Wythoff symbol

In geometry, the Wythoff symbol is a notation representing 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.

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).

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 3 | 2 4 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.

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, p, q, and r, represent the corners of the Schwarz triangle used in the construction, which are ​π⁄p, ​π⁄q, and ​π⁄r radians respectively. The triangle is also represented with the same numbers, written (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:

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

A special use is the symbol | 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

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). 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

3.3.3.3.4
(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.∞
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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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