 TOPICS  # 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