Deriving Uniform Polyhedra Wythoff's Construction

Deriving Uniform Polyhedra
with
Wythoff’s Construction
Don Romano
UCD Discrete Math Seminar
30 August 2010

Outline of this talk
• Fundamentals of uniform polyhedra
– Definitions and properties
– Convex solids
– Regular polyhedra
– Nonconvex solids
– Early research
• Systematic approach for deriving uniform polyhedra
– Spherical tessellations
– Wythoff‘s Construction
– The complete enumeration

Definitions and a few properties
Def. A polyhedron is a finite set of polygons such that every side of
each belongs to just one other, with the restriction that no subset
has the same property
Def. A uniform polyhedron is made up of regular polygons and its
vertices are transitive
– Vertex transitivity means there is an isometry (rotation, reflection)
that takes any vertex to any other
– All vertices are congruent and lie on a sphere
There are 75 uniform polyhedra
– 18 convex, 57 nonconvex and an infinite set of prismatoids

Exhibit at London Museum of Science

Budzelaar Collection

Pawlikowski Collection

Convex Uniform Polyhedra
• 5 Platonic Solids
– Faces are regular polygons of only one kind
– Symmetry groups form basis for all uniform
polyhedra
• Tetrahedral, Octahedral, Icosahedral
– Known since antiquity — Euclid‘s Elements
• 13 Archimedean Solids
– Faces can be of more than one kind (2 or 3)
– Can be derived from Platonics by simple
operations of truncation, rectification, and
cantellation
– Two enantiomorphic pairs (snubs)
– First enumeration by J. Kepler (ca. 1600)
• 2 infinite sets of convex prisms and antiprisms
– Dihedral symmetry

Platonic and Archimedean Solids

Regular Polyhedra
Def. A regular polyhedron is made up of only one kind of regular polygon
and vertices are congruent
– The 5 Platonic solids are regular
– 4 Kepler-Poinsot solids are regular and nonconvex
• 2 have star faces, 2 have star vertices
• Derived by stellating or faceting Platonics
“Wayside shrines at
which one should
worship on the way
to higher things”
– Peter McMullen

Nonconvex Uniform Polyhedra
• Can be derived by faceting Archimedean solids
– Star polygons can be inscribed in faces
– Removing one kind of polygon face and inserting others
• Isomeghethic: same edge set
• Many uniform polyhedra discovered between 1878 -1881
– Edmund Hess (2)
– Johann Pitsch (18)
– Albert Badoureau (37)
• Max Brückner
– Vielecke und Vielflache (1900)
• Isogonal-isohedra a.k.a. ‗noble‘ polyhedra

Brückner’s Noble Polyhedra
and many more . . .

Spherical Tessellations
• Spherical triangles are bounded by
segments of great circles
• The sum of the angles of a spherical
triangle are greater than 180 and less
than 540
• Area of spherical triangle
A = r² E, where E is the spherical
excess, that is, the sum of the angles
minus 180
• Only four ways to cover the sphere
(once) with congruent spherical
triangles

Möbius Triangles
Let the angles of a spherical triangle be
π/p, π/q, π/r where p, q, r are integers
The area of the spherical triangle
[(1/p + 1/q + 1/r) -1] π must be positive
Hence, 1/p + 1/q + 1/r > 1.
Only possibilities for p, q, r are 2, 3, 4, 5 with the restriction
that 4 and 5 cannot occur together
These lead to the four fundamental spherical triangles which are
known as Möbius Triangles:
(2,3 3), (2,3,4), (2,3,5), (2,2,r)
Repeated reflections in sides of triangles will tile a sphere exactly once

The Four Fundamental Spherical Tilings
(2,3 3) (2,3,4)
(2,3,5) (2,2,r)

Tetrahedral Symmetry |g| = 24

Octahedral Symmetry |g| = 48

Icosahedral Symmetry |g| = 120

Dihedral Symmetry |g| = 4n

Schwarz Triangles
Karl Schwarz (1873)
• Proposed and solved problem of finding all spherical
triangles which lead, by repeated reflections in their
sides, to a set of congruent triangles covering the
sphere a finite number of times
• Extension of Möbius triangles where p, q, r are
rational, but not necessarily integral
Still have 1/p + 1/q + 1/r > 1 (positive area)
Admissible values for p, q, r are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, 5/4 with
restriction that numerators 4 and 5 cannot occur together
– Some triplets are reducible
– Some do not cover the sphere a finite number of times
• 44 distinct Schwarz triangles and 2 others of infinite variety
• The density, d, of a Schwarz triangle is the number of times sphere
is covered

Schwarz triangles are composed of fundamental Möbius triangles
π/5
π/5
π/5
π/2
Schwarz triangle (5/2 2 5)
density = 3
π/3
π/5
π/5
Schwarz triangle (5/2 2 3)
density = 7
π/2

Schwarz Triangles — examples
(2, 3, 5)
Angles are π/2, π/3, π/5
d = 1
(3, 5, 5/3)
Angles are π/3, π/5, 3π/5
d = 4

Complete list of
Schwarz triangles
sorted by density
Symmetry Groups
5 Tetrahedral
7 Octahedral
32 Icosahedral
2 Dihedral
Density Schwarz triangle
1 (2 3 3), (2 3 4), (2 3 5), (2 2 n)
d (2 2 n/d)
2 (3/2 3 3), (3/2 4 4), (3/2 5 5), (5/2 3 3)
3 (2 3/2 3), (2 5/2 5)
4 (3 4/3 4), (3 5/3 5)
5 (2 3/2 3/2), (2 3/2 4)
6 (3/2 3/2 3/2), (5/2 5/2 5/2), (3/2 3 5), (5/4 5 5)
7 (2 3 4/3), (2 3 5/2)
8 (3/2 5/2 5)
9 (2 5/3 5)
10 (3 5/3 5/2), (3 5/4 5)
11 (2 3/2 4/3), (2 3/2 5)
13 (2 3 5/3)
14 (3/2 4/3 4/3), (3/2 5/2 5/2), (3 3 5/4)
16 (3 5/4 5/2)
17 (2 3/2 5/2)
18 (3/2 3 5/3), (5/3 5/3 5/2)
19 (2 3 5/4)
21 (2 5/4 5/2)
22 (3/2 3/2 5/2)
23 (2 3/2 5/3)
26 (3/2 5/3 5/3)
27 (2 5/4 5/3)
29 (2 3/2 5/4)
32 (3/2 5/4 5/3)
34 (3/2 3/2 5/4)
38 (3/2 5/4 5/4)
42 (5/4 5/4 5/4)

Wythoff’s Construction
Kaleidoscopic construction by tiling the
sphere with a Schwarz triangles along
with a specific point in the triangle
Willem Wythoff (1907) applied this
method to 4-dimensional problems
• A point is chosen in Schwarz triangle
• Repeated reflections of triangles produce multiple instances of that
point around sphere
• If suitable points are chosen, they will generate the vertices of a
uniform polyhedron

Wythoff — point placements
Points can be chosen in four ways, each with its own Wythoff Symbol
p | q r
Point is at a vertex P of triangle PQR
p q | r
Point is on side of PQ such that it bisects the angle at R
p q r |
Point is at the incenter of triangle PQR (intersection of angle bisectors)
| p q r
Point is the Fermat point and alternating triangles are used

Vertex positions for
polyhedron with Wythoff symbol
2|3 5
Polyhedron with Wythoff symbol
2|3 5

Vertex positions for
polyhedron with Wythoff symbol
2 3|5
Polyhedron with Wythoff symbol
2 3|5

Vertex positions for
polyhedron with Wythoff symbol
2 3 5|
Polyhedron with Wythoff symbol
2 3 5|

The Fermat point
Spherical triangles alternately black and white

Vertex positions for
polyhedron with Wythoff symbol
|2 3 5
Polyhedron with Wythoff symbol
|2 3 5

Wythoff Symbol
p|qr
• Quasi-regular (16 polyhedrons)
• Vertex configuration {q, r, q, r, . . . q, r}
– Regular if r = 2 or q = r
pq|r
• Semi-regular (33 polyhedrons)
• Vertex configuration {p, 2r, q, 2r}
pqr|
• Even-faced (14 polyhedrons)
• Vertex configuration {2p, 2q, 2r}
|pqr
• Snub (11 polyhedrons)
• Vertex configuration {3, p, 3, q, 3, r}

Non-Wythoffian Polyhedron
• Great Dirhombicosidodecahedron
– Discovered by J.C.P. Miller
– ‗Miller‘s Monster‘
• Found by combining both
enantiomorphs of |3 5/3 5/2
• Only uniform polyhedron with 8
faces surrounding each vertex
• Largest number of faces (124) and
edges (240)
• Euler characteristic Χ = - 56
• Has 60 diametral squares that can
be considered snub faces
• Existence indicated no general
reason for restriction to triangles as
snub faces
Vertex figure

Enumeration and Proof of Completeness
H.S.M. Coxeter, M. S. Longuet-Higgins, J.C.P Miller
• “Uniform Polyhedra”, Philosophical Transactions of
the Royal Society of London, 1954
• Complete enumeration of the 75
• Conjectured that list was complete
S.P. Sopov
• “Proof of the Completeness of the Enumeration of
Uniform Polyhedra”, Ukrain. Geom. Sbornik, 1970
J. Skilling
• ―The Complete Set of Uniform Polyhedra‖,
Philosophical Transactions of the Royal Society of
London, 1975
• Computer search examined all possible polygon
configurations for the basic symmetry groups
• ―Skilling‘s Figure‖ was found by relaxing definition of
uniform polyhedron to allow more than two faces at
an edge
Donald Coxeter
John Skilling

Skillings Figure

Facial Intersections (Think inside the box!)
| 3/2 3/2 5/2 {3} (1)
Polyhedral density = 38
Individual surface segments = 3,000

Facial Intersections
| 3/2 3/2 5/2 {3} (4)
“Geometry is a skill of the eyes and
hands as well as the mind.”
- J. Pederson

Caution: Facial Intersections may be hazardous to your mental health!
| 3/2 3/2 5/2 {5/2} (3)

A novice tackles
Miller’s Monster
(ca. 1973)

Uniform polytopes exist in higher dimensions!