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

In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra.

A Wythoff symbol consists of three numbers and a vertical bar. It 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 with Oh symmetry, and as a square prism with 2 colors and D4h symmetry, as well as with 3 colors and symmetry.

With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space.

Contents

DescriptionEdit

In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge. A perpendicular line is then dropped between the generator point and every face that it does not lie on.

The three numbers in Wythoff's symbol,  ,   and  , represent the corners of the Schwarz triangle used in the construction, which are  ,   and   radians respectively. The triangle is also represented with the same numbers, written  . The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following:

  •   indicates that the generator lies on the corner  ,
  •   indicates that the generator lies on the edge between   and  ,
  •   indicates that the generator lies in the interior of the triangle.

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

The one impossible symbol   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.

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.

The Wythoff 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.

Summary tableEdit

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

General Right triangle (r=2)
Description Wythoff
symbol
Vertex
configuration
Coxeter
diagram

 
Wythoff
symbol
Vertex
configuration
Schläfli
symbol
Coxeter
diagram
     
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 r{p,q} t1{p,q}      
truncated and
expanded
q r | p           q 2 | p   t{p,q} t0,1{p,q}      
p r | q           p 2 | q p. 2q.2q t{q,p} t0,1{q,p}      
p q | r           p q | 2   rr{p,q} t0,2{p,q}      
even-faced p q r |           p q 2 |   tr{p,q} t0,1,2{p,q}      
p q (r s) |   - p 2 (r s) | 2p.4.-2p.4/3 -
snub | p q r           | p q 2   sr{p,q}      
| p q r s   - - - -

There are three special cases:

  •   – This is a mixture of   and  , containing only the faces shared by both.
  •   – Snub forms (alternated) are given by this otherwise unused symbol.
  •   – A unique snub form for U75 that isn't Wythoff-constructible.

Symmetry trianglesEdit

There are 4 symmetry classes of reflection on the sphere, and three 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.)

Point groups:

Euclidean (affine) groups:

Hyperbolic groups:

Dihedral spherical Spherical
D2h D3h D4h D5h D6h Td Oh Ih
*222 *322 *422 *522 *622 *332 *432 *532
 
(2 2 2)
 
(3 2 2)
 
(4 2 2)
 
(5 2 2)
 
(6 2 2)
 
(3 3 2)
 
(4 3 2)
 
(5 3 2)

The above symmetry groups only include 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, Euclidean and hyperbolic tilingsEdit

Selected tilings created by the Wythoff construction are given below.

Spherical tilings (r = 2)Edit

(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
               
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
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}
Coxeter
diagram
                                               
Vertex figure pq q.2p.2p (p.q)2 p. 2q.2q qp p. 4.q.4 4.2p.2q 3.3.p. 3.q
 
(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)
 
(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)
 
(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)

Some overlapping spherical tilings (r = 2)Edit

For a more complete list, including cases where r ≠ 2, see List of uniform polyhedra by Schwarz triangle.

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

(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                
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
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}
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)

Dihedral symmetry (q = r = 2)Edit

Spherical tilings with dihedral symmetry exist for all   many with digon faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantillated) are replications and are skipped in the table.

(p 2 2)
Fundamental
domain
Parent Truncated Bitruncated
(truncated dual)
Birectified
(dual)
Omnitruncated
(Cantitruncated)
Snub
Extended
Schläfli symbol
           
{p,2} t{p,2} t{2,p} {2,p} tr{p,2} sr{p,2}
t0{p,2} t0,1{p,2} t1,2{p,2} t2{p,2} t0,1,2{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 (2.2p.2p) (4.4.p) 2p (4.2p.4) (3.3.p. 3)
 
(2 2 2)
V2.2.2
 
{2,2}
 
2.4.4
4.4.2  
{2,2}
 
4.4.4
 
3.3.3.2
 
(3 2 2)
V3.2.2
 
{3,2}
 
2.6.6
 
4.4.3
 
{2,3}
 
4.4.6
 
3.3.3.3
 
(4 2 2)
V4.2.2
 
{4,2}
2.8.8  
4.4.4
 
{2,4}
 
4.4.8
 
3.3.3.4
 
(5 2 2)
V5.2.2
 
{5,2}
2.10.10  
4.4.5
 
{2,5}
 
4.4.10
 
3.3.3.5
 
(6 2 2)
V6.2.2
 
{6,2}
 
2.12.12
 
4.4.6
 
{2,6}
 
4.4.12
 
3.3.3.6
...

Euclidean and hyperbolic tilings (r = 2)Edit

Some representative hyperbolic tilings are 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                
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
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}
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)
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)
 
V4.6.14
 
{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)
 
V4.6.16
 
{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
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)
 
V4.8.10
 
{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)
(6 4 2)
 
V4.8.12
 
{6,4}
 
4.12.12
 
4.6.4.6
 
6.8.8
 
{4,6}
 
4.4.6.4
 
4.8.12
 
3.3.4.3.6
(Hyperbolic plane)
(7 4 2)
 
V4.8.14
 
{7,4}
 
4.14.14
 
4.7.4.7
 
7.8.8
 
{4,7}
 
4.4.7.4
 
4.8.14
 
3.3.4.3.7
(Hyperbolic plane)
(8 4 2)
 
V4.8.16
 
{8,4}
 
4.16.16
 
4.8.4.8
 
8.8.8
 
{4,8}
 
4.4.8.4
 
4.8.16
 
3.3.4.3.8
(Hyperbolic plane)
(5 5 2)
 
V4.10.10
 
{5,5}
 
5.10.10
 
5.5.5.5
 
5.10.10
 
{5,5}
 
5.4.5.4
 
4.10.10
 
3.3.5.3.5
(Hyperbolic plane)
(6 5 2)
 
V4.10.12
 
{6,5}
 
5.12.12
 
5.6.5.6
 
6.10.10
 
{5,6}
 
5.4.6.4
 
4.10.12
 
3.3.5.3.6
(Hyperbolic plane)
(7 5 2)
 
V4.10.14
 
{7,5}
 
5.14.14
 
5.7.5.7
 
7.10.10
 
{5,7}
 
5.4.7.4
 
4.10.14
 
3.3.5.3.7
(Hyperbolic plane)
(8 5 2)
 
V4.10.16
 
{8,5}
 
5.16.16
 
5.8.5.8
 
8.10.10
 
{5,8}
 
5.4.8.4
 
4.10.16
3.3.5.3.8
(Hyperbolic plane)
(6 6 2)
 
V4.12.12
 
{6,6}
 
6.12.12
 
6.6.6.6
 
6.12.12
 
{6,6}
 
6.4.6.4
 
4.12.12
 
3.3.6.3.6
(Hyperbolic plane)
(7 6 2)
 
V4.12.14
 
{7,6}
 
6.14.14
 
6.7.6.7
 
7.12.12
 
{6,7}
 
6.4.7.4
 
4.12.14
3.3.6.3.7
(Hyperbolic plane)
(8 6 2)
 
V4.12.16
 
{8,6}
 
6.16.16
 
6.8.6.8
 
8.12.12
 
{6,8}
 
6.4.8.4
 
4.12.16
 
3.3.6.3.8
(Hyperbolic plane)
(7 7 2)
 
V4.14.14
 
{7,7}
 
7.14.14
 
7.7.7.7
 
7.14.14
 
{7,7}
 
7.4.7.4
 
4.14.14
 
3.3.7.3.7
(Hyperbolic plane)
(8 7 2)
 
V4.14.16
 
{8,7}
 
7.16.16
 
7.8.7.8
 
8.14.14
 
{7,8}
 
7.4.8.4
 
4.14.16
3.3.7.3.8
(Hyperbolic plane)
(8 8 2)
 
V4.16.16
 
{8,8}
 
8.16.16
 
8.8.8.8
 
8.16.16
 
{8,8}
 
8.4.8.4
 
4.16.16
 
3.3.8.3.8
(Hyperbolic plane)
(∞ 3 2)
 
V4.6.∞
 
{∞,3}
 
3.∞.∞
 
3.∞.3.∞
 
∞.6.6
 
{3,∞}
 
3.4.∞.4
 
4.6.∞
 
3.3.3.3.∞
(Hyperbolic plane)
(∞ 4 2)
 
V4.8.∞
 
{∞,4}
 
4.∞.∞
 
4.∞.4.∞
 
∞.8.8
 
{4,∞}
 
4.4.∞.4
 
4.8.∞
 
3.3.4.3.∞
(Hyperbolic plane)
(∞ 5 2)
 
V4.10.∞
 
{∞,5}
 
5.∞.∞
 
5.∞.5.∞
 
∞.10.10
 
{5,∞}
 
5.4.∞.4
 
4.10.∞
 
3.3.5.3.∞
(Hyperbolic plane)
(∞ 6 2)
 
V4.12.∞
 
{∞,6}
 
6.∞.∞
 
6.∞.6.∞
 
∞.12.12
 
{6,∞}
 
6.4.∞.4
 
4.12.∞
 
3.3.6.3.∞
(Hyperbolic plane)
(∞ 7 2)
 
V4.14.∞
 
{∞,7}
 
7.∞.∞
 
7.∞.7.∞
 
∞.14.14
 
{7,∞}
 
7.4.∞.4
 
4.14.∞
3.3.7.3.∞
(Hyperbolic plane)
(∞ 8 2)
 
V4.16.∞
 
{∞,8}
 
8.∞.∞
 
8.∞.8.∞
 
∞.16.16
 
{8,∞}
 
8.4.∞.4
 
4.16.∞
3.3.8.3.∞
(Hyperbolic plane)
(∞ ∞ 2)
 
V4.∞.∞
 
{∞,∞}
 
∞.∞.∞
 
∞.∞.∞.∞
 
∞.∞.∞
 
{∞,∞}
 
∞.4.∞.4
 
4.∞.∞
 
3.3.∞.3.∞

Euclidean and hyperbolic tilings (r > 2)Edit

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
Schläfli symbol (p,q,r) r(r,q,p) (q,r,p) r(p,q,r) (q,p,r) r(p,r,q) tr(p,q,r) s(p,q,r)
t0(p,q,r) t0,1(p,q,r) t1(p,q,r) t1,2(p,q,r) t2(p,q,r) t0,2(p,q,r) t0,1,2(p,q,r)
Coxeter diagram                                                                
Vertex figure (p.r)q (r.2p.q.2p) (p.q)r (q.2r.p. 2r) (q.r)p (p. 2r.q.2r) (2p.2q.2r) (3.r.3.q.3.p)
Euclidean
(3 3 3)
   
 
V6.6.6
 
(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)
    
 
V6.6.8
 
(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)
   
 
V6.8.8
 
(3.4)4
 
3.8.4.8
 
(4.4)3
 
3.8.4.8
 
(3.4)4
 
4.6.4.6
 
6.8.8
 
3.3.3.4.3.4
Hyperbolic
(4 4 4)
    
 
V8.8.8
 
(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
(5 3 3)
    
 
V6.6.10
 
(3.5)3
 
3.10.3.10
 
(3.5)3
 
3.6.5.6
 
(3.3)5
 
3.6.5.6
 
6.6.10
3.3.3.3.3.5
Hyperbolic
(5 4 3)
    
 
V6.8.10
 
(3.5)4
 
3.10.4.10
 
(4.5)3
 
3.8.5.8
 
(3.4)5
 
4.6.5.6
 
6.8.10
 
3.5.3.4.3.3
Hyperbolic
(5 4 4)
    
 
V8.8.10
 
(4.5)4
 
4.10.4.10
 
(4.5)4
 
4.8.5.8
 
(4.4)5
 
4.8.5.8
 
8.8.10
3.4.3.4.3.5
Hyperbolic
(6 3 3)
    
 
V6.6.12
 
(3.6)3
 
3.12.3.12
 
(3.6)3
 
3.6.6.6
 
(3.3)6
 
3.6.6.6
 
6.6.12
3.3.3.3.3.6
Hyperbolic
(6 4 3)
    
 
V6.8.12
 
(3.6)4
 
3.12.4.12
 
(4.6)3
 
3.8.6.8
 
(3.4)6
 
4.6.6.6
 
6.8.12
3.6.3.4.3.3
Hyperbolic
(6 4 4)
    
 
V8.8.12
 
(4.6)4
 
4.12.4.12
 
(4.6)4
 
4.8.6.8
 
(4.4)6
 
4.8.6.8
 
8.8.12
3.6.3.4.3.4
Hyperbolic
(∞ 3 3)
    
 
V6.6.∞
 
(3.∞)3
 
3.∞.3.∞
 
(3.∞)3
 
3.6.∞.6
 
(3.3)
 
3.6.∞.6
 
6.6.∞
3.3.3.3.3.∞
Hyperbolic
(∞ 4 3)
    
 
V6.8.∞
 
(3.∞)4
 
3.∞.4.∞
 
(4.∞)3
 
3.8.∞.8
 
(3.4)
 
4.6.∞.6
 
6.8.∞
3.∞.3.4.3.3
Hyperbolic
(∞ 4 4)
    
 
V8.8.∞
 
(4.∞)4
 
4.∞.4.∞
 
(4.∞)4
 
4.8.∞.8
 
(4.4)
 
4.8.∞.8
 
8.8.∞
3.∞.3.4.3.4
Hyperbolic
(∞ ∞ 3)
   
 
V6.∞.∞
 
(3.∞)
 
3.∞.∞.∞
 
(∞.∞)3
 
3.∞.∞.∞
 
(3.∞)
 
∞.6.∞.6
 
6.∞.∞
3.∞.3.∞.3.3
Hyperbolic
(∞ ∞ 4)
    
 
V8.∞.∞
 
(4.∞)
 
4.∞.∞.∞
 
(∞.∞)4
 
4.∞.∞.∞
 
(4.∞)
 
∞.8.∞.8
 
8.∞.∞
3.∞.3.∞.3.4
Hyperbolic
(∞ ∞ ∞)
    
 
V∞.∞.∞
 
(∞.∞)
 
∞.∞.∞.∞
 
(∞.∞)
 
∞.∞.∞.∞
 
(∞.∞)
 
∞.∞.∞.∞
 
∞.∞.∞
 
3.∞.3.∞.3.∞

See alsoEdit

ReferencesEdit

  • 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 linksEdit