# Wythoff symbol

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 with O_{h} symmetry, and as a square prism with 2 colors and D_{4h} 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.

## 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, , 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.

A special use is the symbol 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)

*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 | 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 |

Fund. triangles | 7 forms and snub | |||||||

(4 3 2) |
3 | 4 2 4 ^{3} |
2 3 | 4 3.8.8 |
2 | 4 3 3.4.3.4 |
2 4 | 3 4.6.6 |
4 | 3 2 3 ^{4} |
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 5 ^{3} |
2 3 | 5 3.10.10 |
2 | 5 3 3.5.3.5 |
2 5 | 3 5.6.6 |
5 | 3 2 3 ^{5} |
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 6 ^{3} |
2 3 | 6 3.12.12 |
2 | 6 3 3.6.3.6 |
2 6 | 3 6.6.6 |
6 | 3 2 3 ^{6} |
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 7 ^{3} |
2 3 | 7 3.14.14 |
2 | 7 3 3.7.3.7 |
2 7 | 3 7.6.6 |
7 | 3 2 3 ^{7} |
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 8 ^{3} |
2 3 | 8 3.16.16 |
2 | 8 3 3.8.3.8 |
2 8 | 3 8.6.6 |
8 | 3 2 3 ^{8} |
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.∞ |

## 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.

## External links

- Weisstein, Eric W. "Wythoff symbol".
*MathWorld*. - The Wythoff symbol
- Wythoff symbol
- Greg Egan's applet to display uniform polyhedra using Wythoff's construction method
- A Shadertoy renderization of Wythoff's construction method
- KaleidoTile 3 Free educational software for Windows by Jeffrey Weeks that generated many of the images on the page.
- Hatch, Don. "Hyperbolic Planar Tessellations".