In geometry, a Wythoff symbol is a shorthand 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.
Contents 
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 
Schläfli symbol 
regular and quasiregular 
q  p r  (p.r)^{q}  q  p 2  p^{q}  {p,q} 
p  q r  (q.r)^{p}  p  q 2  q^{p}  {q,p}  
r  p q  (q.p)^{r}  2  p q  (q.p)^{2}  t_{1}{p,q}  
truncated and expanded 
q r  p  q.2p.r.2p  q 2  p  q.2p.2p  t_{0,1}{p,q} 
p r  q  p. 2q.r.2q  p 2  q  p. 2q.2q  t_{0,1}{q,p}  
p q  r  2r.q.2r.p  p q  2  4.q.4.p  t_{0,2}{p,q}  
evenfaced  p q r   2r.2q.2p  p q 2   4.2q.2p  t_{0,1,2}{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:
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 (2^{3}) possible forms, neglecting one where the generator point is on all the mirrors.
In this notation the mirrors are labeled by the reflectionorder 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 oddnumbered reflected images are ignored. The resulting figure has rotational symmetry only.
This symbol is functionally similar to the more general CoxeterDynkin 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.
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.
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 nonconvex uniform polyhedra.
p4m  p3m  p6m 

*442  *333  *632 
(4 4 2) 
(3 3 3) 
(6 3 2) 
*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.
Selected tilings created by the Wythoff construction are given below.
(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  t_{0}{p,q}  t_{0,1}{p,q}  t_{1}{p,q}  t_{1,2}{p,q}  t_{2}{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) 
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 

t_{0}{p,2}  t_{0,1}{p,2}  t_{1,2}{p,2}  t_{2}{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  p^{2}  (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 

... 
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  t_{0}{p,q}  t_{0,1}{p,q}  t_{1}{p,q}  t_{1,2}{p,q}  t_{2}{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) 
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 
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}  (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 
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  t_{0}{p,q}  t_{0,1}{p,q}  t_{1}{p,q}  t_{1,2}{p,q}  t_{2}{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) 
