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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 Oh symmetry, and 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D2h symmetry.

Contents

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
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 constructable.

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 for the Euclidean plane. A few of the infinitely many for the hyperbolic plane are also listed.

  1. (p 2 2) dihedral symmetry p = 2, 3, 4... (Order 4p)
  2. (3 3 2) tetrahedral symmetry (Order 24)
  3. (3 3 3) *333 symmetry (Euclidean plane)
  4. (4 3 3) *433 symmetry (Hyperbolic plane)
  5. (4 4 3) *443 symmetry (Hyperbolic plane)
  6. (4 4 4) *444 symmetry (Hyperbolic plane)
  7. (4 3 2) octahedral symmetry (Order 48)
  8. (4 4 2) - *442 symmetry - 45-45-90 triangle (Includes square domain (2 2 2 2))
  9. (5 3 2) icosahedral symmetry (Order 120)
  10. (5 4 2) - *542 symmetry (Hyperbolic plane)
  11. (5 5 2) - *552 symmetry (Hyperbolic plane)
  12. (3 3 3) - *333 symmetry - 60-60-60 triangle
  13. (6 3 2) - *632 symmetry - 30-60-90 triangle
  14. (7 3 2) - *732 symmetry (Hyperbolic plane)
  15. (8 3 2) - *832 symmetry (Hyperbolic plane)
Dihedral spherical Spherical
D2h D3h Td Oh Ih
*222 *322 *332 *432 *532
Sphere symmetry group d2h.png
(2 2 2)
Sphere symmetry group d3h.png
(3 2 2)
Tetrahedral reflection domains.png
( 3 3 2)
Octahedral reflection domains.png
(4 3 2)
Icosahedral reflection domains.png
(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
Tile V488 bicolor.svg
(4 4 2)
Tile 3,6.svg
(3 3 3)
Tile V46b.svg
(6 3 2)
Hyperbolic plane
*732 *542 *433
Order-3 heptakis heptagonal tiling.png
(7 3 2)
Order-4 bisected pentagonal tiling.png
(5 4 2)
Uniform dual tiling 433-t012.png
(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.

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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 CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.png CDW dot.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.png CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.png CDW hole.pngCDW p.pngCDW hole.pngCDW q.pngCDW hole.png
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)
Tetrahedral reflection domains.png Uniform tiling 332-t0.png
{3,3}
Uniform tiling 332-t01-1-.png
(3.6.6)
Uniform tiling 332-t1.png
(3.3a.3.3a)
Uniform tiling 332-t12.png
(3.6.6)
Uniform tiling 332-t2.png
{3,3}
Uniform tiling 332-t02.png
(3a.4.3b.4)
Uniform tiling 332-t012.png
(4.6a.6b)
Spherical snub tetrahedron.png
(3.3.3a.3.3b)
Octahedral
(4 3 2)
Octahedral reflection domains.png Uniform tiling 432-t0.png
{4,3}
Uniform tiling 432-t01.png
(3.8.8)
Uniform tiling 432-t1.png
(3.4.3.4)
Uniform tiling 432-t12.png
(4.6.6)
Uniform tiling 432-t2.png
{3,4}
Uniform tiling 432-t02.png
(3.4.4a.4)
Uniform tiling 432-t012.png
(4.6.8)
Spherical snub cube.png
(3.3.3a.3.4)
Icosahedral
(5 3 2)
Icosahedral reflection domains.png Uniform tiling 532-t0.png
{5,3}
Uniform tiling 532-t01.png
(3.10.10)
Uniform tiling 532-t1.png
(3.5.3.5)
Uniform tiling 532-t12.png
(5.6.6)
Uniform tiling 532-t2.png
{3,5}
Uniform tiling 532-t02.png
(3.4.5.4)
Uniform tiling 532-t012.png
(4.6.10)
Spherical snub dodecahedron.png
(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 CDW ring.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.png CDW ring.pngCDW p.pngCDW ring.pngCDW 2.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW 2.pngCDW ring.png CDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW ring.png CDW ring.pngCDW p.pngCDW ring.pngCDW 2.pngCDW ring.png CDW hole.pngCDW p.pngCDW hole.pngCDW 2.pngCDW hole.png
Vertex figure p2 (2.2p.2p) (p.p) 2p (4.2p.4) (3.3.p. 3)
(2 2 2) Sphere symmetry group d2h.png Sphere symmetry group cs.png
{2,2}
2.4.4 4.4.2 Sphere symmetry group cs.png
{2,2}
Tetragonal prism.png
4.4.4
Tetrahedron.png
3.3.3.2
(3 2 2) Sphere symmetry group d3h.png Trigonal dihedron.png
{3,2}
Hexagonal dihedron.png
2.6.6
Triangular prism.png
4.4.3
Triangular hosohedron.png
{2,3}
Hexagonal prism.png
4.4.6
Trigonal antiprism.png
3.3.3.3
(4 2 2) {4,2} 2.8.8 Tetragonal prism.png
4.4.4
{2,4} Octagonal prism.png
4.4.8
Square antiprism.png
3.3.3.4
(5 2 2) {5,2} 2.10.10 Pentagonal prism.png
4.4.5
{2,5} Decagonal prism.png
4.4.10
Pentagonal antiprism.png
3.3.3.5
(6 2 2) Hexagonal dihedron.png
{6,2}
2.12.12 Hexagonal prism.png
4.4.6
Hexagonal hosohedron.png
{2,6}
Dodecagonal prism.png
4.4.12
Hexagonal antiprism.png
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 CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.png CDW dot.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.png CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.png CDW hole.pngCDW p.pngCDW hole.pngCDW q.pngCDW hole.png
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)
Tiling Dual Semiregular V4-8-8 Tetrakis Square-2-color-zoom.svg
V4.8.8
Uniform tiling 44-t0.png
{4,4}
Uniform tiling 44-t01.png
4.8.8
Uniform tiling 44-t1.png
4.4a.4.4a
Uniform tiling 44-t12.png
4.8.8
Uniform tiling 44-t2.png
{4,4}
Uniform tiling 44-t02.png
4.4a.4b.4a
Uniform tiling 44-t012.png
4.8.8
Uniform tiling 44-snub.png
3.3.4a.3.4b
(Hyperbolic plane)
(5 4 2)
Order-4 bisected pentagonal tiling.png Uniform tiling 54-t0.png
{5,4}
Uniform tiling 54-t01.png
4.10.10
Uniform tiling 54-t1.png
4.5.4.5
Uniform tiling 54-t12.png
5.8.8
Uniform tiling 54-t2.png
{4,5}
Uniform tiling 54-t02.png
4.4.5.4
Uniform tiling 54-t012.png
4.8.10
Uniform tiling 54-snub.png
3.3.4.3.5
(Hyperbolic plane)
(5 5 2)
Order-5 bisected pentagonal tiling.png Uniform tiling 552-t0.png
{5,5}
Uniform tiling 552-t01.png
5.10.10
Uniform tiling 552-t1.png
5.5.5.5
Uniform tiling 552-t12.png
5.10.10
Uniform tiling 552-t2.png
{5,5}
Uniform tiling 552-t02.png
4.4.5.4
Uniform tiling 552-t012.png
4.10.10
Uniform tiling 552-snub.png
3.3.5.3.5
Hexagonal tiling
(6 3 2)
Tile V46b.svg
V4.6.12
Uniform tiling 63-t0.png
{6,3}
Uniform tiling 63-t01.png
3.12.12
Uniform tiling 63-t1.png
3.6.3.6
Uniform tiling 63-t12.png
6.6.6
Uniform tiling 63-t2.png
{3,6}
Uniform tiling 63-t02.png
3.4.6.4
Uniform tiling 63-t012.png
4.6.12
Uniform tiling 63-snub.png
3.3.3.3.6
(Hyperbolic plane)
(7 3 2)
Order-3 heptakis heptagonal tiling.png Uniform tiling 73-t0.png
{7,3}
Uniform tiling 73-t01.png
3.14.14
Uniform tiling 73-t1.png
3.7.3.7
Uniform tiling 73-t12.png
7.6.6
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 73-t02.png
3.4.7.4
Uniform tiling 73-t012.png
4.6.14
Uniform tiling 73-snub.png
3.3.3.3.7
(Hyperbolic plane)
(8 3 2)
Order-3 octakis octagonal tiling.png Uniform tiling 83-t0.png
{8,3}
Uniform tiling 83-t01.png
3.16.16
Uniform tiling 83-t1.png
3.8.3.8
Uniform tiling 83-t12.png
8.6.6
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 83-t02.png
3.4.8.4
Uniform tiling 83-t012.png
4.6.16
Uniform tiling 83-snub.png
3.3.3.3.8

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 CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.pngCDW r.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.pngCDW r.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.pngCDW r.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.pngCDW r.png CDW dot.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.pngCDW r.png CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.pngCDW r.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.pngCDW r.png CDW hole.pngCDW p.pngCDW hole.pngCDW q.pngCDW hole.pngCDW r.png
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)
Tile 3,6.svg Uniform tiling 333-t0.png
(3.3)3
Uniform tiling 333-t01.png
3.6.3.6
Uniform tiling 333-t1.png
(3.3)3
Uniform tiling 333-t12.png
3.6.3.6
Uniform tiling 333-t2.png
(3.3)3
Uniform tiling 333-t02.png
3.6.3.6
Uniform tiling 333-t012.png
6.6.6
Uniform tiling 333-snub.png
3.3.3.3.3.3
Hyperbolic
(4 3 3)
Uniform dual tiling 433-t012.png Uniform tiling 433-t0.png
(3.4)3
Uniform tiling 433-t01.png
3.8.3.8
Uniform tiling 433-t1.png
(3.4)3
Uniform tiling 433-t12.png
3.6.4.6
Uniform tiling 433-t2.png
(3.3)4
Uniform tiling 433-t02.png
3.6.4.6
Uniform tiling 433-t012.png
6.6.8
Uniform tiling 433-snub.png
3.3.3.3.3.4
Hyperbolic
(4 4 3)
Uniform dual tiling 443-t012.png Uniform tiling 443-t0.png
(3.4)4
Uniform tiling 443-t01.png
3.8.4.8
Uniform tiling 443-t1.png
(3.4)4
Uniform tiling 443-t12.png
3.6.4.6
Uniform tiling 443-t2.png
(3.4)4
Uniform tiling 443-t02.png
4.6.4.6
Uniform tiling 443-t012.png
6.8.8
Uniform tiling 443-snub.png
3.3.3.4.3.4
Hyperbolic
(4 4 4)
Uniform dual tiling 444-t012.png Uniform tiling 444-t0.png
(4.4)4
Uniform tiling 444-t01.png
4.8.4.8
Uniform tiling 444-t1.png
(4.4)4
Uniform tiling 444-t12.png
4.8.4.8
Uniform tiling 444-t2.png
(4.4)4
Uniform tiling 444-t02.png
4.8.4.8
Uniform tiling 444-t012.png
8.8.8
Uniform tiling 444-snub.png
3.4.3.4.3.4

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 CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.png CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.png CDW dot.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.png CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.png CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.png CDW hole.pngCDW p.pngCDW hole.pngCDW q.pngCDW hole.png
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)
  Great icosahedron.png
{3,5/2}
Great truncated icosahedron.png
(5/2.6.6)
Great icosidodecahedron.png
(3.5/2)2
Icosahedron.png
[3.10/2.10/2]
Great stellated dodecahedron.png
{5/2,3}
Cantellated great icosahedron.png
[3.4.5/2.4]
Omnitruncated great icosahedron.png
[4.10/2.6]
Great snub icosidodecahedron.png
(3.3.3.3.5/2)
Icosahedral
(5 5/2 2)
  Great dodecahedron.png
{5,5/2}
Great truncated dodecahedron.png
(5/2.10.10)
Dodecadodecahedron.png
(5/2.5)2
Dodecahedron.png
[5.10/2.10/2]
Small stellated dodecahedron.png
{5/2,5}
Cantellated great dodecahedron.png
(5/2.4.5.4)
Omnitruncated great dodecahedron.png
[4.10/2.10]
Snub dodecadodecahedron.png
(3.3.5/2.3.5)

See also

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.

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