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Concept
Wythoff symbol
Summary
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.
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.
The Wythoff construction begins by choosing a generator point on a fundamental triangle. This point must be chosen at equal distance from all edges that it does not lie on, and a perpendicular line is then dropped from it to each such edge.
The three numbers in Wythoff's symbol, p, q, and r, represent the corners of the Schwarz triangle used in the construction, which are pi/p, pi/q, and pi/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.
Official source
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