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

In geometry, the Wythoff symbol is a notation representating a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.

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 ${\displaystyle 3\ |\ 4\ 2}$ with Oh symmetry, and ${\displaystyle 2\ 4\ |\ 2}$ as a square prism with 2 colors and D4h symmetry, as well as ${\displaystyle 2\ 2\ 2\ |}$ with 3 colors and ${\displaystyle D_{2h}}$ 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.

## Description

The Wythoff construction begins by choosing a generator point on a fundamental 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, ${\displaystyle p}$, ${\displaystyle q}$ and ${\displaystyle r}$, represent the corners of the Schwarz triangle used in the construction, which are ${\displaystyle \pi /p}$, ${\displaystyle \pi /q}$ and ${\displaystyle \pi /r}$ radians respectively. The triangle is also represented with the same numbers, written ${\displaystyle (p\ \ q\ \ r)}$. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following:

• ${\displaystyle p\ |\ q\ r}$ indicates that the generator lies on the corner ${\displaystyle p}$,
• ${\displaystyle p\ q\ |\ r}$ indicates that the generator lies on the edge between ${\displaystyle p}$ and ${\displaystyle q}$,
• ${\displaystyle p\ q\ r\ |}$ 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 ${\displaystyle p,q,r}$ values are listed before the bar if the corresponding mirror is active.

A special use is the symbol ${\displaystyle |\ p\ q\ r}$ which is designated for 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.

## Example spherical, euclidean and hyperbolic tilings on right triangles (r = 2)

The fundamental triangles are drawn in alternating colors as mirror images; the sequence of triangles (p 3 2) change from spherical (p=3,4,5), to Euclidean (p=6), to hyperbolic (p=7,8,...∞). Hyperbolic tilings are shown as a Poincaré disk projection.

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
Coxeter 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
Fund. triangles 7 forms and snub
(4 3 2)
3 | 4 2

43
2 3 | 4

3.8.8
2 | 4 3

3.4.3.4
2 4 | 3

4.6.6
4 | 3 2

34
4 3 | 2

3.4.4.4
4 3 2 |

4.6.8
| 4 3 2

3.3.3.3.8
(5 3 2)
3 | 5 2

53
2 3 | 5

3.10.10
2 | 5 3

3.5.3.5
2 5 | 3

5.6.6
5 | 3 2

35
5 3 | 2

3.4.5.4
5 3 2 |

4.6.10
| 5 3 2

3.3.3.3.5
(6 3 2)
3 | 6 2

63
2 3 | 6

3.12.12
2 | 6 3

3.6.3.6
2 6 | 3

6.6.6
6 | 3 2

36
6 3 | 2

3.4.6.4
6 3 2 |

4.6.12
| 6 3 2

3.3.3.3.6
(7 3 2)
3 | 7 2

73
2 3 | 7

3.14.14
2 | 7 3

3.7.3.7
2 7 | 3

7.6.6
7 | 3 2

37
7 3 | 2

3.4.7.4
7 3 2 |

4.6.14
| 7 3 2

3.3.3.3.7
(8 3 2)
3 | 8 2

83
2 3 | 8

3.16.16
2 | 8 3

3.8.3.8
2 8 | 3

8.6.6
8 | 3 2

38
8 3 | 2

3.4.8.4
8 3 2 |

4.6.16
| 8 3 2

3.3.3.3.8
(∞ 3 2)
3 | ∞ 2

3
2 3 | ∞

3.∞.∞
2 | ∞ 3

3.∞.3.∞
2 ∞ | 3

∞.6.6
∞ | 3 2

3
∞ 3 | 2

3.4.∞.4
∞ 3 2 |

4.6.∞
| ∞ 3 2

3.3.3.3.∞