Deriving Uniform Polyhedrawith

Wythoff’s Construction

Don Romano

UCD Discrete Math Seminar30 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 PolyhedraDef. 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 TrianglesLet 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 restrictionthat 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

(2,3 3) (2,3,4)

(2,3,5) (2,2,r)

The Four Fundamental Spherical Tilings

Tetrahedral Symmetry |g| = 24

Octahedral Symmetry |g| = 48

Icosahedral Symmetry |g| = 120

Dihedral Symmetry |g| = 4n

Schwarz Triangles

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

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)

Schwarz triangles are composed of fundamental Möbius triangles

π/2 π/2π/3

π/5

π/5

π/5

π/5

π/5

Schwarz triangle (5/2 2 3)

density = 7Schwarz triangle (5/2 2 5)

density = 3

(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

Schwarz Triangles — examples

Density Schwarz triangle1 (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)

Complete list of Schwarz triangles sorted by density

Symmetry Groups

5 Tetrahedral

7 Octahedral

32 Icosahedral

2 Dihedral

Wythoff’s Construction

• 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

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

Wythoff — point placementsPoints can be chosen in four ways, each with its own Wythoff Symbol

p | q rPoint is at a vertex P of triangle PQR

p q | rPoint 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 rPoint is the Fermat point and alternating triangles are used

Vertex positions forpolyhedron with Wythoff symbol

2|3 5

Polyhedron with Wythoff symbol 2|3 5

Vertex positions forpolyhedron with Wythoff symbol

2 3|5

Polyhedron with Wythoff symbol2 3|5

Vertex positions forpolyhedron with Wythoff symbol

2 3 5|

Polyhedron with Wythoff symbol2 3 5|

Spherical triangles alternately black and white

The Fermat point

Vertex positions forpolyhedron with Wythoff symbol

|2 3 5

Polyhedron with Wythoff symbol|2 3 5

Wythoff Symbolp|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 CompletenessH.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 = 38Individual surface segments = 3,000

| 3/2 3/2 5/2 {3} (4)

Facial Intersections “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!