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# Wythoff Symbol

A symbol consisting of three rational numbers that can be used to describe uniform polyhedra based on how a point in a spherical triangle can be selected so as to trace the vertices of regular polygonal faces. For example, the Wythoff symbol for the tetrahedron is . There are four types of Wythoff symbols, , , and , and one exceptional symbol, (which is used for the great dirhombicosidodecahedron).

The meaning of the bars may be summarized as follows (Wenninger 1989, p. 10; Messer 2002). Consider a spherical triangle whose angles are , , and .

1. : is a special point within that traces snub polyhedra by even reflections.

2. (or ): is the vertex .

3. (or ): lies on the arc and the bisector of the opposite angle .

4. (or any permutation of the three letters): is the incenter of the triangle .

Some special cases in terms of Schläfli symbols are

 (1) (2) (3) (4) (5) (6)

Varying the order of the numbers within a subset of , , does not affect the kind of uniform polyhedron. However, excluding such redundancies, the other permutations of Wythoff symbols using "" and the set of nine rational numbers do not always produce new or valid polyhedra as some are degenerate forms (Messer 2002).

Schläfli Symbol, Schwarz Triangle, Uniform Polyhedron

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## References

Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Messer, P. W. "Closed-Form Expressions for Uniform Polyhedra and Their Duals." Disc. Comput. Geom. 27, 353-375, 2002.Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, pp. 8-10, 1989.

Wythoff Symbol

## Cite this as:

Weisstein, Eric W. "Wythoff Symbol." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WythoffSymbol.html