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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

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

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

See also Uniform Polyhedron


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

© 1996-9 Eric W. Weisstein