# Wythoff symbol

Last updated

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:

• p | qr indicates that the generator lies on the corner p,
• pq | r indicates that the generator lies on the edge between p and q,
• pqr | 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 | 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
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)
4 2

43
4

3.8.8
4 3

3.4.3.4
3

4.6.6
3 2

34
2

3.4.4.4

4.6.8
| 4 3 2

3.3.3.3.4
(5 3 2)
5 2

53
5

3.10.10
5 3

3.5.3.5
3

5.6.6
3 2

35
2

3.4.5.4

4.6.10
| 5 3 2

3.3.3.3.5
(6 3 2)
6 2

63
6

3.12.12
6 3

3.6.3.6
3

6.6.6
3 2

36
2

3.4.6.4

4.6.12
| 6 3 2

3.3.3.3.6
(7 3 2)
7 2

73
7

3.14.14
7 3

3.7.3.7
3

7.6.6
3 2

37
2

3.4.7.4

4.6.14
| 7 3 2

3.3.3.3.7
(8 3 2)
8 2

83
8

3.16.16
8 3

3.8.3.8
3

8.6.6
3 2

38
2

3.4.8.4

4.6.16
| 8 3 2

3.3.3.3.8
(∞ 3 2)
∞ 2

3

3.∞.∞
∞ 3

3.∞.3.∞
3

∞.6.6
3 2

3
2

3.4.∞.4

4.6.∞
| ∞ 3 2

3.3.3.3.∞

## Related Research Articles

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in.

A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied by to get 180 degrees. An unlabeled branch implicitly represents order-3.

In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures or both.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.

In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

In hyperbolic geometry, a uniform hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

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.

## 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.