Wythoff symbol

Example Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.

In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.

A Schwarz triangle is a triangle that covers the sphere or the plane a finite number of times by reflection in its edges. The usual representation for the triangle is three numbers - integers or fractions - where π/x represents the angle at that vertex. For example, the triangle (2 3 4) represents the symmetry of a cube, while (5/2 5/2 5/2) is the face of an icosahedron.

Wythoff's construction in three dimensions is by placing a vertex in the triangle, and dropping perpendiculars to each of the edges. The uniform cases must exist in all cases, correspond to when the perpendiculars are either 0 (giving no edge) or 1 (giving a unit edge).

The edges of the triangle opposite an angle are named after the angle, so an edge opposite the right angle would be designated '2'. The symbol then corresponds to a representation of off | on. Each of the numbers p in the symbol becomes a polygon pn, where n is the number of other edges that appear before the bar. So in 3 | 4 2, at the 3 corner of the triangle forms a point (literally something with 3*0 sides), while the 4 gives 4*1 = square the digon 2*1 disappears as an edge.

The special case of the snub figures is done by using the symbol | p q r. This would normally evaluate as the point in the centre of the sphere. The faces of a snub alternate as p 3 q 3 r 3. This gives an antiprism when q=r=2.

Each symbol represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with O_h symmetry, and 2 4 | 2 as a square prism with 2 colors and D_4h symmetry, as well as 2 2 2 | with 3 colors and D_2h symmetry.

It can be applied with a slight extension to all uniform polyhedra, but the construction methods do not lead to all uniform tilings in euclidean or hyperbolic space.

Summary table

The eight forms for the Wythoff constructions from a general triangle (p q r).

There are seven generator points with each set of p,q,r (and a few special forms):

General			Right triangle (r=2)
Description	Wythoff symbol	Vertex configuration	Wythoff symbol	Vertex configuration	Schläfli symbol
regular and quasiregular	q | p r	(p.r)q	q | p 2	pq	{p,q}
	p | q r	(q.r)p	p | q 2	qp	{q,p}
	r | p q	(q.p)r	2 | p q	(q.p)²	t1{p,q}
truncated and expanded	q r | p	q.2p.r.2p	q 2 | p	q.2p.2p	t0,1{p,q}
	p r | q	p. 2q.r.2q	p 2 | q	p. 2q.2q	t0,1{q,p}
	p q | r	2r.q.2r.p	p q | 2	4.q.4.p	t0,2{p,q}
even-faced	p q r |	2r.2q.2p	p q 2 |	4.2q.2p	t0,1,2{p,q}
	p q (r s) |	2p.2q.-2p.-2q	p 2 (r s) |	2p.4.-2p.4/3
snub	| p q r	3.r.3.q.3.p	| p q 2	3.3.q.3.p	s{p,q}
	| p q r s	(4.p. 4.q.4.r.4.s)/2	-	-

There are three special cases:

• p q (r s) | – This is a mixture of p q r | and p q s |.
• | p q r – Snub forms (alternated) are give this otherwise unused symbol.
• | p q r s – A unique snub form for U75 that isn't Wythoff-constructible.

Description

The numbers p,q,r describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of π/p, π/q, π/r. On the sphere there are 3 main symmetry types: (3 3 2), (4 3 2), (5 3 2), and one infinite family (p 2 2), for any p. (All simple families have one right angle and so r=2.)

The position of the vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle. The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2³) possible forms, neglecting one where the generator point is on all the mirrors.

In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The p,q,r values are listed before the bar if the corresponding mirror is active.

The one impossible symbol | p q r implies the generator point is on all mirrors, which is only possible if the triangle is degenerate, reduced to a point. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.

This symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.

Symmetry triangles

There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.)

• (p 2 2) dihedral symmetry, p = 2, 3, 4... (order 4p)
• (3 3 2) tetrahedral symmetry (order 24)
• (3 3 3) *333 symmetry (Euclidean plane)
• (4 3 2) octahedral symmetry (order 48)
• (4 3 3) *433 symmetry (hyperbolic plane)
• (4 4 2) *442 symmetry: 45°-45°-90° triangle
• (4 4 3) *443 symmetry (hyperbolic plane)
• (5 3 2) icosahedral symmetry (order 120)
• (5 4 2) *542 symmetry (hyperbolic plane)
• (6 3 2) *632 symmetry: 30°-60°-90° triangle
• (7 3 2) *732 symmetry (hyperbolic plane)

Dihedral spherical		Spherical
D2h	D3h	Td	Oh	Ih
*222	*322	*332	*432	*532
(2 2 2)	(3 2 2)	( 3 3 2)	(4 3 2)	(5 3 2)

The above symmetry groups only includes the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, and determine the full set of solutions of nonconvex uniform polyhedra.

Euclidean plane

p4m	p3m	p6m
*442	*333	*632
(4 4 2)	(3 3 3)	(6 3 2)

Hyperbolic plane

*732	*542	*433
(7 3 2)	(5 4 2)	(4 3 3)

In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections.

Summary spherical and plane tilings

Selected tilings created by the Wythoff construction are given below.

Spherical tilings (r = 2)

(p q 2)	Fund. triangles	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff symbol		q | p 2	2 q | p	2 | p q	2 p | q	p | q 2	p q | 2	p q 2 |	| p q 2
Schläfli symbol		t0{p,q}	t0,1{p,q}	t1{p,q}	t1,2{p,q}	t2{p,q}	t0,2{p,q}	t0,1,2{p,q}	s{p,q}
Coxeter–Dynkin diagram
Vertex figure		pq	(q.2p.2p)	(p.q.p.q)	(p. 2q.2q)	qp	(p. 4.q.4)	(4.2p.2q)	(3.3.p. 3.q)
Tetrahedral (3 3 2)		{3,3}	(3.6.6)	(3.3a.3.3a)	(3.6.6)	{3,3}	(3a.4.3b.4)	(4.6a.6b)	(3.3.3a.3.3b)
Octahedral (4 3 2)		{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4a.4)	(4.6.8)	(3.3.3a.3.4)
Icosahedral (5 3 2)		{5,3}	(3.10.10)	(3.5.3.5)	(5.6.6)	{3,5}	(3.4.5.4)	(4.6.10)	(3.3.3a.3.5)

Dihedral symmetry (q = r = 2)

Spherical tilings with dihedral symmetry exist for all p = 2, 3, 4, ... many with digon faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantellated) are replications and are skipped in the table.

(p 2 2)	Fund. triangles	Parent	Truncated	Bitruncated (truncated dual)	Birectified (dual)	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol		$\begin{Bmatrix} p , 2 \end{Bmatrix}$	$t\begin{Bmatrix} p , 2 \end{Bmatrix}$	$t\begin{Bmatrix} 2 , p \end{Bmatrix}$	$\begin{Bmatrix} 2 , p \end{Bmatrix}$	$t\begin{Bmatrix} p \\ 2 \end{Bmatrix}$	$s\begin{Bmatrix} p \\ 2 \end{Bmatrix}$
		t0{p,2}	t0,1{p,2}	t1,2{p,2}	t2{p,2}	t0,1,2{p,2}	s{p,2}
Wythoff symbol		2 | p 2	2 2 | p	2 p | 2	p | 2 2	p 2 2 |	| p 2 2
Coxeter–Dynkin diagram
Vertex figure		p²	(2.2p.2p)	(p.p)	2p	(4.2p.4)	(3.3.p. 3)
(2 2 2)		{2,2}	2.4.4	4.4.2	{2,2}	4.4.4	3.3.3.2
(3 2 2)		{3,2}	2.6.6	4.4.3	{2,3}	4.4.6	3.3.3.3
(4 2 2)		{4,2}	2.8.8	4.4.4	{2,4}	4.4.8	3.3.3.4
(5 2 2)		{5,2}	2.10.10	4.4.5	{2,5}	4.4.10	3.3.3.5
(6 2 2)		{6,2}	2.12.12	4.4.6	{2,6}	4.4.12	3.3.3.6
...

Planar tilings (r = 2)

One representative hyperbolic tiling is given, and shown as a Poincaré disk projection.

(p q 2)	Fund. triangles	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff symbol		q | p 2	2 q | p	2 | p q	2 p | q	p | q 2	p q | 2	p q 2 |	| p q 2
Schläfli symbol		t0{p,q}	t0,1{p,q}	t1{p,q}	t1,2{p,q}	t2{p,q}	t0,2{p,q}	t0,1,2{p,q}	s{p,q}
Coxeter–Dynkin diagram
Vertex figure		pq	(q.2p.2p)	(p.q.p.q)	(p. 2q.2q)	qp	(p. 4.q.4)	(4.2p.2q)	(3.3.p. 3.q)
Square tiling (4 4 2)	V4.8.8	{4,4}	4.8.8	4.4a.4.4a	4.8.8	{4,4}	4.4a.4b.4a	4.8.8	3.3.4a.3.4b
(Hyperbolic plane) (5 4 2)		{5,4}	4.10.10	4.5.4.5	5.8.8	{4,5}	4.4.5.4	4.8.10	3.3.4.3.5
(Hyperbolic plane) (5 5 2)		{5,5}	5.10.10	5.5.5.5	5.10.10	{5,5}	4.4.5.4	4.10.10	3.3.5.3.5
Hexagonal tiling (6 3 2)	V4.6.12	{6,3}	3.12.12	3.6.3.6	6.6.6	{3,6}	3.4.6.4	4.6.12	3.3.3.3.6
(Hyperbolic plane) (7 3 2)		{7,3}	3.14.14	3.7.3.7	7.6.6	{3,7}	3.4.7.4	4.6.14	3.3.3.3.7
(Hyperbolic plane) (8 3 2)		{8,3}	3.16.16	3.8.3.8	8.6.6	{3,8}	3.4.8.4	4.6.16	3.3.3.3.8
(Hyperbolic plane) (∞ 3 2)
(Hyperbolic plane) (∞ 4 2)
(Hyperbolic plane) (∞ ∞ 2)

Planar tilings (r > 2)

The Coxeter–Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.

Wythoff symbol (p q r)	Fund. triangles	q | p r	r q | p	r | p q	r p | q	p | q r	p q | r	p q r |	| p q r
Coxeter–Dynkin diagram
Vertex figure		(p.q)r	(r.2p.q.2p)	(p.r)q	(q.2r.p. 2r)	(q.r)p	(p.2q.r.2q)	(2p.2q.2r)	(3.r.3.q.3.p)
Triangular (3 3 3)		(3.3)3	3.6.3.6	(3.3)3	3.6.3.6	(3.3)3	3.6.3.6	6.6.6	3.3.3.3.3.3
Hyperbolic (4 3 3)		(3.4)³	3.8.3.8	(3.4)³	3.6.4.6	(3.3)4	3.6.4.6	6.6.8	3.3.3.3.3.4
Hyperbolic (4 4 3)		(3.4)4	3.8.4.8	(3.4)4	3.6.4.6	(3.4)4	4.6.4.6	6.8.8	3.3.3.4.3.4
Hyperbolic (4 4 4)		(4.4)4	4.8.4.8	(4.4)4	4.8.4.8	(4.4)4	4.8.4.8	8.8.8	3.4.3.4.3.4
Hyperbolic (3 3 ∞)
Hyperbolic (3 ∞ ∞)
Hyperbolic (∞ ∞ ∞)

Overlapping spherical tilings (r = 2)

Tilings are shown as polyhedra. Some of the forms are degenerate, given with brackets for vertex figures, with overlapping edges or verices.

(p q 2)	Fund. triangle	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff symbol		q | p 2	2 q | p	2 | p q	2 p | q	p | q 2	p q | 2	p q 2 |	| p q 2
Schläfli symbol		t0{p,q}	t0,1{p,q}	t1{p,q}	t1,2{p,q}	t2{p,q}	t0,2{p,q}	t0,1,2{p,q}	s{p,q}
Coxeter–Dynkin diagram
Vertex figure		pq	(q.2p.2p)	(p.q.p.q)	(p. 2q.2q)	qp	(p. 4.q.4)	(4.2p.2q)	(3.3.p. 3.q)
Icosahedral (5/2 3 2)		{3,5/2}	(5/2.6.6)	(3.5/2)2	[3.10/2.10/2]	{5/2,3}	[3.4.5/2.4]	[4.10/2.6]	(3.3.3.3.5/2)
Icosahedral (5 5/2 2)		{5,5/2}	(5/2.10.10)	(5/2.5)2	[5.10/2.10/2]	{5/2,5}	(5/2.4.5.4)	[4.10/2.10]	(3.3.5/2.3.5)

See also

• Regular polytope
• Regular polyhedron
• List of uniform tilings
• List of uniform polyhedra

References

• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401–50.
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.  pp. 9–10.

External links

• Weisstein, Eric W., "Wythoff symbol" from MathWorld.
• The Wythoff symbol
• Wythoff symbol
• Displays Uniform Polyhedra using Wythoff's construction method
• Description of Wythoff Constructions
• KaleidoTile 3 Free educational software for Windows by Jeffrey Weeks that generated many of the images on the page.
• Hatch, Don. "Hyperbolic Planar Tessellations". http://www.plunk.org/~hatch/HyperbolicTesselations.