Wythoff Symbol

A symbol used to describe Uniform Polyhedra. For example, the Wythoff symbol for the Tetrahedron is $3\,\vert\,2\,3$. There are three types of Wythoff symbols $p\,\vert\,q\,r$, $p\,q\,\vert\,r$ and $p\,q\,r\,\vert$, and one exceptional symbol $\vert\,{\textstyle{3\over 2}}\,{\textstyle{5\over 3}}\,3\,{\textstyle{5\over 2}}$ used for the Great Dirhombicosidodecahedron. Some special cases in terms of Schläfli Symbols are

$$p\,\vert\,q\,2 = p\,\vert\,2\,q = \{q,p\}$$

$$2\,\vert\,p\,q = \left\{{p\atop q}\right\}$$

$$p\,q\,\vert\,2 = r\left\{{p\atop q}\right\}$$

$$2\,q\,\vert\,p = {\rm t}\{p, q\}$$

$$2\,p\,q\,\vert = {\rm t}\left\{{p\atop q}\right\}$$

$$\vert\,2\,p\,q = {\rm s}\left\{{p\atop q}\right\}.$$

For the symbol $p\,q\,r\,\vert$, permuting the letters gives the same Polyhedron.

See also Uniform Polyhedron

References

Har'El, Z. ``Uniform Solution for Uniform Polyhedra.'' Geometriae Dedicata 47, 57-110, 1993.