Wythoff Symbol

A symbol consisting of three rational numbers that can be used to describe uniform polyhedra based on how a point C 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 3|23. There are four types of Wythoff symbols, |pqr, p|qr, pq|r and pqr|, and one exceptional symbol, |3/25/335/2 (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 PQR whose angles are pi/p, pi/q, and pi/r.

1. |pqr: C is a special point within PQR that traces snub polyhedra by even reflections.

2. p|qr (or p|rq): C is the vertex P.

3. qr|p (or rq|p): C lies on the arc QR and the bisector of the opposite angle P.

4. pqr| (or any permutation of the three letters): C is the incenter of the triangle PQR.

Some special cases in terms of Schläfli symbols are

2|pq={p; q}
pq|2=r{p; q}
2pq|=t{p; q}
|2pq=s{p; q}.

Varying the order of the numbers within a subset of p, q, r 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).

See also

Schläfli Symbol, Schwarz Triangle, Uniform Polyhedron

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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.

Referenced on Wolfram|Alpha

Wythoff Symbol

Cite this as:

Weisstein, Eric W. "Wythoff Symbol." From MathWorld--A Wolfram Web Resource.

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