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The Quarterly of theInternational Society for theinterdisciplinary Study of Symmetry(ISIS-Symmetry)Editors:GyiSrgy Darvas and D~nes Nag¥Volume 5, Number 2, 1994The Miura-oriopened out like a fan

INTERNATIONAL SOCIETY FOR THEINTERDISCIPLINARY STUDY OF SYMMETRY(ISIS-SYMMETRY)PresidentD~nes Nagy, lnslltute of Apphed Physics, University ofTsukuba, Tsukuba Soence C~ty 305, Japan(on leave from Eotvos Lot’find Umve~ty, Budapest, Hungary)IGeometry and Crystallography, H~story of Science andTecbnology, Lmgmsucs]Honorary PresidentsKonstantin V. Frolov (Moscow)andMaval Ne’eman (TeI-Avw)Vice-PresidentArthur L. Loeb, Carpenter Center for the V~sual Arts,Harvard Umverslty. Cambridge, MA 02138,U S A. [Crystallography, Chemical Physics, Visual Art~,Choreography, Music}andSergei V Petukhov, Instnut mashmovedemya RAN(Mechamcal Engineering Research Institute, Russian,Academy of Scmnces 101830 Moskva, ul Griboedova 4, Russia(also Head of the Russian Branch Office of the Society)}B~omechanlcs, B~ontcs, Informauon Mechamcs]Executive SecretaryGybrgy Darvas, Symmetrion - The lnsmutefor Advanced Symmetry StudiesBudapest, PO Box 4, H-1361 Hungary}Theoretical Physms, Philosophy of Science}Associale Edttar.Jobn Hosack, Department of Mathematics and ComputingScience, Unlverstty of the South Pacific, PO Box 1168, Suva, FIji}Mathematical Analysts, PhflosophylRegional Chat,persons / Representatives.AFRICAMozambique Paulus Gerdes, Inst~tutoSuperior Pedag6gico, Ca~xa Postal 3276, Maputo,Mozambique|Geometry, Ethomatb.emaucs, History of Science}AMERICASBrazd: Ubiratan D’Ambrosio, Rua Pe~xoto Gomide 1772, up. 83,BR-01409 S~o Paulo, Brazd[Ethnomathemat~cs]Canada: Roger V. Jean, DEpartement de mathfimatlqueset mformauque, Um~ers~t~ du QuEbec ~ gamouskt,300 allEe des Ursuhnes, R~mouskL QuEbec, Canada G5L 3AI[ B~omathemattcs]U.S.A " William S. Huff, Departlnent of Architecture,State Umversfly of New York at Buffalo, Buffalo,NY 14214, USA.}Architecture. Des~gnlNicholas Toth, Department of Anthropology,Indiana Utaverslty, Rawles Hall 108, Bloomington,IN ~,7405, U.S A.[Preh~storta Archaeology, Anthropology]ASIAChina. t~R. Da-Fu Ding, Shangha~ Institute of Biochemistry.Academia Stoma, 320 Yue-Yang Road,Shanghai 200031, PR China[Theoreucal B~ology]Le~Xiao Yu, Department of Fine Arts. NanjmgNormal Umvers~ty, Nanjmg 210024, P.R China}Free Art, Folk Art, Calhgraphy]lndta. Kirti Trivedi, Industrial Design Cenlre, lndmnInstitute of Technology, Powa~, Bombay 400076, IndialDes~gn, lndmn Art]Israel. Hanan Bruen, School of Education,Umvers~ty of Hallo, Mount Carmel, Haffa 31999, Israel[Educanon]Jim Rosen, School of Physics and Astronomy,TeI-Av~v Umvers~ty, Ramat-Avtv, Tel-Av~v 69978. Israel[Theoretical Physms]Japan. Yasushi Kajfl~awa, Synergel~cs Institute.206 Nakammurahara, Odawara 256, Japan}Design, Geometry]Koichtro Mat~uno, Department of BioEngineering.Nagaoka Umvers~ty of Technology, Nagaoka 940-21, Japan[Theoretical Physms, Blophys*cs]AUSTRALIA AND OCEANIAAustraha Leslie A, Bursill, School of Physics,Umvers~ty of Melbourne,Parkwlle, Vtctorm 3052, Austraha[Physics, Crystallography]F01: Jan Tent, Department of L~leratum and Language,University of the South Pacific,PO Box 1168, Suva, F0t [Lmgmsttcs]New Zealand. Michael C. Corballis, Department of Psychology,Umversily of Auckland, Private Bag, Auckland I, New Zealand[Psychology]Tonga. ’Ilaisa Futa-i-Ha’angana Helu, Director,’Atems~ (Athens) Institute and Umverslly,PO. Box 90, Nuku’alofa, Kingdom of Tonga[Phdosophy, Polynesian Culture]EUROPEBenelux" Pieter Huybers, Facultett der Civlele Techniek,Techmsche Untverstte~t Delft(C~wl Engineering Faculty, Delft Utavers~ty of Technology),Stevmweg I, NL-2628 CN Delft, The Netherlands[Geometry of Structures, Budding Technology}Bulgaria: Ruslan I. Kostov, Geologtcheski lnstltut BAN(Geological Institute, Bulgarian Academy of Sciences),ul Akad G. Bonchev 24, BG-III3 Sofia, Bulgaria[Geology, M meralog3’]Czech Republic: X’bjt~h KopskJ;, Fyz~k~lnt t~stav (~AV(Institute of Physics, Czech Academy of Sciences), CS-180 40Praha 8 (Prague), Na Slovance 2 (POB 24),Czech Republic [Sohd S~te Physics}France: Pierre Sz~kely, 3bts, impasse Vflliers de I’lsle Adam,F-75020 Paris, France [Sculpture]continued inside back cover

.!1..i|CULTURE & SCIENCE I|The Quarterly of the International Society for theInterdisciplinary Study of Symmetry(ISIS-Symmetry)Editors:GyOrgy Darvas and D~nes NagyVolume 5, Number ~ 113-224, 1994SPECIAL ISSUE: ORIGAMI, 2Edited byD~nes Nagy and Gy0rgy DarvasCONTENTSSYMMETRY: CULTURE & SCIENCE¯ Mathematical algorithms for origami design, RobertJ. Lang¯ Mathematical remarks about origami bases, Jacques Justin¯ Evolution of origami organisms, Jun Maekawa¯ Paper sculpture, Didier BoursinSYMMETRIC GALLERY - ORIGAMI¯ Paper sculpture, Didier Boursin¯ Stag beetle 2, RobertJ. LangRESEARCH PROBLEMS ON SYMMETRY¯ Research problem 1, D~nes NagySYMMETR O-GRAPHY115153167179189190197211213SFS: SYMMETRIC FORUM OF THE SOCIETY 219

SYMMETRY: CULTUlCEAND SCIENCE is edited by the Board of the International Societyfor the Interdisciplinary Study of Symmetry (ISIS-Symmetry) and published quarterlyby the International Symmetry Foundation. The views expressed are those ofindividual authors, and not necessarily shared by the Society or the Editors.Any correspondence should be addressed to the Editors:Gy6rgy DarvasSymmetrion - The Institute for Advanced Symmetry StudiesP.O. Box 4, Budapest, H-1361 HungaryPhone: 36-1-131-8326 Fax: 36-1-131-3161E-mail: h492dar@ella.huD6nes NagyInstitute of Applied PhysicsUniversity of TsukubaTsukuba Science City 305, JapanPhone: 81-298-53-6786 Fax: 81-298-53-5205E-mail: nagy@kafka.bk.tsukuba.ac.jpThe section SFS: Symmetric Forum of the Society has an E-Journal Supplement.Annual membership fee of the Society: Benefactors, US$780.00;Ordinary Members, US$78.00 (including the subscription to the quarterly);Student Members, US$63.00;Instituaonal Members, please contact the Executive Secretary.Annual subscription rate for non-members: US$96.00 + mailing cost.Make checks payable to ISIS-Symmetry and mail to Gy0rgy Darvas, Executive Secretary,or transfer to the following account number: ISIS-Symmetry, InternationalSymmetry Foundation, 401-0004.827-99 (US$) or 407-0004-827-99 (DM),Hungarian Foreign Trade Bank, Budapest, Szt. Istv~in t6r 11, H-1821 Hungary(Telex: Hungary 22-6941 extr-h; Swift MKKB HU HB).ISIS-Symmetry. No part of this publication may be reproduced without writtenpermission from the Society.ISSN 0865-4824Cover layout: Gunter SchmitzImage on the front cover. Biruta KreslingThe Miura-ori opened out like a.fan, simulates the mechanism responsible J’or the outstretching of thebeetle’s membraneous hindwin gAmbigram on the back cove~. John Langdon (Wordplay, 1992)Logo on the title page: Kirti Trivedi and Manisha l.eleFot6k~sz anyagr61 a nyomdai kivitelez6st vdgezte:9421768 AKAPRINT Kft. F. v.: Dr. H~czey Lfiszl6n~

3)rmmetry: Culture and ScienceVot 5, No. 2, 115-152, 1994MATHEMATICAL ALGORITHMS FORORIGAMI DESIGNRobert J. Lang7580 Olive Drive Pleasanton, CA 94588, USAE-mail: rjlang@aol.comAlthough hundreds of years old, the Japanese art of origami has only recentlybecome the subject of mathematical scrutiny. In recent years, a number ofmathematical aspects of origami have been published in books and journals. Asampling of the work of mathematical folders is to be found in recent mainstreampublications, e.g., (Kasahara, 1988) and (Engel, 1989); however a large number offolders have attacked the problem of systematic/mathematical origami design. Theyinclude Peter Engel and myself in America, and many folders in Japan, includingHusimi, Meguro, Maekawa, and Kawahata. As befits a young and expanding field,much of the scientific analysis is circulated informally (notably over the origami-Imailing list on the Internet: to join, send the message "subscribe origami-Iyourname" to listserv@nstn.ns.ca).The goal of many origami aficionados is to design new origami figures and formany, the pursuit of origami mathematics is a search for tools leading to ever morecomplex or sophisticated designs. In this article, I will describe two powerfulalgorithms for origami design that I have successfully applied to the design of fish,crustacea, insects, and numerous other origami models. Although I will describethe algorithms in the form I am familiar with, similar techniques have beendescribed by Dr. Toshiyuki Meguro in the (Japanese-language) publication Oru andin the newsletter of the Origami Tanteidan, a Japanese association of origamidesigners.1. THE CIRCLE METHOD OF DESIGNThe first design approach is represented by what I and others call the ’circlemethod’. In the circle method, each flap on the origami model is represented by acircle whose radius is equal to the length of the flap. The goal of the origami designprocess is to place circles representing each flap on the square in such a way thatthe centers of all circles lie within the square (although some part of the circle canextend over the edges of the square) and no two circles overlap one another. This

116R. J. LANGapproach is one that both I and Fumiaki Kawahata have used extensively, although- as happens so often in the sciences - we each developed our methods initiallyunaware of the other’s activities. I am not aware of other Westerners using thecircle method, although the young American folder, Jimmy Schaefer, has developeda successful and related design method, which he has dubbed, ’the method ofisolating squares’, based on concepts similar to the circle method. In Japan, theseconcepts of origami design are more widely known than in the West.The fundamental concepts of the circle method of design and i~s derivatives are_two: first, since most origami models can be broken down into a number of flaps ofvarious lengths, a successful design hinges upon constructing the right number andsizes of flaps. Second, paper must be conserved; any part of the square can be usedin no more than 1 flap at a time. The circle method of origami design consists ofrepresenting the subject as a collection of flaps and allocating a unique circularregion of paper for each flap.For the purposes of origami design, there are three different types of flaps: ’corner’flaps, ’edge’ flaps, and/or ’interior’ flaps, or ’middle’ flaps, as some call them. Thedifferent types of flaps are named for the point where the tip of the flap falls on thesquare. If you take a model that has a lot of flaps, color the tip of each flap, andthen unfold the model, you’ll get a square with a pattern of dots on it. Some dotswill fall on the corners of the square; others on the edges; still others will be in theinterior of the square. Since each dot corresponds to a flap of the model, we canclassify the flap by the location of the dot, which is the location of the tip of theflap. A corner flap has its tip come from a corner of the square, an edge flap has itstip lie somewhere along an edge, and a middle flap, as you would expect, comesfrom the middle of the paper. For example, the four large flaps on a Frog Base arecorner flaps; the four stubby flaps are edge flaps; and the thick flap at the top is amiddle flap.The reason for the distinction between the three different types of flap is that for agiven length, each of the three types of flaps consumes a different amount of paper.One way to see this difference is to fold corner, edge, and interior flaps of exactlythe same size from three different squares as shown in Figure 1 below, where Iillustrated the folding of a corner flap. If you imagine (or fold) a boundary acrossthe base of the flap, then that boundary divides the paper into two regions: thepaper above the boundary is part of the flap, and the paper below the boundary iseverything else. The paper that goes into the flap is for all intents and purposesconsumed by the flap; any other flaps must come from the rest of the square.So, as Figure l(a) shows, if you fold a flap of length L from a square so that the tipof the flap comes from the corner of the square, when you unfold the paper to theoriginal square, you see that the region of the square that went into the flap isroughly a quarter of a circle. Well, technically, it’s a quarter of an octagon. Supposewe made the flap half the width, as shown in Figure l(b) before we unfolded it;

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 117then the flap becomes a quarter of a 16-gon. If we kept making the flap thinner andthinner (using infinitely thin paper!), the boundary of the flap would approach aquarter-circle. Since we may not know ahead of time how thin a given flap will be,we’ll take the circle as a reasonable approximation of the boundary of the region ofthe paper consumed by the flap. A comer flap of length L, therefore, requires aquarter-circle of paper, and the radius of the circle is L, the length of the flap.LFigure l(a): Folding a corner flap of length L from a square.Figure l(b): (Left) Making a narrower flap makes the boundary a quarter of a 16-gon. (Right) The limitof the boundary as the flap becomes infinitely thin approaches a semicircle.Therefore, all of the paper that lies within the quarter-circle is consumed by theflap, and the paper remaining is ours to use to fold the rest of the model.Now, suppose we are making a flap from an edge. How do we do that? Well, if wefold the square in half, then the point where the crease hits the edge becomecomers, and we can fold corner flaps out of one of these new corners, as shown inFigure 2. If we fold and unfold across the flap to define a flap of length L and thenunfold to the square, you see that an edge flap of length L consumes a half-circle ofpaper, and again, the radius of the circle is L, the length of the flap.

1181~ I. I_.ANGFigure 2: Folding an edge flap of length L from a square.Similarly, we can make a flap from some region in the interior of the paper (itdoesn’t have to be the very middle, of course). Figure 3 shows how such a flap ismade. When you unfold the paper, you seen that an interior flap requires a fullcircle of paper, and once again, the radius of the circle is the length of the flap.IIIFigur~ 3: Folding a middle flap of length L from a square.Figure 4: Unfolding a crane (which has four major flap~) reveals that the quarter-circles correspondingto the four [lap~ consume almost all of the paper in the model.So, any given flap in a model consumes a quarter, half, orJfull circle of paper,depending upon whether it is a corner flap, edge flap, or interior flap. It is aninteresting and illuminating exercise to unfold an existing model and draw in the

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 119circles corresponding to the various flaps of the model. For example, the traditionalcrane, made from the Bird Base, has four major flaps. As shown in Figure 4, theyare all corner flaps, and each flap consumes a quarter-circle of the square. (If wecount the pyramid in the middle of the back as a flap, we would have another,smaller circle in the center of the square. However, since real cranes don’t havepyramids in their back, I consider that an ’accidental’ flap and we won’t count it inour tally.) Since well over 3/4 of the area of the square goes into the four majorflaps, we would say that the crane shows an efficient use of paper.What we are doing here is to build up a set of mathematical tools that can be usedto design origami models, making origami design a scientific process. One of thegoals of all scientific endeavors is the concept of unification: describing severaldisparate phenomena as different aspects of a single concept. Rather than thinkingin terms of quarter-circles, half-circles, and full-circles for different kinds of flaps,we can unify our description of these different types of flaps by realizing that thequarter-circles, half-circles, and full circles are all formed by the overlap of a fullcircle with the square, as shown in Figure 5. The concept common to all three typesof flaps is that the paper for each can be represented by a circle with the center ofthe circle lying somewhere within the square. With middle flaps, the circle lieswholly within the square. However, with corner and edge flaps, part of the circlelaps over the edge of the square. (The center of the circle still has to lie within thesquare, though.) Thus, any type of flap can be represented by a circle whose center,which corresponds to the tip of the flap, lies somewhere within the square.Figure 5: All three types of points can be represented by a circle if we allow the circle to overlap theedges o~ the square.The examples above showed how to fold a single flap from a square. Suppose wewant to make more than one flap at a time from the square (which is usually thecase, unless you are designing a worm), and you draw the circles corresponding toeach flap. Is there anything we can say about the circles even before we start? Well,since no part of the paper can be used in two different flaps simultaneously, andeach circle delineates the paper used in each flap, no two parts of the paper can lie

inside two different circles. Therefore, no two circles corresponding to different flapscan overlap on the unfolded square.Although this property seems pretty general, it is in fact quite restrictive. If youwant to fold a model with ten flaps, you know that if you unfold the model anddraw the circles corresponding to each flap, no two of the circles will overlap. So ifyou eliminate all arrangements of points for which the circles overlap, you must becloser to a design solution.In fact, the ramifications of the non-overlapping property are a great deal stronger;if you draw ten non-overlapping circles on a square, it is guaranteed (in amathematical sense) that the square can be folded into a base with ten flaps whosetips come from the centers of the circles. So merely by shuffling circles around on asquare, you can construct an arrangement of points that can be folded into a basewith the same number of points, no matter how complex!Therefore, here is an algorithm for origami design:(1) Count up the number of appendages in the subject and note their lengths.(2) Represent each flap of the desired base by a circle whose radius is the length ofthe flap.(3) Position the circles on a square such that no two overlap and the center of eachcircle lies within the square.(4) Connect adjacent centers to one another with crease lines.The resulting pattern can be foldable into a base with the number and dimension offlaps that you started with.This is a powerful property for origami designers. If you lay out circlescorresponding to the flaps of your subject on a square so they don’t overlap, you areguaranteed of the existence of a folding sequence that can transform the pattern intothe desired base. Finding the folding method may still be a bit of a trick, of course,but by beginning from a valid circle pattern, you certainly eliminate a lot of blindalleys.One thing that is immediately apparent from the circle method of design is thatcorner flaps consume less paper than edge flaps, which consume less paper thaninterior flaps. Turn this property around, and you find that for a given size square,you can fold a larger model (with fewer layers of paper) if you use corner flapsrather than edge flaps, and edges flaps rather than interior flaps. Seen in the lightof the circle method, the traditional crane - and the Bird Base from which it comes- is an extremely efficient design, since all four flaps are corner flaps, and almostall of the paper goes into one of the four flaps. However, add one or two more

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 121flaps, and you are forced to use edge flaps. Once you start mixing edge flaps andinterior flaps, you begin to run into tradeoffs in efficiency. Sometimes, it is evenbetter not to use the corners for flaps if there are additional flaps to be placed onthe square!So, for example, suppose we want to fold a base withfive, rather than four, equallengthflaps. A little doodling with a pencil and paper (or alternatively, you can cutout some cardboard circles and shuffle them around) will reveal two particularlyefficient arrangements of circles, as shown in Figure 6 below.(a)(b)L = 0.707 L = 0.647Figure 6: Two c~rcle patterns corresponding to bases with five equal-length points.Now we have two possible circle patterns. Which one is better? Is there any way toquantify the ’quality’ of a crease pattern?One way of comparing different ways of folding the same base is to compare theirefficiency; that is, from a given size square, how large is the base? A useful measureof efficiency is to compare the size of some standard feature of the base - such asthe length of a flap - to the size of the original square. To facilitate thiscomparison, let’s assume our square is one ’unit’ on a side. If you’re using standardorigami paper, a unit is 10 inches. For the crease patt.erns shown in Figure 6(a), ifall of the circles are the same size, it is fairly easy to work out that the radius ofeach circle, and thus the length of each of the five flaps, is 1/v~, or 0.707. For thepattern in Figure 6(b), it is somewhat harder to calculate but the radius of eachcircle is 0.647, or about 10% smaller. Thus, a five-flap base made from pattern 6(a)will be slightly larger, and slightly more efficient than the pattern made from Figure6(b).These two circle patterns are relatively simple. By connecting the centers of thecircles with creases and adding a few more creases, you can collapse the model into

122 R ].. LANGa base that has the desired number of flaps. As it turns out, there already exists inthe origami literature two bases that correspond to these circle patterns, shown inFigure 7. Figure 7(a) is the circle pattern for the Frog Base, while Figure 7(b) is thecircle pattern for John Montroll’s ’Five-Sided Square’ (Montroll, 1985).(a)(b)Figure 7: Full crease patterns corresponding to the two circle patterns.You can also see a difference between the two bases. In the Frog Base, the fifth flapis a thick middle flap and points in the opposite direction from the four cornerflaps; whereas in the Five-Sided Square, the four edge flaps and the corner flap goin the same direction and can easily be made to appear identical (which, of course,was the original rationale for John’s design). It’s worth a slight reduction in size toobtain the similarity in appearance for all five flaps.While the two solutions for five equal-sized flaps correspond to published bases, Ifind it remarkable that the most efficient base for six equal flaps is not yetpublished. You might wish to try your hand at the following two problems:(1) Find a circle pattern for the largest possible base that has six equal-length flapsand fold it into the base.(2) Find a circle pattern for the largest possible base that has six equal-length flaps,3 on each side of a line of bilateral symmetry, and fold it into the base. (Thesurprising solution has two middle flap!!)As I said before, although the circle method guarantees that a folding sequenceexists to convert the skeletal crease pattern into a base, it doesn’t necessarilyprovide any guidance as to what that folding sequence actually is! (Meguro’s’molecular’ approach of fitting together pre-existing crease patterns, however, helpsfill in this gap.) So even if you work out a circle pattern, with or without acomputer, you (and the computer) still have some work ahead of you to figure outhow to fold the crease pattern into a base. However, it is a big help to start with acrease pattern that is guaranteed to work - you can avoid using a basic symmetrythat is doomed to failure from the start!

MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN123Interestingly, there is a strong connection here between origami design and a wellknownbranch of geometry, that of packing circles. (For an excellent introductionto the latter, see (Gardner, 1992), Chapter 10, Tangent Circles.) For every patternof circle packings in a square there is a corresponding origami base and vice-versa;conversely, many origami design problems my be solved by published solutions todifferent circle-packing problems.)The circle method as described above works very well for models with many flapsthat all come from the same part of the subject’s body. The legs of insects andspiders, for example, all emanate from a single body segment. However, the circlemethod has one enormous liability. Since we consider only the total number offlaps and their lengths in the circle method of design, we have no way ofincorporating information about how those flaps are connected to one another intothe design. In fact, the circle method implicitly assumes that all the flaps areconnected to each other at a single point! In the subject, the head bone may beconnected to the neck bone, and the neck bone’s connected to the chest bone, butwith the circle method, every bone’s connected to every other bone at one spot.That is a severe limitation for origami design. For example, a typical mammal hasthree flaps (head and forelegs) at one end, three (tail and hind legs) at the other,and a body in between. The circle method can produce the head, legs, and tail, butthere is no mechanism to include extra paper between the forelegs and hind legs toform a body. The circle method is not the whole story of origami design, however.There is a more sophisticated algorithm that includes the body, and in fact doesindeed work for arbitrary arrangements of flaps and their connections. Now that wehave established some basic concepts of origami design, we are ready to move on tothe next level of origami design and introduce the ’tree method’. This newalgorithm will be described in the next section.2 TREE METHOD OF DESIGNThe circle method as described above works best for.models that have all of theirflaps emanating from nearly the same place, models whose basic shape is star-like.Quite a few subjects fall into this category - particularly insects, which have legsand wings all emanating from a single body segment, the thorax. (Antenna causeproblems, since they come from the head.) However, the circle method gives lessthan-satisfactoryresults for subjects that don’t have a simple star shape - like mostterrestrial vertebrates. A typical mammal, for example, has a cluster of three flaps(head, forelegs) separated from another cluster of three flaps (tail, hind legs) by anadditional segment (the body). The circle method only deals in flaps and clusters offlaps; we have no way of including segments that connect different clusterstogether.

124 R. I. I../~VGAn extension of the circle method works for a much larger class of models that canhave more complex structure. I call the extended method the ’tree method’.Although the tree method is built upon the ideas of the circle method, it takes asomewhat different form. The fundamental concept of the tree method of origamidesign is that you represent the model by a stick figure (the tree) that has a branchfor each arm, leg, wing, or other appendage. Each branch has a certain length,which you have chosen to be the length of the appendage in the final model. Bymapping the tree onto a square according to a small number of rules, you canconstruct the skeleton of a crease pattern that, like the one you get from the circlemethod, is guaranteed to be capable of being folding into the stick figure and, byextension, into the desired model. Unlike the circle method, which only applied tostick figures that were fundamentally star-like, the tree method works for arbitrarilyconnected graphs.To understand the rules for the mapping, I’ll show how it applies to a realisticdesign problem, a lizard. A lizard is simple enough that we won’t get bogged downin a lot of details, but it’s complicated enough to illustrate the basic approach, andbecause of its long body, it is precisely the type of model that the circle method hastrouble with.So, Figure 8(a) shows a drawing of a lizard. We would like to fold an origami modelof a lizard, that might look something like Figure 8(b). (Actually, I’d hope it looksa lot better than Figure 8(b); I can fold a lot better than I can draw.) I’ll start bydrawing the ’tree’ that represents the lizard, as shown in Figure 8(c). A tree is astick figure. On this stick figure, I have labeled each branch of the tree - whichcorresponds to an appendage or body segment of the lizard - with its desiredlength. The tail and body are 2 units long, the legs are 1 unit long each, and thehead is also 1 unit long. Since I don’t know what size square I’m going to be foldingfrom and I don’t know (yet) how large the lizard is with respect to the square, I’lldefer, for the moment, the question of just how long a ’unit’ is.(a) (b) (c)rear frontfoot foot: headrearfootfrontfootFigure I1: (Left) A real lizard. (Middle) A hypothetical origami lizard. (Right) The tree, or stick figure,corresponding to our hypothetical lizard.My end goal is to lind a crease pattern on a square that can be folded into thelizard. My intermediate goal will be to find a crease pattern that can be folded into

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 125the tree. At first, it seems like I’ve made my life harder by setting the tree as mytarget. Since the branches of the tree are infinitely thin, it would take an infiniteamount of folding (and infinitely thin paper) to fold it exactly. However, think ofthe tree as the lizard stripped of confusing detail. The tree, in its stark simplicity,represents an easier target for origami design than the original subject. Andanyhow, I don’t have to fold the tree exactly; if I fold a shape that closely resemblesthe tree, I’ll have a shape - a base - that is suitable for folding a lizard.In the tree method of design, each branch on the tree corresponds to some flap onthe square. Tree branches that end at a point - which I call ’leaves’ (or terminalnodes, if you prefer) - correspond to appendages of the subject, like the head, tailand legs. Tree branches that are connected to other branches at both endscorrespond to body segments that join groups of appendages.Since I’m trying to establish a link between the square and the hypothetical lizard,let’s start at the lizard and work backwards. If I already possessed a folded versionof the lizard and I made dots at important points - the tips of the legs, head, andtail, and where legs and body come together - when I unfolded the paper to asquare, I could keep track of where those significant points fall on the square. Thepoints where branches terminate or where several branches come together areimportant; I’ll call each of those points a ’node’ and label it with a name,corresponding to its position in the subject. Obviously, all of the nodes must liesomewhere on the square, and a great deal of the structure of the base is tied up inwhere the different nodes fall on the square. We ought to be able to draw the entiretree on the square so that the nodes match up with their corresponding points.There are lots of different possibilities for the position of the nodes with respect tothe square; a few of them are shown in Figure 9.Figure 9: Three different possible arrangements of the lizard tree on the square./..."

It sort of makes sense that we would want the head and tail at opposite corners ofthe square, so let’s suppose for the sake of argument, that we already had asuccessfully folded lizard-like shape, that we marked the locations of the nodes andbranches on the shape, and then unfolded it to a square, giving the arrangementshown in Figure 10.frontfoot~’rontfootFigure 10: One possible arrangement of nodes for the lizard tree.An important issue (actually, one of the most important) is how large the tree iscompared to the size of the square. One of the hallmarks of good origami design isefficiency; the best designs generally are those that waste very little paper. Forexample, if you are folding a 3-pointed shape, you could start from a 4-pointed baseand crumple up one of the flaps to hide it; but that wouldn’t be nearly asesthetically pleasing as to work from a 3-pointed base from the very beginning. It’spoor form to have to hide an unwanted flap, or even to have to make a flapdrastically shorter by folding it in half. In our design, we don’t want to have pointsor flaps that serve no purpose. So, in our design, we should use as much of thepaper as possible for the designed parts of the figure and have no extra paper leftover.Also, the most efficient designs generally have the fewest layers, at least, whencompared to designs of comparable complexity. Generally, the smaller a shape is,the more layers it has. That means that for a given size square, the base that wedesign should be as large as possible; consequently, the size of a unit of length ofthe tree should be maximized.

MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN127Let’s figure out how big the tree is for Figure 10. The distance from the head to thetail is 5 units (2 for the tail, 2 for the body, and 1 for the head). The diagonal of asquare is about 1.4 times the side of the square, and as I’ve shown, the diagonal isequal to 5 tree units; thus one tree unit is 1.4/5 = 0.283 times the side of the square.This quantity - the ratio between a tree unit and the side of the square - is animportant measure of the efficiency of a design, and we’ll call this the ’scale’ of thecrease pattern. The larger the scale is, for a given size square, we get a larger basewith fewer layers, a base that is more efficient and (one hopes) more estheticallypleasing.The main problem with the arrangement of nodes depicted in Figure 10 is that itdoesn’t work. If you try to fold a lizard using this arrangement, you will find that,although the body and tail are easy to make, the legs come out rather shorter thanwe intended. In fact, the back legs will be almost nonexistent. Furthermore, quite alot of paper at the top and bottom corners goes essentially unused. It doesn’t seemright that the base comes out with the wrong proportions and there is unused paperas well! So we can’t just draw the tree to scale on the square and expect things towork out. Obviously, we’re overlooking some crucial concept. There must be someadditional rule to be applied that limits the size of the tree when it is mapped ontothe square.Well, of course with all the folding that goes on between the square and the base,the tree pattern on the square could get rather distorted. We can establish somelimits on the amount of distortion, though. Consider the following thoughtexperiment. Suppose an ant wishes to walk from the tail of the lizard to one of therear feet. On the lizard, she starts at the tip of the tail, walks up the tail to the flapwhere the tail and legs meet, turns, and walks down the leg. The distance the anthas waJked is the length of the tail plus the length of the leg, or, as I’ve drawn inFigure 11, a total of 3 units.frontfoottail .......................................2xheadrearfootfrontfootFigure 11: Picture an ant walking from the tail to the rear leg. She can’t go by the shortest route (as thecrow flies); instead, she must walk up the tail and back down the leg, for a total of three units.

128R..I. L4NGSuppose that just before the ant set out, we dipped her in ink, so that when shewalked, she left a trail of ink soaking through the paper. Now we unfold the baseand look at the various trails left by the ant. Since in most origami bases, each flapconsists of several layers of paper, the ant will probably have left several trailsbetween the two nodes. Depending on the folding pattern for the base, some of thetrails might weave around a bit, while others go more directly from the tail to thefoot. Several possible trails are shown in Figure 12.frontfoottailfrontfootFigurt 12: Three paths from tail to rear foot.Although we may not know what ground was covered by the ant, we do know thatthe ant walked exactly 3 units on the tree. If in the folded base some paper wasdoubled back on itself, then some of the paths on the square might be longer than 3units, but no path can be shorter than 3 units. In particular, the shortest possible inktrail is the one that runs directly between the tail node and the leg node (trail (b) inFigure 5), and it, too, must be at least 3 units long. Thus, we know that a successfulcrease pattern must have the tail node and each leg node separated by a minimumof 3 units.A similar condition exists for every possible pair of nodes. If the ant goes from thetip of the tail to the tip of the head, he travels 2+2+1=5 units; thus the tip of thetail must be separated from the head on the square by 5 units. From one front legto the other is 1+1=2 units, so the two front leg nodes must be separated by 2units; and so on and so forth.This condition must hold for any pair of nodes: the distance between two nodes onthe square must be at least the distance between the two nodes measured along the

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 129branches of the tree. Nodes connected by a single branch must be separated by atleast the length of the branch; nodes connected by two branches must be separatedby the sum of the lengths of the two branches; and so on, for every possible pair ofnodes.Now you see the problem with the tree structure of Figure 10. The way I’ve drawnit, although all nodes connected by a single branch are separated by the properamount, the shortest path between the nodes corresponding to the tail and rearlegs is only v~, or about 2.2 units long, when it should in fact be at least 3 units.The same type of shortfall affects the paths between the front and rear legs. As I’vedrawn them in Figure 10, the front and rear feet on one side of the body areseparated from each other by the body length, or 2 units; but in fact, they need to beseparated from each other by 4 units, since our proverbial ant must walk down oneleg, along the body, and back out the other leg to go from toe to toe. Similarshortfalls afflict the paths between the head and front legs, between the head andrear legs, between the tail and front legs, and between the front and back legs.Figure 13 shows all possible paths drawn in on the tree and all of their minimumlengths.Figure t3: Paths and lengths for all nodes on the lizard tree.In general, many pairs of nodes on the square must be farther apart from oneanother than they are ’as the crow flies’ on the tree. There are two ways we couldovercome this problem. We could keep the arrangement of nodes as we have themin Figure 11 and multiply all distances by some fixed value. A quick check of allpossible paths shows that the paths in Figure 10 in the worse shape are the onesbetween front and rear legs on one side, which must be separated by 4 units. If wesimply double all distances so that the tail node is separated from the hip node by 4units, the leg nodes are separated from the hip nodes by 2 units and so forth, thenthe leg nodes are separated from one another by the required 4 units, and in fact allpairs of nodes meet their minimum separation requirement.

1.30R..1. L4NGBut doing this means making our tree unit smaller. After scaling down the tree by afactor of 2, the diagonal of the square is 10 tree units long, so that the scale of thecrease pattern has fallen from 0.28 to 0.14.But this is awfully wasteful! Although now the path between front and rear legs areequal to their minimum length, all the other paths are longer than they have to be,which means that we’ll be wadding up excess paper to get each point down to itsproper length. What we really ought to do is to increage the size of a unit andrearrange the nodes to allow a larger unit size.Since the lizard is a pretty simple shape, it’s easy to see that we can improve thedesign by moving the nodes corresponding to the feet out to the edges of the squareand, since the tail is longer than the head, moving the rear feet farther from the tailcorner than the front feet are from the head. Figure 14 shows an optimumdistribution of nodes for the lizard. Even for the optimum, most of the paths turnout to be longer than their allowed minima. (In fact, you have considerably freedomin your placement of the nodes corresponding to the hips and shoulders.) Thepaths that turn out to be the limiting paths - which I call the ’critical paths’ - arethe ones from head to forelegs, from forelegs to rear legs, and from rear legs to tail.I’ve made the critical paths heavy in Figure 7. If you work out the geometry (or justdraw it to scale and measure), you find that one tree unit is (vr3-1-5)/3 times theside of the square, which works out to a scale of 0.189 - smaller than what you getfrom Figure 10, but now it’s foldable. Since we know the scale, we can also figureout how big the final model will be: the base we fold from this pattern will make thelength of each leg about one-fifth the side of the square, and from nose to tail, themodel will be almost as long as the side of the square.footfrontfoot2footfoot14: An optimum distribution of nodes on the square for the lizard tree.

MA THEhfA TICA]_, ALGORITHMS FOR ORIGAMI DESIGN131Now, we have a crease pattern that satisfies all of the criteria we have set so far.Can it really be folded into a base of the proportions we have set? The answer isyes, it can. Figure 15 shows all the creases for one of many possible ways to fold thisinto a base. While the base itself is probably what you would fold a lizard from, byrepeated box-pleated sinks, you can transform the base into a pretty fairapproximation of the stick figure, the tree, itself. Of course, you don’t need to gothat far. The useful result of this little exercise is the second figure in Figure 15; thebase, which we construct as a byproduct bf folding the tree.(b)tatlrearfoot(c)rear 1frontfoot,headfrontfootfront~z-_.l foot2 ~ "~headtail - zrear~" " footfootFigure 15: (a) Crease pattern for the lizard base. (b) The lizard base. (c) By repeatedly sinking the lizardbase, you can even make a close approximation of the tree.In fact, just as was the case with the circle method, it can be shown that any patternof nodes you construct according to the rules of the tree method is foldable into thestick figure you started from. What we have here is a rudimentary algorithm fordesigning a large class of origami models. Any subject that can be approximated bya tree diagram - a stick figure - can be designed by (1) identifying the nodes,paths, and their lengths on the tree; (2) Laying out the nodes on a square such thatthe distances between any pair of nodes is larger than the corresponding path onthe tree; (3) Folding the resulting pattern of nodes into a base. I don’t wish to glossover the great difficulty in step 3, of course. This algorithm gives a skeletal creasepattern, not a folding sequence. That you still have to lind yourself. However, thesearch is always easier if you know for certain that the answer exists.

R. J. LANG3 SYMMETRY AND THE TREE METHODIf getting the right number and length of flaps were all that was needed to make asuccessful origami model, then the tree method algorithm as described abovewould be sufficient to do all origami design. However, nothing is perfect, and thereare some limitations to the tree method - some fixable, some not. Although thepattern of nodes you get from the tree method is guaranteed to be foldable into abase with the right number and size of flaps, there is no guarantee that the foldingmethod is (a) symmetric, (b) elegant, or (c) obvious. In fact, in many cases, the treemethod, applied blindly, leads to wild, asymmetric patterns that don’t give verynice-looking models.For example, if we take a simple 5-node tree corresponding to a 4-legged creature,the most efficient skeleton puts the four nodes at the four corners of the square.This crease pattern can be folded into a base in several ways, but one of the mostelegant is the traditional Bird Base as illustrated in Figure 16. The Bird Base isnicely symmetric, both from front-to-back and from side-to-side, and thus it can beused to fold animals that are bilaterally symmetric - animals whose left and righthalves are mirror images of one another.ab2 b2 e deFigure 16: Tree, node pattern, and base for a base with four equal-length points emanating from acommon point.Now, suppose we want the same four flaps in our subject, but we want themseparated by a segment (a body?) one unit long, as illustrated in Figure 17. If youplace the nodes on a square and start figuring out path lengths, you are likely tofind the pattern shown in Figure 17 (or one very similar) as an efficient placementof nodes.The pattern in Figure 17 has a scale [scale=(1 unit of tree)/(side of square)] of1/3=0.333. However, this isn’t the most efficient node pattern possible. The mostefficient distribution of nodes is shown in Figure 18; it has a scale of(vri~-2)/5=0.348, about 5% larger. This means that the base you fold from

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 133Figure 18 would be 5% larger than the base folded from Figure 17 for the same sizesquare, and thus each flap would contain proportionately less paper.ae~c 1 d~ / bqdFigure 17: Tree and node pattern for a base with two pairs of equal-length points separated by a one-unitsegment.dfbFigure 18: The most efficient node pattern for the tree of Figure 17.Efficiency is not the same as elegance, however. Although Figure 18 is moreefficient, Figure 17 makes a better origami animal (that is, if there is an animal outthere with four appendages but no head or tail). Most animals have bilateralsymmetry, but the crease pattern in Figure 18 does not. Figure 18 would be verydifficult to fold into a symmetric-appearing base. Now, for most origami designers,starting from a bilaterally-symmetric crease pattern to fold a bilaterally-symmetricbase would come instinctively. But if we are constructing an algorithm for design,then we must build symmetry into the algorithm explicitly. Although the mostefficient crease pattern for the lizard came out symmetric, we will not always be solucky. I have found tb.at the node patterns for more complicated subjects - spiders,sea urchins, mating mosquitoes - usually are asymmetric and inelegant. Forcingsymmetry of the crease pattern is an absolute requirement for an elegant design.Of course, there are exceptions to every rule. Sometimes you are actually better offusing an asymmetric base, either because your subject is inherently asymmetric,such as a fiddler crab, or because the asymmetry can be disguised, resulting in a

larger overall base. The exception probes the rule, but by definition is less common;so we need to enforce symmetry in most cases.The symmetry shortfall is remedied by recognizing two special types of nodes -those that come in mirror-image pairs (like arms, legs, and feet) and those that liealong a symmetry plane (like the head, shoulders, hips, and tail). We will requirethat for bilaterally symmetric subjects, the nodes corresponding to mirror pairs ofappendages must lie symmetrically about a symmetry line of the square. So, for thelizard, we would require that the node corresponding to the left rear foot be themirror image of the node corresponding to the right rear foot and the nodecorresponding to the left front foot would be the mirror image of the nodecorresponding to the right front foot.There are also nodes that correspond to body parts that lie along a line of symmetryof the subject, and we will simply require that these nodes lie on top of a line ofsymmetry of the square.Note that a square has two different types of lines of mirror symmetry - one fromcorner to corner, and one from edge to edge. For the same tree structure andassignment of symmetry relationships, the two different choices of mirror symmetrywill give different node patterns and consequently, different ways of folding thesame subject. Figure 15 shows the crease pattern form a lizard when we use adiagonal line of symmetry; Figure 19 shows the crease pattern for a similar lizard,but based on a side-to-side line of symmetry. You might find it interesting to try towork out a folding sequence to transform the crease pattern in Figure 19 into abase.(a) rear front (b)foot footrearfoot 3.5 frontfoottail J -’ head tail, headrearfootfrontfootrear 3.5footfrontfoot19: Tree and node pattern for the lizard with rectangular symmetry.

MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN135Two features of this crease pattern are worthy of note: one is that two of thecorners go unused, which just goes to show you that even simple figure don’t alwaysuse the corners. The other feature is that in Figure 19, the path from the front footto the head is no longer a critical path as it was in Figure 15, which illustrates thatas you change the orientation and/or lengths of paths, you need to carefully keeptrack of which paths are critical paths and which are not, because they can change.I mentioned that the crease pattern you derive from the tree method is guaranteedto be foldable into a base with the same number and length of flaps as the originalstick figure tree. However, this property is only true for infinitely thin (zero-width)flaps made with infinitely many creases. When you cut down on the number ofcreases to some finite number - and the jump from infinity down to six or eight isconsiderable - and let the width of the flaps become nonzero, you’ll find that someflaps turn out shorter than they were ia the tree. That is, some of the paper thatmight have gone into ’length’ ends up going into ’width’ instead. This situationhappens whenever two flaps lie side-by-side, rather than one atop the other.For example, if we were making a base for a four-legged animal with two legs onthe left and two legs on the right and no body in between (like Figure 17, it’s agedanken animal), we would start with the tree and crease pattern shown in Figure16, which would lead to a base very like the Bird Base, in which the four flaps comefrom the four corners of the square. According to our tree, each flap would then beas long as half the side of the original square, and indeed, each of the four flaps of aBird Base is half as long as the side of the square from which it is folded.0.35Figure :ZO: Making two symmetric pairs of flaps from a Bird Base.Now, if we want our legs in two side-by-side pairs, the Bird Base doesn’t quite work;it has two flaps side-by-side and the other two one atop the other. We cantransform the four flaps into two pairs of mirror-symmetric flaps by spread-sinkingtwo corners of the Bird Base, as shown in Figure 20. However, when we do this, weget an extra folded edge running across the two flaps that shortens the gap betweenthem. Because the gap has been shortened, the ’free length’ of the flaps has beenreduced. Instead of being 0.5 times the side of a square, the flaps turn out to be only0.354 of the side - a loss of about 30% of their length.

1~ ~. L.ANGWe can lengthen the gap and thus lengthen the flaps with a little more folding. If Iput twice as many pleats running to the tips of the flaps, using the folding sequenceshown in Figure 21, I can increase their length to 0.42, at the expense of doublingthe number of layers, halving the width of the flaps, and adding a whole lot morefolding. You can recover more of the gap by using more and more petal folds andnarrower and narrower flaps, but only in the limit of infinitely many pleats (and aninfinite number of layers) do you recover the full length of the flaps.I0.42Figure 21: Making the gap deeper, at the expense of narrowing the flaps and introducing a lot morecreases and layers.This situation will always occur when you want two flaps to lie side-by-side, asopposed to one atop the other (note that the two flaps in each pair on one side ofthe Bird Base are separated from each another by the proper amount). The way toavoid having to do all of this back-and forth folding is for any pair of flaps that willwind up side-by-side, you add a little extra paper between them. In other words, youincrease the minimum length associated with the path between that pair of flaps.The amount you must add depends on how many back-and-forth pleats you arewilling to tolerate; typically, extending the path by about 40% will do the trick for asingle back-and-forth pleat.But, you’re not ready to place paths yet. After you have placed all the nodes andlengthened some paths, you may find that paths that correspond to importantcreases are oriented at angles close to natural symmetry lines - i.e., at multiples of30 ° or 22.5* - but aren’t precisely at those symmetry lines. For esthetic reasons -

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 137to avoid long, skinny crimps - it may be desirable to require those paths to lieexactly along the natural symmetry lines, even at the expense of a small decrease inefficiency. Thus, some of the nodes will have to be constrained to lie at certainangles with respect to one another.Also for reasons of esthetics, you might want to insure that a particular node landsin a particular place on the square. For example, the most efficient crease patternmight put a particular node in the interior of the paper, whereas you might wantthe node to land on an edge or corner so that it has fewer layers.4 A NUMERICAL IMPLEMENTATION OF THE TREEALGORITHMBy the time you have drawn and labeled the tree, enumerated all of the paths andcalculated their lengths, and noted which nodes need to lie symmetrically, andextended some of the paths, and forced some paths to run at certain angles andforced some nodes to lie on the edges of the square and so forth and so on, thingscan get pretty complicated. The biggest difficulty with applying the tree method ofdesign is keeping track of all of the different paths and the constraints associatedwith each path. If the tree has N nodes, there are N(N- 1)/2 different paths betweenpairs of nodes, each of which has a specific minimum length. The lizard, which has8 nodes, has 28 paths! However, you really only need to keep track of the criticalpaths - the paths that are at their minimum length - and there are only six ofthose in the lizard.The problem with complicated models is that as you try out different arrangementsof the nodes, some paths stop being critical paths and other paths become criticalpaths. In addition, you must also keep track of which nodes lie symmetrically withrespect to one another, and which paths you have extended the length of. This canget very difficult for a person to keep track of, and it is very easy to work out adesign, only to discover that you overlooked a critical path somewhere in the crisscrossingnetwork of paths that doesn’t meet its minimum length requirement andthrows your entire design off.Ah, but what is difficult for a person to keep track of is a snap for a computer. Thetree method of origami design that I have described above can be mathematicallyformalized and implemented as a computer algorithm. The algorithm is simple:You do this to define the shape:(1) Define a set of N nodes and the N-1 branches that connect them. Define thelengths of all of the branches. Give each node a set of coordinates on a square.(This is the setup input to the computer program.)

(2) Define a quantity called the ’scale’, which is the length of a 1-unit branch on thesquare. Assign each node a pair of coordinates on the square. (This is an initialguess at the tree structure, which can be arbitrarily chosen.)You do this to find the crease pattern:(2) Construct all of the N(N- 1)/2 possible paths between pairs of nodes and theirlengths.(3) Maximize the scale subject to the following constraints:(a) The length of each path on the square must be greater than or equal toits length measured along the tree.(b) The coordinates of all of the nodes must lie within the square.(c) Nodes that lie on a symmetry line of the tree must lie on a symmetryline of the square.(d) Nodes that are mirror-image pairs with each other on the tree must bemirror images of each other with respect to the symmetry line of thesquare.(e) Paths close to natural symmetry lines of the square must lie exactly atthe angles of those symmetry lines.To computerize this algorithm, the verbal description of the algorithm needs to beconverted to mathematical equations. I define x i and Yi as the Cartesian coordinatesof the ith node, define lij as the length (in units) of the path between nodes i andj asmeasured along the tree, define m as my scale factor, and define the coordinates ofmy square as lying between -1 and +1 on each axis. The problem becomes:Maximize rn over the variables x i andyi subject to the constraints:(a)(b)(xi -x~’) 2 + (Yi _y~)2 >_m2[2ij foralli, jx i_ -1, ~__-i for alli(c) YiCOSa - ~sin. = 0, for each node i that lies on a symmetry line at anglewith respect to the x axis.(d) (x i - xj)co,sot - (Yi - Yj )sin. = 0, (x~ + ~j. )sinot + (Yi + Y~ ) cos. = 0 fortwo nodes i and j that are symmetric about a line at angle, with respect to the xaxis.(e) (Yi - y~)cosa - (x. - :~ )sin. = 0 for a path between two nodes i andj thatlies at angle a with respect to thex axis.

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 139This is a problem of constrained optimization. In fact, it would be a fairly simpleproblem in linear programming, if it weren’t for the presence of the quadraticterms in letter (a) above. There is no algebraic solution to this system of equations,but the numerical techniques for the solution of such systems have been known foryears. In the early 1970s, several workers contributed their efforts and subsequentlytheir names to the development of a powerful technique for solving generalnonlinear constrained optimizations, known as the Powell-Hestenes-RockafellerAugmented Lagrangian Multiplier algorithm (Rockafeller, 1973).Several months ago, I wrote a computer code that takes a description of a tree,constructs the equations in (a)-(e) above, solves for a local optimum, and displaysthe resulting skeletal crease pattern. The program is called TreeMaker, and itperforms quite well. I’ve now used TreeMaker to design a number of non-trivialmodels, as well as to work out the examples presented in this series. Nowadmittedly, the lizard that I used throughout this series was a pretty simple model,and using a computer program to design it might seem like overkill. While simplemodels can be worked out entirely by hand, I have found that for complex subjects,the computer can find and evaluate efficient crease patterns much faster than I canby hand., including crease patterns that are not obvious to the unaided eye. In thenext section, I’ll illustrate the design process for a challenging model - an insectwith legs, jaws, and antennae - and end with the final folding sequence for thedesign.5 A CASE STUDY IN ORIGAMI DESIGNI’ve always had a fascination for insects as origami subjects. For many years, mostinsects were considered too difficult to be realized in any but the most stylized form(or were realized with cuts and/or multiple sheets of paper). However, with therevolutions in modern origami design, insects have become commonplace. Inorigami, a six-legged, two-winged insect is no longer a challenge; it must have jaws,antenna, multiple color-changed wings, or something else to make it rise above themerely ordinary. Insects, with their skinny, jointed, multi-pointed bodies, areperfect candidates for the tree method - and thus, for computerized design.The insect I chose to make for this article was a stag beetle, a beetle with six legs,jaws, and antennae, for a total of ten flaps. I decided to make the jaws and antennaesomewhat shorter than the legs and to have them emanate from the head. I put thelegs emanating from the thorax along with the abdomen, and make the abdomensomewhat longer than the legs, to give myself some extra paper to make a roundbody and/or pleated wings. This choice of subject also gives me an opportunity tocompare man and machine; I already invented a stag beetle several years ago (it isincluded in my upcoming book, Origami lnsects (Lang, 1994), and is shown inFigure 22. It will be interesting to see if TreeMaker can come up with a betterdesign.

140 R. J. LANGFigure 22: An origami stag beetle.The first step in a design is to take the subject we are working from and convert itinto a stick figure - the tree. Figure 23 shows a drawing of a real stag beetle andthe tree that represents it. I have some choice in how I construct the tree. I’vealready specified that I want jaws and antennae, and of course I need six legscoming from the thorax. If I were being really daring, I could put separate wings onas well, but since beetles almost always keep their wings and elytra (forewings)folded over their body, we’ll allocate a single flap to represent the wings andabdomen, and will plan on trying to suggest both features with pleats, crimps, andother detail folds.Figure 23: A real stag beetle and its equivalent stick figure, or ’tree’.Now we need to assign lengths to the branches of the tree. On the real beetle,although each member of a pair of appendages are the same length, each pair -jaws, antennae, forelegs, midlegs, hind legs - is a slightly different length. When we

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 141make an origami base, however, if all of the flaps are different lengths, then they’llall be slightly different widths where they join the body and each other. Thesedifferences translate into lots of little crimps (if you’re careful) or little bodges (ifyou’re not) to make the model lie fiat; there will also undoubtedly be lots ofmisaligned edges. A base with lots of crimps, bodges, and misaligned edges looksmessy, and a messy base makes a messy subject and a messy folding sequence. Tokeep the base and the folding sequence neat, it helps to make edges line up witheach other whenever possible. Thus, we will stipulate that flaps that areapproximately the same size on the subject should be exactly the same length on thebase. For our stag beetle, we’ll make the six legs all the same length and make thejaws and antenna the same length as each other but a little shorter than the legs.We still need to quantify the lengths of the appendages. If we are trying to make anexact copy of the beetle in the photo, we could measure each leg, jaw, and antennain the photo and assign that length to the corresponding branch of the tree, butthat is a rather brute-force approach. Besides, it overlooks something: it is notenough just to find some folding sequence to make the base; we want to find anelegant folding sequence. ’Elegance’ is a hard-to-define concept; it’s easier todescribe than to define. An elegant folding sequence is one in which all the foldsflow naturally from one step to the next and the edges and creases line up with oneanother. An elegant model is one in which the lines are simple and clean. Elegantfolding sequences arise when the creases and the base arise from naturalsymmetries of the paper. While TreeMaker will find the most efficient creasepattern for any given tree, it is up to the designer to pick the tree that best exploitsthe natural symmetries of the paper.And one of the ways in which natural symmetries appear is in certain combinationsof distances and lengths. Most origami - and in my opinion, the most elegantfolding sequences - arise from exploitation of the symmetries associated with anangle of 22.5 °, which is 1/16 of a unit circle. This angle (and multiples thereof) showup repeatedly in elegant origami bases, as illustrated in Figure 24 in the four’classic’ bases (the Kite, Fish, Bird, and Frog Bases).Along with this ubiquitous angle, there are ubiquitous distances, which are variousalgebraic combinations of integers and the number d;2. If you unfold a model basedon the symmetries of 22.5 ° and calculate the lengths of the major creases, you’ll findthat most of the distances work out to simple combinations of 1, 2, and d2, asshown in Figure 25.

Figure 24: The four classic bases -- Kite, Fish, Bird, and Frog base. The angle between any two creases isa multiple of 22.5 °.Figure 2,5: Typical distances to be found in crease patterns based on the 22.5 ° symmetry.Since in symmetric, elegant crease patterns the same distances show up over andover, if we want our crease pattern to be symmetric and elegant, it makes sense to

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 143use the distances that we know will already be present. Conversely, if we use strangevalues for distances, values that don’t correspond to 22.5 ° symmetries, it’s going tobe really hard to find a folding method that results in an elegant base. We are morelikely to find a symmetric base if we express all the dimensions in terms of’symmetric’ distances.This is really not so hard to do, because there are a lot of symmetric distances tochoose from. It turns out that for any ideal distance, there is an symmetric distancewith nearly the same value. For example, Figure 26 displays the algebraic form ofall the distances shown in Figure 25, along with their decimal value and forcomparison, a line whose length is proportional to that value. No matter whatlength a given flap or appendage is, you can find a symmetric distance that differsfrom the target by only a few percent. Figure 26 is an incomplete list of symmetricdistances associated with the 22.5 ° symmetries - there is a whole other family ofdistances associated with multiples of 15 ° - but you can extend this list to largerand smaller values simply by multiplying or dividing by factors of 2. Any numericalvalue you choose lies fairly close to one of these significant lengths.Algebraic Decimal Linear~ 1.414(1+~/2)/2 1.2071 1.000(3-~/~)/2 0.793(1/~ 0.7072-’~ 0.5861/2 0.500"~-1 0.4141-1/~- 0.293(~--1)/2 0.207Figure 26: Distanc~ found in the eightfold (22.5 °) symmetry. The first column is the algebraic formulafor a length; the second is its decimal value; and the third shows a line segment proportional to thelength.So for our origami stag beetle, we will choose flap lengths that correspond tointeresting lengths. We will define our legs to be 1 unit long each; the jaws andantenna are slightly smaller, so we’ll choose a length of 1/v~=0.707 for those. Theabdomen should be somewhat longer than the legs, to give us some extra paper forcrimps and pleats to form the wings and/or body segments; we’ll make it d’2= 1.414units long. Finally, we need a short segment between the thorax and the head, sothat the jaws and antennae don’t come from the same part of the model as the sixlegs (otherwise, we could just use the circle method). This short segment will be

144R.J. LANGd-2-1=0.414 units long. We now have completely defined the target tree for ourdesign efforts.Now that we have defined our tree, we could start enumerating paths andcalculating their lengths, but since our tree has 13 nodes, there are 78 possiblepaths between nodes that must be considered, which is a lot to try to keep track of.So I will turn now to describe the use of my TreeMaker computer program.TreeMaker has a point-and-click user interface that allows you to enter a treestructure by clicking on a square to create nodes and branches, clicking anddragging to shift them around, and double-clicking on nodes and branches tochange their properties. The tree as I entered it is shown in Figure 27. TreeMakerallows you to name nodes with the part of the subject they correspond to anddisplays the lengths of branches, so I’ve shown those numbers and names for clarity.left jawright jawleftfightantennafightleftbackleg1.4 t4fightbackleg, abdomenFigure 27: The tree I~or a stag beetle, superimposed over a square. The numbers are the desired lengthsof the branches; the names identify which part of the subject corresponds to each node.To give you an idea of the complexity of this model that has 13 nodes and 78 paths,I’ve shown all 78 paths in Figure 28, each represented by a straight line. Every linecorresponds to a constraint on the distance between two nodes. Now, if you or Iwere working this design out by hand, we would figure out which paths are theimportant ones and which could be ignored, but the computer is not so clever, andneeds to consider them all.

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 145Figure 28: Every possible path between two nodes is represented by a line. There are 78 total pathsbetween the 13 nodes.Once we have set up nodes and branches, the computer calculates all possible pathsand the minimum length of each path. To find a possible arrangement of nodes, thecomputer needs to optimize (find the maximum value of) the scale of the tree whileinsuring that all paths meet their minimum distance constraints. At this point, wecould just tell the computer to go through its optimization process. TreeMakerarrives at the following arrangement of nodes:leftaw fight jaw right front legright5 middlelegleftfrontleg/ / /abdomen~8Figure 29: A node pattern that satisfies all of the path constraints for the tree. The circles identify theterminal nodes of the tree. The scale is 0.2012.

146This first go-around was certainly less than successful. One back leg is a middleflap; the other is an edge flap. One jaw is a corner flap; the other is an edge flap.The head is twisted way around to the right. Folding this crease pattern into a basewould be a nightmare. About the only positive thing I can say is that the scale - thelength of one unit compared to the size of the square - is a very respectable 0.2012,meaning that the length of a leg is about 1/5 of the length of the side of the square.Nevertheless, there is no symmetry, no rhyme or reason, and most importantly, noeasy or obvious way to fold this crease pattern into a base! So the first foray intocomputerized design is without doubt, a failure.Well, maybe not quite a complete failure. We haven’t yet imposed any symmetryrequirements, and lack of symmetry is the main problem with the node pattern inFigure 29. We need the legs, jaws and antennae to be mirror images of each otherwith respect to a symmetry line of the square - and they aren’t in Figure 29. Wealso need the head, thorax, and abdomen to lie directly on top of the symmetry line,since they lie on the symmetry line of the subject. Since the shape in Figure 28 isoriented roughly along the diagonal, let’s choose the diagonal of the square to beour line of symmetry. TreeMaker allows you to select either a diagonal or arectangular line of mirror symmetry for the square, to constrain individual nodes tolie on the symmetry line and to force pairs of nodes to be mirror images of eachother about the symmetry line. So, when we establish these symmetry requirementsand restart the optimization, we arrive at the node pattern shown in Figure 30. Thispattern has a scale of 0.1853. The scaled has decreased by about 8%, which meansthat the base folded from this crease pattern will be about 8% smaller than the basefolded from the crease pattern in Figure 28. But the pattern in Figure 30 is morelikely to be foldable into a symmetric base and thus a symmetric model, and so thesmall decrease in scale is an acceptable price to pay.13 54 2Figure 30: Optimized node pattern for the stag beetle with mirror symmet~ invoked. The scale is 0.1853.

MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN147There is one other chore before us; if we wish our finished base to be flattenedfrom top to bottom (as is the beetle in Figure 22), then the legs come in side-bysidepairs. To avoid having to make infinitely many pleats between them, we needto allocate some extra paper by extending the paths between mirror pairs of legsand iaws. To keep the back-and-forth folding to a minimum, I’d like to use a singleback-and-forth pleat between the legs, such as is accomplished in the Frog Base bypetal-folding an edge. For this petal fold, the distance between the two flaps mustbe ::~ times as long as the minimum path, so between side-by-side pairs of legs, I’llneed to extend each path by a factor vr2. Just as you could edit individual nodes,TreeMaker lets you edit individual paths and to increase the minimum separationbetween node pairs. We do this for each pair of mirror flaps, increasing theseparation between the jaws from 1.414 to 2.000, between the antennae from 1.414to 2.000, and between pairs of legs from 2.000 to 2.828. After bumping up thesepaths, I re-optimize. The scale has decreased from 0.1853 to 0.1775, which is stillnot a large decrease, but the resulting pattern, shown in Figure 31 is now closer tosomething that can conceivably be folded into an elegant base.Figure 31: Opttmized node pattern for the stage beetle with mirror symmetry and path extensionsbetween side-by-side flaps. The scale is 0.1775.But wait - there’s more. The crease pattern shown in Figure 31 can in principle befolded into a beetle, but I doubt that the folding sequence is very clean or elegant"because the major crease lines are running at irregular angles. Not only do we wantour distances to be symmetric; we want the angles of major creases also to lie atmultiples of 22.5 ° whenever possible. Thus, we need to set some more constraintsthat force paths to lie at multiples of 22.5 ° . We can’t force all the paths to the sameangles, though; just the ones that correspond to major creases of the base.

Forcing paths to lie at particular angles is another capability of TreeMaker. For anygiven path, there are 8 possible multiples of 22.5 ° to set the angle to, but you mightguess that you’ll perturb the crease pattern the least by looking for paths betweenadjacent nodes that are nearly at multiples of 22.5 ° and setting them to the nearestmultiple. If I do this for the paths marked in Figure 31 and again re-optimize, I findthe following crease pattern in Figure 32, which is more nicely symmetric, andwhich has a scale of 0.1726.Figure 32: Optimized node pattern for the stag beetle with mirror symmetry, path extensions, andangular constraints. The scale is 0.1726.It’s interesting; each time we add some more constraints, the scale, and thus thesize of the model, gets reduced. That is, we are trading off folding efficiency forfolding elegance. This is an esthetic judgment, and requires the active participationof the designer. Computer programs like TreeMaker can simplify aspects oforigami design, but they are no substitute for good design sense.In fact, it might even be considered a bit of a stretch to .say that TreeMaker is’designing’ the base. It isn’t so much creating a useful arrangement of nodes as it iseliminating the enormous number of useless ones. For 13 nodes, there are 27degrees of freedom in the placement of the nodes on the square (two coordinatesfor each node plus the scale~. A given configuration of nodes and scale can bedescribed by a single point in a 27-dimensional space. The various constraints onpath distances and angles can be written as equations that subdivide the 27-dimensional space into spheroidal regions of feasibility (for inequalities) orspheroidal surfaces (for equalities). The combination of all of the equations definean incredibly convoluted hyperdimensional blob of feasible space over which thescale - the merit function - varies in value, and all we are doing is looking for

MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 149those flaps with the maximum value of the scale. The convolution of the surfaceimplies that there are numerous local minima on the surface and there is noguaranteed routine for finding a global minimum. Just because we found the bestconfiguration of nodes from a given starting point, a different starting point mightgive us an even more efficient arrangement of nodes!So, one new starting point we might try is to move the abdomen into the center ofthe paper, rather than using a corner for the abdomen. This would have the effectof moving the legs closer to the corner opposite the head, but exactly what else isnot clear until we re-optimize TreeMaker. This starting point gives the creasepattern shown in Figure 33, which has a slightly larger scale of 0.1745, and wouldgive a barely larger base than the pattern of Figure 22. Significantly, the creasepattern to transform Figure 33 into a base would in all likelihood be entirelydifferent from the pattern for Figure 22! Thus, simply by starting from a differentinitial configuration, you can discover entirely new ways of folding existing subjectssimply by moving the initial distribution of nodes around.Flgur~ 33: Node pattern with the abdomen in the center. The scale is 0.1745.Figure 33 has one other nice feature compared to Figure 32; all of the legs comeout as edge flaps in Figure 33, whereas two of the legs are middle flaps in Figure 32.For an insect, we want the legs to be as thin as possible, and the extra thickness in amiddle flap might be something to be avoided. (Then again, it might not; I know alot of good insect designs that use middle flaps for legs.) If we start with Figure 32,force all of the legs to lie on the boundary of the square and set angles that are near45 ° to exactly 45 °, we get the crease pattern shown in Figure 34, which has a scale of0.1715, still smaller than any of the preceding patterns, but acceptable.

1~o ~ ]. L4NGFigure 34: Node pattern for the stag beetle with all legs constrained to lie on the edge of the square. Thescale is 0.1715.We might also wish the jaws to be edge flaps and allow the legs to be middle flaps.This constraint, plus a few angular constraints gives the very symmetric creasepattern shown in Figure 35. Not only are all the major creases at 45 °, but the centerof the paper is an intersection of two important creases. The scale has decreasedfurther, to 0.167, almost precisely 1/6 of the side of the square.Figure 35: (Left) Node pattern for the stag beetle with all jaws constrained to lie on the edge and withthe antenna and two legs on each side constrained to a 45* line. The scale is 0.1665. (Right) Resultingstag beetle.

MATHEMATICAL ALGORITHMS FOR OR1GAMI DESIGN 151In fact, to my taste, Figure 35 is too symmetric. For years, origami designs weremade from bases such as the Bird and Frog base that are four-fold symmetric aboutthe center of the square, which always resulted in a single thick flap somewhere inthe model - a waste of paper, and a waste of symmetry. Only in rare circumstancedoes a four-fold symmetric base make a good match to a two-fold symmetricsubject. The crease pattern shown in Figure 35 can be folded into the rather stubbystag beetle shown in Figure 35 - you might wish to try to work out a foldingsequence - but the sequence I found is rather predictable, and is four-foldsymmetric for much of the sequence.Of course, we could try something completely different by trying the alternate lineof symmetry - that is, orienting the model along an axis parallel to the side of thesquare, rather than along the diagonal. Setting this line as our symmetry line,optimizing, and forcing a few major crease lines to verticals results in the followingintriguing crease pattern, which clearly has a 30 ° symmetry built in. It also has ascale of 0.1808, the largest of any of the crease patterns we have examined so far.Figure 36: Node pattern for the stag beetle with rectangular symmetiy and path angle constraints. Thescale is 0.1808.Now, all of these crease patterns can be folded into a stag beetle, and the mostinteresting folding sequence for each is going to be very different from thesequence for any of the other crease patterns. So right here we have the beginningsof five or six new models. Of course, the question of how you develop a foldingsequence that takes a crease pattern to a base is no small matter, and is in factworthy of an entire set of articles in its own right. Setting aside the issue of how youfold up a crease pattern into the base, though, you see the boundless possibilitiesthat you can avail yourself up even from a single tree structure! I’ve developed stag

R. J. I.~4~Gbeetles from about half of the crease patterns I’ve shown here. I’ll close this articlewith my favorite, which is derived from Figure 34, and is illustrated in Figure 37.Figure 37: Stag beetles, according to (left) human, (right) computer.You might wish to try to find your own folding sequence for one of these creasepatterns. Better yet - change some dimensions of the tree and start over yourself!Give the beetle extra-long jaws this time, or get rid of the antennae, or add a pair ofopen wings. Whatever you do, you are bound to find new and undiscovered modelslurking in the paper. Whether you use TreeMaker, write your own computerprogram, or work out your design by hand, mathematical origami design is apowerful technique that can lead you to new vistas of origami design.REFERENCES:Engel, P. (1989) Folding the Universe, New York: Random House.Gardner, M. (1992) FractalMusic, Hypercards, andMore, New York: Freeman.Kasahara, IC (1988) Origami Omnibus, New York: Japan~ Publications.Lang, R. J. 1994) Origami Insects, New York: Dover Publications, (to be published.)Montroll, J. (1985) Animal Origami for the Enthusiast, New York: Dover.Rockafeiler, R. T. (1973) A dual approach to solving nonlinear programming problems byunconstrained optiraization, MathematicalProgramming, 5, 354-373.

Symmetry: Culture and ScienceVot 5, No. 2, 153-165, 1994MATHEMATICAL REMARKS ABOUTORIGAMI BASESJacques JustinLITP INSTITUT BLAISE PASCAL4 place Jussieu, 75252 Paris CEDEX 05, France*FOREWORDThis paper is based, with only minor modifications, adjunctions and suppressions,on an handwritten manuscript of 1982 which was sent at this time to some foldersinterested in mathematics. This led to correspondence with at least two of them:professors Kodi Husimi and James Sakoda. The first one has seemingly considereda particular case of what I call a ’perfect bird base’ in his book, in Japanese, Origamino Kikagaku (The Geometry of Origami, Tokyo: Nippon Hyoronsha, 1979). Thesecond one has made use of perfect bird bases in his magazine Mac Origami and inhis book Origami Flowers Arrangement (published by himself, 1992) and has deviseda good approximate method to find the ’origin’ of the perfect bird base.1 INTRODUCTIONWhen we fold the traditional bird base (Orizuru K/so, in Japanese) we find that ithas agreeable properties, for instance the flaps are easy to move. But try with arectangle: the result is less satisfactory. We shall examine the possibility of folding’good’ bird bases with polygons of various shapes. Afterwards we shall try a similarapproach for the frog and the windmill bases.2 DEFINITIONSThe polygons to be studied need not be convex. Let Ar42...A n (Figure 1) be apolygon, and O a point inside it such that the triangles AIOA 2 and so on do not* Mailing address: J. Justin, 19 rue de Bagneux, 92330 Sceaux, France.

154J. JUSTINoverlap each other (the polygon is said to be star-shaped from O; if it is concave,some points O do not work). We call preliminary base the result of bringing OA1,OA 2 .... , OA n together by mountain folds, and origin of the base the point O (Figure2). Of course, when flattening.~_he model_valley folds must appear, say OMI, OM2,... and we have (Figure 1) AIOM 1 = M~IO~ 2 and so on. In the preliminary base, Oand the verticesA~,A2 .... lie on a line, and the flaps like OA2MIA~O can be bookfoldedwith that line as a hinge. Now, let us make a reverse fold on each flap, withthe condition that the creases pass through the corresponding vertices, for instancethrough A 1 and A 2 for the flap OA2MIAIO. We shall call the result a bird base(Figure 3). See Figure 4 for the creases on the unfolded paper, and remark that thesides ,4./1i+ ~ of the polygon have been folded at points E i that differ, generally,from the Mi’s (for ease of notation we consider that ,4n+ 1 = ,41 and so on). Thebird base retains the above-mentioned property of the preliminary base, that O andthe Ai’s lie on the same line A, which is an axis of rotation for the flaps likeOPv4~A20. Now, let us say:Figures 1-3Definition: A bird base (Figure 5) is perfect if:(a) the triangular flap PIA2P2 can be bookfolded with P1P2 as a hinge, and so on forthe other flaps.(b) when P1A2P 2 has been folded ’downward; A 2 is in a new position A’ which lies onA and so on.(c) the flaps like P1A2P 2 consist everywhere of, exact(V, two layers of paper, that is thetriangles A2P1E1 and A2P2E2 of Figure 4 are adjacent without gap or overlap.

MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 155Of course, such properties are among the good ones of the standard bird base,some others being let aside because they are too specific of the square.\ /\\ .\ ’!F~ures5~

156 Z JUSTIN3 PERFECT BIRD BASESFigure 6 shows the hidden reverse folds. By property (c), the points h2, E l, E 2 lieon the same line. Let H 2 be its intersection with PIP2 . By property (a), E 1 and E 2m~t be on...the side of P1P2 which do~ not con_tain A 2. So we have:P1E1A 2

MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 157Theorem: Given a polygon A1A2...An,(a) there exists a perfect bird base with origin 0 if and only if there exist positivenumbers r and e 1 ..... e n such thate i+ei+l=AiAi+l for 1

158 .I. ,JUSTINSo there is at most one perfect bird base for any polygon. We propose, if it exists, tocall its origin the ’point of Loiseau’ of the polygon (L’oiseau: French words for: thebird).Figures 8-9Corollary 1: For any triangle there is exactty one perfect bird base. Its origin can beconstructed with ruler and compasses.Proof: In Figure 9 we have e 1 + e 2 = A1A2, and so on. Then E l, E 2, E 3 are thepoints where the circle inscribed in AIA2A 3 touches the sides. Describe the circle(hl; el) , that is the circle with center A 1 and radius el, and in the same waycircles (A2; e2) , (A3; e3). Then by the relations e i + r = OA i, 0 must be the center ofa circle that touches the three preceding ones. Therefore, O can be obtained byelementary geometry. Here is one possible construction (deduced from inversionswith centers El).Let G 1 on the line A1A 2 be such that (A1; A2; El; G1) is harmonic. Describe thecircle ((71; G1E1). It meets the circle (A3; e3) at H 3 within A1A2A 3. Similarly,construct HI, H 2. Then O is the point of intersection of the lines A1H1, A2H2,A~/-/3.When the triangle is isosceles, AtA 2 = A1A 3 (Figure 10), it is easier to constructthe point of Loiseau. Describe the circle with centerA 1 and radius IA1A 2 - A2A 3 I.It meets the medianA~E 2 at two points, X and Y. Trace the perpendicular bisectoreither ofA2XifA1A2 > A2A3, or of AYif not. It meetsA1E 2 at O.Corollary 2: The quadrangle AIA2AsA 4 has a perfect bird base if and only ifA1A 2 + A3A 4 = A2A ~ + A4A 1. Then the origin 0 lies at the intersection of two

MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 159branches of hyperbolas. In the case where AIA 3 is an axis of symmetry, 0 is the pointwhere the circle inscribed in AIA2A 3 touches A1A3.A,Figures 10-12Proof: If O is the origin of a perfect bird base (Figure 11) we haveAzA 1 -A2A 3 =OA I -OA3 = A4A 1 -A4A 3. This impliesAIA 2 +A3A 4 =A2A 3 +A4A 1.Reciprocally, if that condition holds, then A 2 and A 4 lie on the same branch ofsome hyperbola with focuses A1, A 3. Similarly A 1 and A 3 lie on a branch ofhyperbola with focuses A 2 and A 4. These two curves intersect at some point Owithin the quadrangle. As O satisfies the conditions (3) of the theorem, it is theorigin of a perfect bird base.In the case where hlh 3 is an axis of symmetry (Figure 12), the hyperbola passingthroughA 1 andA 3 becomes the lineA1A3, and then, OA 1 - OA 3 = A2A 1 - A2A 3shows that O is the point of contact ofA~A 3 with the circle inscribed inAIA2,4 3.Remark: If the quadrangle is convex the condition on the sides is equivalent to thecondition that it is circumscribed to some circle.Corollary 3: (a) If the po~gon A1A2...A n, with n even, has a perfect bird base, wemust have:AIA 2 - A2A3 + A3A 4 -...+An_ 1 A n - A n A1 =0"(4)(b) If the po~ygon AIA2...An, with n odd, has a perfect bird base with origin O, then 0can be constructed with rules and compasses.(c) Given a point 0 it is easy to construct irregular polygons with n sides having aperfect b~rd base with origin O.

160 1. JUSTINProof: (a) If n is even, the system of equations e i + ei+ 1 = AiAi+ 1 (1 _< i -< n) hassolutions only if (4) is true.(b) If n is odd, we can find the Ei’s by solving the preceding system for the ei’s.After, the point O can be constructed almost as we did for the triangle.(c) Describe a circle C with center O (Figure 13), then a circle C 1 externally tangentto C, then a circle C 2 tangent to C and C 1, then a circle C 3 tangent to C and C2,always externally, and so on; at last C n tangent to C, Cn_ 1 and C1. Join together thecenters A~, A 2 .... , A n of the circles. We get a polygon having a perfect bird basewith center O.Figure 134 FROG BASESLetA1A2...A n be a polygon, and B i a point on the side A.tAi+ 1 for 1 _< i _< n. Weshall say that a bird base for the polygon AIB1A2..-4nBn is a frog base for thepolygon A1A2...4 n. A frog base is said perfect if the corresponding bird base isperfect. We have the followingTheorem: If 0 is the origin of a perfect frog base for a polygon A1A2...An, then B 1 isthe point where the circle inscribed in AIOA 2 touches A1A:~ and so on.Proof: (Figure 14). The relation (3) applied to the associated perfect bird basegives Br41 - BaA 2 = OAx - 0.4 2 , so that the circle inscribed in A~ OA z touchesA IA 2 at B~.

MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 161FiguresCorollary: For a rhombus, the center of symmetry is the origin of a perfect frog base.Proof: (Figure 15). With O at the center, choose the Bi’s as above (for instancefold the perfect bird base of the rhombus, then the Bi’s are the Ei’s of Figure 7).Then by symmetry OB 1 - OB 2 = A2B a -A 2 B 2 (= 0) so that O is the origin of aperfect bird base for the octagon AIBIA2..~B 4 , that is a perfect frog base for therhombus.5 A PROBLEMFigures 16-17

162Y. JUSTINThe following property was found when trying to make irregular frog bases.Problem: Let OAB be a triangle and M be an arbitrary point onAB. Let I and J bethe incenters of the trianglesAOM and BOM. Show that the circle with diameter IJmeets AB at M (of course) and at a second point, P, which does not depend of M(Figure 16).Hint for a geometrical proof. Project I and J orthogonally on AB at H and K,computeAH andAK and remark that HK and PM have the same middle.Origami solution of the problem. Mountain fold along OI an~d, OJ, then petal foldalong IJ, which gives Figure 17. As PB and PA are adjacent, JPI = ~r/2. So P lies onthe circle with diameter IJ. But PA + PB = AB, and also PA - PB = OA - OB, sothe point P of AB does not depend of M.6 WINDMILL BASESFigures 18-19In current terminology of folders the windmill base made from a square is eitherthe windmill itself, or the quadruple preliminary base obtained by squashing thefour points of the windmill. Here we use the first meaning. Let .A1A2...A n be aconvex polygon (Figure 18), O a point in its interior and M i a point of the side

MA THEMA TICAL REMARKS ABOUT ORIGAMI BASES163A.tAi+I, for 1 _< i _< n. Let us valley fold the polygon in such a way that all the Mi’scome to O. If we pinch the comers and fold them fiat (Figure 19) we shall call theresult a windmill base defined by O and the Mi’s. The main property is that the flapcontainingA 1 can be bookfolded with OD 1 as a hinge, and so on. Figure 18 showsthat D1D 2 is the perpendicular bisector of OM~. The creases D2M ~ and D2M 2correspond to the hinge of the flap A2, as they coincide with D20 when the base isfolded. Last, when we have flattened the flap A2, an extracrease has appeared, sayD2X2, which is the perpendicular bisector ofM1M 2.Now we shall say that the windmill base is perfect if, for 1 _< i _< n, the line D-~" icoincides with the line D.rA i. This amounts to saying that the flap AiXiD~O is atriangle, or consists everywhere of two layers. We have the followingTheorem: Given a convex po~ygon A1A2...An, the points 0 inside and M i on the sidesAiAi+l,for I

164Y. YUS TINsides ofDiD2, soA20 > A2M 1 = e2, and then conditions (2) are satisfied. Last, thepolygon DID2..J) n must be convex with O inside it. So O and D i lie on the sameside of D1D 2 for i # 1, i # 2. So M~/~ > D i O. But D i is the center of the circle(Mi.lMiO). So M 1 lies outside this circle. But if we remark that O and M 1 lie onthe same side of Mi.IM i (opposite to Ai) this is equivalent to the fact that O liesinside the circle (Mi_IMiM1). So conditions (3) are satisfied.(b) Reciprocally, if the conditions (1), (2) and (3) are satisfied, then, by reversingthe arguments above, we see that the perpendicular bisectors of OM 1 and OM 2intersect at some D 2 on the inner bisector of A1 ~"~2A3, and that the polygonD1D2..~D n is convex and contains O. So, by valley folding along D1Dz..J)n weobtain a perfect windmill base.Corollary: lf a convex po~ygon is circumscribed to a circle, then it has infinitely manyperfect windmill bases.(Proof left to the reader).Remarks: (a) the conditions (1) can be satisfied if and only if the lengths A.~li+ 1satisfy some conditions easy to state (solve e i + ei+ 1 = Aiai+ 1 and write that theei’s are positive). It is also equivalent to say that the polygon A1A2...4 n can bedeformed by modifying its angles but not its sides, so that it become circumscribedto some circle.Coordinates: A1 (0, 14), A2 (-16, 8)A3 (-8, 0), A4 (8, 0), A5 (16, 8), O (12, 7)Figure(b) when conditions (1) are satisfied, the region corresponding to conditions (2),that is the intersection of the inside ofAy42...4 n with the outside of all the circles

MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 165(hi; ei) is not empty. Though rather intuitive, this fact is difficult to prove. Maybe itcould be proved by a continuous deformation of the polygon in the way said justabove. In any case, it follows easily from the proposition hereafter which can beproved by methods of Graph Theory.Proposition: Consider a convex polygon AIA2...A n and real numbers a i associatedwith its vertices, such that AiA~+ 1 >_ a i + ai+ 1 for I < i _ a i + ajforall i # j.(c) However it is not always possible to satisfy both (2) and (3), or even (3) alone.Try for instance to fold a windmill base defined by the Mi’s and O of Figure 21.You will obtain interesting results, but not exactly as wanted.7 CONCLUSIONMany geometrical problems arise when one tries to generalize the traditionalorigami bases. We have studied here some natural generalizations. Bases withirregular polygons or with polygons with many sides are probably of little use inOrigami. However some may be amusing. For instance fold a kite base from asquare, then fold the kite into a perfect bird base, then pull out the two corners ofthe square that had been folded first. With this ’kited bird base’ you can fold aflapping bird with long neck, short tail, medium-sized wings and small legs.

Symmetry: Culture and ScienceVot 5, No. 2, 167-177, 1994EVOLUTION OF ORIGAMI ORGANISMSJun MaekawaAddress: 1-7-5-103 Higashi-Izumi, Komae, Tokyo 201, JapanE-mail: maekawa@nro.nao.ac.jpAbstract: This article (based my book Viva! Origami (Maekawa, 1983) will showorigami design as the tiling work. First, the rules of origami and some of the notions oforigami taxonomy will be considered. Furthermore, traditional models will be analyzed;the meaning of ’basic form’ will be discussed; and an introduction to my originaldesigns will be explained.1 ORIGIN OF THE SYSTEMATIC STRUCTUREIn origami, an organism is regarded as a transformed flexible sheet. This sheet isable to be split and fused keeping the distinction between its surface and its reverseside (Kasahara and Takahama). Origami folding begins with a sheet of paper,which is transformed by folds. Of course, there are exceptions in that sometimesseveral sheets of paper are used, and, on occasions, scissors. However, for themoment, we shall consider the strict and traditional rules of origami. Most ofpaper-folders have implicit rules. The following are "the five commandments" byHusimi, arranged by Kasahara (1989). These are typical rules of origami.1. Start with a sheet of square paper.2. Cutting and gluing are forbidden.3. Fold model fiat just before its completion.4. Straight folding is only permitted.5. When constructing a model, bear in mind the physical quali~ies of paper.I think Husimi made his rules from the character of traditional origami. Most classicalorigami models (in documents) violate all of the above rules. Models thatadhere to the rules have been handed down by tradition, and are not described inany extant document. These models are the result of historical selection over a1000 years. (I describe ’over a 1000 years’, but there is no established view whenorigami had its beginnings. This is my guess by the introduction of paper.) It can becompared to natural selection. In this analogy, the most important question to beconsidered is: What is selection pressure?

168 I.. MAEKAWAFish (tradition)Crane (tradition)Frog (tradition)~ Uttle bird (tradition) /Wind mill (tradition)Ff~ure tFigure 1 shows some traditional models and their creases. Their primary characterreveals a sense of ’easiness’. An easy model stands the strongest chance of survival.However, it is very difficult to define the meaning of ’easiness’. It has, at least, twomeanings:1. Easy to make.2. Easy to learn.The former concept relates to technique; the latter to process.Another keyword in traditional origami is ’natural’. Bearing these two key words inmind, we can now explore the rules of origami in more detail.Most of the rules (one sheet, no cutting, no gluing, and fiat folding) encapsulated inthe word ’fold’. These are related to the physical qualities of the paper. We transforma sheet of paper by rolling up, crumpling, folding along curve and foldingalong straight line. Why fiat folding is important in the rule of origami? Miura’sstudies will give us a hint to solve this problem. He has shown us the peculiar fiatfolding as a solution of the strength of materials (Miura, 1989). We can find bits ofthis peculiar fiat folding in the traditional models.

EVOLUTION OF ORIGAMI ORGANISMS 169As for the characters of traditional origami, we shouldn’t ignore the distinctionbetween the surface of paper and the reverse side of it, though it isn’t included inHusimi’s rules. On most of traditional models, surface and reverse side becomeouter inner sides as a result that edges of paper are fitted to another edges. It isnatural and easy process of origami. In biological terms, it correspond to blastula. Itis a wonderful coincidence for me. However, ’origami sheet biology’ is very simple.At present, origami creatures are belonging to a kind of Coelentelata like a jellyfish.I may have overlooked the rules of origami. I think the rules of origami and its systematicstructures are originate from both physical qualities of paper and ’easiness’as selection pressure. These systematic structures lead me to the taxonomy, and my’tiling method’ is based on the taxonomy.2 THE ORGANIZATION OF TAXONOMYThere have been some attempts at producing a taxonomy for origami. There arefour viewpoints as follows.1. Process.2. Symmetry of the complete model.3. Technique.4. Structure.The most famous study on origami taxonomy is that known as ’origami tree’. Thepioneer of this study was probably Ohashi (1977). The origami tree was a systematicmethod used in learning how to construct origami models. Its main concept isthe notion of ’basic form’. Basic forms are simple and geometrical forms which canbe applied to many different kinds of origami designs. In fact, the crease patterns inFigure 1 aren’t actual complete figures, but basic forms of them. There are about 10basic forms. Returning to the biological analogy, the origami tree is a kind of agenealogical tree, for example, the frog’s legs and the Iris’s petals are homologous.It is an interesting viewpoint, but it has a rigid aspect because of its adherence tothe folding processes.The symmetrical analysis of complete models is the second type of taxonomy. Thisview is mentioned by Kihara in his book (Kihara, 1979). He classifies traditionalmodels by point group (a term of crystallography). A new type of work in this fieldis that by Kawasaki who wasn’t aware of Kihara’s book. Kawasaki’s work is called’isoarea folding’ which is a design of 4-times rotatory inversional symmetry(Kasahara and Takahama).The third viewpoint considers the origami design to be an assembly of techniques.This view classifies folding techniques. For example, tsumami ori (pinch fold), nejiri

170 £ MAEKAWAor/(twist fold), sizume or/(push fold) ... At present, this study is only an idea, butthere are many designs which can be explained by their peculiar techniques.Taxonomy by structure will be described in Sections 3 to 5.3 ORDERLINESS OF TRADITIONAL MODELSIn Figure 1, there is a systematic pattern in the fish-crane-frog lines. On the otherhand, the pattern in the windmill is different from the others. The minimum angleof it is 45 degrees. This pattern and its extension have been given the fitting nameof "box pleating" (Lang, 1988; Lang and Weiss, 1990). It has great potential inmaking new structures. Figure 2 is such an example. However, this type of folding isnot always structural because relations between each crease are weak. In short, theyare agglutinative.Rhinoceros (Maekawa)A fundamental shape of the fish-crane-frog system is the right-angle isoscale triaLglewhich is half of the fish base (Fig. 3). I call it the crane unit. The crane base isassembled by 4 units, and the frog base by 8 units. The 8-unit form is not the limitof this system; the other form can be arranged by the same number of units. Figure4 shows these examples. The flower dish is assembled by the same number of unitsas the frog units, whereas the bug is assembled by 16 units.This system has been known for a long time. Uchiyama shows the spider base (4frog bases = 32 crane units) in his book (Uchiyama, 1979), and it can be tracedback more. Figure 5 shows origami designs with cutting from the Edo period (1600-1860’s). The upper figure was introduced in Kayaragusa (Adachi, 1845). In aslightly different sense, the lower figure is an example of renkaku (chained cranes)in Senbazuru Orikata (Rokoan, 1797). I have designed other forms using the craneunit assembly. Figure 6 shows an example.

EVOLUTION OF ORIGAMI ORGANISMS 171Flower dish (tradition)BugFigure 4Crab (tradition)Seigaiha (blue wave)(tradition)

172 ~ MAEKAWACrocodile (Maekawa)The crane unit is a right-angle isoscale triangle. We can extend this unit to arbitrarytriangles. Husimi first explained its geometrical meaning in his book (Husimi andHusimi, 1984).Husimi’s innerpoint theorem states: Any triangle is folded into a form in which allsides are gathered in a straight line. This theorem has been extended to include anyquadrilaterals with inscribed circles, and has been extended to quite arbitraryquadrilaterals (Fig. 7).The inner point theorem (Husimi) The Husimi-Maekawa FoldingThe quadrilateral molecule(Meguro)We can make crooked cranes using Husimi’s theorem. Since I know the arbitrarytriangle unit, I have a tendency to use the right-angle isoscale triangle unit. Usingthe peculiar triangle, we can design new models easily.4 ORIENTATION OF THE ELEMENTARY UNITSThe ’crane unit’ is not an elementary unit (like an atom) of the crane. It is subdividedhere into two types of triangles (Fig. 8). They are elementary units of cranetype origami; the little bird base is also assembled by those units.

EVOLUTION OF ORIGAMI ORGANISMSWe can regard the basicforms as the results ofconditional tiling workusing the elementaryunits. The followingtwo theorems correspondto conditions ofthe tiling work, thoughthey aren’t sufficientconditions (Kawasaki,1989).22.567.5~,~ ~The elementary unitsFigure 8The Maekawa theorem states: At any node of a flat folding, except of those on theedge of the plane, the difference of the number of mountain creases and the numberof valley creases is equal to two (Maekawa, 1983).The Kawasaki theorem states: At any node of a flat folding, except of those on theedge of the plane, the alternate total of angles between the creases is equal to 180degrees (Kasahara, 1989; Kawasald, 1989).Unit (type I) Unit (type 2) Unlt (type3)In designing the new model, I do not use the elementary units themselves, but secondaryunits which are assembled by several elementary units. In short, this tilingwork is hierarchical. Figure 9 shows examples of these secondary units. Of course,the crane unit belongs to the group of the secondary units. Recently, a wellarranged work using those units was made by Meguro (1991-92). He emphasizesthe ’univalency" of the secondary units. In origami, ’univalency’ means the charactershown in the Husimi’s innerpoint theorem, that is to say, that all sides of the figureare gathered in a straight lihe by flat folding. ’Univalency’ is a concept to increaseefficiency of the tiling work.

174 L MAEKAWADevil (Maekawa)Deer (Maekawa)Uzard (Maekawa)Beast (Maekawa)1.S-------) 1AdaptationNew patternAlready Known patternRotational viewAdaptationAdditionAlready known patternSlide view

5 THE ORIGINAL DESIGNSEVOLUTION OF ORIGAMI ORGANISMS 175I have created new designs using the tiling work. Figure 10 shows some of thesedesigns. In designing them, I have used various methods - among them:’adaptation’, ’addition’, ’rotational view’ and ’slide view’. These are illustrated inFigure 11. ’Adaptation’ is a re-introduction of arbitrary triangle folding. ’Addition’and others are extended methods of already known forms.2 times self similar figuresFigureIt is interesting that two peculiar figures appear in those forms. One is a rectanglewhich is square root 2 wide per other side, and the other is a right-angle isoscaletriangle. They are figures that can be divided into two self-similar figures as in Figure12. The crane system is a good example of this pattern. I have achieved someinteresting results by starting with a sheet of this peculiar-ratio rectangle paperinstead of a sheet of square paper (Fig. 13). (This ratio is very common.) Squareroot 2 is the magic number of origami, because we find this ratio everywhere.Giraffe (Maekawa)Fish (Maekawa)Figure 13This magic number is significant under the conditions that folding angles arerestricted within multiples of 22.5 degrees. If we use other angles, we will find othermagic numbers or will not see it.

176 1.. MAEKAWAUnit anqleTilin.qTriancjles(l~)x90degrees45~ 4575(l~)x90degrees18’,l/4)x90degrees(lf3)x90degreesTable ITable 1 shows an extension of the elementaq¢ units. The hexasection of the rightangle is productive. I have tiled the hexasectional units on a square field as in Figure14. The trisection has a possibility on regular hexagons and rectangles: the ratiobetween the width and the length is an integer division or a multiple of the squareroot of 3. The pentasection can be used on regular pentagons and the Penrose tiles,but I have not accomplished presentable designs to date.

EVOLUTION OF ORIGAMI ORGANISMS 177REFERENCESAdachi, K. (1845) Kayaragusa [The Collection, in Japanese],(not published), A part of copy in: Kasahara, K., Origami5, Tokyo: Yuki-Shobo, 1976.Husimi, K. and Husimi, M. (1984) Origami no Kikagaku,[Geometry of Origami, in Japanese], enlarged ed.,Tokyo: Nihon-Hyoron-Sha.Kasahara, K. (1989) Origami Shin-sekai [The New World ofOrigami, in Japanese], Tokyo: Sanrio.Kasahara, K. and Takahama, T. (1987) Origami for theConnoisseur, Tokyo: Chuo-Kouron-Sha.Kawasaki, T. (1989) On relation between mountain-creasesand valley-creases of a flat origami, In : Huzita, H., ed.,Origami Science and Technology, Proceedings of theInternational Meeting of Origami Science andTechnology, Ferrara, pp. 229-237.Kihara, T. (1979) Bunshi to Uchuu [Molecule and Cosmos, inJapanese], Tokyo: lwanami.Lang, R. (1988) The Complete Book of Origami, New York:Dover.Lang, R. and Weiss, S. (1990) Origami Zoo, New York: St.Martin’s Press.Maekawa, J. (1983) Viva ! Origami, Kasahara, K., ed., Tokyo:Sanrio.Meguro, T. (1991-1992) Jitsuyou origami sekkeihou [Practicalmethods of origami designs, in Japanese], OrigamiTanteidan Shinbun [The Origami DetectivesNewsletter], 5-36-7 Hakusan Bunkyou-ku Tokyo:Gallery Origami House, Nos. 7-12, 14.Miura, K. (1989) Map fold a la Miura style, its physicalcharacter and application to the space science, In:HuTJta, H., ed., Origami Science and Technology,Proceedings of the International Meeting of OrigamiScience and Technology, Ferrara, pp. 39-49.Ohashi, K. (1977) Sousaku Origami [Creative Origami, inJapanese], Tokyo: Bijutsu-Shuppan-Sha.Rokoan, (1797) Senbazuru Orikata [Folding Forms of 1000Cranes, in Japanese], Republished: Kasahara K.,Origami 2 -- Senbazuru Orikata, Tokyo: Subaru-Shobo,1976.Uchiyama, K. (1979) Junad Origami [The Pure Origami, inJapanese], Tokyo: Kokudo-Sha.Mantis (Maekawa)Rail (Maekawa)Fibre 14

Syr, vnetry: CuRu,’e and ScienceVo~ 5, No. 2, 179-188, 1994PAPER SCULPTUREDidier Boursin17 rue Sainte Croix de la Bretonnerie, F-75004 Paris, FranceOrigami, or paper folding, is always practised with both hands. This reflects thesymmetry implicated in paper folding. This symmetry is accompanied by a furtheraspect; that of the way the folding is performed: the folds made on the paperintroduce successively different points of reference; points around which thesymmetry turns, and folded lines which act as the axes of symmetry.The marks into the space are also dualistic operations: left - right, above - under,up - down, right-side - wrong-side, and so on... Symmetry is ever present though itsform is dependent on which model do you wish to construct. For instance, inorigami, animals generally have an axis of symmetry along a single axis; plants andflowers, on the other hand, have their symmetry organised around a central point.All the traditional bases and techniques are also objects of analysis from the pointof view of symmetry: the water-bomb base, the preliminary base, the windmill base,the bird base, the fish base; the reverse fold, the squash fold, the rabbit’s ear.The paper shape used is often the square, which has many axes of symmetry, likepolygons for example.Symmetry is the foundation of any equilibrium. It puts the mind ease. It is theorigin of aesthetic. It allows the glance not to be lost. The symmetry is a glance on ahuman being, who is symmetrical too. In the sense of origami, a human being issymmetrical and is, moreover, represented in a closed surface. We start from apiece of square paper, and, through successive symmetrical folds, reduce it while weget a closed, completed origami model. The most important feature of origami isnot the final object created, but the operations that are performed to obtain thatfinal product. That final product may become elegant and simple, for it to be seenas interesting and aesthetically pleasing. If not the best techniques are applied ortoo much paper is used in the construction of a model, it will be seen as inelegantand not very artistic. The application of paper in successive layers is consideredunaesthetic and spoiling its artistic and symmetrical charm. Ideally, origami shouldinvolve delicate folds, each of which should be reflected upon, like the moving ofchesspieces on a chessboard.

180 D. BOURSIN2 Fold in front andA behind 4 Fold and unfold 5 Fold twiceand behindjoin the dots6 Unfold and returnI///)9 Fold angles inhalf and unfold~ / / ~XX N7 Join the dots 8 Join the dots( / [ ~ and return folding in half~ ~~ 10 Open;fold in halfConnect t/" /5 pieces~14 Lock by thislast fold insideFigure !: Star by D. Boursin (1992).

PAPER SCULPTURE 181t’,’ I,/Figure 2: Star in 3 dimensions by D. Boursin (1992).

182 D. BOURSIN1 Take a A4 paper and fold it in quartersand cut it in half2 Fold all the quartersexcept two parts andcut until the middle3 Fold each triangle then fold it in3 Dimensions5 The cube where appearsV’ and V z4 Fold the both pieces of A4 and connectroger her.Figure 3: Cube structure by D. Boursin (1990).

PAPER SCULPTURE 1831 pher les 4 pointes d’un cartejusqu’au centre pu~s marquerles phs au t~ers dansles 2 sens2ouvrir completementet retourner:~ plier les 2 trianglesderriere et marquerderniers ptis commeindiquOs puis retourner4ghsser les cbtesI’un darts I’autre5rabattre les po=ntes ~. I’~nt~r~eurpuls remonter le tout ~ la verticale6 embo~ter I’un dans I’autre"~ le cube terrnln~Figure 4: Cube by D. Boursin (1990).

184 D. BOURSINThe model’s symmetry should explain the essence itself of the paper and symmetry:minimum of matter for a maximum of expression should be one of the fundamentalrules to follow in creating a new aesthetic of symmetry and generally origami.To practice the folding, it’s to sculpt the matter trying to explain tae bestexpression. Personally, I prefer simple origami; I try to be simple in my origamiworks, however it’s very difficult.For me, origami is most of the time, a wink of life, generally very ephemeral. Nowthe task is to fold different irregular sizes, or polygon shapes of paper from theusual square-shaped paper. A4 size paper owns interesting proportions: 21 x V~ =29.7 (~/~-is the diagonal of a square, ~ is the diagonal of an A4 size paper andalso the diagonal of the cube). The relation between 2 and 3 dimensions is alsointeresting to explore and can be used for folding different sorts of cubes andpolyhedrons with this paper size.Summing up, symmetry cannot be dissociated from a sense of equilibrium,proportion, elegantness and aesthetic beauty.

PAPER SCULPTURE 1851 A4 paper : fold and unfold asindicates2 Hold the paper with both handsand fold dot to dot3 Make the creases then makes j’ reverse folds ~4 Fold inside the littletriangles (4) then foldthe right side down tothe dot.Repeat on the other side.5\\// Put the left triangle insidethe central one6 Fold and unfold themiddle on the bothsides.For making in3 dimensions,pull outthe opposite corners.7 The octahedron finishedFigure 5: Octahedron by D. Boursin (1992)

186D. BOURSINFisure 6: Folding frame: The structure (D. Boursin).

PAPER SCULPTURE 187Figure 7: Folding Frame: Details (D. Boursin).

188 D. BOURSINTHE AUTHOR’S PUBLICATIONS ON ORIGAMI:Bout-sin, D. (1983) Le ticketplid.Boursin, D. (1988) Manuelpractique d’origami, ed. Celiv.Boursin, D. (1989) Papiers plids, des idees plein les mains, ed. J’ai lu.Bout’sin, D. (1990) Pliages en mouvernent, ed. Dessain et Tolra.Boursin, D. (1991) Pliage des serviettes, ed. Dessain et Tolra.Boursin, D. (1992) Pliages en libertY, ed. Dessain et Tolra.Boursin, D. (1994) Pliagespremierspas, ed. Dessain et Tolra.

Symmetry: Culture and ScienceVoL 5, No. ~ 189-210, 1994SYMMETRIC GALLERYORIGAMI

190 D. BOURSINSymmetry: Culture and ScienceVoL 5, No. 2, 190-196, 1994Shooting Stars, one piece paper with curve, (Didier Boursin).

PAPER SCULPTURE -- GALLERY 1913 Horns, it is a work on curving paper then continue and discontinue, (Didier Boursin).

192 D. BOURSINTetrahedron, from A4 sheet of paper, (Didier Boursin, photo: Fabrice Besse).Cubes, from A5 sheet of paper, (Didier Boursin, photo: Fabrice Besse).

PAPER SCULPTURE -- GALLERY 193Octahedron, (Didier Boursin, Photo: Fabrice Besse).Polihedra, (Didier Boursin, photo: Fabrice Besse).

194 D. BOURSIN5Points Star, (Didier Boursin, photo: Fabrice Besse).Star in 3 Dimettsions, (Didier Boursin, photo: Fabrice Besse).

PAPER SCULPTURE -- GALLERY 195Growing Plant, it contains 9 meters o[ paper in one piece, (Didier Boursin).Folding Frame. It is impossible to fold directly by hands. The artist has different flames like this with aminimum of foldings; this is one of the best solutions, (Didier Boursin, photo: Fabrice Besse).

196 D. BOURSINMobile, (Didier Boursin, photo: Fabrice Besse).

198 R..L LANG15. Fold the paper in halfalong the diagonal and rotate1/8 turn counterclockwise16, Fold and unfold 17. Reverse-fold the edge18. Fold and unfold19. Open-sink the corner. 20. Spread-sink the edgesymmetrically21. Fold the edge upward22. Fold the upper edge downto align with a hidden edge,crease, and unfold,23. Pull the paper out from underthe pleat and sink a square regionof the paper outlined by thecreases you made m step 2124. Flatten the model 25. Repeat steps 23--24 b~hmd 26. Asymmemcally squash-foldthe right side Note that thetwo circled crease intersectionscome together

STAG BEETLE 2 -- FOLDING INSTRUCTIONS 19927. Reverse-fold the edgeupward28. Bring two layers of paperto the front.29. Reverse-fold the far edge30. Fold and unfold along ahorizontal crease, which linesup with a hidden edge31. Fold and unfold along acrease perpendicular to theadjacent folded edge32. Pleat the nght side,plVOtlng the bottom corner sothat its edges align with theraw edges on the far layersThe model will not lie fiat33. Reverse-fold the corner 34. Full out a single layer ofpaper (open unsmk) to makethe pleat symmetric35. Close up the model andflatten completely36. Mountain-fold the narrowflap behind to make the modelsymmetric from front to back37. Simultaneously squash-foldthe left edge andmountain-fold the bottomunderneath, tucking tt into thepocket Repeat behind38. Lift up one edge and bringthe bottom corner up so thatstands out away from themodel

2OOR]. LANG39. Pull a colored comer 40. Repeat steps 38-39out from lnstde the pocket, behtndsquash the top down, pullthe sides out, and flatten41. Reverse-fold the edges 42. Fold and unfoldRepeat behLnd43. Fold and unfold, 44. Fold the comer down 45. Fold the comer backRepeat behind whde squash-folding the up but keep the squashedge underneathfold in place46. Pull out a raw edge.47. Reverse-fold the edge 48. Reverse-fold the edge 49. Reverse-fold the edge 50. Repeat steps 42-49back out again behind5;1. Fold one layer up, 52. Fold the raw edge overrepeat behindto the right and ptnch theexcess paper upwardRepeat behind53. Reverse-fold the whitepoint to the left54. Wrap one layer fromthe mstde to the front.Repeat behind.

STAG BEETLE 2 -- FOLDING INSTRUCTIONS 20155. Fold and unfold, repeat 56. Open-sink the corner, 57. Ptnch the narrow pointbehind repeat behind m half and swing itdownward Repeat behind59. Stretch and narrow the 60. Now we’ll work on one 61. Fold and unfoldpoint and swing ttside at a nme for a whiledownward Repeat behindFold and unfold62. Reverse-fold 63. Reverse-fold on the 64. Reverse-fold againcrease you made in step 6165. Reverse-fold the 66. Pull out a raw edge 67, Reverse-fold the edgeremaining edgehalfway between existingcreases

202 R.£ LANG68. Reverse-fold on an 69. Reverse-fold the edge 70. Repeat steps 66-69existing crease back down to hne up with behtndthe other edges71. Fold two edges down 72. Lift up one layer 73. Reverse-fold the ms,deto the lower left and shghtly edgespread-squash ~e tophorizontal edge74. Close tt up 75. Fold a mbb*t ear from a 76. Fold two layers 77. Fold a rabb*t ear fromsingle layer upward the remaining layer78. Reverse-fold the pointupward; the reverse goesbetween the topmost edgeand the remaining edges79. Sink the edge 80. Mountain-fold twoedges together, forming asmall swivel fold at thebase of the point

STAG BEETLE 2 -- FOLDING INSTRUCTIONS2O381. Repeat steps 60~0 82. Fold and unfold,behind83. Fold and unfold84. Reverse-fold two 85. Fold and unfold Thepointscoming step will be easter tfyou mountain-fold the paperthrough the thick layers aswell Repeat behind86. Closed-sink the cornerRepeat behind87. Fold the left edge to he 88. Pull out some loosealong the diagonal creasepaper and squash the excessRepeat behindupward Repeat behind89. Open out the flap90. Pleat the flap upward 91. Pull out some loosepaper92. Squash-fold the edge andswing the excess paper to theright

2O493- Pull out some moreexcess paper94. Detml of the left half ofthe model Squash-fold theflap95. Petal-fold the edge96. Pull one of the corners ofthe square all the way outfrom ms,de97. Squash-fold the cornersymmetrically98. Reverse-fold the edges99. Fold and unfold100. Fold the point up to theleft101. Pull out some paper102. Squash-fold the paper 103. Pull out some paper 104. Outside-reverse-fold d~eover to the rightcorner

STAG BEETLE 2 -- FOLDING INSTRUCTIONS 205105. Fold and unfold106. Fold one flap over to 107. Fold and unfoldthe left108. Reverse-fold the corner109. Enlarged viewMountatn-fold the whitecorner underneath and swingthe two potnts down110. Squash-fold the point111. Inside petal-fold theedge112. Fold two points up 113. Open-sink both corners 114. Fold two potnts down115. Fold two points upward 116. Close up the model, 117. Reverse-fold two edgesincorporating a reverse fold

206118. Pull out some loosepaper119. Valley fold the mw edgedown Repeat behind120, Fold half of the toplayers forward and ball of thebottom layers behind121, Carefully spread-stnk thecorner, spreading all of the layerssymmetrically122. Closed un-smk the pocket 123. Grasp two edges of the pointyou lUSt pulled out and stretch themto the left, the point disappears inthe process124. In progress 125. Fold and unfold

STAG BEETLE 2 -- FOLDING INSTRUCTIONS 207126. Closed-sink the edge on the creaseyou lUSt made It helps to open the modelout from the underside while you do this127. Spread-sink the edge symmetrically128. Sink the bottom two corners andsqueeze the excess paper upwnrd Becareful to keep the paper from filbng uptbe vertxc:al sht between the sides t Repeaton the top129. Bring one layer m front130. Mountain-fold one layer behind ontop and bottom131. valley-fold a double edge to thecenter hne of the model above andbelow132. Undo two reverse folds 133. Turn the model over

2O8/~J. LANG134. Pleat the abdomen 135. Mountain-fold the edges of thepleated par~136. Crimp the thtck (tuner) points so thatthey stand straight out from the body137. Mountain-fold the edges and tuckthem into pockets Repeat behind138. Next wews am a close-up of theright side of the model139. Fold and unfold about 2/5of the pomt140. Fold and unfold Note thatthe vemcal crease h~ts the topedge at the same place the lastcrease d~d141. Camp the point downward 142. Pull out two layers on the 143. Wrap one layer to theon the existing creases top and bottom front Repeat behind

STAG BEETLE 2 -- FOLDING INSTRUCTIONS 209144. Pull out some loose paperRepeat behind145. Reverse-fold the point 146. Reverse-fold the corner147. Reverse-fold the h~ddencorner148. Repeat steps 138-147 onthe lower flap149. Pinch the two flaps in halfthrough all layers150. Turn the model over 151. Mountain-fold the bluntcorner inside152. Pinch the two small pointsm half153. Ptnch the two small points 154. Pull out two points 155. Turn the model back overin half

210 R.Z LANG156. Crimp the antennae to 157. L~ke th~sthe ngbt158. This shows the entire model Crimp{he forelegs (on the rJght) Reverse-fold theh*nd legs160. Turn tbe model over159. Reverse-fold the hind feetReverse-gold each m~d leg twice,161. Round the abdomen and pleat *t down{he m(ddle Make the body 3-D162. F*mshed Stag Beetle

Syrametry: Culture and ScienceVoL 5, No. 2, 211-212, 1994RESEARCH PROBLEMS ON SYMMETRYRESEARCH PROBLEM 1Two-sided wallpaper groups of periodic origami patternsOrigami is useful to make infinite periodic patterns, more precisely, to represent afinite part of them. These periodic patterns are not 2-dimensional in a strict sense,because they are not totally fiat, but rather occupy a narrow layer. We may callthem 2.5-dimensional patterns, continuing Coxeter’s joking remark that the 7frieze-groups (strip-groups) are 1.5-dimensional.The 2.5-dimensional periodic patterns can be analyzed with two-sided wallpapergroups (layer groups). These were firstly enumerated by C. Hermann (1929): thereare exactly 80 types. In the same year L. Weber (1929) illustrated all of them byblack-and-white patterns where the colors refer to the front and the back sides,respectively (see, e.g., Shubnikov and Koptsik, 1972, where Weber’s figures arereprinted). From these 80 types only 46 ones are really interesting, while theremaining 34 types are degenerate cases (i.e., the 17 strictly 2-dimensionalwallpaper patterns, as those cases where only one side is used; and those further 17cases where the above mentioned 17 patterns are mirrored to the back side). These46 patterns are also known as two-colored (black-and-white) patterns. Thisinterpretation with colors is useful in both fields: crystallography (where the colorsrefer to different physical properties, e.g., positive and negative magnetism) anddesign. Indeed, these patterns were discovered and studied not only by thementioned crystallographers, but also by the textile engineer H. J. Woods (1936).His patterns are reprinted in a more recent monograph on pattern analysis ofornamental arts (Crowe and Washburn, 1988).Question: How many of these 46 types can be represented by origami without usingcuts?REFERENCESHermann, C. (1929) Zur systematischen Strukturtheorie: 3. Ketten und Netzgruppen, [About asystematic structure-theory: Chain- and net-groups, in German], Zeitschriflfiir Kristallographie, 69,25O-270.

212 RESEARCH PROBLEMS ON SYMMETRYShubnikov, A. V. and Koptsik, V. A. (1972) Simmetriya v nauke i islo~sstve, [in Russian], Moskva:Nauka, 339 pp.; English trans., Syrmnctry in Science andArt, New York: Plenum Press, 1974, xxv +420 pp. [See Chap. 8].Washburn, D. K. and Crowe, D. W. (1988) Symmetn’es of Culture: Theory and Practice of Plane PatternAnalysis, Seattle, Wash.: University of Washington Press, x + 299 pp.; Paperback ed., ibid., 1991.[See Chap. 3].Weber, L. (1929) Die Symmetrie homogener ebener Punktsysteme, [Symmetry o1~ homogeneous planarypoint-systems, in German], ZeitschtiftffirKristallographie, 70, 309-327.Woods, H. J. (1936) The geometrical basis of pattern design: Part 4, Counterchange symmetry in planepatterns, Journal of the Tepaile Institute, Transactions, 27, T305-T320.D6nes Nagy

Symm~ay: Culture and ScienceVot 5, No. 2, 213-218, 1994SYMMETR O-GRAPHYSection Editor: Ddnes Nagy, Institute of Applied Physics,University of Tsulazba, Tsukuba Science City 305, Japan;Fax: 81-298-53-5205; E-mail: nagy@kafka.bk.tsukuba.ac.jpBOOK REVIEWBoursln, Didier, Pliages en mouvement, [Folding in Movement, in French], Paris:Dessain et Tolra, 1990, 79 pp.; Reprint, ibm, 1991.The author of this book is the President of the Mouvement Franfais des Plieurs dePapier (French Association of Paper Folders). The work starts with a one-pageintroduction with an historic survey, including some interesting data about thespread of paper-folding in France. A group of Japanese touring in France actuallydemonstrated the ’real’ origami in 1860. The author emphasizes, however, thatthere was in Europe an independent tradition based on folding table napkins,which originated in the courts of Henry III and Louis XIV. The book has five mainparts: Animals, Magic, Gravity, Sounds (i.e., instruments producing sounds), Cards.The contents has an important feature: the items are classified into three categoriesof very simple (*), simple (**), and difficult (***). This is followed by the detailedexplanation of the visual symbols used in paper folding. In each case of thepresented ’paper compositions’ we find double illustrations: a color photograph ofthe completed work and the detailed explanation how to make it, with very wellmade drawings. The quality of the photographs, made by Fabrice Besse, represent ahigh level of artistic skill. In many cases the installation was made by the Japanesewife of the author, Setsuko, a fashion designer. Although this book is addressed toa broad public, it is also useful for specialists. Thus Boursin gives credit to theinventor of each individual piece, including such details that, for example, hedeveloped a frog (p. 16) using the idea of K. Kasahara. Note that many of the pieceswere invented by Boursin himself. One of the items should be made of cloth (pp.24-25). We also have some symmetry-related remarks. Boursin gives credit to LuisaCanovi of Italy for inventing a closed chain of tetrahedra that provide an excitingflexible object (pp. 36-37). This item was, however, discovered earlier outside of theworld of origami. An early discovery of this idea is due to Paul Schatz in 1929,which was probably independently rediscovered by Schattschneider and Walker inthe 1970s, see their book M. C. Escher Kaleidocycles (New York, 1977). Anotherrelated object is discussed by Boursin under the name ofFleragon (p. 39). Indeed, itis a very expressive name, because the flexible chain of units form a hexagonal

214SYMMETRO-GRAPHYshape. Boursin remarks that we can make with this object kaleidoscopic images.From a didactical point of view, it would be better to reverse their order: to explainfirst the Flexagon and then to turn to the more complex chains of tetrahedra. Thepiece Equilibrium (pp. 50-51), designed by the author, has an interesting connectionwith the topic of symmetry. Balancing toys were popular in earlier ages: each ofthese objects, usually dominated by a U shape turned upside-down, has asupporting point higher than its center of gravity. Thus, these toys can balance onthe top of various sharp objects. It is exciting to see the origami version of thismechanical toy and the photograph where it balances on the top of a bottle.Another item, a beautiful periodic pattern (pp. 44-45), inspired us to formulate aresearch problem about symmetry groups of this type of structure. Note thatRobert Harbin’s name is misspelled on p. 38 as "Herbin". Illustrations: it isbasically a picture book with very many drawings and photographs. Address:Cr~ati0n Setsuko, 17, rue Sainte Croix de la Bretonnerie, F-75004 Paris, France.D6nes NagyHarglttai, Istv~in and Hargittai, Magdolna, Symmetry, A Unifying Concept,[Symmetry in general], Bolinas, Calif.: Shelter Publications, Inc., 1994, 222 pp.Istv~in and Magdolna Hargittai have published several books, partly as editors,partly as author, on the topic of symmetry. They are chemists, who started theirinvestigations from the symmetry of molecules to discover further symmetries ofthe invisible and visible world. As scientists, they explain the regularities, patterns,and laws of hidden symmetries, while as fans of photography, they present thebeauty of the visible ones’ to the reader.One meets symmetry in sciences, arts, and even in everyday life, nevertheless towrite a book on the topic conceals traps: to aim at a broad public, frommathematicians to biologists, from physicists to linguists, from chemists to artists,from students to professors limits the scientific level; to keep high scientificstandards limits the widths of the audience. The Hargittais manoeuvre skilfullybetween these Scylla and Charybdis, well known for any interdisciplinary author.They ferry the reader from the point-group symmetries through the examples forantisymmetry to the space-group symmetries. They do not sacrifice the wideunderstandability for deep scientific exactness. The basic symmetry concepts areintroduced mainly by a visual way, special terminology is mainly avoided, in othercases simple definitions are given. Clear drawings help the understanding. Laymanreaders can discover new connections, new patterns in the world they had thoughtthey knew. One explores, step by step, the way to the most recent symmetryinspireddiscoveries like the quasicrystals showing fivefold symmetry which hasbeen not usual in the inanimate nature. The book is well illustrated with hundredsof black-and-white photos, reproductions, and drawings.Gy0rgy Darvas

216SYMMETRO-GRAPHYSYMMETRIC REVIEWS 5.2The "Symmetric Reviews" (SR), as a regular subsection, publish brief notes aboutbooks and papers. These are not conventional reviews; their main goal is to emphasizethe connections with symmetry and, in same cases, the required backgrourwLCorrespondence should preferab~v be sent to both the section editor (for reviewing) andthe Symmetrion in Budapest (for the data bank).SR 5.2 - 1 (Origami: popular)Boursin, Didier, Pliages en mouvement, [Folding in Movement, in French], Paris:Dessain et Tolra, 1990, 79 pp.; Reprint, ibid., 1991.SR 5.2 - 2 (Interdisciplinary book: popular)Hargittai, Istv~in and Hargittai, Magdoina, Symmetry: A Unifying Concept,Bolinas, Calif.: Shelter Publications, Inc., 1994, 222 pp.The authors of this books are Hungarian chemists, husband and wife. Istv~inHargittai is a Honorary Member of ISIS-Symmetry; see the list of his symmetryrelatedbooks in this quarterly, Vol. 3, No. 3, p. 324. The new book of the Hargittaiscovers various aspects of symmetry in art and science. It is addressed to a very broadpublic using the advantage of large number of illustrations. This book is reviewedin more details; see the section "Book review". Illustrations: 850. References: tobooks 29, to illustrations 177. Address: E0tv0s Lorhnd University, P.O. Box 117,Budapest, H-1431 Hungary.SR 5.2 - 3 (Origami: popular)Pataki, Tibor, Hajtogatni j6, [Folding is Exciting, in Hungarian], Budapest:Gyorsjelent~s Kiad6 Kft., 1993, 79 pp.This book belongs to the rare category of Western books on origami where themajority of the pieces are designed by the author himself, and there are only veryfew adaptations. In those cases the author refers to the sources, which are eithertraditional origami items or the works of modern inventors. The book starts with atwo-page preface by a noted Hungarian author, G~ibor N6gr~di. Pataki himself iseven shorter: his introduction is just a half-page. The rest of the book is dominatedby figures with brief notes. First of all Pataki explains the notations and presentsthe four basic folding operations. After these, he creates almost an entire origamizoo with very many animals. There are a few other objects: a sailing boat, a piano, aSanta Claus, a decoration for Christmas-tree, a steam boat, a fighter, an envelope,three masks, a geisha, a flying machine, a clown, and a rocket. Altogether there are40 pieces in the book. Some of the animals feature in more than one work. Forexample, the book presents a running rabbit, then a sitting one. It is followed by afox, and we think that children may immediately create stories with these animals.In Hungary origami paper is not widely available, therefore the book includes atthe back 12 sheets. The front side of each these square-shaped origami papers is

SYMMETRO-GRAPHY217yellow, while the back side is white. These colors help the author to better explainthe process of folding. Specifically, the drawings clearly indicate when we shouldwork with the yellow front side of the paper, and when with the white back side. Inthe beginning the author consistently uses this helpful ’color coding’, but later hefrequently violates it: thus, the back side is sometimes gray, while in other cases thebackground (the table) is yellow and the two sides are distinguished by white andgray colors. In a possible new edition of the book, we suggest being more consistentin coloring the figures, which is, indeed, a very useful idea. The book includes colorphotographs of the objects: these are collected in a section at the middle. Thecaptions of these photographs clearly refer to the page numbers where the foldingprocesses are explained. One may claim that it would be better to have thephotographs of the pieces and the corresponding drawings of the folding processesside by side, but that arrangement would make the book much more expensive. Wesuggest supplementing any possible new edition of the book with a table ofcontents. Illustrations: about 800 drawings and 30 color photographs. Address:Budapest, Kir~ily u. 67. V.em. 3., H-1077 Hungary.D6nes NagyPeacock by T. Pataki. (See the folding instructions in Symmetry: Culture and Science, 5 (1994), 1, 88-89.)

218 SYMMETRO-GRAPHYSparrow by T. Pataki. (See the folding instructions in Symmetry: Culture and Science, 5 (1994), 1, 94-95.)Tree-frog by T. Pataki. (See the folding instructions in Symmetry: Culture and Science, 5 (1994), 1, 92-93.)

Syraraetry: Culture and ScienceVoL 5, No. 2, 219-223, 1994SFS: SYMMETRIC FORUM OF THE SOCIETY(BULLETIN BOARD)All correspondence should be addressed to the editors: G~yOrgy Darvas or Ddnes Nagy.ANNOUNCEMENTSSECOND INTERNATIONAL MEETING OFORIGAMI SCIENCE AND SCIENTIFIC ORIGAMINovember 29 - December 2, 1994Otsu, Shiga, JapanSCOPE AND TOPICSOrigami no longer means only orizuru, a folded paper crane, the mostrepresentative example of a classic style work, but is the proper subject ofmathematics and involves the field of science and technology, which havedeveloped the new idea of origami art. The first International Meeting of OrigamiScience and Technology, held at Ferrara, Italy, in 1989, was the result of thisinvolvement.Because of this good first step, it was desired to have the second meeting in Japan,the home of Origami. Therefore, it has been decided to have this event in the CityOtsu, near Kyoto, from November 29 to December 2, 1994.The aim of this meeting is to bring together specialists in origami science andscientific origami. It is intended to provide the participants with information aboutthe important topics.The contributed papers may cover origami science and scientific origami. However,’science’ includes wider fields, in particular, mathematical foundations of origami,natural forms and origami, classical vs. modern origami, origami art, origami andindustrial design, Euclidean geometry and origami axiom, origami and algebra,education and therapy by origami, mathematical design of origami, paper materials,curve folds modular origami, definition of origami, etc.

220 SFSParticipating OrganizationsArs +, Asociacion Espanola de Papiroflexia, British Origami Society, CentroDiffusione Origami (C.D.O.), Dansk Origami Center, ISIS-Symmetry, KoreaJongie Jupgi Association, Mouvement Francais des Plieurs de Papier (M.F.P.P.),Nippon Origami Association, Origami Deutschland (O.D.), The Board ofEducation, City of Otsu, Shiga Prefecture, The Form and Culture Society, TheFriends of the Origami Center of America, The Society for the Science of Form.CALL FOR PAPERSSYMPOSIUM: Origami: East and Westwill be organized in the framework of the following congress and exhibition:S YMME TRY."NATURAL AND ARTIFICIALThird Interdisciplinary Symmetry Congress and Exhibition of theINTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OFSYMMETRY (ISIS-SYMMETRY)August 14- 20, 1995Old Town Alexandria (near Washington, D.C.) U.S~4.CALL FOR PAPERS, WORKSHOP TOPICS, ANDEXHIBITION ITEMSFIELDS OF INTERESTSYMMETRY: NATURAL AND ARTIFICIALThe congress and exhibition present a broad interdisciplinary forum where the representativesof various fields in art, science, and technology may discuss and enrichtheir experiences. The concept symmetry, having roots in both art and science, helpsto provide a ’common language’ for this purpose. The new ’bridges’ between disciplinescould inspire further ideas in the original fields of participants, as well asfacilitate the adaptation of existing ideas and methods from one fieM to another.The title of the congress emphasizes the presence of symmetry (dissymmetry, brokensymmetry) both in nature and in the objects created by artists, scientists, and engineers.

SYMMETRIC FORUM OF THE SOCIETY 221Exhibition: Ars ScientificaThere have been several exhibitions representing the specific impact of certainfields of science and technology on art, but ISIS-Symmetry has initiated a regularforum for a broader interface of art and science. The exhibition will consist of twoparts: a professional exhibition and an informal one, based on the objects illustratingthe lectures given by the participants. Some workshops will be conducted in theexhibition rooms. SPECIAL INTERESTS ofArs Scientifica are, among others: kaleidoscopes,polyhedra, model designs, new media.CALL FOR PAPERSA lecture proposal should include a maximum 4-page extended abstract in a cameraready version. Keeping in mind the interdisciplinary goals of the congress and thecomposition of the participants, please try to help the readers outside of your maindiscipline e.g., by explaining some special concepts, using intuitive approaches, orgiving comprehensive tables and illustrations. The extended abstracts should either(a) describe concrete interdisciplinary ’bridges’ between different fields of art, science,and technology using the concept of symmetry; or (b) survey the importanceof symmetry in a concrete field with an emphasis on possible ’bridges’ to otherfields. Note, please, that the central topic of the present congress Symmetry:Natural and Artificial opens a wider door towards technological applications.Papers discussing links between any form of symmetry-asymmetry phenomenon orlaw in nature on the one side, and artistic, technical achievements on the other, arepreferred. Please consider tlxat the meetings of ISIS-Symmetry are informal and donot substitute for the disciplinary conferences, only supplement them with abroader perspective.The extended abstracts should be submitted in 2 copies, mailed 1 each to G. Darvasand D. Nagy, on A4 or letter size pages, printed on one side of each sheet, with atleast 2.5 cm (1 inch) margins both sides, top and bottom, double spaced, 12-pointcharacters.Sample:TITLE WITH CAPITAL LE’Iq"ERS[two line-spaces]Joe Symmetrist and Josephine AsymmetristDepartment of Dissymmetry, Fibonacci UniversitySan Symmetrino, SY 12358, SymmetrylandE-mail: symmetrist @ fibonacci.edu[two line-spaces]The text should be printed in one column. Figures (black-and-white only) mayinterrupt the text. Please avoid using any other heading (e.g., ’Extended Abstract’,’submitted to ...’). Page numbers should be marked with pencil.References [at the end]: Alphabetical order, full bibliographic description.

222 SFSFor more details refer to the "Instructions for contributors" on pp. 110-111.CALL FOR EXHIBITION ITEMSItems for the exhibitions should be introduced in the same form as lecture-abstractson A4 or letter size sheets, in black-and-white camera ready, reproducible form.Please mark with pencil at the top of the sheet: (EXHIBITION). A short descriptionand/or explanation of the items, as well as the connection to the main theme ofthe congress and exhibition, is preferred. Please give the dimensions of each item.Art works, models, demonstration materials, etc. are welcome, e.g., in the followingsections: Kaleidoscopes, Polyhedral symmetry, Origami, The beauty of molecules,Aesthetics of man-made constructions, Mechanical structures inspired by nature:Artificial and natural structures, Design principles, New Media. Proposals forfurther sections are encouraged.CALL FOR WORKSHOP TOPICSPlease give the approximate title, short description (how do you plan to organizethe workshop), other expected/proposed contributors, etc. Proposals emphasizinginteractions, mediated by symmetry, between different disciplines; science, art, andtechnology; cultural origins, and relying upon the interest of participants with differentbackgrounds, are preferred.CALL FOR PROPOSALS FOR EVENING ACTMTIES OR PERFORMANCESMusic, dance, video, laser, paper folding, etc. programs are welcome. Please submityour proposals, similar to the lecture abstracts, with descriptions of the feasibilityand the technical requirements. Please mark with pencil at the top of the sheet:(PERFORMANCE), (VIDEO), etc., respectively. (Formal requirements are thesame as above for papers.)DEADLINESfor application and short description of contribution and other proposals:December 15, 1994;for submitting final (camera ready) versions of the emended abstracts:March 31, 1995.

SYMMETRIC FORUM OF THE SOCIETY 223THE FORMAT OF THE CONGRESS AND EXHIBITIONThe tradition, initiated by ISIS-Symmetry, to facilitate interdisciplinary dialoguesamong scientists, engineers, and artists will be continued. There will be no parallelsections (which would lead to disciplinary separation of the participants), but eachmorning there will be plenary sessions, while the main ideas will be discussed anddeveloped in afternoon workshops. For the evenings there are scheduled performancesand informal meetings, including recreational, and ars scientifica programs.The working language of the congress is English.The Scientific Advisory Committee of the Congress and Exhibition is the Board ofISIS-Symmetry (see inside front and back covers).CONTACT PERSONS (for the Congress and Exhibition)Martha Pardavi-Horvath, Site CoordinatorGeorge Washington UniversityDepartment of Electrical Engineering and Computer ScienceWashington, D.C. 20052, U.S.A.Phone: 1-202-994-5516; Fax: 1-202-994-5296; E-mail: pardavi@seas.gwu.eduGy0rgy Darvas, Executive Secretary, ISIS-SymmetrySymmetrion - The Institute for Advanced Symmetry StudiesP.O. Box 4, Budapest, H-1361 HungaryPhone:36-1-131-8326; Fax: 36-1-131-3161; E-mail: h492dar@ella.huD6nes Na~, President, ISIS-SymmetryInstitute of Applied Physics, University of TsukubaTsukuba Science City, Ibaraki-ken 305, JapanPhone: 81-298-53-6786; Fax: 81-298-53-5205; E-mail: nagy@bk.tsukuba.ac.ipAPPLICATION FORMName: ......................................................................................................................................Affiliation: ..............................................................................................................................Mailing Address: ....................................................................................................................City: ............................................................... State/country: ...............................................Fax: ................................. Phone: ................................. E-mail: ............................................I intend to: © attend the Congress © submit a paper © exhibitTentative title of my contribution: .....................................................................................

224 AIMS AND SCOPEThere are many disciplinary _periodicals and symposia in various fields of art, science, and technology, but broadiaterdisciplioary forums for the connections tmtween distant fields are ve~, rare. Consequently, tl~e interdisciplinarypapers are dispersed in very different j.ournals and proceedings. Th~s fact makes the cooperation of theauthors dilficult, and even affects the ability to locate their papers.In our ’split culture’, there is an obvious need for interdisciplinary journals that have the basic goal of buildingbridges (’symmetries’) between various fields of the arts and sciences. Because o[ the variety of topics available,the concrete, but general, concept of symmetry was selected as the focus of tbe journal, since it has roots in bothscience and art._SYl~t~Rlq,_ CULTURF.. AIgD SCIENCE is the quarterly .of the It¢’r~_I~C,~TIOI¢~_SO~.I~. ]~oR ~ INTERDISCI~PLIN~_ y¯ ~’TUDYOFb’YJO~TRY (abbreviation: ISIS-Symmetry, shorter name: ~ymmemy ~’o¢iety). ISIS=s]t, mmetry was roundedduring the symposium Symmetry of Smacture .(First lnterdiscipFauzry Symme_ tty Sympos~ura and Exhibition),Budapest, August 1:3-19, 1989. The focus of ISIS=Symmetry is not only on the concept of symmetry, but also itsassociates (asymmetry, dissymmetry, antisymmetry, etc.) and related concepts (proportion, rhythm, invariance,etc.) in an ~nterdisciplinary and intercultural co.ntext. We may refer to this broad approach to the concept assyrn~trolog),. The suffix -1o~ can be associated not only with knowledge of concrete fields (cf., biology, geology,philologyj psychology, sociology, etc.) and discourse or treatise (cf., methodology, chronology, etc.), butalso with the Greek terminology of proportion (cf., logos, analogia, and their Latin translations ratio,i~rol~o~io).The basic goals of the Soci¢o~ are(1) to bnng together artists and scientists, educators and students devoted to, or interested in, the researchand understanding of the concept and application of symmetry (asymmeh’y, dissymmetry);(2) to provide regular information to the general public about events in symmetrology;_,(3.) to ensure a regular fornm (including tr~e organization of symposia, congresses, and the publication of aperiodical) for all those interested in symmetrology.The Society organizes the triennial Interdisciplinary S_ym/~.Fy Congress and Exhibition Cstarting with the symposiumof 1989) and other workshop, meetings, and exhibitmns. The forums of the Society are informal ones,which do not su~titute for the disciplinary conferences, only supplement them with a broader perspective.The Quarterly - a non-commercial scholarly journal, as well as the forum of ISIS-Symmetry - publishes originalpapers on symmetry and related questions wliich present nov results or nov connections between known results.The papers are addressed to a broad non-speciahst public, without becoming too general, and have an interdisciplinarycharacter in one of the following senses:O) they describe concrete interdisciplinary ’bridges’ between different fields of art, science, and technologyusing the concept of symmetry;,(2) they survey the importance of symmetry in a concrete field with an emphasis on possible ’bridges’ to otherfields.The Quarterly also has a special interest in historic and educational questions, as well as in symmetry-relatedrecreations, games, and computer programs.The regular sections of the Quarterly:Symmetry: Culture & Science (papers classified as humanities, but also connected with scientific questions)Symmetry: Science & Culture (papers classified as science, but also connected with the humanities)¯ Symraetry In Education (articles on the theory and practice of education reports on interdisciplinarypro’ects)¯ SF~: Symmetric Forum of the Society (calendar of events, announcements o[ ISIS-Symmetry, news frommembers, announcements of projects and publications)¯ Symmetro-graphy (biblio/disco/sogtware/ludo/historio-graphies, re, cloys of books and papers, notes onanniversaries)Additional non-regular sections:¯ Symmetrospective: A Historic View (survey articles, recollections, reprints or English translations of basicpapers)¯ Symmetry: A Special I~o~us on ... (round table discussions or survey articles with comments on topics ofspecial interest)Symmetric Gallery (works of art)Mosaic of Symmetry (short papers within a discipline, but appealing to broader interest)¯ Research Problems o/~ Symmetry (brief descriptions of opei~’problems)¯ Recreational Symmetry (problems, puzzles, games, computer programs, descriptions of scientific toys;for example, filings, polyhedra, and origami)¯ Reflections: Letters ~o the Editors (comments on papers, letters of general interest)Both the lack of seasonal references and the centrosymmetric spine design emphasize the international characterof the Society; to accept one or another convention would be a ’symmetry ~iolation’. In the first part of theabbreviation ISIS-Symmetry all the letters are capitalized, while the centrosymmetric image iSIS! on the spine isflanked by ’Symmetry’ from both directions. This convention emphasizes that ISIS=symmetry and its quarterlyhave no direct connection with other organizations or journals which also use the word Isis or ISIS. There aremore than twenty identical acronyms andmore than ten such periodicals, many of which have already ceased toexist, representing various fields, including the history of science, mythology, natural philosophy, and orientalstudies. ISIS-Symmetry has, however, some interest in the symmetry-related questions of many of these fields.

German). FR Andreas Dress, Fakuhat ~ur Mathemauk,Umvers=tat Blelefeld,D-33615 Btelefeld I, Fosffach 8640, F R Germany[Geome{ry, Mathemauzatmn of Sc,ence]Then Hahn, lasutut fur Kristallographle,Rhem~sch-WestF;ihsche Technische Hochschule,D-W-5110 Aachen, FR Germany[Mineralogy, Crystallography]Hungary. M~h~tly Szoboszlai, l~pit6szm~rnok~ Kar,Budapesu M,3szak~ Eg2/etem(Faculty of Arch*tecture, Techmcal Umvers*ty of Budapest),Budapest, P.O Box 91, H-1521 Hungary[Architecture, Geometry, Computer Aided Arch.ectural Des*gn]Italy Giuseppe Caglioli, Ist.uto d~ Ingegneria Nucleate -CESNEE Pohtecmco d* Mdan, Vm Ponzm 34/3.1-20133 Mllano. Italy[Nuclear Pb.3,sics, Visual Psychology]Poland" Janusz R~;bielak, Wydzml Arch*tektury,Pohtechmka Wmc~aw~ka(Department of Architecture, Techmcal Umvers*ty of Wrc.ct’aw),ul B. Pmsa 53155. PL 50-317 Wrocl’aw, Poland[Architecture, Morphology or" Space StructuresJPortugal Josd Lima--de-Faria. Centro de Crtstalografiae Mmeralogla, Iasntuto de Invest~ga¢:~o Cientifica Tropical,Alameda D Afonso HennqOes 41.4.°Esq , P-1000 Lisbon,Portugal[Crystallography, Mineralogy. H,story of Sc*ence]Romama" Solomon Marcus, Facultatea de Matematica,Unlvegsihatea dm Bucure~tl(Faculty of Mathematms, Umversity of Bucharest),Str Academ*el 14, R-70109 Bucure~it~ (Bucharest), Romama[Mathematical Analysm. Mathematical Lmgmstms and Poetics.Mathemahcal Scm~oucs of Natural and Sccla} Sciences}Russia" Vladimir A. Koptsik, F*z*chesku fakultet,Moskovskll gosudarstvenny, umversltel(Physmal Faculty, Moscow State Umver~,b, )117234 Moskva. Russ*aI Crystalphysms]Scandinavia. Tuft 9,’~ster, Sksvelaborat~,net,Baerende Konstruktioner, Kongelige DaaskeKunstakadem, - Ark~tektskole(Laboratory for Plate Structures, Department of StructuralSmence, Royal Damsh Academy - School of Arch.ecture),Peder Skramsgade 1, DK-1054 Kobenhavn K (Copenhagen),Denmark [Polyhedral Structures. Biomechamcs]Swttzerland. Caspar Schwah¢. Ars GeometncaRfim,strasse 5, CH-8024 Zhnch, Sw~taerlandIArs GeometricalU K " Mary Harris, Maths ra Work Project,Insutute of Educauon, University of London,20 Bedford Way. London WCIH 0AL. England[Geometry, Ethnomathematies, Textile Design]Anthony Hill. 24 Charlotte Street. London WI. England[Visual Arts. Mathematms and Art]Yugoslavm: Sla’vik V. Jablan, Matemau~.ki mstnut(Mathemattcal Institute), Knez Mlhaflova 35, pp 367,YU-II001 Beograd (Belgrade), Yugoslavia[Geometry, Ornamental Art, Anthropology]Cha~rperso~ ofArt and Sctence F_xhtbittons L.’iszl6 Beke,Magyar Nemzett Gal6rta (Hungarran Nauonal Gallery),Budapest, Buda~ri Palota, H-1014 HungaryItsuo Sak~ne, Faculty of Env*ronmentalInformation, Keio Umversity at Shoran Fujtsawa Campus,5322 Endoh, Fuj,sawa 252, JapanCognmve Science Douglas R. Hofstadter, Center for Researchon Concepts and Cognition. Ind~ara Unlvers*ty,Bloomington, lndmnz 47408, U S,AComputmg and Apphed Mathemattcs Sergei P. Kurdyumov.Insutut pr~kladnot matematlk~ ~m. M V Keldysha RAN(M V Keldysh lastilute of Apphed Mathematrcs. RussmnAcademy of Sciences). 125047 Mosk’~a, Miusskaya pl. 4, RussmEducatton. Peter Klein, FB E~ehungsw~sseaschafi.Umvers~tat Hamburg, Von-Melle-Park 8,D-20146 Hamburg 13. F R. GermanyHzstory and Phdosophy of Science" Klaus Mainzer,Lehrstuhl ~ur Philosoph~e, Umversltat Augsburg,Umvers~mLsstr. 10, D-W-8900 Augsburg, F R GermanyProJect ChatrpersonsArchttecture and Mustc Emanuel Dimas de Melo Pimento,Run T~eruo Galvan, Lute 5B - 2 °C, P-1200 Lisbon, FonugalArt and Biology Werner Hahn, Waldweg 8, D-35075Gladenbach, FR GermanyEvolutton of the Umverse: Jan Mnzrzymas, [nstytut Flzyk~.Umwersytet Wroct’awski(Insmute of Theoreucal Physics, Umvers~ty of Wroctaw),ul Cybulskiego 36, PL 50-205 Wrocl’aw, Pc, landH~gher-D~menstonal Graphics Koji Miyazak~,Department of Graphms, College of L~beral Arts.Kyoto Umvers~ty, YosMda, Sakyo-ku, Kyoto 606. JapanKnovdedge Representatton by Meta.strucmres Ted Goranson,S~nus Incorporated. 1976 Munden Point, V~rgmm Beach,VA 23457-1227. U.S A.Pattern Mathemattcs Berl Zaslow,Department of Chemistry, Arizona State Umverstty. Tempe,AZ 85287-1604, U S A.Polyhedral Transformations Haresh Lalvani,School of Arch.ecture, Pratt Insutute, 200 Wdloughby Avenue,Brooklyn, NY 11205, U S AProportion and Harmony m Arts S. K, Hemnger, Jr.Department of Enghsh, Umvers~ty of Norlh Carolina at ChapelHdl, Chapel Hill. NC 27599-3520, U S A.Shape Grammar" George Stiny, Graduate School of Architectureand Urban Planmng, Umvers~ty of Cahforma Los Angeles,Los Angeles, CA 90024-1467, U.S.A.Space Structures. Koryo M~ura, 3-9-7 Tsurukawa, Mach~da,Tokyo 195, JapanTibor Tarnai, Techmcal Umvers~ty of Budapest,Depanmem of C~vtl Engineering Mechamcs.Budapest, Mfegyetem rkp. 3, H-IIII HungaryLiaison PersonsAndra Akers (Internattonal Synergy Institute)Stephen G. Davies (Journal Tetrahedran Assyrnmetry)Bruno Gruber (Symposta Symmetries m Science)Alajos K~lm~in (lnternattonal Umon of Crystallography)Roger F. Malina (Journal Leonardo and lnternauonal Society forthe Arts, Sciences, and Technology)Tohru Ogawa and Ryuji Takaki (Journal Forma and Socmty [’orScmnce on Form)Dennis Sharp (Curule6 lnternauonal des Cn.quesd’A rch~tecture)Erz.~bet Tusa (INTART Socmty)

ISiS-SymmetryBudapest, P.O. Box 4H-1361 HungaryCALL FORPAPERS AND WORKSHOP TOPICSSYMMETRY:NATURAL AND ARTIFICIALThird InterdisciplinarySymmetry Congressof theINTERNATIONAL SOCIETY FORTHE INTERDISCIPLINARYSTUDY OF SYMMETRY(ISIS-SYMMETRY)August 14-20, 1995Old Town Alexandria(near Washington, D.C.)U.S.A.aq~ .~ouo!qq!qx~[ Laomm£SLn;u!ld!os!p~muI P~!qz:A~LLCJIAIINASS V~J~ I NOLLKtlHX~~oa ~qvD~6unHf, xo8 "O’d ’lsedepn~]fu~,em LuAS-S~S~