Wythoff symbol

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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, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a kaleidoscopic construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane.

The Wythoff symbol gives 3 numbers p,q,r and a positional vertical bar (|) which separate the numbers before or after it. Each number represents the order of mirrors at a vertex of the fundamental triangle.

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.

1 Summary table
2 Description
3 Symmetry triangles
4 Summary spherical and plane tilings
5 See also
6 References
7 External links

[edit] Summary table

The 8 forms for the Wythoff constructions from a general triangle (p q r).

There are 7 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
regular and quasiregular	q \| p r	(p.r)^q	q \| p 2	p^q
	p \| q r	(q.r)^p	p \| q 2	q^p
	r \| p q	(q.p)^r	2 \| p q	(q.p)²
truncated and expanded	q r \| p	q.2p.r.2p	q 2 \| p	q.2p.2p
	p r \| q	p.2q.r.2q	p 2 \| q	p.2q.2q
	p q \| r	2r.q.2r.p	p q \| 2	4.q.4.p
even-faced	p q r \|	2r.2q.2p	p q 2 \|	4.2q.2p
even-faced	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
snub	\| 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 constructable.

[edit] 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 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.

[edit] Symmetry triangles

There are 4 symmetry classes of reflection on the sphere, and two for the Euclidean plane. A few of the infinitely many for the hyperbolic plane are also listed.

(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 3) *433 symmetry (Hyperbolic plane)
(4 4 3) *443 symmetry (Hyperbolic plane)
(4 4 4) *444 symmetry (Hyperbolic plane)
(4 3 2) octahedral symmetry (Order 48)
(4 4 2) - *442 symmetry - 45-45-90 triangle (Includes square domain (2 2 2 2))
(5 3 2) icosahedral symmetry (Order 120)
(5 4 2) - *542 symmetry (Hyperbolic plane)
(5 5 2) - *552 symmetry (Hyperbolic plane)
(3 3 3) - *333 symmetry - 60-60-60 triangle
(6 3 2) - *632 symmetry - 30-60-90 triangle
(7 3 2) - *732 symmetry (Hyperbolic plane)
(8 3 2) - *832 symmetry (Hyperbolic plane)

Dihedral spherical		Spherical
D_2h	D_3h	T_d	O_h	I_h
*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 uniform polyhedrons.

Euclidean plane			Hyperbolic plane
p4m	p3m	p6m
*442	*333	*632	*732	*542	*433
(4 4 2)	(3 3 3)	(6 3 2)	(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.

[edit] Summary spherical and plane tilings

An selection of tilings created by the Wythoff construction are given below.

[edit] Spherical tilings (r=2)

(p q 2)	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}	t_0,1{p,q}	t₁{p,q}	t_1,2{p,q}	t₂{p,q}	t_0,2{p,q}	t_0,1,2{p,q}	s{p,q}
Coxeter-Dynkin diagram
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(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)

[edit] 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)	Parent	Truncated	Bitruncated (truncated dual)	Birectified (dual)	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol	$\begin{Bmatrix} p , 2 \end{Bmatrix}$	$t\begin{Bmatrix} p , 2 \end{Bmatrix}$	$t\begin{Bmatrix} 2 , p \end{Bmatrix}$	$\begin{Bmatrix} 2 , p \end{Bmatrix}$	$t\begin{Bmatrix} p \\ 2 \end{Bmatrix}$	$s\begin{Bmatrix} p \\ 2 \end{Bmatrix}$
Extended Schläfli symbol	t₀{p,2}	t_0,1{p,2}	t_1,2{p,2}	t₂{p,2}	t_0,1,2{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)	2^p	(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
...

[edit] Planar tilings (r=2)

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

(p q 2)	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}	t_0,1{p,q}	t₁{p,q}	t_1,2{p,q}	t₂{p,q}	t_0,2{p,q}	t_0,1,2{p,q}	s{p,q}
Coxeter-Dynkin diagram
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Square tiling (4 4 2)	{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)	{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

[edit] 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)	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	(q.2r.p.2r)	(r.2q.p.2q)	(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

[edit] 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)	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}	t_0,1{p,q}	t₁{p,q}	t_1,2{p,q}	t₂{p,q}	t_0,2{p,q}	t_0,1,2{p,q}	s{p,q}
Coxeter-Dynkin diagram
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(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)

[edit] See also

[edit] 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.

[edit] External links

Eric W. Weisstein, Wythoff symbol at 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.

Categories: Polyhedra | Polytopes