserre, 4, and Kappraff, 3].
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Figure 3: The‘Parthenon scale’; the 480 – 600 – 640 – 720 – 800 – 960 is the reverse Dorian
scale, the ratio between 840 and 666 2/3 represents 3 √2; the ratio between 666 2/3 and 560
represents 4 √2; and the ratio between 623 1/3 and 440 represents √2.
In conclusion, by creating the ‘Parthenon scale’, simple ratios are revealed between
numbers, of which 72 – 96 – 108 – 120 – 144 are exactly the same as the tones in the
reverse Dorian scale. There are more similarities between the ‘Parthenon scale’ and
musical scales; In Pythagorean musical scales, the tetrachord is the framework and the
ratios between other tones could vary, depending on the ‘mode’ that was adhered to.
With these other tones in between, there came Pythagorean commas, Pythagorean
leimnas, etc, which in essence are ‘left-overs’ that had to cover the distance to reach the
octave. This is also the case in the height measurements of the ‘Parthenon scale’. The
height of some building elements are literally ‘what is left over’ after subtraction
between large measurements. The difference between historic musical scales and the
‘Parthenon scale’ is that the ‘Parthenon scale’ incorporates ratios, which turn out to be
excellent approximations for the roots of 2, whereas in the Pythagorean musical scales,
√2 is the geometric mean of the octave, but it could never be reached by a coherent
system of simple ratios. (The 126/100 ratio is less than 0.1 % off the real value of 3 √ 2,
while the 17/12 ratio is 0.17 % off the real value of √2.)
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It needs to be stressed that the ‘Parthenon scale’ was not made out of the blues! On the
contrary, every single number of the scale accurately refers to a height measurement of
the Parthenon. And as it is shown at length that the height of each building element
could be determined by these twelve measurements and their interrelated ratios, I
suggest that the architects had exactly the same measurements and ratios in mind in
order to design the heights of all the building elements of the façade and flank.
This article shows how the Parthenon design is rooted in Pythagorean music, whereby
Kappraff and McClain contributed greatly to the understanding of ancient musical scales
and theory. Concerning the Parthenon, the design issue was to marry a set of
measurements exalting roots of 2 with a set of measurements, which were known to form
part of the Dorian musical scale. When the 666 2/3 Dactyl measurement was normalized
as 100, it provided the means to merge the two sets together in such a handsome way,
that even the complete sequence of the ratios 10/9, 9/8, 8/7,…, 2/1 was the result. In the
next article, then, it will be explained why and how the architects incorporated the roots
of 2, especially 3 √2, in the Parthenon’s plan and elevations.
References
Bulckens, A..M. (1999) The Parthenon’s Main Design Proportion and its Meaning. [Ph.D. Dissertation],
Geelong: Deakin University, 269 pp.
Bulckens, A..M. (2001) The Parthenon’s Symmetry, in Symmetry: Art and Science,
(Fifth Interdisciplinary Symmetry Congress and Exhibition of the ISIS-Symmetry, Sydney, July 8-14, 2001),
no. 1-2, 38-41.
Kappraff, J. (2002) Anne Bulckens’ System of Proportions of the Parthenon and its Meaning. A presentation
at the Carleton University School of Architecture, Ottawa.
Lasserre, F. (1964) The Birth of Mathematics in the Age of Plato. London: Hutchinson.
According to Lasserre, Plato used the equation x² – 2y² = ±1, to find ratios between integers x and y that
closely approximate √2. He further asserts that Plato, when using this equation, was Pythagorising.
Since 17 x 17 – 2 x 12 x 12 = 1, it follows that 17/12 approximates √2, and thus according to Lasserre,
the Pythagoreans knew this.
According to Kappraff, the ratio 17/12 was also used in Roman architecture (Ostia), in order to approximate
√2.
McClain, E. (1976) The Myth of Invariance: The Origin of the Gods: Mathematics and Music from the Rg
Veda to Plato. York Beach: Nicolas-Hays.
McClain, E.(1997) The 'Star of David' as Jewish Harmonical Metaphor, International Journal of
Musicology, no. 6, 43-47.
Penrose, F.C. (1888) An investigation of the Principles of Athenian Architecture. 2nd ed., Washington,
D.C: McGrath.
229

230

ARTISTIC DESIGNS BY MEANS OF ALGEBRAIC
STRUCTURES
Ruiz, F. and Peñas, M., Department of Didactic of Mathematics,
University of Granada
Names: Francisco Ruiz and María Peñas
Address: Department of Didactic of Mathematics. University of Granada, Campus de Cartuja. 18071 Granada
(Spain) tel. (34)958243951 fax: (34)958246359.
Email: fcoruiz@ugr.es and mtroyano@ugr.es
Fields of interest: Geometry, representations, numerical thinking, mathematics and art.
Publications and/or Exhibitions:
Ruiz, F.: Organizer of the exhibitions:
- "M. C. Escher, entre la Geometría y el Arte" (M.C. Escher between Geometry and Art). University of
Granada and Cordon Art. Granada. May 1990.
- "M. C. Escher". Museum of Contemporary Art. University Complutense of Madrid. July 1990. Madrid. July
1990.
Publications:
- Ruiz, F. (1990). La simetría en la obra de M.C. Escher (Symmetry in the work of M.C. Escher). Catalogue of
the Exhibition "M.C. Escher, entre la Geometría y el Arte". (M.C. Escher between Geometry and Art)
University of Granada. May 1990. (pp. 1-3). Published by the University of Granada (Spain).
- Ruiz, F. (1990). M.C. Escher. Catalogue of the Exhibition M.C. Escher. University Complutense of Madrid.
July 1990. (pp. 9-11). Published by the University Complutense of Madrid (Spain).
- Ruiz, F. & Rico, L. (2001). Visualization of Numerical Patterns by Means of Congruence Relation. Fifth
Interdisciplinary Symmetry Congress and Exhibition of the International Society for the Interdisciplinary
Study of Symmetry (ISIS-Symmetry). Intersections of Art and Science. Sydney, Australia. 2001.
Abstract
The study of algebraic structures usually presents difficulties because of its degree of
abstraction. The visualization of elements of these structures can facilitate their
recognition and the understanding of some properties. In order to visualize the additive
and multiplicative groups of the residual classes Zn we substitute the numeric symbols
for certain colored visual patterns in the corresponding Pythagorean table, in order to
231

highlight the regularities in the groups. This way, the resulting table maintains the
original structure and provides new geometric elements to reduce the degree of
abstraction and to obtain creative designs, which emphasize the symmetry. The students
work, carried out manually or by means of computers, become motivational elements
for the study of the algebraic structures.
1. Introduction
Visual mathematics has become an important reference point in researching
mathematical education. Sensory perception is an important way to access knowledge.
It is appropriate to consider the intervention of the senses in the transmission,
acquisition and construction of any type of mathematical knowledge. Visual
information as a generator of images and mental objects provides an important function
in learning mathematics. The capability to visualize any mathematical concept or
problem requires experimentation to interpret and understand figurative information
about the concept and be able to manipulate it mentally, as well as to express it visually.
The study of algebraic structures usually presents difficulties because of the degree of
abstraction. Visualization of these structural elements can facilitate recognition and
understanding of some of their properties.
Since the Spanish curricula for Secondary Education have given up on the teaching of
“Modern Mathematics”, algebraic structures are taught only to university students in
some scientific careers, while Secondary students have never heard of the structure of
group or field.
In spite of the difficulties that the notion of algebraic structures has for many teaching
students, we believe that teaching them contributes to enriching the view of future
teachers about mathematics.
232

2. The aims of the experience
We aim to contribute to correcting the difficulties of this process of abstraction by
means of:
- a visual form of mathematics derived from the existing connections between algebra
and geometry and the use of representations,
- motivational elements for the students like the use of resources such as materials and
computers.
- strengthening the creativity of students by means of drawing artistic designs.
233

First of all we consider that the fields of knowledge that compose mathematics are
related to one another, rather than being isolated subjects. In Mathematical Education
we should make an effort to highlight these connections since they are not evident for
our students. It is usual for students not to use their knowledge of algebra when they are
studying statistics, or their knowledge of geometry when studying analysis, etc. For this
reason we should promote experiences that relate the different fields of mathematics,
and let future teachers make connections between algebra and geometry, by creating
geometric patterns that highlight algebraic properties and relationships.
A second objective of this experience is to take into account the elements of motivation
in teaching. Mathematics is not the most popular subject for teaching students, which is
why motivation becomes especially necessary in this case. The use of physical and
technological methods can help to increase the pleasure of learning mathematics for
students who have difficulties with abstract concepts.
A third objective is creativity. Creativity has a decisive influence on the acquisition of
certain learning skills that ought to be acquired gradually This creativity, in our
experience, is linked to the perception of shapes and patterns, and it facilitates
establishing a connection between numbers and geometric figures. To make
compositions aesthetically combining shapes and colors involves an effort in the use of
regularities, geometric figures, transformations, etc. until the students achieve a
composition that satisfies their aesthetic expectations.
3. The study
The aim of our study was to answer questions such as the following:
· What do we achieve by replacing traditional numeric symbols with geometrical
ones?
· Is visualization of the properties of algebraic structures facilitated in this way?
· Can we perform geometric transformations upon these tables?
· What is the role of computers in these activities?
The experience involved two groups of primary student teachers at the University of
Granada in two different sessions. The first one took place during the 1992/93 academic
year. The lecturer made a short presentation with transparencies in black and white, and
the work was done by hand, using a photocopier and several kinds of paper or
transparencies. Only one student used color and another used a computer program. The
second experience took place during the year 2001/02. The same short presentation was
made by the lecturer but on this occasion he used a computer, with the same figures but
234

in color.
In a first stage we aim for the students to come into contact with the mathematical
knowledge that they will be working with. They are explained the mathematical
structures they are going to use in order to generate patterns. We go into details of the
properties of the elements of these structures and the operations of an algebraic system.
In this case our purpose is that the students:
• work to obtain the residual classes (Zn)
• determine the properties of an additive group
• check under what conditions the multiplicative structure provides the
structure of a group
• be able to operate with these new elements
• familiarize themselves with the use of Pythagorean tables.
• use geometric figures and apply them in geometric transformations.
In order to introduce the residual classes module n to the students, we show some
transparencies like the following, which is a sort of “machine” that schematizes Z4.
235

In a second phase the students receive information about
the activities they have to develop, and they may be shown
some examples made previously by the lecturer, and some
materials such as transparencies, photocopies, grids, etc.
We define the addition of classes as usual, and we display
the addition and multiplication of classes on a
“Pythagorean table (fig. 2).
In order to represent numbers, mathematicians use symbols, which are special drawings
with very well known meanings.
Artists make different kinds of drawings without any restriction. What would happen if
we substituted numerical symbols with geometrical drawings that have no previously
agreed upon meaning?
The students must assign a geometric form to each element of the residual class
suggesting the idea that inverse elements should have complementary forms. Then, the
students make the corresponding Pythagorean tables of these elements with the sum and
the product.
1 8
2 7
3 6
4 5
0
+
0
1
2
3
4
5
6
7
8
(Z9, +)
Fig. 3
236
0
0
1
2
3
4
5
6
7
8
1
1
2
3
4
5
6
7
8
0
2
2
3
4
5
6
7
8
0
1
3
3
4
5
6
7
8
0
1
2
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
Fig. 2
4
4
5
6
7
8
0
1
2
3
5
5
6
7
8
0
1
2
3
4
6
6
7
8
0
1
2
3
4
5
7
7
8
0
1
2
3
4
5
6
8
8
0
1
2
3
4
5
6
7

An example shown to the students belongs to (Z9, +) in figure 3.
In such a table, we can see numeric regularities, such as symmetry, along the main
diagonal line (commutative property), or along parallel secondary diagonal lines (filled
with the same number). After substituting each numeric symbol with its corresponding
geometric figure, we obtain an isomorphic table (figure 4). (Forseth, S. & Troutman,
A., 1974).
This table can be modified by geometric transformations such as reflections or rotations
(figure 5).
Now, the students must go into in the algebraic structures working on their own, to find
properties and regularities, identify structures visually and carry out artistic designs
using different variables:
• Additive or multiplicative structure
• Module
• Shapes of each element
• Color
• Shapes of the table (different grids)
• Geometric transformations (line symmetries, rotations, glide reflections,
translations to make tessellations, etc.)
• Deformations using computers programs
Figure 4
237
Figure 5

4. Students work
Here are some examples from the first group of students, most of them done by hand, in
black and white:
The shapes chosen for classes are squares, but the cells of the grid have different
shapes. Then, this student uses mirror symmetries.
238

In this case the shapes for classes and the grid are triangles, but the cells of the grid are
triangles and quadrilaterals.
The resulting shapes using mirror symmetries (Fig. A) or rotations (Fig. B) are
hexagons:
A B
239

In this example the author uses the square and the arrow as a base for making his
drawing. Most of the students did their work in black and white, because in this
experience the introductory examples shown had this characteristic.
The only student who used color started from these simple shapes. By applying line
symmetries he got the following drawing:
240

The only student who used a computer program made a drawing based on (Z3, +) the
pattern for both classes 1 and 2 being complementary.
The final result after mirror symmetries is rather simple:
He then created distortions with the computer, obtaining the following outputs:
241

Some examples from the second group of students:
We found a great variety in the students work. A few of them made almost an exact
copy of what they saw in the classroom, for example these two:
242

They both correspond to an additive group, but there are no relations between the
symmetrical classes.
The following used a greater variety of geometric shapes, including triangles, squares,
parallelograms, etc. Since the work is framed with the neutral element, we can
appreciate that its structure is a multiplicative one.
In the next output, the student used the square and the 3-D effect of MS Office drawing
tools. She could not achieve a perfect symmetry with the elements. After trying
repeatedly, she realized that the mirror reflection of these tools affects only the shape
and not to the contents of the square that she made.
243

Some other examples follow:
Tessellation made with (Z10, x), framed in black.
The following are three variations of the previous structure.
244

Another three variations of the same structure are:
245

In spite of the fact that the two following pieces of work look very different to each
other, they have the same mathematical source. The first one reminds us of the mosque
of Cordoba (Spain).
246

Next is an example of how to use colors to highlight the structures visually. Since the
module increases, we can see how the colors and the shapes also increase. On the other
hand we can appreciate the differences between these tessellations based upon the
additive structure and the multiplicative one.
247

(Z2, +) (Z3, +)
(Z4, +) (Z5, +)
Additive structures
248

5. Conclusions
(Z2, x) (Z3, x)
(Z4, x) (Z5, x)
Multiplicative structures
The main conclusion of this experience is that it motivates students, and by means of
these activities students are able to:
• perceive and visualize algebraic structures and their properties
249

• operate with:
algebraic structures
geometric figures
geometric transformations
• The use of computers:
• Facilitates the design of geometric figures
• drawings, color, duplication, special effects
• Improves the application of geometric transformations
• Increases the students motivation and level of implication in the
tasks
• Improves the quality of the students’ output.
We want to emphasize that our main purpose was not for the students to learn more
mathematics, but to begin changing their attitude to mathematics. Many of them said
that mathematics is not so awful as they thought previously, and they really enjoyed
creating designs using mathematical concepts.
6. References
Forseth, S. & Troutman, A. (1974): Using mathematical structures to generate artistic designs. Mathematics
Teacher, vol. 67, nº 5.
Ruiz, F. & Rico, L. (2001) Visualization of Numerical Patterns by Means of Congruence Relation. Fifth
Interdisciplinary Symmetry Congress and Exhibition of the International Society for the
Interdisciplinary Study of Symmetry (ISIS-Symmetry). Intersections of Art and Science. Sydney,
Australia
250

ENDLESS FORM-WORLD GENERATED BY INTEGER
PERMUTATION
GEORGE LUGOSI
Name: George Lugosi, director of R&D&I (b. Budapest, Hungary, 1936)
Address: R&D&I, 2 Union Street, Kew 3101, Victoria, Australia
E-mail: g.lugosi@hfi.unimelb.edu.au
Field of interest: Patterns, forms, analogical thinking, crystallography, geometry; (sailing, gliding, “sci-fi”).
Publications: Benapozás vizsgálata, [Investigating the Sun-path in a room, in Hungarian] Művészet, 1977, 3
Szabadág = semmibe vett szükségszerüség? [Is freedom=ignored necessity? in Hungarian] Művészet, 1978, 6
Patterns based on permutations, Symmetry: Culture and Science, Vol. 3 No. 2 (1992)
More investigation on permutation-generated patterns, Symmetry: Culture and Science, Vol. 6 No. 2 (1995)
Folk art and symmetries of permutation-generated signs, Symmetry: Art and Science, Vol. 1 No. 1-2 (2001)
Abstract: The permutation generated patterns, which are created in square matrices,
consisting of “n” rows and “n” columns are naturally arranged in tables. Selecting an
”n”, the number of basic forms (or signs) is n! x n!. This “world” offers really “crystallike”
symmetries, and beyond this, several of its elements we can find in the folk art of
several South American, Middle American, Asian and European countries. It would be
a far reaching and fruitful task of etnomathematic to make serious research-work on
this field.
1. Introduction
How to generate the above-mentioned signs? The simplest rule is: the M(c, r) n x n
matrix has to be filled so that if the columns are c(1)=n, c(2)=n-1,…c(n)=1 (from left to
right), the rows are r(1)=1, r(2)=2,…,r(n)=n (from up to down), and n(c) + n(r) > n,
than M(c,r) = 1, otherwise it is 0. When we filled the first sign, than we take the next
permutation of the rows and leave the columns intact, fill the next sign, etc. When we
have no more possibility for the new permutation of the rows, keeping the r=1 at the
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first place (which means that we filled (n-1)! signs), then we start a new row of the
signs, starting with 2 at the first place.
If we started with the n=4 and the sequence of the columns are 4,3,2,1, of the rows
1,2,3,4, then we permute the rows as 1,2,4,3; 1,3,2,4; 1,3,4,2; 1,4,2,3; and 1,4,3,2, all
together 3! =6 signs. Then we cannot keep 1 at the first place, start a new row of the
signs with 2,1,3,4 row-sequence, and continue with 2,1,4,3; 2,3,1,4; 2,3,4,1; 2,4,1,3;
and 2,4,3,1. The next line of the signs will start with 3,1,2,4, and the last with 4,1,2,3.
The 24 signs, arranged in four rows and six columns, will create the image of Fig. 1. It
is the first table. We leave a gap, the size of a unit-square, between the signs, and the
size of a sign between the tables.
Fig.1 Fig. 2
After this we have to change the column sequence from 4,3,2,1 to 4,3,1,2 and start
again with the row 1,2,3,4. Similarly to the previous one, we can fill the second table,
etc. When we finished the sixth table (the first row of the tables), we have to start the
next column-sequence with 3,4,2,1. The third row of the tables will start with 2,4,3,1
column arrangements, the fourth (the last, if n=4) with 1,4,3,2.
Finishing this, we can see that in the full block we will have n x (n-1)! number of tables
with n x (n-1)! number of signs in each of them, which gives n! x n! number of signs.
The number of signs in the tables (equally, the number of tables in a block) are growing
so rapidly that we meet with a very interesting phenomenon: let us select n=5, and the
size of the unit square=1 mm, which fits a sign in a 5 x 5 mm square. If we wish to print
out the full block, then we will realize that the printout is more than 3.5 meters long. It
means that if n > 5 then we cannot see the full block from one standing point. To be
able to “walk around” and to see the generated form-world we need the help of a
computer. We can imagine that we can move the screen around as a special window
within our permutation-generated “world”.
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2 The basic symmetries
Using the n=4 as an example, we can find within the block as well as within the tables
several symmetry connections. Fig. 2 shows these connecting lines. The famous saying
“as above so below” is true here also. The basic symmetries in the block are the same
ones as in the tables. Regardless of the “n”, in one table there are two and only two
diagonal-symmetric signs. When we investigate the signs in a table and the tables in the
block, we can find symmetries, which are the same within a table and within the block.
The lines are connecting those places where we can find the same diagonally
symmetrical signs (in rotated positions) and in the same relative positions from these
signs we will find the same surroundings in the connected tables also. These are
important information about the so-called “logical places”.
3 Some important signs
It is a remarkable fact that in the folk art we can find certain signs, which are connected
to each other. The “stairs” (the first and last signs of the first table) many times depicted
together, in Hungarian cross stitch symbols as well as in North American Hopi Indian
images or on Aztec shield and woven materials from Peru, with the “spirals”. We can
find these forms together even as a clay vessel in Peru. Fig. 3 shows a Hopi-Indian sign
and Fig. 4 are Hungarian cross-stitch elements from the region of the river Tisza.
It is remarkable, that on this Hopi sign we can see two different types of spirals beside
the stairs.
Fig. 3 Fig. 4
4 Symmetry of symmetries
In the folk art we can find Hanti shirts, which shows elements of the same “logical
places” when “n” = 3, 4, and 5, and these are on the same item. (See Fig. 5). Naturally,
it would be a rather silly idea to suppose that the maker of this item had any idea about
253

a sign system. The elements are coming from the folk soul. However, it is really
interesting that on the stole we can see the “n”=4 system 11 th table 3 rd sign and under
the “V” (which is at the end of the stole) we can find the “n”=5 version of it.
Every country or area has some characteristic patterns, which they use as their formelements.
It would be an interesting task of etnomathematic to investigate these
patterns, and find out, why can we find certain sign-combinations (e.g. stairs and spirals
together) in different areas of the world, thousands of kilometers from each other.
Fig. 5
The best way to see, understand and enjoy the symmetry-world of this endless and
amazing system is to “step into it” and create the signs ourselves. It can strongly
improve our form-recognition also.
References
Ditfurt, H., (1973), A világegyetem gyermekei, [Children of the Universe, in Hungarian], Budapest
McIntyre, Loren, Mystery of the Ancient Nazca Lines, National Geographic, May, 1975
Molnár, V. J., (1999), Világ – Virág, [World-Flower, in Hungarian], Budapest
Purce, Jill, (1980), The Mystic Spiral, Thames and Hudson, Singapore
Spinden, H. J., (1975), A Study of Maya Art,, Dover Publications, Inc., New York.
254

VISUALIZATION VS. VERBALIZATION, INSIGHT
INTO THE MORPHOLOGY OF POLYHEDRA
IRIT WERTHEIM AND NITSA MOVSHOVITZ-HADAR
Name: Irit Wertheim.
Address: Technion – Israel Institute of Technology, Department of Education in Science and technology,
Haifa 32000, ISRAEL.
E-mail: weririt@techunix.technion.ac.il.
Fields of interest: morphological approach to 3-D geometry.
Name: Nitsa Movshovitz-Hadar.
Address: Nitsa Movshovitz-Hadar, Ph.D., Head of Kesher Cham - National Center for Mathematics
Education, Director of the Israel National Museum of Science, Planning, and Technology, Technion 32000,
ISRAEL.
E-mail: nitsa@techunix.technion.ac.il.
Abstract: The impact of visual representations on understanding, and even more so, on
actively doing mathematics, has been intensively researched and is widely recognized.
This is particularly true for the study of 3-d geometry. What about the role of verbal
descriptions? Are they necessarily needed? Simply redundant? Or are they
inappropriate, may be even disturbing? In this paper we attempt at demonstrating the
crucial role verbalization plays as a complementary mode to visualization, for dealing
with 3d geometry tasks, understandably and insightfully. Our claim is that neither
visualization alone nor verbalization in itself suffice for meaningful conceptualisation.
The cognitive processes involved will be demonstrated in the context of polyhedra.1
Introduction - One picture is worth a thousand words, or is it?
In their struggle towards obtaining a sense of meaning for the hypercube (a 4-d cube),
Davis & Hersh describe vividly, the important role played by manipulating a visual
representation:
"...I was impressed by ... the sheer visual pleasure of watching it. But I was
disappointed; I didn't gain any intuitive feeling for the hypercube... I tried turning the
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hypercube around, moving it away, bringing it up close, turning it around another way.
Suddenly I could feel it! The hypercube had leaped into palpable reality, as I learned
how to manipulate it..." (Davis & Hersh, 1981).
"One picture is worth a thousand words". Underlying this well-known saying is the
widespread experience that a relatively simple visual representation can replace a lot of,
and save hours of talking.
The impact of visual representations on understanding, and even more so, on actively
doing mathematics, has been intensively researched in the past decades and is widely
recognized nowadays. This is particularly true for the study of 3-d geometry (e.g.,
Parzysz, 1999; Barwise & Etchemendy, 1991; Zimmermann & Cunningham, 1991).
Actually, the employment of visual representations in the study of spatial geometry
comes very natural and handy. It is commonly agreed that 2-d drawings and 3-d
concrete models are necessary tools, which provide comprehensive understanding of 3d
configurations. What about the role of verbal descriptions? Are they necessarily
needed? Simply redundant? Or are they inappropriate, may be even disturbing???
In this paper we attempt at demonstrating the crucial role verbalization plays as a
complementary mode to visualization, for dealing with 3d geometry tasks,
understandably and insightfully. Neither visualization alone nor verbalization in itself
suffices for meaningful conceptualisation. This claim is supported by Kosslyn (1980)
and by Stigler (1984) who suggested that visual and verbal representations of the same
information have distinct attributes augmenting one another.
To elaborate on the issue we focus our attention to the elementary processes of
watching a solid or a 3-d configuration, followed by articulating its structure and
properties. We'll show that the latter, promotes insight and understanding, which might
not be gained through sheer observation nor by manipulating the visual representations.
This we do in the context of convex Deltahedra - - Polyhedra with equilateral triangular
faces.
To conclude this introduction, here is a personal anecdote by the first author:
The three Platonic polyhedra made out of equilateral triangular faces, were old
"acquaintances" when I was first introduced to the set of convex Deltahedra (Note that
if we do not restrict ourselves to convex deltahedra, there are infinitely many
deltahedra, e. g., the stellated dodecahedron). The deltahedral dipyramids were quite
easy to grasp (a dipyramid is a solid made of two congruent pyramids with a common
base). However, there were three other deltahedra, D12, D14 and D16, which remained
altogether "strangers". It was difficult for me to gain any intuitive feel for them - I was
confused. I got a hold of them, only as I was able to construct a verbal description for
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each (!). "Suddenly I could feel them! They had leaped into palpable reality", just like
Davis' experience with manipulating the visual representation of the hypercube.
2 How many convex deltahedra possibly exist?
An n-face deltahedron is a polyhedron with n congruent equilateral triangular faces,
conventionally denoted Dn. The name comes from the Greek capital letter delta, which
has a triangular shape. An interesting problem is to find how many such convex
polyhedra there are.
In trying to exhaust the set of convex deltahedra, we note, first of all, that this set
includes three of the five regular/Platonic solids, D4 - the tetrahedron; D8 - the
octahedron; D20 - the icosahedron.
Other convex deltahedra might be obtainable by attaching several triangles side to side
in different spatial configurations. Our question is: How many convex deltahedra can
thus be obtained?
2.1 How many faces can convex deltahedra possess?
To answer this question we first observe that in each vertex 3, 4, or 5 triangular faces
only, can meet. These numbers define the valence of a vertex. Clearly, the valence of all
vertices in any given deltahedron is not necessarily the same. Denote V3, V4, V5 - the
number of vertices of valence 3, 4, 5 respectively.
Next, note that
(i) The minimum number of faces is four, as in the tetrahedron.
(ii) Since no vertex can have more than five equilateral triangles meeting in it (as six
give a flat 3600 angle sum around the vertex), the largest possible convex Deltahedron
is a solid having 5 equilateral triangles meet in each of its vertices, i.e., the icosahedron.
Thus twenty is the maximum number of faces for any Deltahedron.
Therefore, The total number of faces of any existing convex Deltahedron is in the range
between 4 and 20.
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2.2 Candidates for existing deltahedra.
First observe that the number of faces in any Deltahedron must be even. To establish it
note that each face has three edges and each edge is shared by two faces. Using Euler’s
formula for simple polyhedra, F(aces)+V(ertices)=E(dges)+2 , we get F=2(V-2) which
implies the evenness of the total number of faces. Thus, we conclude that there are only
9 candidates for deltahedra – those having 4, 6, 8, 10, 12, 14, 16, 18 and 20 (triangular)
faces.
To determine the various possible deltahedra, we employ Descartes’s formula for the
total angular deficit of a convex polyhedron. First observe that:
(i) The angular deficit of a vertex of valence 3 is 180 o ;
(ii) The angular deficit of a vertex of valence 4 is 120 o ;
(iii) The angular deficit of a vertex of valence 5 is 60 o .
Descartes’s formula for the total angular deficit of a convex polyhedron yields
or
180 o V3 + 120 o V4 + 60 o V5 = 720 o ,
3V3 + 2V4 + V5 = 12.
Every Deltahedron must satisfy this (Diophantine) equation. This is a necessary but yet
insufficient condition for the existence of a Deltahedron.
2.3 The solution
Every solution of the above mentioned (Diophantine) equation, which takes into
account the evenness of the total number of faces, is a candidate for an existing
Deltahedron. Altogether, there are nineteen possible solutions to the equation. They are
listed in Table 1 (Lichtenberg, 1988). However, the only solutions that correspond to
actually existing deltahedra are 1, 3-7, 16, and 19, total of eight convex deltahedra (a
deltahedra is a polyhedra with equilateral triangular faces).
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3. Visual and verbal description of the eight deltahedra.
As we have seen, altogether, there are exactly eight convex deltahedra. This set
includes:
• The Platonic solids: The tetrahedron D4, the octahedron D8, and the
icosahedron D20;
• Two dipyramids - D6, and D10. (D8 which is also a dipyramid is not included
here because it is mentioned above).
D6 - triangular dipyramid - two attached tetrahedral.
(D8 - square dipyramid - two attached square pyramids, actually the
Octahedron).
D10 - pentagonal dipyramid - two attached pentagonal pyramids.
• And - three rather more complicated deltahedra:
D12 - snub disphenoid. This solid can be thought of as a tetrahedron
split into two wedges each made of two adjacent faces, which are then
joined together by a band of eight triangles.
D14 - triaugmented triangular prism - three square pyramids, each
attached to a square side face of a triangular prism.
D16 - gyro elongated square dipyramid - two square pyramids attached
to a square antiprism.
Note that an antiprism has two congruent regular n-side polygons, as bases, in parallel
planes. One base is twisted so that each of its vertices is midway between two vertices
of the other, to each of which it is joined. The side faces are triangles. While twisting
one of the prism's bases, the square side faces of the prism are folded into triangles, and
the result is an antiprism.
3.1 D14 and D16 – Can you tell the difference?
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It is quite difficult to distinguish D14 and D16 from each other by their visual images.
Holding their concrete real models and looking at them from various angles does not
help much either. On the other hand verbal descriptions make it quite clear:
D14 is a triangular prism with a square pyramid attached to each of its side faces;
D16 is a square antiprism with a square pyramid attached to each of its bases.
3.2 “Seeing” and believing
We observed high school and college students attempting to describe verbally these two
solids, after they saw their actual models and various drawings. The evidence collected
suggest quite clearly, that only after one is able to give a coherent verbal description,
one reaches an understanding of the structure of each solid, and is able to “see” the
differences between the two solids. - - As if these solids leaped into palpable reality, in
Davis’s (ibid) language.
4. A Brief Discussion
Each of us tends to go through an initial "grasping" in which we understand key
concepts but cannot converse about them fluently. As our exposure to the material
increases, we are able to shape our comprehension through questions, tentative
verbalizations, informal talks with others, reorganization of notes, and so forth.
Through language, then, we gradually extend the meaning and gain understanding. We
not only recognize the structure of the subject, but also verbally manipulate its ideas,
expressing its orderliness in personalized and unique ways (Suhor, 1984). Mathematics
educators have been busy studying mathematical discourse in the past two decades. It is
commonly agreed within this community of researchers that intuition and operational
experimentation serve successfully the development of higher level understanding,
however, reaching this level necessitate reflective thinking which is anchored in verbal
discourse. (E.g., Skemp 1973, Hiebert, and Carpenter 1992, Sfard 1994 and others)
There is a tendency to communicate about visible entities by visual representations, and
about formal mathematics notions, by symbols and verbal arguments. Although 3-d
geometrical entities are real and visible, the examples given above illustrate that in
order to develop a sense of meaning for their structures, their properties and the interrelationships
among them, it is necessary to combine language and pictorial
representation. "... thought unexpressed remains immature and eventually dies out.
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Language therefore, is not just an expression of otherwise independent and fully formed
thought, but rather is a necessary form of the thought's realization." (Vygotsky, 1986).
While the visual representation provides an intuitive infrastructure, it is the language
that guarantees the transition from the intuitive perception to an analytic-synthetic
understanding, and even to innovative findings. For example, while describing in words
the process of truncating (cutting off) the vertices of a tetrahedron through mid-edges,
students observed that truncating the “top” vertex first, yielded a triangular face parallel
to the “basis” of the original tetrahedron. This observation led to the insight that by
truncating the other three vertices, a triangular antiprism is obtained. A second thought
brought them to the conclusion that the resulted 8 triangular face polyhedron is actually
an octahedron, and from here the way was short to the insightful finding that the
triangular antiprism and the octahedron are nothing but two sides of the same coin. The
property of the octahedron as an antiprism emerged as a result of the discourse among
the group of students who had a tetrahedron in front of them and were challenged to
study the results of various ways of truncating it. The students arrived at their
discovery, totally on their own based upon the combination of the mental image they
were able to operate on, and their linguistic ability to follow their imaginary action of
truncating and connect it to a recently acquired concept of the antiprism.
Visual representations and mental (pictorial) images, being wholesome and compact
supported students’ overall structural conception, that can be grasped at one glance,
while verbal encoding enabled a process of decomposition of the whole into its parts,
analysing the inter-relationship among the parts and the complete new structure.
Language played an important role in coming to grips with spatial relationships and
structural properties. Using 'literary' forms in making connections, helped students in
making sense of mathematical constructs, and in remembering it. Language is the
medium within which the creation of new concepts takes place. We are not questioning
the power of visualization. As Sfard (2000) noted: "Availability of visually
manipulability means, either actual or only imagined, underlies our ability to
communicate on the objects and operate them discoursively". On the other hand, we
suggest that verbal representations for visual entities are crucial for the understanding
of those entities. The concepts we are dealing with belong to the domain of concrete
objects or processes, usually, accessible to us through perceptual experience. As these
concepts are more complicated, the role of language and verbalization becomes more
crucial.
Students come to recognize the properties of a solid, by describing them verbally.
While verbalizing, students organized their visual impressions by comparisons:
distinguishing which aspects of the shape are to be noticed and which ones are to be
ignored, which properties are similar to already known objects, and which are different.
The immediate implication is that whoever strives to become a professional, must
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develop the ability to express verbally, visual observations and to visualize verbal ideas
mentally.
Describing an object verbally is a process that involves the decomposition of the whole,
followed by sequential reassembling of the parts. As early as 1969, Paivio observed that
visual imagery has many of the properties of a spatially parallel system, whereas verbal
processes are better suited for handling sequential, serial information (Paivio 1969).
It takes cognitive processing to make sense of visual information, be it a 2d
representation or a concrete 3-d model. The interplay between visualization and
verbalization is the key to cognitive processing of that sort. The enhancement of this
combination is therefore at the heart of a literate approach to any profession that has to
do with spatial relations - - Architecture, Engineering, Mathematics and Arts.
Finally, to balance the claim made so far, we leave you with a dilemma we started this
paper with: Is verbal description always appropriate? Is it necessarily needed with no
exception? Or is it sometimes simply redundant? Possibly disturbing???
References
Barwise, J. & Etchemendy, J. (1991). Visual Information and Valid Reasoning. In (Ed) Zimmermann, W. &
Cunningham, S. Visualization in Teaching and Learning Mathematics. Mathematics of America. USA
Davis, P.J. & Hersh, R. (1981). The Mathematical Experience. Birkhauser Boston. U.S.A.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.
Kosslyn, S.M. (1980) Image and Mind. Harvard University Press.
Lichtenberg, D.R. (1988). Pyramids, Antiprisms and Deltahedra. Mathematics Teacher. Vol. 81 Num. 1 pp.
261-265.
Paivio, A. (1969). Mental imagery in associative learning and memory. Psychology Rev. 76 pp. 241-3.
Parzysz, B. (1999). Visualization and Modelling in Problem Solving: From Algebra to Geometry and Back.
Proceedings of the 23rd Conference of the PME.
Sfard, A. (2000). Steering (Dis)Course Between Metaphors and Rigor: Using Focal Analysis to Investigate an
Emergence of Mathematical Objects.
Sfard, A. (1994). Reification as a birth of a metaphor. For the Learning of Mathematics, 14(1), 44-55.
Skemp, R.R. (1973). Relational Understanding and Instrumental Understanding. Mathematics Teaching.
Stigler, J. (1984). Mental Abacus: The effect of abacus training on Chinese children's mental arithmetic.
Cognitive Psychology 16 145-176.
Suhor, Charles (1984). Thinking Skills in English and across the Curriculum. ERIC Digest. Eric Product
(071); Eric Digests (selected) (073).
Vygotsky, L. (1986). Thought and Language. The MIT Press. Cambridge. Massachusetts.
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Zimmermann, W. & Cunningham, S. (Eds). (1991). Editors' Introduction: What is Mathematical Visualization.
In: Visualization in Teaching and Learning Mathematics. Mathematics of America. USA
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264

EXPANDABLE POLYHEDRAL VIRUSES –
KINEMATICS IN BIOLOGY
FLÓRIÁN KOVÁCS
Name: Kovács Flórián, Civil engineer, (b. Budapest, Hungary, 1973)
Address: Research Group for Computational Structural Mechanics, Hungarian Academy of Sciences,
Budapest, Műegyetem rakpart 3, H-1521 Hungary.
E-mail: kovacsf@ep-mech.me.bme.hu
Fields of interest: Kinematics, deployable structures, geometry (also geography, maps and classical music).
Publications and/or Exhibitions:
Kovács, F., Hegedűs, I., Tarnai, T. (1997), Movable Pairs of Regular Polyhedra, in Proceedings of
International Colloquium on Structural Morphology (eds. J. C. Chilton, B. S. Choo, W. J. Lewis and
O.Popovic), Nottingham, 123-129.
Kovács, F., Tarnai, T. (1998), Foldable Bar Structures on a Sphere, in Proceedings of 2nd International PhD
Symposium (ed. G. L. Balázs), Budapest, 305-311.
Kovács F. (2000), Foldable Bar Structures on a Sphere, in IUTAM-IASS Symposium on Deployable
Structures: Theory and Applications (eds. S. Pellegrino, S. D. Guest), Solid Mechanics and its
Applications, Vol. 80, Kluwer Academic Publishers, Dordrecht, Netherlands, 221-228.
Kovács F., Tarnai T. (2000), An expandable dodecahedron, in Bridge between Civil Engineering and
Architecture – Proceedings of International Colloquium on Structural Morphology (ed. J. M. Gerrits),
Delft, Netherlands, 227-234.
Kovács F., Tarnai T. (2001), An expandable dodecahedron, HyperSpace, 10, no. 1, 13-20.
Abstract: The word “mechanism” is commonly associated with technical sciences.
Altogether, there are some bright examples of them among living organisms such as in
certain polyhedral viruses that are able to change their diameter by means of a
symmetric rotational-translational motion. Mechanical modeling of this mechanism can
provide deeper insight to natural kinematical processes and can give hints for
structural engineering. First stage of modeling is the physical one that was built to
simulate the observed motions, and then a numerical modeling follows done by
computer analysis. Compound symmetric structures with multiple sets of independent
infinitesimal displacements – as occur in this analysis – require efficient tools to
interpret the motions in a physically plausible form: in our practice the group theory
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was explored. Finally, some general conclusions are drawn about infinitesimal mobility
of the physical model, extended to a family of slightly different polyhedral mechanisms.
Fig. 1: Cardboard model of an expandable dodecahedral virus, closed and expanded.
A
C
B
i
Fig. 2: Two pentagonal prisms and their connection
266
D
E
F

1 PHYSICAL MODEL
Biologists recently discovered some virus species that represent real mechanisms in
nature. These structures are built of pentamers and hexamers (pentagonal and hexagonal
faces) that are able to rotate and translate about their own axis if the pH of the
environment changes. All this highly symmetric displacement is constrained by double
protein links between two adjacent faces (Speir et al., 1995).
In terms of structural mechanics, the question is how to model the protein links in these
viruses in order to get the same type of mechanism like the original one? Our cardboard
model is one of the possible answers where double links are substituted by single but
planar ones, composed of triangular and rectangular elements. For the sake of
simplicity, only twelve pentamers are used that produced a dodecahedral arrangement
(Fig. 1). All members except for the hinges are supposed to be rigid, so all mechanisms
must be inextensional.
It is quite straightforward that linking structure between adjacent faces essentially
determines the kinematical behavior of the structure. In this particular case, adjacent
faces are connected to each other by means of a linkage composed of four (“fold”-type)
hinges and a triangle (ABC), a rectangle and again a triangle (DEF) among them
(Fig.2). Following just this one-parameter finite motion shown in Fig. 1, a direct
relationship can be established between the amount of rotation of each face and the
diameter of the circumscribed sphere of the structure: to the rotation of π/5 (from the
thoroughly closed configuration to the open state) belongs a 1.77 times larger diameter.
Note that a fully expanded configuration means a truncated icosahedral form – like a
soccer ball – and means a point of kinematical bifurcation, but the real range of
expansion for the viruses does not include this extreme configuration.
It can be verified on the physical model, however, that the structure has other –
probably not finite – mechanisms: it is possible, for example, to lift infinitesimally just
one face prism while the others remain fixed. The existence of this “face mechanism”
can be easily proved through elementary geometrical arguments: it is evident that face
mechanism means a pure translation in closed state, but pure rotation at the bifurcation
point.
2 NUMERICAL ANALYSIS
Due to the design principles, the structure is able to imitate the expanding motion of
viruses. At the same time, however, the structure might have other kind of mechanisms,
coupled to the previous one. To discover these additional motions, a detailed cinematic
267

analysis was done in two steps (Kavas & Tania, 2001). First: just by counting cinematic
constraints and unknowns it turned out that the structure must have at least six
infinitesimal mechanisms (rigid body motions are not considered among them), second:
the singular value decomposition of the compatibility matrix has shown nine more
mechanisms.
Regarding the possibility of coupling, it is quite complicated to interpret such a big
amount of independent systems of displacements. To make this task easier, some
considerations on symmetry were introduced by means of group theory (Kangwai,
1997). The result is that twelve of the fifteen mechanisms come from the individual
infinitesimal translation-rotation of each face, while the remaining three ones belong to
a certain symmetry group (T3) (Fowler et al., 2002) but their physical meaning is not
clarified yet – they are very likely to be also infinitesimal that would mean that only one
of the mechanisms (the totally symmetric expansion) is finite, as experimented in
nature. Note that in this case many of independent self–stress-states must occur in the
structure that are not investigated till the moment.
Interestingly enough, the analysis above can also be carried out for other expandable
structures based on trivalent polyhedra, i. e. tetrahedron and cube (Fowler et al., 2002):
according to the results, the six mechanisms by counting and face mechanisms plus
three T3-mechanisms by matrix analysis are unchanged that supposes a more general
cinematic property behind this kind of virus models.
Acknowledgements
Hungarian-British Intergovernmental Scientific and Technological Cooperation
Programme (GB-15/98), OTKA grant no. T031931 and FKFP grant no. 0177/2001 are
gratefully acknowledged.
References
Fowler, P. W., Tarnai, T., Guest, S. D., Kovács, F. (2002) Expandable polyhedra (to be published).
Kangwai, R. D. (1997) The Analysis of Symmetric Structures using Group Representation Theory [Ph.D.
Dissertation], University of Cambridge, Department of Engineering, 162 pp.
Kovács, F., Tarnai, T. (2001) An expandable dodecahedron, HyperSpace, 10, no. 1, 13-20.
Speir, J. A., Munshi, S., Wang, G., Baker, T. S. and Johnson, J. E. (1995) Structures of the native and swollen
forms of cowpea chlorotic mottle virus determined by X-ray crystallography and cryo-electron
microscopy. Structure, 3, 63-78.
268

Location.
Day 4
At the Horta van Eetvelde Hotel, 2002 April 12.
The location for the Friday talks was the Hotel van Eetvelde. It is described in the book
Great Architecture of the World:
"The Hotel van Eetvelde in Brussels was designed in 1898 by
Victor Horta, undoubtedly the key European Art Nouveau
architect. While most other architects flirted with the new style,
Horta found it gave the best expression to his ideas. His skill is
demonstrated in his ability to slip his domestic designs into
narrow constricted sites. The interiors become of great
importance as centers of light, which permeates through the
filigree domes and skylights—usually in the center of the
building. The Hotel van Eetvelde is a remarkable example of the
way Horta handled the situation and used it to highlight the
imposing staircase, which leads up to the first-floor reception
rooms. "
The superb house can only very occasionally be visited. During the Monument Days in
September, the house is open for the public, but there are lengthy cues. Owned by the
federation of gas suppliers of Belgium, FIGAZ, the Mat mium organization could get
access to the building thanks to colleague Arch. Jan Bruggemans, of the Brussels' Sint-
Lukas Archief.
269

Special Feature: mad math cartoons.
Patrick Labarque printed computer outputs on old
fashion slides first, to make sure the output would
be fine.
Virpi Kauko's talk was not a piece of pie, and the
cartoonist was impressed.
At the Hotel van Eetvelde Emiel De Bolle, a professional
artist put his impressions about the Mat mium meeting and
its participants on paper. It stressed the informal character of
the meeting, at its final day. Note that these drawings are
copy right protected, and by explicit wish of the author
presented in reduced format to show they are not in their final
finished and polished form.
270
Using a too ROUGH MathCad set-up, Labarque
obtained FINE art work.
Virpi's Mandelbröt sets were impressive.

Rebielak had a proof for a new type of roof, but it
was not waterproof.
Van Maldeghem’s talk was about spirals.
Encarnacion Reyes talked about making polygons
by folding paper.
271
Rebielak used plenty of arcs in his roof
constructions
Van Maldeghem’s spiral could lead to an overdose,
thought the cartoonist
The young architects from the 51N4E group talked
about the restoration of the so-called Lamot
building, a former brewery.

Tomasz Michniowski told the participants how to
lie with mathematics.
About Radmila Sazdanovic’ tilings
Were Schindler's dimensions too small?
272
Prof. Ogawa had a top-level talk about the true
mathematical notion called ... frustration.
And again about Radmila Sazdanovic’ tilings …
Han Vandevyvere’s subject was about measuring
Gothic town halls.

Dirk Frettloeh’s subject was tiling design.
Some general cartoons
273
Vera de Spinadel's topic was not about the lottery,
but the cartoonist saw a lot of 'golden' numbers.
It was an interdisciplinary conference;
mathematicians, architects and artists sometimes
spoke different languages.

274

PATTERN DESIGN BY IMPROPER USE OF
MATHCAD
PATRICK LABARQUE
Name: Labarque Patrick, architect, (b. Kortrijk, Belgium, 1945).
Address: Hogeschool voor Wetenschap & Kunst, Sint Lucas Architectuur, Paleizenstraat 65-67 BRUSSELS
E-mail: plabarque@archb.sintlucas.wenk.be
Fields of interest: colour theory; the 4-colourproblem; geometry in general such as polyhedra, projection
systems, geometric optics...
Awards: Laureate of the urban design competition "Europa Kruispunt - Carrefour de l'Europe" Brussels 1970
Abstract: If one plots a function in Mathcad (or another math program) one has to
define the x-values of the points to be plotted. These points are then successively joined
by a line. When plotting a repetitive function, with the point spacing bigger than its
repetition length, a nearly erratic plot is the result. The same can be done with contour
plots. Here the contours are interpolated as well but they depend on the xy-values of the
grid points. Some pattern designs extracted from the animation resulting from this
improper way of plotting are shown.
INTRODUCTION
The pattern design started from a camera obscura or pinhole camera. One of the
disadvantages of this camera is the very long exposure time it needs. A solution for this
problem may be the use of a zone plate. Such a plate can be plotted with any mathprogram,
printed, photographed and reduced on microfilm to the appropriate dimension.
The plotted function is shown below as a surface plot (middle) and as contour plots.
The contour plot with 2 levels (left) is a Fresnel zone plate. A Gabor zone plate (right)
is the result when we plot it with an infinite number of levels from white to black,
resulting in continuous greyscales from high to deep.
275

The illustration below explains the geometrical optics of the Fresnel zone plate, based
on the interference of light. The ray in black has constructive, the one in white has
destructive interference in the focal point with the central ray.
In order to define the pattern for the zone plate we start from the focal point. We divide
the distance D by the wavelength and express it as a cosine function to obtain the
amplitude in the z direction.
276

⎛
⎞
⎜
x²
+ y²
+ f ²
amplitude = cos
⋅ 2π⎟
⎜ λ ⎟
⎝
⎠
If the function is positive, this ray has constructive interference, if not destructive
interference with the central ray in the focal point. If we eliminate now the rays with
destructive interference by use of black rings we have an amplification of
monochromatic light in nearly the same way as with an ordinary lens (in fact things are
more complicated than explained here). A Fresnel zone plate with only black and
transparent rings has not a sharp focal point for a given wavelength. A Gabor zone plate
corrects this.
However, a disadvantage of such a zone plate is the big chromatic aberration, as its
focal distance is to a high degree depending on the wavelength. The use of a green filter
gives a substantial amelioration, because green is one of the additive main colours
(RGB), and these main colours give a more monochromatic light. Besides, as there is a
lot of green light in normal white light, we lose less light with a green instead of a red
or blue filter.
PATTERN GENERATION
The surprise at first was big when the result below, with 5 levels was obtained.
277

Afterwards the explanation was very simple; the contour plot above was made with a
9x9 matrix (for a "quick" result). The plot program then interpolates the contours (see
the last figure in the paper). For a more correct plot a grid with much higher density was
needed (and a lot more of rendering time).
The first disillusion turned to curiosity; the lens design switched to pattern design.
Experimenting with exactly the same formula, but with other grid values (11x11, 13x13
and 15x15 matrix), patterns as illustrated below are obtained.
Only by changing the grid density, a nearly infinite number of different patterns can be
generated from one and the same formula.
The question arises whether the patterns can continuously be transformed into each
other to make animations. This is impossible by changing the number of grids, as we
need integers to build the matrix for the contour plot. But gradually changing the λvalue
can do the job (also "f" can do it).
The next illustration shows such a continuous transformation of a pattern with the same
grid density by changing the λ−value step by step.
278

The patterns look too homogeneous. Therefore the amplitude is scaled from centre to
edge. The figure below illustrates the principle of the erratic plot and the result of the
scaling.
279

After careful experimentation with different step and scaling values, the final animation
"MAT MIUM 2002" was made. The mini-animation on the Matomium website is
made for internet use, as the original animation was 80Mb!
References
........,...Mathcad7, ©1986-97 by MathSoft, Inc.; MathSoft Kernel Maple ©1994 by Waterloo Maple; File
Filters ©1985-97 by Circle Systems, Inc.
Feynman, Richard P. (1988) QED. De zonderlinge theorie van licht en materie. Amsterdam, Aramith
Uitgeverij. ISBN 90 6834 037 9. Translated from: (1985) QED. The Strange Theory of Light and
Matter. Princeton N.J., Princeton University Press
Ernst, Bruno, (1986) HOLOGRAFIE: Toveren met licht (in Dutch). Amsterdam, Aramith Uitgeverij. ISBN 90
6834 011 5.
Iovine, John (1990) Homemade holograms: the complete guide to inexpensive, do-it-yourself holography.
©TAB Books, a division of McGraw-Hill, Inc. ISBN 0 8306 7460 8.
280

GROWING SYMBOLIC TREES and
BUILDING POLYTOPES
Virpi KAUKO
Name: Virpi Kauko, Mathematician, Licentiate in Philosophy
Address: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FIN-
40014 Jyväskylä, Finland. e-mail: virpik@maths.jyu.fi
Fields of interest: Complex dynamics, geometry (also evolutionary biology)
Publications: Shadow Trees In Mandelbrot Sets (in preparation)
Abstract: Abstract mathematical concepts, fractals in particular, may carry somewhat
unexpected similarities to the natural world. Generalized Mandelbrot set Md
consists of those complex parameters c for which the orbit of zero under iteration of
polynomial f(z) =z d + c (of degree d ≥ 2) remains bounded. Combinatorial properties,
like the tree structure, of these fractal sets can be studied by viewing them
as subspaces of a larger, abstract symbol space. Not all such symbolic sequences
refer to actual parameters in Md. This gives rise to a visual interpretation of the
symbol space: the Mandelbrot set lying flat on the complex plane and ”nonexistent”
component trees branching off it into another dimension, rather like peculiar shadows.
– The connection between symbolic sequences and tree structures also gives an
analogy to studying evolutionary trees of living organisms by finding mutations in
their DNA sequences. – Visualizing objects with dimensions higher than three may
be challenging, but possible. As an example, we construct 4-dimensional polytopes
using the classical 3-dimensional Platonic solids, or regular polyhedra, as building
blocks.
1 FRACTALS and TREES
Mandelbrot sets (defined above) are simply connected and compact, infinitely complicated
fractal sets on the complex plane with dihedral symmetry groups. Each
Md contains hyperbolic components, connected by branching, thin threads [4],
[8]. We will find an abstract space which turns out to have a natural, similarly
tree-like structure.
281
1

1.1 Abstract Mandelbrot set
The combinatorial properties of polynomials f : z ↦→ zd + c (and hence also
Mandelbrot sets) are based on the fact that to raise a complex number into power
of d means to multiply its angle by d (adding c affects little for large |z|). Thus we
study mappings of angles on the circle [1]; we are particularly interested in angles
that are periodic under multiplication by d (modulo full turns). For example,
25
72 ↦→
125
72 =
53 49 29 1 5
↦→ ↦→ ↦→ ↦→
72 72 72 72 72 ,
so these angles are six-periodic under five-tupling; note that 25/72 = 5425/(56−1). Angles can be turned into symbolic sequences as follows: given an angle α, divide
the circle into d equal sectors at angles α/d, (α +1)/d, . . . and label them
0, 1,...,d − 1 startingfrom the sector containingangle 0 = 1. The kneading sequence
lists the labels of sectors where the iterated angles α, dα, d2α, . . . sit. For
example, K(25/72) = 244201 0244201 0244201 0 ...= 244201 0 .
Identifyingangles with equal kneadingsequences gives rise to a “pinched-disk”
model [6]. It consists of “pawprints” with d − 1 toes, connected by branching
threads. For each degree d, this tree-like structure is actually “same” as the Mandelbrot
set!
1.2 Symbolic sequence space
Each pawprint in the (abstract) Mandelbrot set has its own kneadingsequence.
A natural question to ask is, whether the Mandelbrot set contains a “pawprint”
(or hyperbolic sector, actually) with any given sequence of appropriate form as its
kneadingsequence. The answer is no; there are sequences which are not realized.
For example, the sequence 12112∗ non-exists in M3 because no angle of the form
a/3 6 − 1=a/728 has it as kneadingsequence. Hence the abstract Mandelbrot set
is a proper subset of a larger symbol space Ad consistingof all sequences a1a2a3 ...
with a1 = 0.
1.3 Growing trees
The definition of kneadingsequence implies an important result [5], [7]:
282
2

When the angle θ moves counter-clockwise around the circle, the nth
entry in its kneading sequence changes from j to j +1precisely when
θ crosses a rational angle of the form (rd + j)/(dn − 1).
Given a pair of kneadingsequences, A and B, one can thus find the minimal period
of angles separating them from each other and from the origin. This information
can be used to figure out how the corresponding pawprints are arranged: either the
paths leading to each from origin diverge, or one path is contained in the other.
This algorithm [7] works for all sequences, realizable or not. Therefore the space
Ad also has a natural tree structure, an extension of the abstract Mandelbrot set.
One can also see from the sequence what other sequences there are “ahead” when
lookingaway from the origin. Given a pawprint C ∼ c1 ...c (nk), we find the “visible
trees” as follows (however, it may happen that some pawprints obtained this way
are nonexistent even though the base C is not):
• for each q ∈ N and s ∈{1,...,d−1}, B ∼ (c1 ...c (nk)) q−1 c1 ...(c (nk) + s) =
b1 ...bn corresponds to a satellite of C
• for each B already in the tree, check if b1 ...bl = b (n−l+1) ...bn for some l. If
so, then A ∼ a1 ...am = b1 ...(bn−l + r) isaboveB for all r ∈{1,...,d−1}.
1.4 Evolutionary trees
A somewhat similar method of translatingsymbolic sequences into tree structures
is used by molecular biologists when they reconstruct history of life by studying
283
3

stretches of DNA molecules or proteins coded by them.
Suppose we have three extant species of animals X, Y, Z, such that the latest
common ancestor of two of them has lived more recently than the latest common
ancestor of all three of them. Then there are three possible ways they may have
evolved. If they have (fictional) DNA sequences
X: ...AAA AAC CCT GTG TGT GTT CGT CGC TCG GTC GTC ATA...
Y: ...AAG AAC CCT GTG TGT GTC CGT CGC TCG GTC GTC ATA...
Z: ...AAG. AAC CCT GTG TGT GTC. CGT CGC TCG A.TC GTC ATA...
we see that two mutations separate Y and Z from X, whereas only one mutation
separates Z from X and Y. Therefore the hypothesis that the lineage of X branched
off earlier – and thus Y and Z are more closely related to each other than either
of them is to X – seems more credible than the two alternavive hypotheses.
2 REGULAR POLYTOPES
The first section of this paper dealt with visualizingan abstract mathematical
object in two and three dimensions. Polytopes, on the other hand, may in general
have any dimension by definition; because the human brain has evolved in a threedimensional
world, visualizinghigher dimensions is not easy.
2.1 Regular polytopes of dimensions 2–3
Regular polygon {p} can have any number p of vertices. Its corner angle is
π(1 − 2
p ).
Regular polyhedron {p, q} has q regular p-gons meeting at each vertex; the
midpoints of their edges are the vertices of {q}. This is possible exactly when
qπ(1 − 2
p ) < 2π, or(p− 2)(q − 2) < 4, so we have the five Platonic solids
{3, 3} tetrahedron (4, 6, 4)
{4, 3} cube (8, 12, 6) – {5, 3} dodekahedron (20, 30, 12)
{3, 4} octahedron (6, 12, 8) – {3, 5} icosahedron (12, 30, 20)
where (V,E,F) are the numbers of vertices, edges and faces, respectively. The
dihedral angle (between the planes of adjacent faces) is 2 arcsin cos π
π
q /sin p .
2.2 Regular polytopes of dimension 4
Polytope {p, q, r} consists of a number (C) of cells {p, q}, r ≥ 3 of them around
each edge [2], [3]. Hence r times the dihedral angle must be less than 2π, so
cos π π π
q < sin p · sin r . It follows that there are six possibilities:
{3, 3, 3} simplex (5, 10, 10, 5) – {3, 4, 3} (24, 96, 96, 24)
{4, 3, 3} hypercube (16, 32, 24, 8) – {5, 3, 3} (600, 1200, 720, 120)
{3, 3, 4} “co-cube” (8, 24, 32, 16) – {3, 3, 5} (120, 720, 1200, 600)
284
4

The midpoints of all edges meeting at a vertex are the vertices of a regular polyhedron
{q, r}, the vertex figure.
For example, a vertex-tetrahedron can accomodate four cells with q =3.Wenow
construct the polytope {5, 3, 3}. Startingwith one dodekahedron, we first add
twelve cells at each face. Then we add a second layer of cells, one into each dent
left in between; there are twenty of them. Now the twelve outmost faces of the
first layer are still visible; we cover these with a third layer. The fourth layer of
cells must be added one edge down, and there are thirty of them. Continuing this
way, we see that we need 120 dodekahedra for this polytope; hence its other name,
120-cell.
References
[1] Atela, P. (1991) Bifurcations of Dynamic Rays in Complex Polynomials of Degree Two,
Ergodic Theory and Dynamical Systems 12, pp. 401–423
[2] Berger, M. (1977/1987) Geometry I–II (Translation of Géométrie), Springer-Verlag
[3] Coxeter, H.S.M. (1969) Introduction to Geometry; Wiley, New York
[4] Carleson, L. and Gamelin, T.W. (1993) Complex Dynamics; Universitext, Springer-Verlag
[5] Kauko, V. (2000) Trees of Visible Components in the Mandelbrot Set, Fundamenta Mathematicae
164, pp. 41–60
[6] Keller, K. (2000) Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot
Set; Lecture Notes in Mathematics #1732, Springer-Verlag
[7] Lau, E. and Schleicher, D. (1994) Internal Addresses in the Mandelbrot Set and Irreducibility
of Polynomials, Preprint #19, Institute for Mathematical Sciences, Stony Brook
[8] Peitgen, H.-O. and Richter, P.H. (1986) The Beauty of Fractals, Springer-Verlag
285
5

286

VIRTUAL AND REAL STATES.
INNER STRUCTURE OF THINGS AND OBJECTS
TOMASZ MICHNIOWSKI
Name: Tomasz A. Michniowski (b. Radomsko, Poland, 1963)
Address: Division of Physics, Catholic University of Lublin, Raclawickie Ave 14, 20-038 Lublin, Poland, email:
pool@kul.lublin.pl
Fields of Interest: relativistic and Quantum Cosmology and Astrophysics, Philosophy of Science,
Methodology of Physics
Main publications (in reverse order):
"Mathematical Representations of Physical Reality in Quantum and Relativistic Model", a part in a book:
Philosophical and Scientific Elements of the Universe's Description. Part 2, ATK, Warszawa, Poland 1998,
p.121-39
"Geometry of Space-time; Groups of Symmetry as the Representations of Physical Laws in Cosmological
Models", ISIS-Symmetry, 1999
"Questions and Working Hypotheses in Recognising the Universe", The Man and the Nature 6, p.93-108,
Lublin, Poland 1997
"Basic Research Schemes in Constructing the Models of Universe", Yearbook of Philosophy of Catholic
University of Lublin, Part: Philosophy of Nature 44 , 3, p.75-87, Lublin, Poland 1996
"Inflationary Universes and Serial Model's Conception", KUL, Lublin, 1993
Abstract: Constructing the models of physical reality takes place in science with use of
specific method. This method is based on general mathematical features of the Nature,
called 'mathematization' and 'idealisation'. Thanks to it, there is possible to construct
quite good emulators of reality, when having only limited knowledge about itself. The
models are cognitive efficient, what means, that they are able to product verifiable
predictions on Nature's behaviour. Symmetry is a tool for all formal verifications of
these predictions and, as well, a tool for planning experiments for empirical
verification. In the article there is shown, how in practice the method is being used in
science. An example of advanced specific speculation with use of some features of main
mathematical models of reality is being shown. It is pointed, what kind of strict
scientific results may we expect when using the method. Cognitive importance of the
procedure is being emphasised. Figures illustrating the content are included.
287

1. SCIENTIFIC METHODS ANS COGNITIVE
RESULTS
Scientific recognising the universe is possible thanks to a specific feature of the physical
world called 'mathematization'. Well-known Einstein's joke about non-malicious way of
creating the world by God identifies main idea of 'mathematization'. In simple words, the
feature consists in being able to construct mathematical structures, which are able to
emulate behaviour of physical objects up to the high level of conformity. These
structures, called models of reality (or, shortly: models), allow calculating inside
themselves. Results (often called 'predictions') derived in analytical way, are fully
experimentally verifiable. Sets of predictions derived form models, until being checked
experimentally, are usually called 'theories'. Sets of results verified positively in
experimental way are known as 'scientific knowledge'.
These relationships are obvious at all levels of scientific recognition. Both in the
extremely simple, separate cases, as well as in very complicated physical situations,
recognition with use of mathematical models goes always the same way. For example,
simple, local model, created for emulating the process of falling down small balls (and
being able to predict only point and/or moment of the fall of dropped ball), uses the
same mathematics for emulating the phenomenon, as does it each one global
cosmological model, constructed for finding behaviour of groups of galaxies in the far
space. The only difference lies in the scale of mathematical complexity of both models
and subtlety of accepted assumptions.
Constructing models, of course, is not the easy task. It usually starts from very few
separate information or only intuitions we have on the area of physical reality we intend
to discover. The method of science (see Fig.1) allows us finally to reach the model and
to derive its predictions (testable in the experimental way), which may lead us to new
knowledge we hadn't got earlier.
Figure 1: Scientific method.
Discovering unknown areas of physical reality in this way may be efficient enough
thanks to special feature of mathematical models called sometimes 'idealisation'. It
concerns in making models cognitive efficient even if their structures are not especially
reliable. This means, that there may exist many models emulating the same set of
288

phenomena in a quite good way. All of them may be useful for discovering new physical
facts in the same area of investigated reality. For better explanation see Fig. 2. Here are
shown cosmological evolutions of the universe predicted by two different models:
deSitter's and Opened Friedmanian ones. Both of them have been constructed in the
same mathematical environment, but with use of different sets of dynamical
assumptions. The model of deSitter (from 1917) assumes, that space-time of the
universe is empty (no matter at all). Friedmanian model (from 1920) accepts the
presence of matter of isotropic distribution inside the space-time. Both models predict
expansion of the universe. It is easy to perceive, that both predictions unify for far future
of cosmic evolution, when the average density and pressure of matter in Opened
Friedmanian world vanishes and becomes close to assumed in deSitter's model.
Figure 2: Cosmological evolutions of Friedman and deSitter.
Another non-trivial property of mathematical constructions and mathematics as a whole
is impossibility of lying. In a well-known example [i. e. A = B ⇒ 3A = 3B ⇒ 3A – B =
2B ⇒ 2A – B = 2B – A ⇒ 2A – 2B = B – A ⇒ 2 (A – B) = – (A – B) ⇒ 2 = –1.
Notice, that (A – B) = 0, because A = B (first term of calculation)], there appears a trial
of banned operation (dividing by zero). As a consequence of it, in last term there
appears a hard contradiction, which is not allowing any further continuation of
calculation.
All these mentioned (and others) features of mathematical structures being an element of
scientific methods, allow us to feel relatively safe in our trials of discovering unknown
physical facts. The knowledge reached in this way (including final experimental
verification of all predictions of models) is really reliable and unquestionable. Usually is
yet much wider and richer than we expect. Except of having answers for what we wanted
to know, we get much more. Large part of the knowledge obtained in this way is
completely unexpected, mostly incomprehensible and – at least at the very beginning –
seems to be incoherent with the rest of science. What is the reason?
289

2. MAPS AND SCIENTIFIC KNOWLEDGE
Mapping the land in geography is not easy operation. One must create the model
(usually graphical one) of chosen area, respecting all details of its surface. Of course, it's
possible only when we know exactly each one point of it. If not, on the map there exist
'blank spots' (for example see old maps of African interior from the beginning of 20-th
century).
In science, situation is quite different: 'blank spots' never occur. If we try to construct the
'map' of investigated area, we are not able to represent all details of reality by
appropriate mathematical objects, because we don't know too much about them (and
usually are even not conscious their existence). We rather try to create the model with
use of only a few facts we suspect we know and are able to represent by mathematical
objects. These objects, existing in their specific mathematical environment, thanks to the
features of 'mathematization' and 'idealisation', do fix the main structure of the model (i.
e. the 'map'; in mathematics, a ‘map’ is each opened sub-set and a function, which
represents the sub-set in Banach space. The set, together with all sets of maps being
modelled in Banach space, is a manifold. Manifolds are natural environments for smooth
functions, essential for emulating phenomena in all realistic models of physical spacetime).
If we are really lucky, we sometimes can guess a set of assumptions and boundary
conditions for this structure, appropriate enough to allow the model quite good
emulation of reality. This means (see Fig.3), that 'map' we constructed, shows us
investigated area of reality, but only approximately, not exactly (as geographical maps
should do).
Figure 3: world and maps.
290

As luckier we were finding mathematical representations (for facts we had chosen as
representative for investigated reality) and defining the set of conditions for the model,
as better our 'map' emulates details of world it should do. This means, that the 'map'
allows us to spot physical facts (mathematical details on the 'map') we couldn't find in
any other way at all (note that these facts, being formally predictions of the model, must
be verified in experiments until being accepted as elements of scientific knowledge).
Of course, less luck or formal errors we could do in the process of constructing models
imply that our 'map' emulates reality in a wrong way, up to the level of full fiction. Such
situation means, that the 'map' shows reality, which doesn't exist within the space-time of
our universe. Practical usefulness of such model disappears than.
3. SURPLUSAGE OF MATHEMATICS
The cognitive limitations for scientific model's domain are the consequence of
assumptions we must accept in the process of constructing each one model. Arbitrary
conditions we bring, don't deprive the mathematical structure of the model of possibility
of emulating much wider areas than same model is intended to do. All models we use in
physics and natural sciences have been constructed with use of the mathematics of very
special kind, namely highly regular one. The general structure of manifolds we
permanently use for emulating 4-dimensional space-time, is very comfortable. It assures
the best possible environment for smooth enough functions (thanks to it analytical
calculations are possible) and, as well, requires no more general environment for itself
(in mathematics, manifolds often have been called self-contained structures).
Very interesting question is - what do emulate mathematical structures of less regular
kind than manifold is? Fig. 4 shows the situation. Manifold-like structure is good enough
to emulate relativistic 4-dimensional world. But in whole body of mathematics, the area
of regularity is of a very small measure. It would be strange, if the physical world would
be created in such primitive way to have to be represented by such small part of
mathematics only. Increase of scientific knowledge during the 20-th century shows, that
physical phenomena have got much more complicated roots than classical only.
Mathematical models of their mechanisms have got quite different nature. For example,
model of quantum mechanics needs allowing, in general, the complex multidimensional
spaces, instead of real ones.
291

Figure 4: Physical universe and its mathematical model.
4. QUANTUM AND REAL STATES
Scientific experiences of last century teach, that mechanisms of all physical phenomena
we know are, in general, not explainable at the ground of classical paradigm only. This
means, that everything what we can perceive and measure in physical universe, is only a
part of much more complicated system of dependencies, not accessible to our senses.
More simplified illustration of what we speak, is each quantum system; for example:
quantum model of an atom. Fig. 5 shows measurable levels of energy in the atom.
Understanding and possibility of making predictions of its behaviour is being possible
thanks to Schroedinger equation only, solved for the specific set of boundary conditions.
The mathematical environment of this model is not classical one. So, speaking clearly,
we do use mathematics of complex spaces for predicting real behaviour of physically
perceived system, called atom.
The good question in this context is "how really the atom looks like"? It is hard to call
"atom" the diagram shown at Fig. 5. But we really haven't got any other, different in
quality, visualisation!
It seems, visible world and phenomena in it, are only one of possible cross sections of
much richer 'super reality' (see Fig. 6). Knowledge about it is highly desirable in context
of our scientific and technological needs. Such statement, one hundred years ago was
shocking, now presents itself as serious difficult technical and analytical scientific
problem only. It is namely extremely hard to construct and to derive mathematical
structures of unknown and (even potentially) not measurable areas of reality. It is
difficult to solve them and to localise their separate predictions as well. But it isn't
impossible!
292

Figure 5: levels of energy in the atom. Figure 6: Quantum phenomenon.
5. PRECAUTIONS: EXPERIMENT AND SYMMETRY
There are two important problems connected with practical applying discussed method
of science. The first one is: "how far may we be sure, mathematics is not showing us
not-existing worlds?" The answer is simple: as far, as our physical experiments are able
to verify predictions of models. Fig. 6 shows what part of 'super reality' may be checked
by experiment. Until experimental methods are applicable (Nobody says it's easy task.
Sometimes experiments seem to be more complicated than predictions of models, which
should be verified), the danger of loosing scientific dimension of recognition doesn't
exist. Up to this level, science is not being imminent by philosophy.
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The second concern, connected with previous, is: "how can we differentiate predictions
of physical meaning from other coincidences or interferences, without any cognitive
dimension?" In this case, the solution lies in symmetries. Careful exploration of
mathematical structures for identifying symmetrical structures leads us to tools, which
allow us practically to find essential (in cognitive sense) objects among the formulas and
equations of local importance (more about this was explained in a previous ISIS-
Symmetry World Congress in Haifa 1998, see conference papers for details).
6. IDEAS
Cognitive efficiency of discussed scientific method manifest in practice. Historically,
this efficiency had been tested many times with good result. It is interesting whether this
method's opportunity of applying is of temporal or universal kind. For test, let's have
now some speculations. For the moment we would try to apply our method for finding
results of cognitive (or not) dimension, using the scientific facts from 'the edge' of
present knowledge. In simple words, we would try to discover 'something' we don't
know yet. If result is satisfactory, we will be able to accept main trains of our
presentation.
Let's take into account the problem of entropy. Equations of physics, including the most
fundamental ones (for example, mentioned Schroedinger equation), don't distinguish the
arrow of time. Therefore, physical phenomena ought to be reversible in time.
Unfortunately, in real world we don't observe it: broken glass is not going to consolidate
again and old houses don't transform themselves into new ones. For justifying the
situation, we define the quantity called entropy, understood as the measure of sets of
permissible configurations of the system. Second Law of thermodynamics says, that
entropy of adiabatic system never decreases, what implies, that only some configurations
of the system are permissible in physical processes (consequence of it is the arrow of
time).
This interpretation is a source of serious problems of modern physics. Steady rise of
entropy forces the whole universe to be of zero-like entropy kind at the beginning, what
is probabilistically negligent. What more, rise of entropy is not going to reverse after
turning the arrow of time away (Exact explanations of this mechanism one can find in
Penrose's books, for example "Emperor's new mind). All trials of justifying the effect by
respecting expansion of space-time lead to other problems, for example to non-existence
of 'white holes' (After turning away the arrow of time, black holes should spit the energy
and matter away - to inverse themselves into 'white holes' -, what doesn't happen) or lack
of the effect of reverse of time in very strong gravitational fields. This suggests, that rise
of entropy is not connected with expansion of space-time, what was conceptual included
into the definition of the term 'entropy'. It seems, such model produces contradictory
predictions and mayn't be accepted for emulating evolution of the universe in a whole.
294

7. SPECULATIONS
Let's agree, that both models: relativistic and quantum ones, are simplifications of the
more general structure (Finding it exactly, is a dream of probably all physicists in the
world). Hoping, that 'super reality' in which our world does exist is physically and
mathematically uniform, and, in general, it is able to create their models, we will try to
'guess' and to verify in formal way the mechanism of entropy. We'll do it using the
scheme of discussed method, accepting separate elements of relativistic and quantum
models as not contradictory.
According to what we said before, we will now consider the 4-dimensional space-time
of our universe as being the quasi-continuous (outside of singularities, of course) and
smooth enough ensembles of quantum sub-space-times. Local and global curvatures of it
will be interpreted as non-zero gravitational fields. The observer, travelling through this
space-time, always perceives one real stratification (sub-world) from the quantum set of
them.
There are two quantum procedures of modelling systems. The first one describes the
general evolution of quantum systems and is not breaking the symmetry of time. The
second one, used for determining possible real manifestations of the system (levels of
energy, by which the system is able to manifest their existence in physical world), is
breaking symmetry of time permanently. It seems, the arrow of time comes out only in
the act of reduction to one of real sub-worlds.
Our observer, walking through the space-time, admires the reality of sub-worlds.
Independently on the geodesic line he chooses, he always, with each step along the line,
must perceive something. Each act of such perceiving is equal to observer's reduction to
one of sub-worlds, what is connected with breaking the time-symmetry. Number of acts
of breaking is proportional to the value of entropy measured by the traveller. Here we
have another sense of the term 'entropy'. The quantity may be understood now as a
parameter of reduction procedure. Let's check whether such interpretation helps in
aborting mentioned cosmological problems connected with time-symmetry braking.
Let's suppose that our observer starts his travel through the universe from the point very
close to Big Bang. The space-time there is being described by extremely high values of
Ricci tensor. This implies rapid expansion of the universe (Thanks to a specific shape of
effective potential of vacuum and repulsive character of forces, existing in this symmetry
of space-time). But each expansion ends very quickly in the Big Crunch singularity or in
deSitter epoch, if only in the closest area of Big Bang there exist any, even the smallest,
curvature of Weyl-type (Riemanian geometry distinguishes two types of curvature:
Ricci- and Weyl-like). The second one is being understood as gravitational tide-force.
To allow our observer travelling through the non-trivial world, we must accept the fact,
that the whole area close to Big Bang is permanently flat in the Weyl sense (No world,
which doesn't satisfy this condition, is able to exist much longer than the period
295

comparable to Planck time, i. e. 10 -44 s.) Any measurable Weyl-like curvatures are
possible only quite far from Big Bang.
Penrose suggests, that reduction procedure may happen when local Weyl-like curvature
is more than so-called Planck mass (i.e. local gravitational gradients are strong enough).
If he's right, our hypothetical observer will be able to notice growth of entropy as faster,
as further from Big Bang he travels. At the beginning of his journey, in the almost
completely flat universe, the traveller has nothing to observe, because reductions to real
sub-worlds are not possible. Further from beginning singularity, where the universe
comes not to be so flat, reductions are permissible and the observer is able to admire
reality around, measuring in the same time the growth of entropy with his each next step
along the geodesic line he had chosen. Let's notice, that if our observer is trying to
reverse the line he came, he won't be able to visit again sub-worlds he admired before
(reductions to the same sub-worlds are probabilistically impossible). Still the entropy he
is permanently measuring, will decrease again when he approaches to Big Bang. This
happens, because each reduction that observer notices, eliminates rich set of reductions,
he potentially would be able to suffer a step earlier. Approaching to Big Bang by any
possible geodesic line restores amount of sets of potentially possible reductions in
different (not flat) areas of space-time. In this visualisation, 'further' to Big Bang means
'later' in the observer's time measure. This is equal to the growth of entropy.
The effect of growth of entropy will happen as well each time when the observer
approaches to local singularity (for example black hole) or other strong gravitational
field source. Responsible for it is the fact, that strong fields reduce rapidly number of
geodesic lines in the area (and even, after passing the Chandrasekhar border, cut off all
the geodesic curves at all), what imposes strong limitations for set of potentially
accessible reductions the observer could suffer.
8. RESULTS
As being announced, we tried to discuss importance of the term 'entropy', which is, in its
classical concept, a source of serious problems of scientific and cognitive dimension.
For clarifying the problem, we used traditional scientific method. We shaped the general
outline of the new model, assuming only general conformity of well-known scientific
models and mathematical character of investigated 'super reality'. We used as well some
elements of presently accepted scientific knowledge and one hypothesis of quite nice
likelihood of truthfulness.
As expected results we had:
another and more universal understanding of the term 'entropy'
296

elimination of conceptual and model contradictions, which were the consequence of
traditional definition of entropy
preservation of old models of thermodynamics, which work in the same way for both
definitions of entropy.
nothing discovered here is contradictory with what we know up to now
Unexpected, but, as always, occurring results (scientific predictions) of physical
importance, accessible for experimental verification, are:
thermodynamical evolution of the universe satisfies symmetry of time (entropy increases
after inverting the arrow of time),
as closer to beginning singularity, as smaller is probability of virtualisation (reduction)
the energy in real state,
irreversible nature of reduction procedures is obvious,
entropy increases near all sources of strong gravitational fields,
Weyl-like flatness of space-time near the Big Bang singularity is necessary for existence
of the universe,
limitations for tracing time-like geodesic lines through the area of quantum stratification
of the universe are of the Weyl-like curvatures shape.
9. COMMENTS
Let's finally collect everything what we discussed and discovered. The result is being
shown at Fig.7. Phase space of vacuum curvatures, with diagrams drown for opened and
closed Friedman-like evolutionary scenarios, is valuable and verifiable scientific result.
Occurrence of it confirms correctness and cognitive efficiency of discussed scientific
method.
Explaining advanced problems of modern science seems to be, again, impossible with
use of formalisms and complete, systematic knowledge only. This is a manifestation of
Goedel statement (it is not possible to prove each one theorem, which can be formulated
in specific system, with use of logical tools of this system only). Fascinating features of
reality – mathematization and idealisation – give us opportunity of having deep insight
into the nature of phenomena of our universe, even if we are not pretty sure what are we
looking for.
297

References
Figure 7: phase space of curvatures of Friedman universes.
Penrose R., "Emperor's new mind", Oxford University Press 1989
Hawking S., Penrose R., "The Nature of Space and Time", Princeton University Press 1986
Michniowski T., "Cognitive function of physical models", in: Philosophical and Scientific Elements of the
Universe's Description. Part 5, UKSW, Warszawa, Poland 1998, p.121-39.
298

TESSELATIONS OF EUCLIDEAN,
RIEMANNIAN AND HYPERBOLIC PLANE
RADMILA SAZDANOVIĆ, MIODRAG SREMČEVIĆ
Name: Sazdanović, R., Mathematician, (b. Belgrade, Serbia, Yugoslavia,1977)
Address: The Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade,Yugoslavia
E-mail: seasmile@galeb.etf.bg.ac.yu
Fields of interest: Geometry, Differential Geometry, Topology (Computer Science).
Publications and/or Exhibitions:
1.Knezević, I., Sazdanović, R., Vukmirović, S., (2002) L2Primitives, Userguide, Mathematica® package,
http://www.mathsource.com/Content/WhatsNew/0211-879
2. Knezević, I., Sazdanović, R., Vukmirović, S., (2002), Visualization of the Lobachevskian Plane, Visual
Mathematics, electronic journal, http://members.tripod.com/vismath7/sazdanovic/home.htm
Name: Sremčević, M., Physicist, (b. Kragujevac, Serbia, Yugoslavia, 1976)
Address: Institute of Physics, University of Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany
E-mail: msremac@agnld.uni-potsdam.de
Fields of interest: Polyhedra, Planetary sciences
Abstract: The very first human perception of this world was plane. Then one sphere
among others in the universe. Or is it just a small part of the hyperbolic space?
Different features of Euclidean, Elliptic and Hyperbolic geometry are presented
through all types of planar edge-to-edge tessellations with regular polygons as tiles (not
all congruent) and vertices of the same type, including regular, uniform, non-uniform
and coloured tessellations. The most intriguing, hyperbolic geometry is the main topic
of the algorithm implemented in Mathematica 4.0® that provides tools for creating and
drawing tessellations and their animation under different isometries, in both Poincare
models and Klein disk model. The package provides additional data such as: geometry
and type of tessellation, number of all possible realizations, angles of tiles and
transformation rules between vertices upon which the tessellation is constructed.
1 INTRODUCTION
According to the Oxford Dictionary the word tessellate means to form or arrange small
squares in a checkered or mosaic pattern. Being familiar or not, with this or any other
(more) formal definition, everyone can recognize tilings like brick walls and tile floors.
299

Tessellation as an art predates human history- Platonic polyhedra were well-known in
antiquity and a toy regular dodecahedron was found in Padua in Etruscan ruins dating
from 500 B.C. Excellent examples of tessellations can be found in Moorish architecture
in Spain and Islamic architecture in the Middle East; not to mention Japanese and
Chinese designs. Artists such as Albrecht Durer and Pierro della Francesca made
drawings of many of the semi-regular polyhedra. Johannes Kepler was the first to give a
complete description of such figures in his Harmonices Mundi which appeared in 1619.
Except this initial study, some formal mathematical investigation took place before the
end of 20th century. We must mention Grunbaum, Shephard, Robin, Sommerville,
Andreini, M.S. Escher and that much was done also by chemist and crystallographers .
2 DEFINITIONS
Def. 1. A plane tiling is a countable family of polygons P1, P2,... called tiles such that:
1. Their union is the entire plane
2. Interiors of the tiles are pairwise disjoint
Def. 2. An edge-to-edge tiling is plane tiling where each two tiles intersect along a
common edge, common vertex or not at all.
Def. 3. A vertex of a tiling is said to be of type n1, n2,…,nN or that (n1, n2,…,nN) is the
vertex type (configuration) if the polygons about this vertex in cyclic order are an n1gon,
an n2-gon, . . . and an nN-gon where N is the number of polygons at each vertex.
Def. 4. An edge-to-edge tiling is called Archimedean if
1. Each tile is a regular polygon (not all congruent)
2. All vertices are of the same type.
Def. 5. A regular tiling is an edge-to-edge tiling whose tiles are congruent to a single
regular polygon. The other name is monohedral tiling with regular polygons as tiles.
Def. 6. A semi-regular or uniform tiling is an Archimedean tiling whose symmetries are
transitive on its vertices.
Figure 1. (4,4,4,6): two uniform (a, b) and non-uniform realization (c): colours make
distinction between polygons that can not be mapped to each other by global
symmetries.
300

The following array shows the relations between different types of tessellations:
Regular ⊂ Uniform ⊂ Archimedean ⊂ Edge-to-edge ⊂ Plane Tessellations
For the rest of this paper we restrict our attention to the tessellations whose tiles are
regular polygons and vertices are of the same type. This gives us an opportunity to
define tessellations with their vertex configuration, although this correspondence is not
always one-to-one. The vertex symbol uniquely determines the type of a tiling in
Euclidean and Elliptic plane (except (3,4,4,4) which has two realizations:
rhombicuboctahedron and a non-uniform polyhedra, Fig. 3). However, non-congruent
tessellations in the hyperbolic plane may have the same vertex symbol (Figs 1 and 2).
3 THEORY AND ALGORITHM
The general solution for constructing tessellations is based on a few rather simple ideas
that we shall briefly explain. The algorithm can be divided in two logical parts:
Step 1. Vertex configuration →→→→ transformation rules for neighbouring vertices
Step 2. Transformation rules for neighbouring vertices →→→→ graph →→→→ tessellation
Step 1. The starting point of the algorithm and the only input is the vertex configuration.
In order to find all possible realizations for a given vertex configuration we developed a
very general algorithm. Our initial assumption is that each polygon belongs to a different
equivalence class (each presented with different colours in our figures) under the
symmetries of the whole tessellation (global symmetries) and then we, if necessary,
decrease the number of classes. In each iteration we look for transformation rules i.e.
mappings of the corresponding vertices and check if they really do form a tessellation.
In this way we are able to find both uniform and non-uniform tessellations i.e.
all Archimedean tessellations including enantiomorphic realizations (Fig 4) and all
coloured realizations (Fig. 2.a, 2.b).
Figure 2. (3,3,3,3,3,4): uniform realization (a) and its coloured version (b), and one of
the non-uniform realizations (c).
Step 2. Obtained transformation rules posses two important properties: they are
uniquely defined and sufficient for creation of the whole tessellation starting from any
301

vertex. So, once we have transformation rules we construct the graph and tessellation at
the same time, using following ideas:
♦Vertex configuration imposes geometry onto the graph. Hence, the graph can be
realized only on a 2-dimensional, simply connected, unbounded, complete Riemannian
surface of constant curvature, i.e. Euclidean, Elliptic or Hyperbolic plane [2].
♦In constructing tessellations problems arise on local and global level [5]. On the
local level we consider only those images of the original tile obtained by a sequence of
symmetries that move it in a way that the tile and its image are not totally disjoint. So
the local tessellation problem is the question if these adjacent tiles cover a certain area
completely enclosing the original one without gaps and overlapping. Global tessellation
problem is the same question involving the whole 2-dimensional space. The main point
is that in order to determine if the tessellation exists or not, we need to solve only the
local problem.
Figure 3. (3,4,4,4): uniform tessellation (a, b) and non-uniform (c, d).
This is the very core of the algorithm and additional calculations are necessary for
drawing tessellations [1, 3, 7]. The algorithm contains methods for drawing Euclidean
and Elliptic tessellations (wired and coloured model or a corresponding polyhedra, Figs
3, 4c) and for hyperbolic tessellations Mathematica® [6, 8] package L2Primitives [4] is
used (both Poincare’s and Klein disk model). Furthermore, it is possible to apply various
isometries and obtain interesting animations. As a side effect, algorithm provides
additional useful information such as geometry of tiling, type of tessellation, number of
all possible realizations, angles of tiles [3] and complete listing of transformation rules.
The detailed description of the algorithm and the theory behind will be published
elsewhere.
4 GALLERY
302

Figure 4. Uniform enantiomorphic tessellations: (3,3,3,3,6) Euclidean (a, b), (3,3,3,3,5)
Riemannian (c), and (3,3,3,3,7) hyperbolic in Klein disk model (d).
Figure 5. Tessellations with a single realization: uniform (4,8,10) and (3,6,4,6) (a, b)
and non-uniform (3,4,3,4,4) (c). Tessellation with multiple realizations (3,3,3,4,3,4): two
uniform (d, e) and a non-uniform (f).
References
1. Coxeter, H.S.M., (1988) Non-Euclidean Geometry, The Mathematical Association of America
2. Goodman-Strauss, C., Personal communication.
3. Har'El, Z., Uniform Solution for Uniform Polyhedra, Geometriae Dedicata 47 (1993), 57-110
4. Knezević, I., Sazdanović, R., Vukmirović, S., (2002) Visualization of the Lobachevskian Plane, Visual
Mathematics, e-journal, http://members.tripod.com/vismath7/sazdanovic/home.htm
5. Magnus, W., (1974) Non-Euclidean tessellations and their groups, Academic Press
6. Maeder, R., Programming in Mathematica, (1991) Addison-Wesley Publishing Company
7. Paraslovov, V.V., Tihomirov,V.M., (1997) Geometrija, MCNMO
8. Wolfram, R., The Mathematica Book, Mathematica Version 4, Wolfram Media & Cambridge University
Press
303

304

A REMARKABLE SPIRAL
ANNIE VAN MALDEGHEM
Name: Annie Van Maldeghem, Mathematician, (b. Gent, Belgium, 1948).
Address: Hogeschool voor Wetenschap en Kunst, Departement Architectuur, Sint-Lucas Gent, Belgium.
E-mail: a.vanmaldeghem@archg.sintlucas.wenk.be
Fields of interest: Ruled surfaces, minimal surfaces.
Abstract: This paper is the result of some mathematical amusement. In Dutch we call
this with a German word: ‘spielerei’. The spiral that is subject of this talk is completely
anonymous. At the occasion of Mat mium it makes sense to call it ‘atom-spiral’. This
double spiral has a very interesting symbolic meaning and a lot of variations on its
elegant shape are possible. All these spirals are related to the Archimedes’ and
hyperbolic spirals. Moreover, they provide the possibility to design the character “e”
surrounded by an appealing curl, comparable to the snail @. All the spiral drawings
are computer generated by executing an Autolisp program (AutoLISP runs within the
environment of AutoCAD). The subject is in harmony with the decorative composition in
the superb Hotel Van Eetvelde. The elegant curved lines, the sweeping twists and twirls,
the curls and the swirls that are characteristic of Art Nouveau are creating a perfect
setting to put mathematical spirals into the scene.
1. THE ATOM – SPIRAL
The atom-spiral is depicted in fig.1. This drawing is
computer generated by executing an AutoLISP program.
AutoLISP is a parametric programming language, the LISP
variant, which is running within the AutoCAD environment.
To write a program for this graph I used its polar equation:
θ
+
r = k , k ∈ R 0 (1). This polar equation is plotted with
θ −1
respect to a pole o and a polar axis X.
305
o
Fig. 1
X

The program calculates the polar co-ordinates of a large number of points (r, θ) on the
spiral (r = directed radius and θ = polar angle; θ-values chosen between 0 and 1 result in
negative r-values). Then the program draws line segment joining two consecutive
points, joins all these line segments together to one polyline and finally changes the
polyline into a spline. The range of θ is R \ 0 and the function defined by equation (1) is
a monotone (decreasing) function. This means that equation (1) leads to a spiral that
consists of two branches. It is easily verified that the spiral has two asymptotes: a
straight line and a circle. One end of the interior branch tends to one half of the straight
line, while the other end approaches the circle as limit from the inside; one end of the
exterior branch tends to the other half of the straight line, while the other end
approaches the circle as limit from the outside. I first noticed this spiral in 1966 in a
mathematical magazine for youngsters (see references). I fell for this spiral because of
its remarkable property to possess a circle as an asymptote: both branches of the spiral
are winding themselves around the pole coming closer and closer to the asymptotic
circle without ever touching it, from either side. And when we agree to consider a circle
as a symbol for perfection we can see this spiral as a symbol for the never-ending search
for perfection, which will never be achieved. In other words, the curve seems to tell that
the hunt for perfection is to try for the impossible.
2. GENERAL SET AND ITS SUBSETS
mθ
+ n
Equation (1) has following form: r = , m, n, p, q∈
R; (p, q) ≠ (0, 0) (2) . This
pθ
+ q
equation, in 4 homogenous parameters m, n, p and q describes a set of spirals, which
falls apart in 4 subsets.
2.1 Subset 1 ↔ circles
m n
If the parameters m, n, p, q fulfil = 0 ∧ (p, q) ≠ (0, 0), then equation (2) changes
p q
into r = k (k ∈ R). This equation leads to a set of concentric circles (circle with radius =
zero included), if k runs through R. So, a first subset of the set described by (2) consists
of circles.
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2.2 Subset 2 ↔ Archimedes’ spirals
If the parameters m, n, p, q fulfil
n
≠ 0 ∧ p = 0 ,then
q
equation (2) changes into: r = k (θ − a), k ∈ R0, a ∈ R (2.2).
For any value of a (k is used to scale), if the range of θ is R,
this equation represents a double Archimedes’ spiral which
is directly similar to the one depicted in fig. 2.2. So, a
second subset of the set described by (2) consists of (double)
Archimedes’ spirals. The pole is a point of any spiral in this
subset and the parameter a can be used to rotate.
2.3 Subset 3 ↔ Hyperbolic spirals
k
r = , k ∈ R0, b ∈ R (2.3). For any value of
θ − b
b, (k is used to scale), if the range of θ is R \ ⎨b⎬,
this equation represents a double hyperbolic spiral
which is directly similar to the one depicted in fig.
2.3. So, a third subset of the set described by (2)
consists of (double) hyperbolic spirals. Any spiral
in this subset has a straight line as an asymptote.
Moreover, the pole is an asymptotic point, which
Fig. 2.3
means that any curve of this subset spirals down to
the pole without ever reaching it; the parameter b can be used to rotate.
m
p
m n
If the parameters m, n, p, q fulfil ≠ 0 ∧ m = 0 , then equation (2) changes into:
p q
2.4 Subset 4 ↔ variations on the atomspiral
m
If the parameters m, n, p, q fulfil
p
n ⎧m
≠ 0
≠ 0 ∧ ⎨ ,
q ⎩p
≠ 0
θ − a
then equation (2) passes into: r = k
θ − b
, k ∈ R0, a, b
307
Fig. 2.2
Fig. 2.4 a
k > 0
b = 0

∈ R, a ≠ b (2.4). For any value of a and b (k is used to scale), if the range of θ is R \
⎨b⎬, this equation represents a spiral which consists of two separate branches. All
spirals described by equation (2.4) have two asymptotes: a straight line and a circle. So,
a fourth subset of the set described by (2) consists of double spirals, which are all of the
same type as the atom-spiral (a = 0, b = 1). The pole is a point on the interior branch of
any spiral in this subset. Fig. 2.4 a shows one of the many variations on the atom-spiral
(a = − π/2, b = + π/2). Fig. 2.4 b shows spirals corresponding to random values for the
parameters a and b: they are (in general) not at all similar.
Fig. 2.4 b
r = k
θ + 1
θ −1.
5
θ − 0
r = −k
θ +
. 618
0.
382
k > 0
3. Variations on the atom-spiral: properties
r = k
θ − 0
r = −k
θ +
θ −1
θ − 0.
5
. 618
0.
382
But if ⎪ a − b ⎪= constant: all spirals are directly or indirectly similar. Fig. 3.1 a, fig. 3.1
b and fig. 3.1 c show 3 spirals all corresponding to the same k-value. If they were
represented with respect to the same pole and pole axis the spirals in fig. 3.1 a and fig.
3.1 b would coincide up to a rotation; the spirals in fig. 3.1 a (or fig. 3.1 b) and fig. 3.1 c
would coincide up to a rotation and a reflection.
a − b = − 1
a − b = − 1
Fig. 3.1 a Fig. 3.1 b Fig. 3.1 c
308
a − b = + 1

The smaller ⎪a − b⎪, the larger the loop and the better the asymptotic circle is suggested
(see fig. 3.2).
a − b = − 8
In addition, there is the amusing fact that if a − b > 0, the corresponding spirals
described by (2.4) can be used to design the character e with an appealing curl just like
in the snail @ (see fig. 3.3).
Fig. 3.3
Any variation on the atom-spiral, a (double) Archimedes’ spiral and a (double)
hyperbolic spiral all belong to a set of spirals described by equation (2) while the
m
parameters fulfill the same condition
p
3.4. Following spirals were chosen:
n
≠ 0
q
. This relationship is visualized in fig.
θ + 2π
* variation on the atom-spiral: r = k θ ∈ R0 (3.4 a)
θ
* Archimedes’ spiral: r = k’ (θ − 2.77), θ ∈ R (3.4 b)
* hyperbolic spiral:
" k
r = , θ ∈ R0 (3.4 c)
θ
k'
k k"
The scaling factors k, k’ and k” fulfill: = = . For each diagram the range of θ
1 10 100
is restricted to subsets of the given domains in order to optimize the similarity between
309
Fig. 3.2
a − b = + 0.3

the interior branch and the Archimedes’ spiral and between the exterior branch and the
hyperbolic spiral.
Fig. 3.4 a
Interior branch
Exterior branch
Spiral (3.4 a) was chosen because one of the polar angles which corresponds to the
1+
5
double point of the loop is − Φ*π radians where Φ = (golden number). This
2
angle corresponds to an angle ≈ − 291°. The line that joins the pole and this double
point makes an angle ≈ 69° with the polar axis (see fig 3.5 a). One of the polar angles,
which corresponds to the double point on the Archimedes’ spiral (3.4 b), is (2.77 − π/2)
radians or (159° − 90°). So, the line, which joins the pole and the double point on the
Archimedes’ spiral (3.4 b), also makes an angle ≈ 69° with the polar axis (see fig 3.5 b).
310
Fig. 3.4 b
Archimedes’ spiral
Fig. 3.4 c
Hyperbolic spiral

Remark
Equation (2), in polar co-ordinates, alludes of course to a similar equation in Cartesian
co-ordinates:
mx + n
y = , m, n, p, q∈
R; (p, q) ≠ (0, 0) (4) . This equation defines the
px + q
homographic function, which describes a straight line in case the parameters fulfil
either:
m
p
n
= 0 ∧ (p, q) ≠ (0,0) or:
q
m
p
n
≠ 0 ∧ p = 0 . If the parameters fulfil
q
m n
≠ 0 ∧ p ≠ 0 , the homographic function
p q
describes a rectangular hyperbolar. The two
congruent branches of a rectangular hyperbola
(see fig. 4) explain the word ‘homographic’:
‘homos’ (Greek) = the same + ‘grapheini’
(Greek) = to write.
Reference
− 291° 69°
Fig. 3.5 a
Exlibris en Wiskunde, Pythagoras (Dutch mathematics journal for youngsters, Editor Bruno Ernst) (1966), 5,
no.5, (pages 103-105).
311
159°
x + 6
y =
x − 2
− 6
69°
− 3
Fig. 3.5 b
Fig. 4

312

RUDOLPH M. SCHINDLER’S BRAXTON HOUSE:
THE FIBONACCI AND LUCAS SEQUENCE.
Name: Jin-Ho Park, Ph.D., Professor of Architecture.
Address: University of Hawaii at Manoa, USA.
JIN-HO PARK
Abstract: Most writings on the proportional study of Rudolph Michael Schindler’s
houses have been descriptive observations rather than in-depth analyses using
computational techniques or making pedagogical inferences. At times, the significance
of the architect’s proportional system has been blatantly ignored. Architectural
historians, like David Gebhard (1971), considered Schindler’s system merely one
practical tool for wood frame structure and construction. It is true that Schindler’s unit
system was derived from practical needs for his space architecture, but it may be a
mistake to consider the system without theoretic implications. Recently, March (1993)
interpreted Schindler’s proportional method as an analogy with classical proportion,
much more than a mere construction tool. When he used a musical analogy to examine
the proportional design of the How house, in his paper, “Dr. How’s Magical Musical
Box,” he seems to portray “architecture as frozen music.” Schindler left behind
numerous built and unbuilt projects, which demonstrate extraordinary spatial
complexity as well as variety. To achieve such spatial sophistication, Schindler argues
that the architect needs to not only improve his mental image of the space but also
possess a system. Consequently, Schindler proposed a proportional system of space
reference frame. With this simple technique, the forms of space are freely conceived and
precisely measured in the architect’s mind. Forms of space are envisioned “by being
inside of it” like musicians imagine and articulate their music with notes. By 1928,
Schindler had sufficient command of his method in organizing his space and space
forms to propose a design, called the Braxton House. This paper first introduces
Schindler’s proportional system, “Reference Frames in Space,” outlines the spatial
organization of the house, and finally interprets Schindler’s use of proportion in the
house in relation to the Fibonacci series and Lucas sequence.
313

1. REFERENCE FRAMES IN SPACE
Although Schindler expressed his early interest in a proportional system as early as
1916, it was not clearly expressed until 1946 when Schindler published a
comprehensive summary of his proportional system in an article, “Reference Frames in
Space.” In the article, his idea of proportion is well defined as follows: “Proportion is an
alive and expressive tool in the hands of the modern architect who uses its variations
freely to give each building its own individual feeling.” The system for Schindler is
indispensable to the creation of his notion of space architecture as a doctrine of spatial
organization. Schindler argued that he started using the system as early as 1920. In fact,
an analysis of Schindler’s Free Public Library project of 1920 has demonstrated his use
of the system. (Park, 1996)
The reasons for his using this system are two fold. First, all locations and sizes of the
parts with respect to the whole are precisely identified during the construction process.
Thus, no obscure or arbitrarily unrelated measurements are involved in the unit system.
Second, the unit grid system offers the means to visualize ‘space forms’ in three
dimensions. He argued, “ … it must be a unit which he can carry palpably in his mind in
order to be able to deal with space forms easily but accurately in his imagination.”
Although there are few exceptions, like the Schindler Shelter project where he used a 5foot
unit module, Schindler recommended 48 inches (4-foot) as the basic unit, to be
used with simple multiples and with 1/2, 1/3, and 1/4 subdivisions. Among the
subdivisions, with only a few exceptions, 1/3 and 1/4 are used for vertical modules in
his works. This single unit module with its multiples and subdivisions form the basis of
all dimensions of rooms.
This choice has two reasons. First, the unit must be related to the human figure to satisfy
all the necessary sizes for rooms, doors, and ceiling heights; second, for practical
reasons, the 48-inch module fits the standard dimensions of materials and common
construction methods available in California at that time. Pueblo Ribera Court of 1923
is a typical example. This multiple housing project, built in San Diego, is superimposed
on 4 foot by 4-foot grid lines. Its vertical module is based on a 16-inch dimensioning
system, which controls not only the height of the room but also of all elements including
built-in furniture, chair, table, windows, doors, and clerestory, providing, in Schindler
words, “a uniform scale.”
He utilized his unit system in a square grid pattern. Numbers and letters are laid out on
the grid on the floor plans in sequence and the vertical module is identified with an
elevation grade. This pattern was original to Schindler. The grid was presented on
drawings and on the house in his earlier designs, yet they disappear from the house and,
at times, from the drawings. However, this does not mean he abandoned his system; on
the contrary, his system remains embedded in the designs as underlying principles.
314

Thus, every location of the buildings is identified accurately in the convenience of
composition and construction. In addition, Schindler used the system to measure room
sizes. In his preliminary sketches, his room sizes with the whole numbers on drawings
are commonly presented. These numbers are increments of unit multiples with its
subdivisions.
2. THE HOUSE
The Braxton house was designed for Shore brothers on the beach in Venice, CA by
1928-30 but never built. Gebhard (1980) argued Schindler’s interlocking forms of
building are indebted to the De Stijl mode, saying, “The year 1928 marks Schindler’s
full commitment to de Stijl ... the use of intersecting rather than singular volumes to
establish their forms. This feature is what most strongly differentiates Schindler’s work
from that of the closely knit Internationalists – Gropius, Mies, Le Corbusier (Pre-1935)
and Neutra.” Then, he continued, “the Braxton house shows the architect was slowly
giving up his laissez-faire view of Southern California and its climate, and was
demanding an increased control over nature.” Giella (1985) points out some features of
the house, ‘the use of Wrightian windows,’ ‘balanced asymmetry,’ ‘cantilevered
structure.’
There are two major elements of the house: one for the house and the other for a garage.
They are connected by a horizontal structure. A courtyard covered with sand occupies
the space between. The main building is lifted up from the ground floor. The rectilinear
box form dominates its spatial outlook, in Gebhard’s words, “ …[a] rectangular box into
and out of which secondary volumes projected,” and in Giella’s words, “a slightly topheavy
composition is created by cantilevering out each successive story from bottom to
top.” Various box forms are projected, recessed, and interlocked along the longitude
axis. It is Schindler’s usual practice and signature of his designs.
The ground level preserves an open playground and patio for the natural beach sand,
forecourt, outdoor fireplace, the garage, a guestroom, maid’s quarters, and furnace. The
ground level is an independent unit and there is no interior staircase connecting the
ground to the upper level. The major portion of the house is raised above the lot, which
might be ‘indigenous to all beaches’. The option of the lifted structure may be suitable
for the beach house by the necessity of allowing sun and air to reach the entire floor,
preserving some part of site beneath for a playground and providing an open outlook to
the ocean. On the main level, the architect created a living room, two stories in height. It
is open to the Pacific Ocean and enclosed by individual bedrooms on the balcony floor
overlooking the living area, and below is the living room, kitchen facilities, dining room
where porches are adjacent to dining room and kitchen. Inside the main floor, there is
315

little built-in furniture shown on the drawing compared to his other projects, so that the
inner space remains visually unobstructed. And also, like in the Lovell house, finely
proportioned window mullions are inset facing the courtyard, and at the same time, door
mullions face the ocean side. Three bedrooms on the balcony floor are disposed equally
overlooking the Pacific Ocean. The composite roof is used as a terrace; a portion of it is
used for sunbathing, a sleeping porch and a small closet.
The whole building is supported by means of five horizontal structures, different from
the five reinforced vertical concrete frames in the Lovell house (1929). Rather than
using concrete, these structures are made of wooden frames. All space volumes of the
house are interlocked along the five major structures.
Figure 1. The Braxton house (1928): a quarter-inch scale model fabrication.
3. PROPORTIONAL ANALYSIS OF THE HOUSE
Despite its novel character, curiously enough, the Braxton house has remained to date
unexplored in any depth, unlike its contemporaries the Wolfe house (1928) and the
Lovell house (1929). In particular, whereas the proportional study of the Lovell house
316

was attempted earlier by August Sarnitz (1986, 1988), that of the Braxton house has
never been investigated. Sarnitz’s analysis of the Lovell house plans and elevations is
based on his belief that simple square and double square determine the overall
proportional system of the house. Sarnitz also applied the same method in the analysis
of the Free Public Library project.
There are various sets of drawings of the house in the Schindler Archive at the
University of California at Santa Barbara, including preliminary sketches, structural and
sash details, construction drawings, and well-organized presentation drawings. These
sources provide valuable information for the proportional analysis of the house.
On the drawings, dimensions and placements of various spatial forms and details of the
house are controlled by Schindler’s unit system. The 4-foot unit system is clearly
identified in plan with numbers and alphabets, and the 16-inch vertical module in
elevation with grades. All major space, details, and structure underlie its subdivisions
and multiples where all parts are related to each other in terms of simple unit relations to
produce a coherent unity. Most of the major rooms in the construction drawings are
measured in whole numbers and written on the drawing as is usually in other designs.
Although rooms are frequently not a simple rectangular form, Schindler approximated
those to the whole numbers. It appears that simple whole numbers with respect to the
space reference frame is easy to grasp its size, as Schindler implied in his space
reference frame system.
Given the evidence of the floor plan, the dimensions of the rooms include the following:
a guestroom 9-foot x 15-foot, maid room 10-foot x 10-foot, a bathroom 5-foot x 8-foot,
a furnace 6-foot x 8-foot on the ground level; the living room 24-foot x 27-foot, the
kitchen 10-foot x 16-foot and a porch in front 6-foot x 10-foot, and the entrance 8-foot x
8-foot on the second level; his room 11-foot x 12-foot, her room 14-foot x 16-foot,
another bedroom 10-foot x 16-foot on the third floor. The rafters of the horizontal
wooden structures are regulated in a 2-foot distance – half of the unit module.
However, dimensions written on the construction drawings are found to be inconsistent
with the real dimensions of rooms. The construction documents of Schindler provided
actually consist of drawing which other architects might consider schematic like the
instruction to the builder. Two reasons might be considered: First, this difference relies
on their stages of design development, either conceptual or practical, that are common
practice in the architectural field. Early schematic designs are primarily conceptual
basis, and evolve depending on specific practical requirements in the design process and
during fabrication as well as construction. Thus, room sizes shown on the drawings do
not always accord with dimensions measured from the drawings, although the former
might be what the architect had initially in his mind; rather, the latter is a result of some
adjustments. For example, in the Braxton house, the dimension of the guestroom shown
317

on the drawing is 15-foot by 9-foot, yet the real measure of the drawing of the room is
14-foot by 11-foot. Before arriving at a final scheme of a project, exercises of various
possible schematic layouts are an almost universal procedure employed by architects,
including Schindler. For example, in the Gibling house (1925-26), Schindler provided
four different stages of schematic sketches for the plan development until he arrived at
the final scheme. Each scheme offers different room sizes from another.
Second, various rooms are not rectangular in shape but interlocked, overlapped, and, at
times, zigzagged. In this case it is not possible to present room ratios as a:b, other than
approximation. In fact, Schindler approximately measured the room sizes in a rational
manner with his unit system to determine its size of each room and probably their
proportional relations.
Although these inconsistencies may cause confusion for this analysis, the collection of
room dimensions serves as an excellent basis for further explorations of his proportional
designs. This proportional analysis solely relies on the architect’s room dimensions
written on the drawings, since it maintained the architect’s original intent of room sizes
and their relations to each other before the architect’s transfer of the ratios to the
practical necessities. Here is a list of all rooms and their relations:
Floor Room Names Room
Dimensions
Ground
Floor
Second
Floor
Space
Reference
Frame
318
Ratios Decimal
Value of
the Ratios
Guestroom 15 x 9 3 ¾ x 2 ¼ 5 : 3 1.67
Maid room 10 x 10 2 ½ x 2 ½ 1 : 1 1.00
Bathroom 8 x 5 2 x 1 ¼ 8 : 5 1.60
Furnace 8 x 6 2 x ½ 4 : 3 1.33
Living room 27 x 24 6 ¾ x 6 9 : 8 1.125
Kitchen 16 x 10 4 x 2 ½ 8 : 5 1.60
Porch 10 x 6 2 ½ x 1 ½ 5 : 3 1.67
Entrance 8 x 8 2 x 2 1 : 1 1.00

Third
Floor
His room 12 x 11 3 x 2 ¾ 12 : 11 1.09
Her room 16 x 14 4 x 3 ½ 8 : 7 1.14
Bedroom 16 x 10 4 x 2 ½ 8 : 5 1.60
Bathroom 10 x 6 2 ½ x 1 ½ 5 : 3 1.67
There are some interesting results from the analysis as shown in the table. First, among
three fractions that Schindler used which include1/2, 1/3, and 1/4, 1/3 of 48 inches (16inch)
is not used in room dimensions of the plan. Instead, the fraction is used only for
the vertical module. The16-inch vertical module measures the room height of the house.
It also governs window mullions, door mullions, and the thickness of a series of
horizontal structures. Accordingly, it is inferred that Schindler used those fractions
separately: 12” increments for plans and those of 16” for elevations.
The door height is his typical 6 feet 8 inches. The room heights vary, but they are
subdivisions and multiples of 16 inches vertical module. The room height of the ground
floor including the maid, guestroom, furnace, and garage is 8 feet. But the height of the
open playground is 10 feet. The height of the kitchen and the dining room in the second
floor is 8 feet. The height of the two story high open living room is 14 feet 16 inches.
The height of His room and another bedroom is 8 foot 16 inches in the third floor. The
height of Her room is 9 foot 8 inches, but that of the bathroom and the dressing room
next to Her room is exceptionally 8 feet 10 inches. A series of five horizontal structures
are apart 8 feet, except for the height of the first floor, which is 8 feet 9 inches.
Second, these surprisingly few room ratios are worthy of the profound study of their
interrelationship. In the house seven different ratios: 1:1, 4:3, 5:3, 8:5, 8:7, 9:8, and
12:11 are collected. There might be a certain relationship between these ratios. In order
to deduce relationship of ratios, a particular method as an effective tool is introduced to
derive all these ratios. Subsequently, these ratios are identified with the extreme and
mean ratio geometrically incorporated in the regular pentagon. The ratios suggest a
possible relationship with the Fibonacci sequence, in which each successive number is
equal to the sum of the two preceding numbers, 1, 1, 2, 3, 5, 8, 13, 21 … and the Lucas
sequence, 2, 1, 3, 4, 7, 11, 18, 29 … The ratios of successive numbers of both series
converge to approximately 1.618, which is known as the extreme and mean ratio. As
goes on to the right in these sequences, the ratios get close to the Golden Ratio, which is
1.618.
319

Figure 2. A graph showing its successive Fibonacci sequence closes to 1.618.
Its proportion with a regular geometrical pentagon can be identified in the following
diagram.
320

Figure 3. A double square and pentagon with root 5.
Ratios including 1:1, 4:3, 5:3, and 8:5 are certainly among these. Geometric
constructions using existing ratios on the drawings derive new proportional ratios. By
computing other ratios of the two series together, the remaining three ratios can be
constructed: 7/4 ÷ 2/1 = 8/7, 4/3 ÷ 3/2 = 9/8, and 18/11 ÷ 3/2 = 12/11. All ratios are
derived directly from other ratios previously used. The implication of these newly
constructed ratios with old ones is that all room ratios become associated each other,
thus a ratio consistency is achieved. Schindler might deliberately incorporate sizes of
each room in his projects. This computation is delineated in the following diagrams.
321

Figure 4. Geometric construction of the room ratios.
Thus, all room ratios of the project are suggestive of methods of computation coming
out of the geometric and proportional construction. By this method, all other room ratios
are constructed with regard to each other. Schindler might be aware of such ideas as he
wrote, “We must realize that ‘proportion’ is not any more a simple mathematical
relationship (Golden Rule, etc.) which can be applied universally in all buildings as it
was in classical times.” However, it is incorrect to suppose Schindler derived his room
ratios this way or mathematically. Perhaps, some sort of computing procedure may have
been involved in the architect’s mind. The architect could choose the whole number of
room dimensions in the project with reasons to reference proportions derived from
simple geometry or to practical solutions to the problems of a particular design.
322

4. SUMMARY
The proportional design of Schindler’s Braxton house has been analyzed with regard to
the Fibonacci or Lucas sequence. It is unlikely that while designing the project,
Schindler calculated those numbers and their ratios with deliberate intent relying on the
particular computational method. As discussed, the architect had no particular belief of
“a simple mathematical relation” or “Golden Rule.” Thus, the analysis has based on the
hypothetic speculation rather than documented evidence of the architect. Also it is hard
to believe Schindler favored these individual room proportions as a whole or their
relation to one another.
That does not mean, however, that the analysis is useless. Even if the architect never
claimed that his theory of proportion is based on a particular mathematical ground, there
are some implications inherent in his designs where each room associates with others in
terms of their ratios. Schindler wrote, “The house of the future is a symphony of ‘space
forms’ - each room a necessary and unavoidable part of the whole.” In the design, the
architect’s approach was extremely simple. The composition of various rooms did not
have to follow mere mathematical play of rules but have to express its relations in size
between rooms so as to form an “organic unity”: a whole composed of related parts in
orderly arrangement.
323

Figure 5. 3-D representation of rooms with their ratios hung on the horizontal structures.
Although speculative, the proportional relations of those ratios examined in this article
represent a valid method for analyzing proportional relationship of the rooms in the
house, their harmonic use in the spatial organization of the design. Consequently, it
could be said that the house is composed of simple room ratios and their relations in its
entirety. Their relations in 3-D space are constructed to illustrate its integrity.
Therefore, the observation leads to a conclusion that proportional relations of these
ratios tie together in a cohesive whole in a single design, thus creating such a complex
design. Whether the design was intentional or arrived at by chance, R. M. Schindler
must have a splendid sense of proportional eyes to project such a simple and rational
system to a very complex spatial composition.
5. ACKNOWLEDGEMENT
The author is indebted to professor Lionel March for his advice in carrying out this
proportional analysis.
REFERENCES
Gebhard, D. (1971). Schindler, Thames and Hudson, London.
Giella, B. (1985). R.M. Schindler's Thirties Style: Its Character (1931-1937) and International Sources (1906-
1937), Ph.D. Dissertation, New York University, New York, NY.
March, L. (1993). “Dr. How 's Magical Music Box,” RM Schindler: composition and construction, Eds. L.
March and J. Sheine, Academy Edition, London.
March, L. (1993). “Proportion is an Alive and Expressive Tool …” RM Schindler: composition and
construction, Eds. L. March and J. Sheine, Academy Edition, London.
March, L. (1998). Architectonics of Humanism, Academy Editions, London.
Park, J-H. (1996). “Schindler, Symmetry and the Free Public Library, 1920”, Architectural Research
Quarterly, 2(2) pp. 72-83.
Sarnitz, A. “Proportion and Beauty – the Lovell Beach House by Rudolph Michael Schindler, Newport Beach,
1922-26” JSAH, December 1986, Vol. XLV, No. 4.
August Sarnitz (1988) R. M. Schindler-Architect, 1887-1953, New York: Rizzoli.
Schindler, R.M., (1946). “Reference Frames in Space”, Architect and Engineer, San Francisco 165: pp10, 40,
44-45.
324

IN SEARCH OF ELWIN BRUNO CHRISTOFFEL;
IS THIS FAMOUS MATHEMATICIAN NOT SO
WELL-KNOWN?
ALEXA and ROBERT WILLEM VAN DER WAALL
Name: Dr. H. A. van der Waall (*1970, Nijmegen), Dr. R. W. van der Waall (*1941, Den Haag),
mathematicians.
Address: (H. A.): Simon Fraser University, Dept. of Mathematics and Statistics, 8888 University Drive,
Burnaby, BC, Canada V5A1S6;
(R. W.): KdV-Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV
Amsterdam, The Netherlands.
E-mail: awaall@cecm.sfu.ca; waallr@science.uva.nl
Fields of interest: (H. A.): algebraic number theory, computer algebra, differential Galois theory (architecture,
art, design, skiing, traveling).
(R. W.): number theory, group theory, geometry, coding theory, history of science (chess, genealogy,
architecture, old scripts like hieroglyphs, linear B, maya)
Awards: (R. W.): 1954, first prize in school education at the Gymnasium Haganum, The Netherlands;
1989: vita published on invitation in: “Who is who of intellectuals”, Melrose Press, Cambridge, U.K., ISBN
0948875305;
1991: award for Dutch paleography at Utrecht, The Netherlands.
Publications and/or Exhibitions: (H.A.) Lamé equations with finite monodromy; dissertation Utrecht; 2002
145 pages, ISBN 90-393-2927-3.
(R.W.) 1) (with E. B. Kuisch) Homogeneous character induction II; Journal of Algebra 170 (1994), 584-595.
2) (with E. B. Kuisch) Modular Frobenius Groups; Manuscripta Mathematica 90 (1996), 403-427.
3) (with R. C. Lindenbergh) Ergebnisse über Dedekind-Zeta-Funktionen, monomiale Charaktere und
Konjugationsklassen endlicher Gruppen, unter Benutzung von GAP; Bayreuster Mathematische Schriften 56
(1999), 79-148.
4) (met L. de Clerck) Kijk op kegelsneden (Dutch) (View on conics); volume 21 in the Doe-Boek reeks (dobook
series) of the society "Vierkant voor Wiskunde", for development of mathematics skills for high-schoolstudents;
36 pages; publisher CWI-Amsterdam; 2000.
5) On iterated group actions and direct products; Journal of Algebra 223 (2000), 57-65.
Abstract: While on vacation in Monschau (Germany) we suddenly stood in front of a
commemorative plaque on behalf of the 150est birthday of the famous mathematician
Elwin Bruno Christoffel (1829-1900). The talk at the Mat mium 2002 conference
325

presents an overview of his work and his influence and of the color locale of places
where he used work and to teach. It will be embedded in the historical surroundings of
the cities of Aachen, Monschau, Berlin, Zürich, and Straßburg. A survey of our
investigations is due to appear in the column The Mathematical Tourist of the journal
The Mathematical Intelligencer. The talk will be accompanied with lots of slides,
photographs, and journal pages.
326

1 INTRODUCTION
The subtitle of this contribution seems to contain a contradiction. Indeed, is it possible
that someone is famous whereas it looks as if that very same person is not so well
known? Well, here we present a way to get rid of this riddle.
It appears that the mathematics of Elwin Bruno Christoffel (born 1829, Montjoie -
deceased 1900, Straßburg) is well known, no doubt about that! Many of you, dear
mathematically inclined readers, will be aware of the so-called Christoffel-Darboux-
Einstein summation formulae, Christoffel symbols, or of the Christoffel-Schwarz
mapping theorem. There is more, of course, much more. Among other things
Christoffel's thirty-five printed papers deal with subjects like function theory including
conformal mapping theory, propagation of electricity, Gaussian quadrature, continued
fractions, dispersion of light, movements of points in periods, continuity conditions as
to differential equations, minimal surfaces, theory of invariants, geodetic triangles,
geometry and tensor analysis, orthogonal polynomials, shock waves, potential theory,
Riemann's integrals, Jacobi's theta-sequences, irrational numbers, all yielding a
beautiful lot of 19th century mathematics; here you should consult his Collected Papers
(8).
There also exist contemplative contributions on the mathematical works of Christoffel.
To start with, view (9) and (12), being about a century old. On the other hand,
Christoffel as a person acting in his society and habitat, his social, mathematical and
intellectual acquaintances in Berlin, Zürich and Straßburg (the places where he used to
lecture) happened to be another story at our initial sight, i.e. without making use of the
so-called electronic search-machines of today. But around the year 1981 it turned out
already, that three elucidatory sources, other than (9) and (12) were in existence, as we
discovered in 1999, in a rather surprising way.
2 A MEMORIAL
During a short holiday in the summer of that year 1999 in the neighborhood of the socalled
"drielandenpunt" (meaning: the meeting point of the three countries The
Netherlands, Belgium, Germany) both of us took a stroll through the colorful and
picturesque city of Monschau (until 1918: Montjoie), where suddenly our attention was
fixed upon a plaque on one of the walls of the building at the Rurstraße 1.
The inscription on it reads as follows:
Dr. Elwin Bruno
CHRISTOFFEL
327

Professor der Mathematik
in Zürich, Berlin, Straßburg
* 10.11.1829 in Monschau
+ 5. 3.1900 in Straßburg
GEBURTSHAUS
Figure 1. The memorial.
So here it is. It concerns the building in which Christoffel was born. More precisely, as
we found out, the original home in which he was born, was burned down in 1835. The
plaque contains a sculpture of his face en face too; it is derived from the only known
photograph for which one is absolutely sure that it represents Christoffel (he is then
about 40 years of age).
328

Figures 2-3. Left: the only known photograph representing Christoffel. Right: the
(German) subscript in (6) to the second photograph indicates that it might be another
picture of Christoffel, at older age, as physiognomy suggests.
Nowadays the building at the Rurstraße 1 comprises the Parfumerie-Foto-Drogerie
"Servaes" at the bottom floor. Its proprietor Mrs. Ingrid Hermanns did tell us that the
plaque has been unveiled in the year 1979 in commemoration of the 150th birthday of
Christoffel. She also showed an invaluable item in connection to all aspects of life and
work of Christoffel. It turned out to be a three-issues-in-one volume of a journal dealing
with the local history of the city of Aachen and its surroundings; view (6). That source
(not so well-known to mathematicians we presume) did trigger our investigations on
Christoffel. We daresay, that the eighty pages of (6) are thoroughly filled will all sorts
of profound and important information; as such it is an indispensable toolkit. It provides
a lot of photographs (Lejeune-Dirichlet, Geiser, Prym, Rost, Hilbert, Riemann, Einstein,
Weierstrass, Klein, von Laue, Reuleaux, Ricci-Curbastro to mention a few) besides a
description of the social environment of Monschau and a genealogy of Christoffel, a
history on his life and work, his school years in Cologne, mathematics and politics
combined with intrigues in Zürich, Berlin and Straßburg. Of course, one finds there the
list of all of his printed publications.

Figure 4. View from dormer window of the building at the Rurstraße 1, in 1889, at the
place where Christoffel was born.
3. ADDITIONAL INFORMATION AND CONCLUSION
Be aware that in (6) all of the beautiful and clarifying material, presented over three
columns per page and supplied with an enormous amount of (pragmatic) references, is
brought to you in the German language. The reader not in command of the German
language should not be discouraged, however. Our investigations revealed, indirectly
from source (6), that a conference took place in Aachen and Monschau (both in
Germany) in November 8-11 of the year 1979, on the influences and aspects of the work
of Christoffel into our era. Much is to be learned from the twelve lectures given at the
conference and from forty-five papers written on invitation; see (7). The beginning of
(7) yields an English version of some of the papers presented in (6), be it in a form
slightly more condensed in comparison to the ones in (6). It is nevertheless clear that the
starting part of (7) provides a good impression of time and circumstances in which
Christoffel used to work. An accurate, short but complete, overview of the contents of
the conference is also to be found in the very handy non-mathematical source (1).
Above we have told our story how we did unveil Christoffel's life and work up to the
year 1981. An electronic search-machine investigation provides relevant and additional
material of later date (read up Christoffel's geometry in (11) for instance). The so-called
"Riemann example" of a continuous non-differentiable function is scrutinized in dept in
the remarkable paper (4). Furthermore one finds that mathematics used to be a vivid
affair in the region of the "drielandenpunt" during the last 1200 years (view (3) and (5)),
330

whereas mathematics in Zürich and Berlin over the ages has been described in (10) (see
the contributions of G. Frei and E. Knobloch therein, respectively).
In summary, it has come to light that an almost conclusive overview of life and work of
Christoffel in his days and its impact into our days, has been described in (1), (3), (5),
(6), (7) and (11), while its relevant mathematics is presented in the greater part of (7)
and (of course) in (8). Besides that, modern electronic data comprise source (14) and
above all, the unsurpassed source (13). Hundreds of papers have been published since
1980 in which Christoffel's mathematics plays the leading part.
All this being said, we hope to have convinced you, that the question brought up in the
subtitle of this mathematical tourist contribution, has to be answered in the negative,
i.e., Christoffel is famous and well known!
331
Figures 5-6. View on the house and the
memorial plate, shown by the authors.

Acknowledgement
We are indebted to Mrs.Ingrid Hermanns and her husband in providing an original issue
of (6); without it, this note would never have been written.
References
1. Bericht über das Internationale Christoffel-Symposium in Aachen und Monschau, 8.-11.November 1979;
von P. L. Butzer and F. Fehér; in: Berichte zur Wissenschaftgeschichte 3 (1980),193-201.
2. P. L. Butzer- An outline of the life and work of E. B. Christoffel (1829-1900), in: Historia Mathematica 8
(1981), 243-276.
3. P. L. Butzer- Mathematics in the region Aachen-Liège-Maastricht from Carolingian times to the 19th
century, in: Bull. Soc. Roy. Liège 51 (1982), 5-30.
4. P. L. Butzer and E. L. Stark- "Riemann's example" of a continuous non differentiable function in the light
of two letters (1865) of Christoffel to Prym; in: Bull. Soc. Math. Belge, Sér. A, 38 (1986), 45-73.
5. P. L. Butzer- Scholars of the Mathematical Sciences in the Aachen -Liège-Maastricht region during the past
1200 years, an overview (with the assistance of Helga Butzer Felleisen), pages 43-90 in: P. L. Butzer
(ed) et all., Karl der Große und sein Nachwirken: 1200 Jahre Kultur und Wissenschaft in Europa. Band
2: Mathematisches Wissen, Turnhout Brepols (1998); ISBN 2-503-50674-7.
6. Elwin Bruno Christoffel (10.November 1829, Monschau - 5.März 1900, Straßburg), Professor der
Mathematik in Zürich, Berlin und Straßburg - Gedenkschrift zur 150.Wiederkehr des Geburtstages, in:
Heimatblätter des Kreises Aachen, Jahrgänge 34/35 (1978,3/4 und 1979,1) - 80 Seiten (in einem Band).
7. E. B. Christoffel - The influence of his Work on Mathematics and the Physical Sciences; edited by Paul Leo
Butzer und Franziska Fehér (International Christoffel Symposium in honor of E.B. Christoffel on the
150 Anniversary of his Birth, 1979, Aachen and Monschau (Germany)); 1981 - Birkhäuser Verlag,
Basel-Boston- Stuttgart; ISBN 3-7643-1162-2; XXV + 761 pages.
8. E. B. Christoffel- Gesammelte Mathematische Abhandlungen; unter Mitwirkung von A. Krazer und G.
Faber herausgegeben von L. Maurer; zwei Bände, 1910 - Teubner Verlag, Leipzig und Berlin.
9. C. F. Geiser und L. Maurer- Elwin Bruno Christoffel, in: Mathematische Annalen, 54 (1901), 329-341.
10. Jahrbuch Ueberblicke der Mathematik 1994 (S. D. Chatterji (ed) et all.); Vieweg - Braunschweig; ISBN 3-
526-06578-8; VII + 265 pages.
11.K.E.B.Leichtweiß- Christoffels Einfluß auf die Geometrie, Seiten 93-103 in: Sitzungsberichte der Berliner
Mathematischen Gesellschaft, Jahrgänge 1972-1987 (1.10.1971-31.10.1987); Berlin 1987, 185 Seiten.
12.W. Windelband- Zur Gedächtnis E. B. Christoffel; in: Mathematische Annalen 54 (1901), 341-344.
13. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Christoffel.html
14. http://www.stetson.edu/~efriedma/periodictable/html/CF.html
332

Name: 51N4E, Brussels (Belgium).
SPACE PRODUCTION.
51N4E
Abstract: 51N4E is a Brussels based office working on matters of space production, and
visual aspects of buildings. Space production implies the creation of an adapted reality.
51N4E does not consider reality as a mere given or context, instead, it provides a new
uncanny context – a necessary luxury for the production of successful space. Advanced
domestics, European space, scaling and outsmarting building regulations are but a few
tools in the quest of mastering space production. Today, 51N4E’s main activities focus
on: E.D.E.N. (public park Luxembourg Brussels), CELL (a mass producible minidwelling),
LAMOT (production centre for an urban culture) and €-space.
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FRUSTRATION: SOURCE OF COMPLEXITY
E-mail: Ogawa-t@koalanet.ne.jp
TOHRU OGAWA
An important role of fundamental science is to prepare some views and concepts, which
are available in extending the scope of science. The attempt in this paper is based on this
philosophy. Cellular automata are often studied from a similar point of view, in which
simple basic laws governing local processes give rise to highly complex phenomena.
These studies are very important in the extension of physical science in the future even
though the correspondence with the real physical world may be at most metaphorical or
suggestive. For further development along this line, it is desirable to try to include some
conservation laws and frustration phenomena in models of cellular automata. A
frustrated system is very interesting and important from this point of view because of
abundance of states and phenomena appearing in it. We may bear in mind that, in living
organisms, a wide variety of phenomena are contained within a very narrow energy
range. We may expect the study of a frustrated system to provide an insight into more
complex systems than treated hitherto.
The cellular automaton models are usually deterministic. However it is very difficult to
construct a purely deterministic model with conservation and frustration. Therefore a
kinetic model containing stochastic (probabilistic) terms is studied in this paper.
The antiferromagnetic Ising model on a triangular lattice is typical of such systems. The
frustration produces a degeneracy of the ground-state. The number of degrees of
freedom of the ground-states is exp[0.3231] = 1.381 per site. The residual entropy of the
system, which was rigorously calculated by Wannier, is proportional to the logarithm of
this value. Almost nothing is known of the details of the degenerate ground-states. Some
years ago, the author studied their cluster structure in relation to some maze-like patterns
observed in the monodispersive latex system in a (2+e)-dimensional space.
Though the present investigation is on similar lines, its purpose is more abstract. It is the
so-called "glass transition" that is the closest physical effect to the present research. It is
supposed that there is no essential difference between a liquid and a glass so far as their
351

static structures are concerned. Or, at least, no abrupt change in the static structure can
be expected. The difference in structure between a liquid and a glass may be expected to
lie in the dynamics of their phase spaces. If so, the present model is the simplest one that
can be investigated in detail.
Reference
T. Ogawa and Y. Nakajima; “Frustration, Degeneracy, and Forms: A View of the Antiferromagnetic Ising
Model on a Triangular Lattice” Progress of Theoretical Physics, Supplement No. 87 (1986), pp. 90-
101.
It is reprinted at http://members.tripod.com/vismath5/ogawa1/forms.htm (VisMath)
352

DIGITAL SHAPING OF SPATIAL STRUCTURES
JANUSZ REBIELAK
Name: Janusz. Rebielak, architect, (b. Bierutow, woj.wroclawskie., Poland, 1955).
Address: Department of Architecture, Wroclaw University of Technology, ul. B. Prusa 53/55, 50-317
Wroclaw, Poland. E-mail: j.rebielak@wp.pl
Fields of interest: Architecture, morphology of space structures, roof covers, high-rise buildings, formex
algebra ( history of science, art and culture).
Awards: Distinguished Leadership Award of the American Biographical Institute (2001).
Publications and/or Exhibitions:
1. Rebielak Janusz: Bar Space Structures - Rules of Shaping, in: Symmetry: Natural and Artificial,
Proceedings of the Third Interdisciplinary Symmetry Symposium and Exhibition, Washington, USA, August
14-20, 1995, Symmetry: Science & Culture, Quarterly of the ISIS-SYMMETRY, Vol. 6, No 3, pp. 442-445.
2. Rebielak Janusz: Space structures shaping and visualisation of their digital models by means of Formian,
in: Proceedings of the Fifth Interdisciplinary Symmetry Congress and Exhibition, Sydney, Australia, July 8-
14, 2001, Symmetry: Art and Science, Quarterly of the ISIS-SYMMETRY, pp. 158-161.
3. Rebielak Janusz: Structural systems of cable domes composed of concentric spatial hoops, in: ed. H.
Kunieda, IASS 2001 Extended Abstracts, International Symposium on Theory, Design and Realization of
Shell and Spatial Structures, October 9-13, 2001, Nagoya, Japan, pp. 328-329.
4. Rebielak Janusz: General morphology of new types of tension-strut systems, Lightweight Structures in
Civil Engineering, Local Seminar of IASS Polish Chapter, Micro-Publisher Jan B. Obrebski Wydawnictwo
Naukowe, Warsaw- Wroclaw, 7 th December, 2001, pp. 50-55.
5. Rebielak Janusz: Prismatic space frames as the main support structures for high-rise buildings,
Lightweight Structures in Civil Engineering, Local Seminar of IASS Polish Chapter, Micro-Publisher Jan B.
Obrebski Wydawnictwo Naukowe, Warsaw-Wroclaw, 7 th December, 2001, pp. 81-89.
Abstract: The paper will present the proposals of the space structure shaping as the
structural systems for lightweight roofs of large spans and as the main support for the
high-rise building. It will be presented the concept of the special type of the retractable
dome. The author recently invented some new types of these systems are by and for the
main types of them he has prepared several programmes written in the programming
language Formian. These programmes determine the digital models of the proposed
forms of the space structures systems. Owing to the suitable application of the systems
the buildings designed by means of them may obtain the interesting and the individual
architectonic views.
353

1 INTRODUCTION
The aesthetic perception of the form of many types of buildings has been radically
changed after the wide application of the space structures in architecture and in the civil
engineering during the 20th century. The specific way of the arrangement of the
component parts in spaces of these structures causes, that they have their own
homogeneous architectonic views having an extremely great imprint to the architecture
of the objects, in which they are applied. It is one of the most important factors justified
the highest interest of many architects in processes of their shaping. Many evidences
show that the develop potential of the space structures is very great and makes possible
to discover their new and numerous applications in the future. Because of the basic rules
of the theory of structures the general aspects of symmetry or sometimes asymmetry
have to be taken into consideration during the design processes of the space structures.
The are many propositions given by the author in the area of shaping of the structural
systems devoted for various types of the roof covers and for the high-rise buildings.
Many of them are of very complex shape therefore the design process of them is
complicated and it needs special tools. One of the very useful and powerful tool in this
field is the programming language Formian, Nooshin & others (1993). It is invented by
Professor Hoshyar Nooshin and developed by his team in the Space Structures Research
Centre at the University of Surrey in England on the basis of the r a bra , also
developed by Prof. H. Nooshin, Nooshin (1984). It was intended as a tool for improving
these processes easier and faster and in order to make the better co-operation between
the architects, the civil engineers and the other designers involved in the project.
Formian makes possible the fluently changes of the geometry of the structure at any
stage of the design. The visualisation of any, even the very complex form of a space
structure, is by means of Formian relatively simple.
2 SHAPING OF TENSION-STRUT SYSTEMS
The tension-strut structures are in the last few decades used in the design and
realisations of various types of the lightweight roof covers. Some types of these covers,
particularly the cable domes developed by D. Geiger, are of the significant importance
in the developing of the structural solutions in this domain of the engineering activity.
The structural systems, proposed by the author for these purposes, were developed as a
result of suitable transformations of the bar arrangement of the dome, the structural form
of which was proposed earlier by the author, Rebielak (1996). The tetrahedron and the
octahedron were assumed as the bases in these processes, Rebielak (1999, 2001). One
on the results of the shaping processes presents Figure 1. The digital model of this
structure was defined by means of Formian and than it was transported to Corel, the
software designed for preparing the visualization processes. The proposed structural
system may be applied for the designing of very lightweight covers s of large spans.
354

Figure 1: Perspective view of the one of the tension-strut systems proposed by the author
for the lightweight structures of large span covers.
The appropriate usage of the symmetry formulas make the processes of preparation of
the digital models of these structure much easier and shorter than before. The structure
as itself and its digital model have to be built according to the basic rules of symmetry.
3 RETRACTABLE AND FOLDABLE ROOF STRUCTURE
Figure 2 shows the basic schemes of the structural system proposed by the author for the
foldable and retractable structure of e.g. the dome cover. The system consists of the
triangular bar sets, see Figure 2a. The chosen bars, marked there by grey colour, build
the main girders located onto the perpendicular surface marked by the dash lines. These
a
V
Figure 2: General schemes of structural system proposed by the author for the foldable
and retractable structures of e.g. the dome cover.
355
b

girders are the main ribs of e.g. the dome cover and they are focused in its crown-node.
This basic configuration of the structural module shows Figure 2b. Almost all of these
bars are of the fixed lengths, only the vertical, marked in Figure 2a by the letter symbol
V, have to be able to change their lengths. These bars can be made as the struts of equal
lengths with the appropriate screws, what enables the necessary changing of the bar set
shapes. The mechanisms of the retractable and foldable covers are of the high technical
requirements, Ishii (2000). These types of structures and the assembly processes of them
have to be designed according to the rules of symmetry. The special bars, marked by
symbol V in Figure 2a, are perpendicular to the middle surface of the planned roof.
Owing to the suitable length changes the entire space structure can be folded and
refolded. By means of the length controlling of these bars the structure is retractable and
it can obtain the planned shape. The proposed system can be used in the design and in
erection of the structure being at the same time the foldable and the retractable structure.
The digital models of these systems, prepared in Formian, will be used to carry on many
comprehensive analyses in order to estimate their suitability to the proposed purposes.
References
Nooshin, H. (1984) Formex Configuration Processing in Structural Engineering, Elsevier Applied Science
Publisher, London and New York.
Nooshin, H., Disney, P., and Yamamoto, C. (1993), Formian, Multi-Science Publishing Co. Ltd, Brentwood,
England.
Ishii, K., (2000) Structural Design of Retractable Structures, WITPress, Southampton, Boston.
Rebielak, J. (1996) Examples of Shaping for Large Span Roofs and for High-Rise Buildings, International
Journal of Space Structures, special issue on Morphology and Architecture, Volume 11, Nos 1&2, pp.
241-250.
Rebielak, J. (1999) Cable Dome Shaped on the Ground of the {T – T} Double-Layer Space Structure.
Example of Formian’s Application in Creation of Numerical Model of a Structure, Lightweight
Structures in Civil Engineering, Local Seminar of IASS Polish Chapter, Micro-Publisher Jan B.
Obrebski Wydawnictwo Naukowe, Warsaw, pp. 86-87.
Rebielak, J. (2001) Examples o Space Structures Shaping and Visualisation of Their Digital Models by
Means of Formian, Proceedings of the Fifth Interdisciplinary Symmetry Congress and Exhibition,
Sydney, Australia, July 8-14, 2001, Symmetry: Art and Science, Quarterly of the ISIS-SYMMETRY,
pp. 158-161.
Rebielak, J. (2001) Structural systems of cable domes composed of concentric spatial hoops, ed. Kunieda, H.,
IASS 2001, Extended Abstracts, International Symposium on Theory, Design and Realization of Shell
and Spatial Structures, October 9-13, 2001, Nagoya, Japan, pp. 328-329.
Rebielak, J. (2001) General morphology of new types of tension-strut systems, Lightweight Structures in
Civil Engineering, Local Seminar of IASS Polish Chapter, Micro-Publisher Jan B. Obrebski
Wydawnictwo Naukowe, Warsaw- Wroclaw, 7 th December, 2001, pp. 50-55.
Rebielak, J. (2001) Tension-strut systems built by means of modules with V-shaped bar sets, Lightweight
Structures in Civil Engineering, Local Seminar of IASS Polish Chapter, Micro-Publisher Jan B.
Obrebski Wydawnictwo Naukowe, Warsaw-Wroclaw, 7 th December, 2001, pp. 78-80.
356

PROPORTIONS AND DISSECTIONS IN
POLYGONS
ENCARNACION REYES IGLESIAS
Name: Mª Encarnación Reyes Iglesias, mathematician, ( Burgos, SPAIN., 1954).
Address: Departamento de Matemática Aplicada Fundamental, E.T.S. Arquitectura. University of Valladolid
SPAIN.
E-mail: ereyes@maf.uva.es.
Fields of interest: Geometry, proportion, symmetry (Art and Architecture, Mathematics with Paper Folding,
Phyllotaxis, and Mathematics in Daily Life).
Publications and/or Exhibitions:
- Proporciones y Arquitectura. ICME, (Congreso Internacional de Educación Matemática) Sevilla 1996.
- Proportions mathématiques et leurs applications à l' architecture. Bulletin IREM (Institut de recherche pour
l'enseignement des mathématiques de Toulouse) Toulouse. France, 1998, 39 pp.
- Papiroflexia y proporciones dinámicas en rectángulos. JAEM (Jornadas Nacionales para la Enseñanza y
Aprendizaje de las Matemáticas.) 1999
- Mathematics: a determinant strategic for the development of an architectonic project. Symposium on
Symmetry. Alhambra 2000, Granada (Spain)
-Paper folding and proportions in polygons. ISIS Symmetry Congress & Exhibition. Sydney 2001.
Abstract: In this work I will focus mainly on some proportions and dissections in
polygons, particularly hexagons and octagons. The classical method of the
decomposition of polygons into smaller parts that can be rearranged to form other
polygons is a pedagogical tool to understand the geometry of polygons and their
properties. An interesting problem to be solved en each case is to find an equivalent
polygon by using the minimum number of pieces. I will show some dissections of the
hexagon and the octagon with their equivalent figures. In the other hand, teaching
mathematics through paper folding is an alternative way to understand them. I will
present in this exposition some constructions of the hexagon and the octagon by using
these techniques. My interest focuses on researching the underlying mathematics
involved in the paper folding process. In each case (hexagonal and octagonal) it will be
calculated the relationships of proportion between diagonals and sides the same in
convex polygons as in their stars. I will establish also the concept of polygonal
proportion (PP) using it to calculate the hexagonal and octagonal global proportion.
357

We define the proportion of the regular polygon Pn, written p(Pn), as the quotient
between the area A( Pn
) of the polygon and the square of the length of its side L:
A(
Pn
)
p(Pn) = .
2
L
If H6 is a regular hexagon with side L,
this definition gives us
p(
H
6
3L
3
A(
H6
)
) = =
2
L
2
2
L
3 3
= .
2
In the hexagon we also have the
2
D
following proportion: = 3 . We can construct a hexagon and its star 6/2 by using
L
paper folding. We get a strip of paper with the same proportion than the rectangle
D π
= 3 . Rotate the strip an angle of around its center and repeat this operation once
L
3
more.
358

In case of the regular octagon O8 with side L we
A(
O8
)
have: p(
O8
) = = 2θ
= 2(
2 + 1)
where θ
2
L
denotes the silver number.
2 2
= 2 x L = D
In the figure, we have L 2x
; + ,
solving this system, we obtain: x =
L
;
2
⎛ L ⎞ D
therefore 2 ⎜
⎟ + L = D ⇔ = 2 + 1 = θ .
⎝ 2 ⎠ L
π
By rotating the silver rectangle (a rectangle with proportion θ ) an angle of around
4
its center, we can obtain an octagon and its star 8/3.
RATIOS OF LENGTH OF SIDES IN HEXAGONS AND
HEXAGRAMS 6/2
Let { Li} be the decreasing sequence of the
length of the sides of the convex hexagons
(see figure). This sequence is a geometric
series of ratio
Ln
+ 1 1
= and whose sum is
L 3
n
359

1 3 + 3
S = = If we consider the sequence { A i}
of lengths of the hexagrams, we
1
1−
2
3
also obtain the same relation.
An
+ 1 1
= .
A 3
n
RATIOS OF LENGHTS OF SIDES IN OCTAGONS AND
THEIR STARS 8/3
Let { L i}
be the decreasing sequence of
the length of sides of convex octagons
(see figure). This sequence is a
geometric series of ratio
L
L
n + 1
n
1
=
θ
The sum of the series { L i}
is:
1 θ 1
S = = = 1+
. If we
1 θ −1
θ −1
1−
θ
consider the decreasing sequence of
lengths of sides { A i}
in the star
polygon 8/3, we also obtain the same
relation.
A
A
1
=
θ
n + 1 .
n
360

In the following dissections of the hexagon and the octagon we can rearrange the pieces
to form equivalent rectangles.
This interesting dissected octagon appears in
CEAC (Center for the Study of Contemporary
Art) in Barcelona (Spain); it is conceived by J.
L. Sert. This kind of division of the octagon into
nine parts is also useful in pavements, because it
can be constructed using only two types of tiles:
a square and its half. However, such an octagon
is irregular.
361

References
Boltianski, V.G. Book. Equivalent and Equidecomposable Figures. D.C. Heath, Boston, 1963
Fernández, I, Reyes E. Construcciones y disecciones del octógono, SUMA, 38, (2001) 69-72
Kappraff, J. Connections.Book. The geometric bridge between art and science. Mc Graw Hill, New York,
1991.
Kappraff, J. Systems of Proportion in Design and Architecture and their Relationship to Dynamical Systems
Theory. http://members.tripod.com/vismath/kappraff/kap4.html
Reyes, E. Papiroflexia y proporciones dinámicas en rectángulos. JAEM (Jornadas Nacionales para l
Enseñanza y Aprendizaje de las Matemáticas.) 1999.
Reyes, E. Paper folding and proportions in polygons .Symmetry: Art and Science. Congress. Sydney, 2001
Vera W. De Spinadel The metallic means family and multifractal spectra. Non Linear Analysis, 36 (1999)
721-745.
Vera W. De Spinadel Book. Title: From the Golden Mean to Chaos, Ed. Nueva Librería, Buenos Aires,
Argentina,1998.
ADDENDUM: QUESTIONS ABOUT THE EXPOSITION
It is possible the construction of the regular pentagon by using a DIN A4 sheet of
paper?
The answer was:
A Din A 4 sheet of paper is not suitable for the construction of a regular pentagon
because the proportion of a DIN A4 is the irrational number 2 , and the proportion of
1+ 5
the regular pentagon involves the golden number .
1
2
The right construction of the regular pentagon starting with a
DIN A 4 sheet of paper requires to cut a small strip ≈ 8 mm
(7,96 mm), making the long side of the sheet of paper shorter.
The proof is the following:
540 º
The internal angle of a pentagon is = 108º.
5
Starting from a DIN A4
tg α= 2 ⇒ α = actg 2 ≈ 54,74º.
If we fold through the transversal line we obtain the angle 2α ≈
109,47 º, that which is different from 108º.
108
We need another rectangle with β = = 54 º, tg54º = 1,376,
2
therefore if we denote by “y” the long side of the rectangle A4,
y
tgβ = ⇒ y = 210. tg54º = 289,04 mm.
210
362
√ 2
y
β
α
α
210
β

And since the length of the long side of a DIN A4 is 297 mm, we must cut: 297 - 289,04
=7,96 mm. ≈ 8mm.
Therefore,
-Cut 8 mm to the long side of a DIN A4 format. (Then, cut along the short side, see
figure)
-Join the two opposite vertices of paper and fold it.
-Fold the figure along its axes of symmetry
-Fold the smallest sides once more until they coincide with the axis of symmetry.
THE REGULAR PENTAGON IS OBTAINED
Another question was: What about the heptagon? Has it been constructed?
The answer was:
So far no heptagon construction using paper folding has been executed. The only regular
polygons whose construction was studied by the author were those that can be drawn by
ruler and compasses.
363

364

THE GEOMETRY OF FLEMISH GOTHIC TOWN
HALLS
HAN VANDEVYVERE
Name: Han Vandevyvere, engineer-architect, scient. coll. KU.Leuven, Belgium (b. Brugge, Belgium, 1966).
Address: Departement Architectuur, Stedenbouw en Ruimtelijke Ordening, K.U.Leuven, Kasteelpark
Arenberg 1, B – 3001 Leuven (Belgium).
E-mail: han.vandevyvere@asro.kuleuven.ac.be
Fields of interest: Architecture, building & engineering, geometry (also arts, philosophy).
Publications: Vandevyvere, H., “Het stadhuis van Leuven: een geometrische analyse”; in Jaarboek van de
geschied- en oudheidkundige kring voor Leuven en omgeving, 39 (2000), pp 171-192; and Vandevyvere, H.,
"Gothic Town Halls in and around Flanders, 1350-1550: A Geometric Analysis", Nexus Network Journal,
vol. 3, no. 3 (Summer 2001), http://www.nexusjournal.com/Vandevyvere.html.
Abstract: Can we find evidence for the application of design rules in the plans of the
gothic town halls in Flanders? This question was dealt with in a recent investigation
into the layout of some medieval town halls of the southern low countries.
The research was initiated following a series of accidental discoveries. These indicated
that not only medieval churches were built according to a symbolic geometry, but
certain civic buildings as well. Starting from a graphical analysis of the building plans
of the town halls under investigation, an attempt was made to reconstruct the scenario
of their design.
As a result, we may put forward a number of conclusions. A particular system of
‘applied mathematics’, characteristic of the gothic tradition, was used to set out the
building plans. The geometry is based on manipulations of the compass and the
carpenter’s square. A taste for symbolic series of numbers and figures appears. Acting
together with the local measurement systems that were used to set out the plan, we often
obtain a subtle game of numbers in the entire composition, expressed in local feet or
rods.
If the medieval building master is known to have applied symbolic geometries in
religious buildings, thereby referring to the metaphysic views of his time, then we may
now state that he did this also in the civic architecture of the town halls. The medieval
city had at that time become a powerful entity, challenging the established orders, and
the town hall was in its design the very expression of this new reality. As such it
365

confirmed the consciousness of the citizen in way proper of the Middle Ages: by
symbolic representation.
Recent research on the design of a number of gothic town halls in the southern low
countries indicates that not only medieval churches were built according to a symbolic
geometry, but certain civic buildings as well. Starting from a graphical analysis of the
building plans of a number of representative town halls in and around Flanders, an
attempt was made to reconstruct the scenario of their design.
Methodology
The research was started from a selected set of premises, after a building module in the
design of the Leuven town hall was discovered rather accidentally.
A first assumption deals with the validity of a research on hidden design and proportion
systems in a building. Do we have reasons to look for such a system, and what is the
probability rate of the findings? As there is a good deal of evidence about the existence
of medieval design systems in general, a closer look to the town halls under
consideration was at least defendable. As a further constraint, only those schemes would
be accepted that are relatively simple, evident, and robust. The last condition means that
conclusions should have a tolerance margin, accounting of e.g. survey errors in the
plans, and building particularities.
Carrying on, a set of design principles has been selected for consideration. We can
summarize them as follows:
• Buildings are set out in the local measurement units, in use at the time of
construction. Documentation about the measures in use in Flanders during the
middle ages is available, with a high degree of reliability;
• Following some medieval writings, we should consider that what is to be found
in plan, is reflected in elevation [1]: we have to find a three-dimensional
approach based on ‘pulling up’ the plan;
• Symbolic, simple integer number series, such as to obtain simple ratios between
dimensions, are typical of medieval architecture and sculpture. So they may
well occur in the current research subject;
• An applied geometry, based on manipulations of the compass and the
carpenter’s square, is likely to be a major design canon for similar reasons.
Emphasis was put on a graphical analysis of the buildings as they stand; research on
literary sources was limited to building history information necessary to reconstitute, as
much as possible, the medieval situation. It should be noted that in the 18th and 19th
centuries a lot of “correcting” restoration was performed on medieval buildings, often
creating an idealized “reconstruction” of something that never existed before. Doors
were moved, balustrades added, and so on.
Finally, used building plans should have a reasonable degree of accuracy for the
purposes of the research. In the case of Leuven, a new survey of some cardinal points of
366

the building was made, confirming the satisfactory precision of the survey plans at
disposition.
In what follows, we bring forward some results of the research that point towards the
existence of a number of generally applied design principles. The buildings that were
investigated are situated in Brugge, Oudenaarde, Brussel, Leuven (B), Veere (NL),
Arras and Saint-Quentin (F).
Issue 1: the cubic grid
For the buildings situated in present Flanders (B), it could be verified that they are set
out in the local, medieval measurement units of the city where they stand. All of the
four cities had a different basic foot, and a rod composed of a different number of these
local feet.
Moreover, for the buildings outside actual Flanders, it was possible to find a similar
modularity in the overall dimensioning of the composition.
We can conclude that the contouring volumes of the buildings are simple rectangular
parallelepipeds, with the ribs made up of a simple number of rods. For example, in
Brugge we find a contouring volume of 3 x 5 x 7 Brugge rods.
As an illustration, we give two graphical examples of buildings where this cubic grid is a
major characteristic of the design. Figure 1 shows Leuven, figure 2 shows Veere.
367

Figure 1: Leuven grid
Figure 2: Veere grid
For Leuven, the basic cube of the grid has a side of two Leuven rods, or 5.71 meter. See
further on for comments on this dimensional module.
For the buildings outside the actual Belgian borders, it should still be checked if the
apparent module coincides in a similar way with a precise number of local rods or feet.
368

Issue 2: the contouring square
In connection with the first finding, the front elevation of the town halls, including the
projection of the roof, is dictated by a contouring square. This square may be put on a
base in some cases.
As an illustration, we show the contouring square for Leuven. Further on the ones for the
other town halls will appear in connection with other design elements.
Figure 3: the principle of the contouring square (Leuven)
Issue 3: circle, triangle and square; or the numbers 1, 3
and 4
Not only do the front elevations of the town halls display the contouring square, but
often, this square is also embedded in a symbolic construction composed furthermore of
a circle and its inscribed equilateral triangle. So we have a composite figure created
from a 1-sided, 3-sided and 4-sided polygon. At this point it is interesting to notice that
the construction of this figure also gives good approximations for the division of a side
of the square into 3rd and 7th parts.
369

Figure 4: symbolic circle, triangle and square. The fat lines give the exact proportions
of 1 to 3 and 1 to 7. The intersections of the square and the triangle are very good
approximations of these positions.
The symbolic meaning of this figure could probably be explained by tracing
“compagnonesque” knowledge, but by now we may assume already that a medieval
verse is stating this kind of construction as follows: “A point in a circle – And that can
be situated in the square and the triangle – Do you know the point? All is for the better
– Don’t you know it? All is in vain”. [2]
Moreover we find, corresponding with this figure, a number series that is especially
confirmed in the design of the town hall at Oudenaarde. The series is as follows: 1 – 3 –
4 – 7 (i.e. 4 + 3) – 12 (i.e. 4 x 3). One Oudenaarde rod is 21 (i.e. 3 x 7) Oudenaarde
feet, hence a number game in all of the town hall’s composition.
Together with the scheme applied for Oudenaarde, we give a few other significant
examples of this design canon in the other buildings. Note that entrance doors often
reach until the 1:7 level of the square side, and roof balustrades until the 4:7 or 5:7 level.
In two cases, Oudenaarde and Saint-Quentin, the scheme can be repeated half of the
radius of the circle upwards, i.e. putting the base of the second triangle through the
center point of the first circle. Doing this we find the position of other cardinal points of
the front elevation.
In Brugge, by contrast, the scheme seems to have been repeated half a Brugge rod, or
seven Brugge feet, upwards.
370

Figure 5: facade composition for Oudenaarde with shifted scheme (R/2), relations
between plan & elevation.
371

Figure 6: facade composition for Brussel; relations between plan & elevation. Note that
for Brussel only the oldest front wing of the building is considered. The more recent
half of the building, right of the central gate and tower (not on plan), does not seem to
obey to the design system of the first building concept.
372

Figure 7: facade composition for Brugge; relations between plan & elevation, shifted
scheme (7 feet)
373

Figure 8: facade composition for Arras; relations between plan & elevation.
374

Figure 9: facade composition for Saint-Quentin, shifted scheme (R/2).
Some considerations about symmetry.
In our times, we tend to interpret a composition as being symmetrical when it can be
perfectly mirrored around an axis or a point. Through the research, it can be illustrated
that the concept of compositional symmetry had a different interpretation in the Middle
Ages. Often, the design of a building and of its facades is (slightly) asymmetric, due to
the program of use, the situation of the building site, or for other reasons. It was not
considered a problem that, e.g. the main entrance door did not sit centrally in the front
facade. As such, it was at first found strange that a notorious historian of the 16th
century, Justus Lipsius, praised the beautiful symmetry of the building: “Magnitudo iusta
est, symmetria bellissima” [3].
Denes Nagy has pointed out that we indeed should refer in this case to a different
meaning of the word symmetry, which lasted until the 17th-18th century. It goes back to
the ancient Greek concept of symmetria, which refers to “commensurability” and stands
375

for the correspondence between the part and the whole, for the common measure
between the different parts of the composition and their relation to the whole.
It is moreover very significant that the main facades of some town halls were modified
in the 18th century, to obtain the “new” kind of symmetry. In Leuven, one front door
was added beside the existing one, and in Bruges one of the two existing doors was
moved, in both cases to obtain a strictly axial symmetry.
But as mentioned higher, we should consider a different approach for the medieval
building master. A look back to the ground plans of Bruges, Brussels and Veere
indicates already the ease with which these plans were adapted to the particularities of
the building site.
In the facades at Leuven, we find a particular game of “asymmetry” as well, for example
in the series of balustrade motives (figure 10), or in the east gable facade. In the last
case, a stair tower imposes a subtle shift of the axial positions of different building
elements. However, these translations are combined in such way that a very balanced
overall composition is obtained (figure 11).
Figure 10: roof balustrades of the town hall in Leuven
376

Figure 11: east elevation of the town hall in Leuven. Note the asymmetry, caused by the
lower left tower drum, provoking eccentric positions of the gable and the corner towers
as well.
A final note on medieval measurement units
Considering the different measurement systems that were in use in the towns (each
Flemish town had its own privileges, jurisdiction, and even measurement units), Leslie
Greenhill has suggested that those units would probably be re-combinations or reinterpretations
of ancient measurements.
For the Leuven rod, which is known to be 5.7102 meters, he considers that this length
corresponds with 225 inches in the British Imperial system. 5.7102 meters at 39.37
inches a meter brings us indeed to 224.8121 inches, which gives a deviation of only
0.1% from the ‘symbolic’ 225 inches. That measure should be seen as 15 x 15 inches,
or as 6.25 (2.5 squared) yards. Square or cube numbers tend indeed to indicate
377

symbolic design. The British system is on its turn based on older, antique measurement
units, but this last issue is still under investigation.
The appearing of the number 225 is especially interesting in relation to antique design,
as e.g. the Parthenon in Athens is known to exhibit the 225:100 proportion (length to
breadth of the top of the stylobate). This ratio is often expressed as 9:4. But the use of
the number 225 occurs also in other places, and would further go back to Egyptian
times.
Here we probably come on the track of the building lodges of the “free masons”, which
have their roots in antiquity as well. We may mention the Greek Pythagorian
brotherhoods, with their three degrees of initiation, as forefathers of the medieval
companions. A consistent body of knowledge that was transferred through the centuries
is the most probable assumption to start from when we want to further investigate this
subject.
References
[1] Shelby, L.R., Gothic design techniques – the fifteenth-century design booklets of Mathes Roriczer and
Hanns Schuttermayer, Southern Illinois University Press, Carbondale and Edwardsville, 1977, p. 168,
citation from "Die Satzungen des Regensburger Steinmetzentages nach dem Tiroler Huttenbuche von
1460", in Zeitschrift fur Bauwesen, 46 (1896).
[2] Ghyka, M., Le nombre d’or: rites et rythmes pythagoriciens dans le développement de la civilisation
occidentale, Gallimard, Paris, 1969, p. 72; and also Ballegeer, J., Stadhuis Brugge, De Gulden Engel,
Wommelgem, 1987, p. 20
[3] Justus Lipsius, Lovanium: sive opidi et academiae eius descriptio. Libri tres, Antwerp, 1605, p. 88 (2nd
edition, Antwerp, 1610, p. 90).
378

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384

SYMMETRY GROUPS IN MATHEMATICS,
ARCHITECTURE AND ART
VERA W. DE SPINADEL
Name: Vera W. de Spinadel, Mathematician, (born Buenos Aires, Argentina, 1929).
Address: Centre of Mathematics & Design, Faculty of Architecture, Design and Urban Planning, University
of Buenos Aires, José M. Paz 1131 – Florida (1602) – Buenos Aires, Argentina.
E-mail: vspinade@fibertel.com.ar; vwinit@fadu.uba.ar; maydi@cvtci.com.ar
Fields of interest: Fractal Morphology, Onset to Chaos, Number Theory (Continued Fraction Expansions of
Real Numbers).
Awards: Several national Research and Development prizes.
Publications and/or Exhibitions:
1) “The Metallic Means family and forbidden symmetries”, International Mathematical Journal, vol. 2, Nr. 3,
pp. 279-288, 2002;
2) Exhibition “Fractal Art”, Fifth Interdisciplinary Symmetry Congress and Exhibition of the ISIS-Symmetry,
July 14, 2001, Sydney, Australia;
3) “Continued fraction expansions and Design”, Proc. of Mathematics & Design 2001 – The third
International Conference, July 3-5 2001, Deakin University, Geelong, Australia;
4) “The family of Metallic Means”, Visual Mathematics, vol. I, Nr. 3, 1999.
http://members.tripod.com/vismath/;
5) “From the Golden Mean to Chaos”, book edited by Nueva Librería, ISBN 950-43-9329-1, 1998.
Abstract: The word “symmetry” has two meanings. A symmetric object is well
proportioned but the concept is not restricted to concrete objects; the synonym
“harmony” refers to its use in Acoustics and Music. The second meaning is that of the
geometric bilateral symmetry, the symmetry so evident in superior animals, especially
in men. From the mathematical point of view, a whole symmetry theory can be
considered for applications. From this theory, we have chosen the “symmetry group of
the square” to present interesting uses in Architecture and art.
385

1. Symmetry in Culture
The word symmetry comes from the Greek symmetria, meaning “the right proportion”.
From the historical point of view, the term symmetry has denoted many meanings,
depending on the field of human knowledge where it was used. Notwithstanding,
“symmetry is a unifying concept”, as Hargittai’s, Magdolna and István, have proved in
their beautiful and unique book “Symmetry”. Indeed, the concept of symmetry can
provide a connecting link among many different fields of endeavor, perhaps the best and
more appropriate link to protect human studies from the increasing and separating
compartmentalization within our scientific world.
Going back to the year 27 B.C., we found a monumental work: the 10 books written by
the roman architecture Vitruvio (probably Marco Vitruvio Pollione) and dedicated to the
Emperor August. Architecture, says Vitruvio, depends from order, disposition,
eurhythmy, property, symmetry and economy. These terms have today, completely
different meanings. E.g., order, says Vitruvio, confers the appropriate measure to the
elements of a certain building, when considered separately and symmetry, gives
concordance to the proportions of the different parts of the construction. This approach
to the meaning of symmetry is quite similar to the mutually corresponding arrangement
of the various parts of a human body around a central axis, producing a proportioned
balanced form.
Vitruvio dedicated much time to the study of the proportions in the human body in his
considerations on symmetry. Symmetry, says Vitruvio, comes from proportion, that is,
from a correspondence between the dimensions of the parts of a whole and of the whole
with respect to a certain part selected as a model, the module. Such a selection of parts
of the human body as a module, initiated probably by Vitruvio, was the very beginning
of a historical ergonomic chain linking Vitruvio, Albrecht Dürer, Leonardo da Vinci and
many, many other artists, including the modern contemporary architect Le Corbusier.
2. Symmetry in Mathematics
In Euclidean geometry, a “symmetry” of a figure is a rigid motion that leaves the figure
unchanged. And what is a rigid motion? A “rigid motion” of the plane is any way of
moving all the points in the plane such that
• The distance between points stays the same
• The relative position of the points stays the same.
386

This concept of symmetry in the plane is easily generalized to symmetry in threedimensional
space.
The simplest rigid motion is “translation”. In a translation, everything is moved by the
same amount and in the same direction. We specify a translation by drawing an arrow.
Another rigid motion of the plane is “rotation”. A rotation fixes one point and
everything rotates by the same angle around that point. A third rigid motion is
“reflection”. A mirror line determines a reflection and it is easy to prove that
• points on the mirror line are unchanged by reflection;
• the distance from a point to the mirror is the same as the distance from the
image of that point to the mirror.
Finally, combining reflection with translation, we get a “glide reflection”, that is, a
mirror reflection followed by a translation parallel to the mirror. To make the list
complete, we add the “do-nothing” operation as another rigid motion of the plane.
All these rigid motions of the plane generate different types of symmetries:
Translation symmetry or repetition: it means shifting and repeating the same motif,
producing a “periodic” pattern. In ornamental art, this type of symmetry is called
“infinite ratio”, in which case it is a glide reflection, like in the frieze of Persian archers
at Dare’s Palace, in Susa (at present Khuzistan, Iran), depicted in Fig. 2.1 or the Palace
of the Doje in Venice, Italy (Fig. 2.2).
Fig. 2.1 Fig. 2.2
Rotational symmetry: it is an operation that iterated, brings the configuration again to its
original position. To avoid ambiguity, it is normally assumed that counterclockwise
rotations are positive. On the plane, the simplest figures that exhibit rotational symmetry
are regular polygons. In three-dimensional space, a body has rotational symmetry with
respect to an axis r if every rotation around r, takes the configuration back to its original
387

position. E.g., a four-blade pinwheel has rotational symmetry and if all its petals are of
different colors, it is easy to prove that after a whole turn, we are back at the starting
position. This is called a “4-fold rotational symmetry”.
Mirror symmetry or bilateral symmetry: it occurs when two halves of a whole are each
other’s mirror images. This symmetry was also called “heraldic symmetry”, because the
old inhabitants of Sumer (southern part of Mesopotamia) were the first ancient people
using heraldic drawings, having its use being extended later on to Persia, Syria and
Byzance.
To introduce a mathematical language and redefine symmetry, let us consider a spatial
configuration F. We shall say that the motions that leave F invariant -- or unchanged --,
form an algebraic structure called a “group of transformations” G, defined in the
following way:
Let G be a set together with a composition law ° which associates to each pair g,h ∈ G
another element g ° h ∈ G, called the “product” of g and h. Suppose that this product
satisfies the properties:
a) I ∈ G (the identity is in G)
b) if s ∈ G then s -1 ∈ G (the inverse transformation is in G)
c) if s and t ∈ G then the composition s ° t ∈ G.
Then the set of transformations G forms a “group”.
This group of transformations describes exactly the symmetries of the figure F.
To describe three-dimensional space, the notion of “congruence” is very useful. Two
spatial regions are congruent if a rigid body in two different positions can occupy them.
Congruent transformations form a group and, obviously, the simplest types of
congruencies are translations, reflections and rotations. The symmetry of any figure in
three-dimensional space is described by a subgroup of the group G.
Example 1: Let us consider the symmetries of a square. The eight distinguishable spatial
transformations, which comprise this group, are four quarter-turns and four reflections
(see Fig. 2.3):
d1 : Do-nothing
d2 : Rotation by 1/4-turn
d3 : Rotation by 1/2-turn
d4 : Rotation by 3/4-turn
388
d5 : Reflection in mirror m1
d6 : Reflection in mirror m2
d7 : Reflection in mirror m3
d8 : Reflection in mirror m4

Fig. 2.3
By direct inspection of the figure, it is evident that a square has four rotation symmetries
and four reflection symmetries. This is a property that can be extended to all regular
polygons, and it is easy to prove the following general result: The regular polygon with
n sides, has exactly n rotation symmetries and n reflection symmetries.
Adopting the usual composition of transformations as the fundamental operation, we
may write the composition table called Cayley diagram, that gives the results of the
composition dj ° dk , for j = 1,...,8; k = 1,...,8 and describes the group of symmetries of a
square.
389

j
1 2 3 4 5 6 7 8
1 1 2 3 4 5 6 7 8
2 2 3 4 1 8 7 5 6
3 3 4 1 2 6 5 8 7
k 4 4 1 2 3 7 8 6 5
5 5 7 6 8 1 3 2 4
6 6 8 5 7 3 1 4 2
7 7 6 8 5 4 2 1 3
8 8 5 7 6 2 4 3 1
The symmetries of a square form a group (non-commutative, as is easily verified by the
non-symmetrical table) and this result is generalized in the following sense: The
collection of symmetries of any figure will always be a group.
Furthermore, it is easy to prove that
d3 = d2 2 ; d4 = d2 3 ; d6 = d2 2 ° d5 ; d7 = d2 ° d5 ; d8 = d2 3 ° d5 .
Notice that the elements d6 , d7 and d8 can be expressed by the compositions of d2 and
d5 alone.
Example 2: Another interesting example is the famous pentagonal star with which Faust
exorcised Mephistopheles (Fig. 2.4). It coincides with itself by performing 5 rotations of
angles 72°, 144°, 216°, 288°, 360° and the 5 reflections with respect to the lines linking
the center of the figure with the 5 vertices of the pentagon. These 10 operations form a
group and this group indicates what sort of symmetry this star has.
390

Fig. 2.4
Note: The most fascinating and comprehensive analysis of all sort of symmetries, can be
found in the book written by the mathematician D. Schattschneider (“Visions of
Symmetry”) who has devoted much of her research to studying the work of the famous
graphic M. C. Escher. In this beautifully illustrated volume, she presents a detailed
symmetry inventory of Escher’s periodic drawings with one motif, two or more motifs
and non-interlocking patterns.
3. Symmetries in Architecture
An interesting application of the above mentioned point group symmetry of the square,
has been recently presented by Jin-Ho Park. Two housing plans have been analyzed by
him to show how the mathematical methods of symmetry operations are employed as
thematic elements in the unit design as well as the variations in the planning of the site.
The first example was Frank Lloyd Wright’s social housing project, called the
Quadruple Building Block. Wright (1867-1959) used a standard unit plan for the
project, based on his earlier design of his own Home & Studio, Oak Park, Illinois, 1889.
A version of this project was originally published as a vignette in the ‘Ladies Home
Journal’ article of 1901 on his “Small House with Lots of Rooms in It”. Whereas the
unit itself was asymmetric, various local symmetries were involved in the unit. The
assembly in their site layout included two types: one is the pinwheel type and the other is
the mirrored reflection type. Each house was set on four equally subdivided lots, sharing
a common backyard in the center for all four houses. The whole complex of the
housings was laid out in a way that their elevation could be varied according to their
arrangement. The Quadruple Block plan was a scheme with which Wright was actively
391

concerned from 1902 till the next 10 or 15 years. We cannot be sure that his purpose
was a practical one at all because they are not inexpensive row houses designed for mass
production; they are substantial dwellings on half-acre plots. Unfortunately, he never
found either four families who wanted identical dwellings or a real state investor who
could believe that such blocks would constitute a salable commodity. But he never
abandoned this revolutionary idea of urban design. Wright’s next housing designs share
a common compositional theme: the pinwheel type of symmetry, characteristic of the
point group symmetry of the square (Figs. 3.1 and 3.2).
Fig. 3.1 Fig. 3.2
The second example was Rudolf Michael Schindler(1887-1953) unexecuted Shelter
project, developed from 1933 to 1942. In this project, the overall floor plan was based
on a 5-foot square grid. Along with the square grid, the symmetry governed the internal
structure of the spatial composition in each shelter unit as well as the unit variations. The
internal organization was subdivided by the removable closet partitions for spatial
flexibility and set along the pinwheel type of rotational symmetry. Then the garage was
added to any side of the house as a separate unit. But instead of providing only a
standard unit with fixed layouts, a sequence of variations were provided, variations that
shared the same underlying organization system. Using the symmetry operations of the
square, new eight different shelter designs were produced. Taking advantage of the
possibilities of combining units into groups, Jin-Ho Park presented an experimental
grouping exercise using the shelter unit corresponding to the eight transformations of the
symmetry of the square in an abstract level, quite similar to what Wright envisioned in
his Quadruple project.
392

4. Forbidden symmetries
Quasicrystals, which belong to a class of quasiperiodic systems, gave diffraction patterns
that show local n-fold rotational symmetry forbidden in Crystallography: n = 5, 8, 10,
12. In this context, quasilattices can be thought as mathematical discrete sets supporting
Bragg peaks or atomic sites. They play the same role as lattices do for crystals.
Quite recently, discrete sets of numbers, the β-integers Z β , have been proposed as
numbering tools for coordinating quasicrystalline nodes in 1, 2 or 3 dimensions, and
also the Bragg peaks in diffraction patterns. In the observed cases:
Golden Mean (penta- or decagonal quasilattices):
1+ 5 π
β = φ = = 2 cos
2 5
Silver Mean (octagonal quasilattices)
π
β
= σAg
= 1+
2 = 1+
2 cos
4
Subtle Mean (dodecagonal quasilattices)
β = φ
3
= 2 +
An important mathematical common characteristic is that the continued fraction
expansions of these irrational numbers are the following:
3 [ 1 ] ; σ = [ 2 ] ; = [ 4 ]
φ Ag
= φ
3 = 2 + 2 cos
π
12
where the usual notation […] for continued fractions is used. All of them are purely
periodic continued fraction expansions.
The relevant scale factor β is a quadratic Pisot-Vijayaraghavan number or more simply,
a PV number, i.e. an algebraic integer β > 1 which is solution to equations of the type
393

2
x = ax ± 1 ( a ∈{
1,
2,
4}
)
such that all respective second roots β’ (called Galois conjugates of β) have modulus
strictly smaller than 1.
These PV numbers are a subset of the Metallic Means Family (MMF), introduced by the
author. The more outstanding member of this family is the well known Golden Mean,
then comes the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean and
many others. These positive quadratic irrational numbers intervene in the determination
of the quasi-periodical behavior of non-linear dynamical systems, being therefore an
invaluable key in the search of universal roads to chaos. Besides, the members of the
MMF satisfy simultaneously many additive and geometric properties, having been in
consequence advantageously adopted as bases of many architectonic systems of
proportions.
The role played in lattice theory by the ring of integers Z and the planar rotational
compatibility condition ρ = 2cos(2π/n) ∈ Z is in quasilattices replaced by ρ ∈ Z β .
In terms of the members of the MMF, quadratic Pisot numbers have the following
equivalences
394

n ρ β β ’
5 2 cos 2π
/ 5 = 1/
φ 1 + cos 2π
/ 5 = φ
1 − φ
8 2 cos 2π
/ 8 = σ Ag − 1 1 + 2cos
2π
/ 8 = σ Ag
1 − 2 = 2 − σ A
12 2 cos 2 / 12 = 3
π 1+2 cos 2π
/ 12 = 1 + 3 = [ 2,
1,
2]
1 − 3 = [ 1,
2]
12 2 cos 2 / 12 = 3
π 2 + 2 cos 2π
/ 12 = 2 + 3 = [ 3,
1,
2]
2 − 3 = [ 1,
1,
2]
The discovery of quasi-crystals with crystallographically forbidden symmetries is one of
the most striking examples where a pure symmetry analysis determines mathematically
forbidden symmetries appearing in a new solid state of matter.
Interesting to mention, Silver films with a close-packed structure modulated by a “Silver
Mean” quasi-periodic sequence have been experimentally obtained. This represents a
new type of quasicrystal that is fundamentally different from the commonly known
quasicrystals, which possess at least two-dimensional rotational order.
5. Symmetry and fractals
Fractals are geometric configurations with a built-in self-similarity, that is,
configurations that remain invariant in the presence of “scale changes”. More simply,
they possess borders, surfaces or internal structures with patterns that zoomed and
zoomed until infinity, are invariant, exactly or statistically. Fractal structures are
important to study because they are intrinsically related to the important notion of
“chaos”. A physical process is said to be “chaotic” in relation to its dynamics that
means, when it is impossible to make any type of prognosis about its future evolution,
since it is verified that very similar initial conditions give rise to system behaviors that
differ enormously among them. To be able to find the connection between fractal
structures and chaotic processes, it is necessary to geometrize the system dynamics.
Using computerized graphics, it is feasible to detect fractal patterns that dynamically are
considered as “universal scenarios of the roads to chaos”.
395

Fig. 5.1
Let us consider some variations of a classical geometric fractal: the so-called Sierpinski
gasket, depicted in Fig. 5.1. Beginning with a square, it is possible to set three
contractions that reduce the initial square by a factor of 1/2 and place the resulting
square appropriately
⎛ x y ⎞
v1(
x,
y)
= ⎜ , ⎟
⎝ 2 2 ⎠
⎛ x + 1 y ⎞
v2
( x,
y)
= ⎜ , ⎟
⎝ 2 2 ⎠
⎛ x y + 1⎞
v3(
x,
y)
= ⎜ , ⎟
⎝ 2 2 ⎠
Returning to the 8 symmetry transformations of a square, we may define a family of
patterns specifying a triplet w1, w2, w3, where each wi is given by the product of
transformations wi = vi dk for k = 1,2,...,8 and i = 1,2,3. This makes altogether 8 3 = 512
different triplet’s w1, w2, w3, and each one describing a specific pattern. To illustrate this
amazing variety of fractal patterns, all of which are close relatives of the Sierpinski
gasket, let us consider 8 different sets of transformations, which are symmetric with
respect to the diagonal and the corresponding fractal patterns (see Fig. 5.2).
396

Fig 5.2
This theoretical description is based on the so-called “Hutchinson operator”,
introduced in 1981 by the mathematician J. Hutchinson. Using this operator, M. F.
Barnsley was able in 1985, to devise an IFS (Iterated Function System) with which he
succeeded in getting fractal images that were so near from a natural image as desired.
6. Symmetry and visual design
Ljubisa Kocic, a Serbian mathematician, has developed a simple technique for
converting real numbers into linear ornaments and vice versa. To achieve this
visualization of real numbers, he has recently presented two algorithms A and B, based
on the continued fraction expansion of a real number.
Algorithm A
1. Take the continued fraction expansion of a real number [ a0
, a1,
a2
, ]
the sequence { a k } up to the (m + 1)-th term to obtain m a a a , , , 0 1 .
397
x = . Cut
2. Replace a i by ) 8 (mod a i = ai
. The number x will be represented by
a
′
a
′
a
′
0 , 1 , m , m , where a i ∈{
0,
1,
, 7}
and it is denoted as
=
′ ′ ′
m a a a x , , , ) ( α 0 1 m ..
( )

3. Starting from ( 0 , y0)
= ( 0,
0)
{ ( , y ) , k = 1,
2,
, m + 1}
x create a sequence of points
xk k
following the rule
xk
+ 1 = xk
+ p
yk
+ 1 = yk
+ q
where p and q are given in the following table
a
′
i
1 2 3 4 5 6 7 0
P 1 1 0 -1 -1 -1 0 1
Q 0 1 1 1 0 -1 -1 -1
4. Draw the polygonal line x) { ( x , y )( , x , y ) , m,
( x , y ) }
β ( = 0 0 1 1 m+
1 m+
1 .
Note: This table encodes the direction of the path of a moving particle according to the
compass rose:
E, NE, N, NW, W, SW, S, SE.
Curiously, this pseudo-random Brownian walk was used by Berthelssen et al. to
investigate the quantity of information hidden into the DNA molecule chain.
It is easy to see that Algorithm A, applied to the Golden Mean
1 + 5
φ = = [ 1,
1,
m]
,
2
will produce a horizontal line. The same will happen with all the Metallic Means that
have a purely periodic continued fraction expansion, of the form [] n . But there is a lot of
very interesting Metallic Means that originate very nice Brownian friezes, so called
because the algorithm produces a kind of Brownian motion trajectory. In fact, since the
members of the Metallic Means family are defined as the positive solutions of quadratic
equations of the form x 2 - px - p = 0 (p, q natural numbers), i.e.
p + p + 4q
σ ( p,
q)
=
,
2
it is possible to apply Algorithm A to produce beautiful Brownian friezes, like the
shown at Fig. 6.1 that correspond to the following Metallic Means
398
2

σ
σ
σ
σ
3 + 21 1 + 21
( 3,
3)
= = 1 + = 1 + [ 2,
1,
3]
2
1 + 33
( 1,
8)
= = [ 3,
2,
1,
2,
5]
2
1 + 129
( 1,
32)
= = [ 6,
5,
1,
1,
2,
3,
2,
1,
1,
5,
11]
2
1 + 2313
( 1,
578)
= = [ 24,
1,
1,
4,
1,
5,
5,
5,
1,
4,
1,
1,
47]
2
2
Notice that all these Metallic Means are periodic of the form [ , , ] n
, 1 2 n m and these
periods are “palindromic” i.e. the periods are symmetric about their centers, except for
the last number of the period, which equals 2m − 1.
Algorithm B
Fig. 6.1
1. Take the continued fraction expansion of x.. Cut the sequence { a k } up to the
(m+1)-th term to get m a a a , , , 0 1 m .
ak , am−
k ; k = 0,
1,
m,
.
Ak = ak
, am−k
counterclockwise about the origin and
draw this part of the polygonal line m A A A m x , , , ) , ( 0 1 m = γ .
2. Create a sequence of pairs { } m
3. Rotate each point ( )
The graph γ ( x,
m)
is a symmetric figure with respect to the y-axis and is called the
“symmetrogram” of x.
The reason of introducing this second algorithm is that the “Brownian frieze” method
produces friezes that are not univocal, that is, two different numbers may have the same
graph.
Let us apply Algorithm B to irrational numbers containing e = 2.71828…, the basis of
natural logarithms.
399

For example:
γ ( e,
50)
γ ( e
γ ( e
γ ( e
2
3
1
, 65)
, 30)
/ 4
, 70)
produce the symmetrograms of Fig. 6.2.
Fig. 6.2
400

Summarizing, the application of these two algorithms for visualization of real numbers
produce patterns and ornaments that range from straight lines over periodic and cyclic to
chaotic ones.
7. A visual notation for rational numbers
The Australian mathematician Julie Tolmie has presented at the ISIS-Symmetry Fifth
Interdisciplinary Symmetry Congress and Exhibition held at Sydney, Australia, July 8-
14, 2001, an interesting methodology to construct a visual notation for rational numbers.
Rational numbers mod 1 are represented as equivalence classes of pairs (p,q). The torus,
with its two directions of rotational symmetry is used as a phase space; its longitudinal
cycle for the denominator and its meridian cycle for the numerator. Using cyclic motion
and a discrete map of {1,2,3,…,37} into the RedGreenBlue RGB-color space, the
numerator is encoded as cyclic permutations of color coded dots. These configurations
are placed in the appropriate meridian slices (in radial vertical planes), being the result a
three dimensional navigable object contained in a torus (a visual abstract is provided at
the site http://www.ozemail.com.au/~jatolmie/isis0.html).
As is well known, the rational numbers mod 1 have a Farey tree, structure, which is
called Stern-Brocot tree because it was independently, discovered by the German
mathematician Moritz Stern (1858) and the French clockmaker Achille Brocot (1860).
To introduce rotational symmetry, the Farey tree is made circular and embedded in the
longitudinal plane of the torus. Rational numbers are then defined as curve segments,
which span the longitudinal region, bounded by its Farey parents. These curve segments
are cut from curves, which wind around the torus at a constant integer velocity and in
this way, a three-dimensional navigable object is made.
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Semi-groups”, ed. J. Patera, Fields Institute Monograph Series, volume 10, Amer. Math. Soc.
Barnsley M. F. (1988), “Fractal modelling of real world images” in “The Science of Fractal Images”, H.-O.
Peitgen and D. Saupe (eds.), Springer-Verlag, New York.
Berthelsen C. L., Glazier J. A. and Skolnick M.H. (1992), “Global fractal dimension of human DNA
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“Pisot and Salem numbers”, Birkhäuser Verlag.
Bogomolny A., “Farey Series”, http://www.cut-the-knot.com/blue/Farey.html 1996-2001.
401

Burdik C., Frougny C., Gazeau J. P. and Krejcar R. (1998), “Beta-Integers as Natural Counting Systems for
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Elser V. (1985), “Indexing problems in Quasicrystal Diffraction”, Phys. Rev. B32, 4892-4898.
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Gazeau J. P. (1997), “Pisot-cyclotomic Integers for Quasicrystals”, The Mathematics of Aperiodic Long
Range Order (ed. R. V. Moody) NATO-ASI Proceedings, Waterloo 1995, Kluwer Academic
Publishers.
Gazeau J. P. (2000), “Counting Systems with Irrational Basis for Quasicrystals”, editors: F. Axel, F. Dénoyer,
J. P. Gazeau. In From Quasicrystals to more Complex Systems, No. 13. Les Houches School
Proceedings, Springer-Verlag, Heidelberg, Berlin.
Gazeau J. P. and Lipinski D. (1997), “Quasicrystals and their Symmetries”, Symmetries and structural
properties of condensed matter, Zajaczkowo 1996. Ed. T. Lulek. World Scientific: Singapore.
Hargittai István & H. Magdolna (1960), “Symmetry: a unifying concept”, Shelter Publications Inc., Bolinas,
California, 1994.
Hutchinson J. (1981), “Fractals and self-similarity”, Indiana University Journal of Mathematics, 30, pp. 713-
747.
Ikezawa K. and Kohmoto M. (1994), “Energy spectrum and the critical wavefunctions of the quasiperiodic
Harper equation – the Silver Mean case – “, J. of the Phys. Soc. of Japan, 63, Nr. 6, pp. 2261-2268.
Kocic Ljubisa (1999), “Portraits of Numbers”, Visual Mathematics, vol. 1, No. 4,
http://www.mi.sanu.ac.yu/vismath/pap.html
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M&D-2001, 3-5 July 2001, Deakin University, Australia.
Pedoe Dan (1976), “Geometry and the liberal arts”, Penguin Books Ltd.
Schattschneider Doris (1990), “M. C. Escher: Visions of symmetry”, W. H. Freeman & Co., New York.
Shih Chih-Kang, “Growing atomically flat metal films on semiconductor substrates”,
http://www.aps.org/BAPSMAR98/abs./S278002.html
Smith A. R., Chao Kuo-Jen, Niu Qian, Shih Chih-Kang (1996), “formation of atomically flat Silver films on
GaAs with a ‘Silver Mean’ quasi periodictiy”, Science, 273, pp. 226-228.
Spinadel Vera W. de (1998), “From the Golden Mean to Chaos”, Nueva Librería, Buenos Aires.
Vitruvio (1960), “Los 10 libros de Arquitectura”, Editorial Iberia S.A.
Weyl Hermann (1952), “Symmetry”, Princeton University Press.
402

ADVANTAGES OF DECENTRALIZED ELECTRICITY
AND HEAT SUPPLY FOR BUILDINGS, USING FUEL
CELLS.
ERICO SPINADEL
Name: Erico Spinadel, Industrial Engineer ( born Vienna, Austria, 1929).
Address: Argentine Wind Energy Association AAEE, Universities UBA, UNLu, EST-IESE, UFASTA. José
María Paz 1131, RA1602 Florida, Buenos Aires, Argentina.
E-mail: espinadel@fibertel.com.ar ; gencoeol@cvtci.com.ar ; homepage
www.ecopuerto.com/organizaciones.asp
Fields of interest: Wind Energy, Renewable Energies, Electric Energy, Hydrogen Technologies.
Awards: Several national & international R&D prizes
Publications (2001): 1) “ Mathematical Model for Optimizing Sizes of PEM Fuel Cells in Combined Natural
Gas and Electricity Energy Supply”. Proc. M&D2001, The Third International Conference, Deakin
University, Geelong, Australia. ISBN 0 7300 2526 8, pp. 166 – 173. 2) “ Stand Alone Energy Islands”. Proc.
HYPOTHESIS IV, The 4th. International Symposium on Hydrogen Power, University of Applied Sciences,
Stralsund, Germany. ISBN 3-9807963-0-2, Vol I, pp. 40 – 47. 3) “Patagonian Wind Exported as Liquid
Hydrogen”. Proc. HYPOTHESIS IV, The 4th. International Symposium on Hydrogen Power, University of
Applied Sciences, Stralsund, Germany. ISBN 3-9807963-0-2, Vol I, pp. 88 – 92. 4) “ Feasibility Study for the
Hacienda Project (Wind-H2 Stand Alone Energy System)”. Proc. HYPOTHESIS IV, The 4th. International
Symposium on Hydrogen Power, University of Applied Sciences, Stralsund, Germany. ISBN 3-9807963-0-2,
Vol III, pp. 446 – 450. 5) “Definición de las Características de un Generador Eoloeléctrico para Alimentación
de Electrolizadores Alcalinos”. Revista Ingeniería Militar, Argentina. ISSN 0326-5560, year 18, No.43, Jan-
Jun.2001, pp. 36-37. 6) “ Hay conciencia de que la energía eólica existe, pero la legislación aún no es
suficiente”. Revista Electrogremio, Argentina. ISSN 0329-3009, year 15, No.l45, Jan. 2001, pp. 74-75. 7) “
Celdas de combustible para uso domiciliario”. Revista Megavatios, Argentina. ISSN 0325-352X, year 24,
No.242, Jun 2001, pp. 102-106. 8) “Condicionantes socio-económicos para un mayor aprovechamiento de la
fuente energética primaria eólica en la Argentina”. Revista Megavatios, Argentina. ISSN 0325-352X, year 24,
No.242, Jun 2001, pp. 120-142. 9) “ Energía eólica, una inversión a futuro”. Revista Tecnoil, Argentina. RPI
324-856, year 23, No.232, Oct. 2001, ps. 54-60.
Abstract. The fast development of fuel cells technologies helps to consider their use in
decentralized energy supplies. They have an advantage over solar power: In addition to
investments by a handful of start-ups, several corporations are spending more than $1
billion annually on fuel cell research. This technology is tout as a way to wean nations
403

off foreign oil, as fuel cell systems extract their hydrogen from natural gas, and to rid
the countries of coal-fired and nuclear power plants. Fuel cells are also money-savers.
Given the cost benefit of natural gas over gasoline and the fuel cell's more efficient
extraction technology, experts predict that fuel cell systems will become attractive when
they reach $1,000 a kilowatt in the next seven to 10 years. Fuel cells will then save
money in places where utilities are charging more than 10 cents per kilowatt-hour,
offering energy security, energy quality, and environmental quality.
1. General description of the proposal in countries with
access to Natural Gas NG resources.
For several reasons, in certain countries natural gas is commercialized at very low
prices. Argentina is one of those countries. For long years, electricity had been
generated in Argentina by hydro energy (some 50%), by thermal energy (some 40%)
and by nuclear energy (some 10%). The resulting mix was commercialized at the gross
electricity market at about 32 mils per MWh (US$ 0,032/kWh). Nowadays, using the
combined cycle generation (gas turbines plus steam turbines using the former wasted
process heat from the gas turbine cycle) in about 33% of the total argentine active
generation park of some 17 GW, the resulting mix can be commercialized at only 28
mils.
Normally, small households as well as big office or apartment houses, are connected as
well to the electric grid as to the natural gas distribution system, using simultaneously
both energy services, for lightning and powering electrical devices the former and for
heating and cooking purposes the later. Consequently, as analyzed in some papers by
different authors, a typical middle class household pays for both energy services
together, as a yearly average, some US$ 120 per month.
The expected presence of PEM fuel cells on the market (to be used in combination with
reformers producing hydrogen from natural gas, in small 10kW units with an output
capacity of some 7 kW thermal and 3 kW electricity and at prices in the order of about
US$5.000 per unit) will substantially modify the commercial energy demand. As
analyzed by the same authors, the mentioned typical middle class household will be able
to reduce the energy services bills to only some US$ 70 per month, using only the gas
connection and the combination of PEM fuel cell and reformer described later.
The same criteria may be applied to the energy consumption prognosis for large
buildings. In this case, a very careful study must be done in each case, analyzing the
required balance between thermal and electricity needs, in order to optimize the size of
404

the fuel cells to be installed, and, consequently, the overall costs of the energy provision
to the building.
Some guidelines for developing a mathematical model to be used in each case are also
proposed, contemplating the specific thermal and electricity demand and assuming
current costs for the kWh of electricity and for the cubic meter NG, in order to obtain
the price-optimal system (fuel cell - commercial energy provision).
2. Strategic advantages.
A flexible, decentralized energy system allows the simultaneous generation of both
electricity and heat, exactly covering the user’s demand. A service based on fuel cells
fed by NG is the first step to an integrated Hydrogen Energy System. The second step
would be the use of renewable primary energy sources, for instance, wind energy, to
produce pure H2 by electrolysis. This alternative is of particular interest for countries
with the possibility of implementing large wind farms (onshore or offshore). The
strategic advantage becomes evident once the no dependence of fossil fuels for
electricity generation is considered. Besides that advantage, hydrogen may be used as an
energy accumulator as well as an energy vector. If hydrogen is produced using a
renewable primary energy source and stored as pressurized gas, or in the liquid phase or
by means of metallic hydrides, the entire prime energy offer may be used time
independent and electricity and heat may be generated only when really needed. And
last but not least, the possibility of a decentralized electricity generation system
eliminates the necessity of transmission lines. Resuming, no more dependence of fossil
fuels coming from abroad, as well as a more rational use of electricity, as it is generated
in accordance with the real demand and without losses due to transmission lines.
3. Increased anti-terrorist security
Doubtless, high pressure NG-ducts are less vulnerable to sabotage as power lines are, or
as any nuclear or thermal powerhouse. It is relatively easy to produce severe damage to
a power line and black out even several big cities for a considerable time extension. In
the case of an underground high-pressure gas duct, not only the sabotage would not be
so easy, but also the rehabilitation of the service would demand less time. The same
could be said for a distribution network.
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4. Schematic lay out of a fuel cell.
In the next section, different fuel cells operating nowadays will be presented. All of
them act according to the same principle, shown in the following diagram (Fig.1).
Cathode and anode are separated by an electrolyte and are covered on the inner face by
a catalyst. On the anode, molecular hydrogen is atomized, adsorbed by the catalyst and
split into a proton H + and an electron e -. On the cathode, oxygen, in most cases coming
from the air input, is also atomized and adsorbed by the catalyst. It recombines then with
the proton H + that traveled through the electrolyte and the electron e - that traveled
through the electric load. Surplus air or oxygen as well as water vapor is liberated at the
cathode and excess hydrogen is liberated at the anode. Also a given amount of heat is
liberated and may be used.
Figure 1.
5. Different types of fuel cells.
The different types are presented in (Fig.2). Values and composition of the input gases
are given, as well as the average operating temperature, type of electrolyte used in each
case and their average efficiency. Historically, the alkaline cell, AFC, is the oldest
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developed and has been used and is still widely used in quite different applications. The
first space missions had been equipped with this type of cells.
The Proton Exchange Membrane Cell, PEM, (Fig.3) is the other known low temperature
cell. Several automobile producers are experimenting with this type of cells and their
cars will most probably be in series production in the very near future. This type of cells
is also of great interest for its use in the theme of this presentation. The BEWAG utility
of the City of Berlin in Germany has a 250 kW unit of this type currently in operation.
Figure 2.
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Figure 3.
In the range of medium temperatures, the PEM cell may also be used fueled with CNG
in cars, provided that they have a reformer unit on board, for splitting the methanol into
H2 and CO2.
It should be noted that the Proton Exchange Membrane is made of very special
materials, with different names and covered by patents by their manufacturers,
characterized by its property of being only permeable to protons. In this type of cells,
the catalyst is Pt, very sensitive to CO and CO2 , gases that should be avoided in order to
improve the lifetime of the cell.
The membrane, covered by patents by their manufacturers, are, as shown in Fig. 4,
molecules containing only one hydrogen atom, loosely linked to the molecule and
therefore permitting an easy hydrogen transport through the electrolyte.
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Figure 4.
The Phosphoric Acid (PAFC) is another medium temperature cell, already extensively
used for combined electricity and heat production. The trouble is the high aggressivity
of the phosphoric acid at the working temperature; the assumed lifespan of this type of
cells is therefore of no more than some four years.
In the high temperature range, lot of research is done nowadays with the Molten
Carbonate (MCFC) and the Solid Oxide Fuel Cells (SOFC). As at these high
temperatures both the emitted CO and theCO2 are cracked, coal gas as well as biogas
may also be employed as fuel, with no reasonable fear of contaminating the catalyst with
these gases.
6. Cell and System Costs.
First of all (Fig.5), the definition of efficiency in fuel cells has to be considered. It is also
necessary to keep in mind the impossibility of generating electricity or heat separately,
as both of them are produced simultaneously.
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Figure 5.
At present, costs are relatively high as fuel cells are produced one by one. The goals to
be attained in the next years are:
• A power-to-mass ratio of some 1 kg/kW.
• A specific cost around 100 US$/kW for the cell stack.
• A service life of at least 2.500 operating hours (in the case of mobile use, 5
years with an average of 30.000km/year).
The current cost of the membrane foil (in the case of a PEM cell stack of only 1kW
electric power, about 1 m 2 ) is around 500 US$.
If the mentioned goals were attained, the decentralized use of those devices for
simultaneous electricity and heat supply will be able to compete favorably with the
current commercial electricity and heat prices. It is not necessary to mention the
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important reduction of the environmental impact for any given amount of commercial
energy used, specially the reduction of the green house effect due to the less intensive
CO2 emissions.
For each case, the demands of electricity and heat must be carefully analyzed, in order to
optimize the design of the system, reducing the number of fuel cells required for each
particular application. The peak seasonal demands of electricity and of heat
(respectively during summer and winter) must be carefully studied. A mathematic model
should be developed in order to perform this optimization.
7. Examples of uses in buildings in a decentralized energy
supply.
The typical complete fuel cell power system is shown in Fig.6. In certain applications,
some of the components may loose significance and could not to taken into account.
Also, not all of the emissions should necessarily be evaluated in a first approach.
Figure 6.
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8. Steps to be followed for an adequate system proposal.
Normally, one should analyze the possible utilization of fuel cells for the decentralized
co-generation of electricity and heat, connected to the natural gas net. After studying
different extreme summer and winter conditions, the requirements for a mathematical
modeling to optimize the quantity of fuel cells should be stated. This optimization is of
an economical nature, since its main aim is to obtain the price-optimal system (fuel cell -
commercial energy provision) for different values of the variables.
The following items must be taken into consideration for establishing this mathematical
model. First of all, it is to point out that this is really one of the first historical analyses
of a combined decentralized electricity and heat supply for buildings. No significant
previous experiences are therefore available.
A careful balance of the electric and thermal demand as a function of time must be
established. Proper use of the eventual excess of electricity or heat must also be
carefully studied. As previously pointed out, both energy forms, electricity and heat, will
always be produced simultaneously by the fuel cell, and this is why the ENERGY
demand must be optimized. But, at the same time, peaks in the electricity and heat
demands must be diminished, in order to optimize also the POWER demand.
The designer must also take into account different possible technological alternatives
related to each energy demand (for instance, refrigeration, lightning, etc.).
Lots of variables appear simultaneously in the problem; all of them are easy to analyze
one at a time, but sometimes very difficult to evaluate while acting together, due to their
interactions. Optimizing means that both, energy offer and energy demand, need to be
simultaneously dimensioned, in accordance one with the other.
References.
Barlow R. “Residential Fuel Cells: Hope or Hype?” Published by Homepower Magazine Nº 72, August 1999.
Available under http://www.humboldt1.com/~michael.welch/extras/homefuelcells.pdf.
Hart David L. “Hydrogen Power – The commercial future of the ´ultimate´ fuel”. Published by Financial
Time Energy Publishing, London, UK, 1997.
Lehmann J. “Costs an niches – hydrogen prices today”. Proc. HYPOTHESIS IV, The 4th International
Symposium on Hydrogen Power, University of Applied Sciences, Stralsund, Germany September
2001.
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Lehmann J. “Fuel cells and hydrogen”. Proc. Fuel Cell Seminar, Universidad Tecnológica Nacional, Unidad
Académica Concordia, Argentina, March 2002.
Ogden Joan M. and Nitsch J. “Solar Hydrogen”, Published as Chapter 22 of “Renewable Energies – Sources
for fuel and electricity” by Island Press, Washington DC, U.S.A., 1993.
Ogden Joan M. and Williams R.H. “Solar Hydrogen – Moving beyond fossil fuels”, Published by the World
Resources Institute, Washington DC, U.S.A., October 1989.
Reidpath M.M. “Fuel Cell Technology Overview”. Federal Energy Technology Center, U.S. Dept. of Energy,
U.S.A., August 1999.
Schnurnberger W. “Technologies for a sustainable energy economy: renewable energy resources and
hydrogen”. Proc. HYPOTHESIS IV, The 4th. International Symposium on Hydrogen Power,
University of Applied Sciences, Stralsund, Germany, September 2001.
Spinadel E, Gracia Nuñez S.L., Maislin J., Wurster R. and Gamallo F. “Domestic Electricity Generation by
means of hydrogen and fuel cells”, Proc. Conference of Technological Innovations in Electricity
Distribution Systems, School of Electrical Engineering, Catholic University of Valparaiso, Valparaiso,
Chile, October 1999.
Spinadel E., Spinadel V., Gamallo, F. “Mathematical Model for Optimizing Sizes of PEM Fuel Cells in
Combined Natural Gas and Electricity Energy Supply”. Proc. M&D2001, The Third International
Conference, Deakin University, Geelong, Australia, July 2001.
Sun Media GmbH “H2 Tec – The Magazine for H2 and Fuel Cells”. Published by Sun Media GmbH,
Hannover, Germany, April 2001.
Tetzlaff K.H. “The better way to an energy economy without emissions”. Proc. HYPOTHESIS IV, The 4th.
International Symposium on Hydrogen Power, University of Applied Sciences, Stralsund, Germany,
September 2001.
10. Acknowledgment.
To Florencio Gamallo, my first assistant in the GENCo R&D Group at the University of
Buenos Aires, board member of the AAEE and co-author of several of my publications.
IN MEMORIAM OF PROF. SABAS LUIS GRACIA NUÑEZ
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Day 5
Study visit to Bruges, 2002 April 13.
The final day led the participants to Bruges, where
Ann Ratinckx organized a guided walk (the
charming girl on the right of the picture on the left).
It led to the statues of Simon Stevin (below), the
Flemish da Vinci, and Van Eyck, the master of
Flemish art.
An exclusive visit to the new concert hall was
equally part of the program (left, second photo from
top). One of the architects of the building, Hilde
Daem, was the guide. This remarkable concert hall
was an exponent of the Bruges 2002 art festival,
when the city of Bruges was the Cultural Capital of
Europe. The brand new concert hall is hoped to play
a role similar to the Guggenheim museum in Bilbao.
More information can be found on the site
http://www.concertgebouw.be/browser_ok.asp
Finally, the visit led to the Memling museum,
located at the "St. Jans Hospitaal" (picture down left;
see http://www.trabel.com/brugge-m-memling.htm).
Hans Memling (1435/40(?) - 1494) is one of the
most important Flemish primitives. One of the
masterpieces is the Shrine of St.Ursula, and another
the Altarpiece of Saint-John the Baptist and Saint-
John the
Evangelist
(painting
down right).
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