Wythoff symbol

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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. Wythoffian construction diagram.svg
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). Wythoff construction-pqr.png
The eight forms for the Wythoff constructions from a general triangle (p q r).

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.

Contents

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 (pqr). The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following:

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 | pqr 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 22 q|p2 |pq2 p|qp|q 2pq| 2pq 2 ||pq 2
Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Vertex figure pqq.2p.2pp.q.p.qp.2q.2qqpp.4.q.44.2p.2q3.3.p.3.q
Fund. triangles7 forms and snub
(4 3 2)
Octahedral reflection domains.png
4 2
Uniform tiling 432-t0.png
43
4
Uniform tiling 432-t01.png
3.8.8
4 3
Uniform tiling 432-t1.png
3.4.3.4
3
Uniform tiling 432-t12.png
4.6.6
3 2
Uniform tiling 432-t2.png
34
2
Uniform tiling 432-t02.png
3.4.4.4

Uniform tiling 432-t012.png
4.6.8
| 4 3 2
Spherical snub cube.png
3.3.3.3.4
(5 3 2)
Icosahedral reflection domains.png
5 2
Uniform tiling 532-t0.png
53
5
Uniform tiling 532-t01.png
3.10.10
5 3
Uniform tiling 532-t1.png
3.5.3.5
3
Uniform tiling 532-t12.png
5.6.6
3 2
Uniform tiling 532-t2.png
35
2
Uniform tiling 532-t02.png
3.4.5.4

Uniform tiling 532-t012.png
4.6.10
| 5 3 2
Spherical snub dodecahedron.png
3.3.3.3.5
(6 3 2)
Tile V46b.svg
6 2
Uniform tiling 63-t0.png
63
6
Uniform tiling 63-t01.png
3.12.12
6 3
Uniform tiling 63-t1.png
3.6.3.6
3
Uniform tiling 63-t12.png
6.6.6
3 2
Uniform triangular tiling 111111.png
36
2
Uniform tiling 63-t02.png
3.4.6.4

Uniform tiling 63-t012.svg
4.6.12
| 6 3 2
Uniform tiling 63-snub.png
3.3.3.3.6
(7 3 2)
H2checkers 237.png
7 2
Heptagonal tiling.svg
73
7
Truncated heptagonal tiling.svg
3.14.14
7 3
Triheptagonal tiling.svg
3.7.3.7
3
Truncated order-7 triangular tiling.svg
7.6.6
3 2
Order-7 triangular tiling.svg
37
2
Rhombitriheptagonal tiling.svg
3.4.7.4

Truncated triheptagonal tiling.svg
4.6.14
| 7 3 2
Snub triheptagonal tiling.svg
3.3.3.3.7
(8 3 2)
H2checkers 238.png
8 2
H2-8-3-dual.svg
83
8
H2-8-3-trunc-dual.svg
3.16.16
8 3
H2-8-3-rectified.svg
3.8.3.8
3
H2-8-3-trunc-primal.svg
8.6.6
3 2
H2-8-3-primal.svg
38
2
H2-8-3-cantellated.svg
3.4.8.4

H2-8-3-omnitruncated.svg
4.6.16
| 8 3 2
H2-8-3-snub.svg
3.3.3.3.8
(∞ 3 2)
H2checkers 23i.png
∞ 2
H2-I-3-dual.svg
3

H2 tiling 23i-3.png
3.∞.∞
∞ 3
H2 tiling 23i-2.png
3.∞.3.∞
3
H2 tiling 23i-6.png
∞.6.6
3 2
H2 tiling 23i-4.png
3
2
H2 tiling 23i-5.png
3.4.∞.4

H2 tiling 23i-7.png
4.6.∞
| ∞ 3 2
Uniform tiling i32-snub.png
3.3.3.3.∞

See also

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In geometry, a Goursat tetrahedron is a tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-space. Coxeter named them after Édouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangles for Wythoff constructions on the sphere.

In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle,, defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles. Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror. Up to 3 face types exist centered on the fundamental triangle corners. Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling.

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