Wythoff Symbol

A symbolic representation of a polyhedron by reference to a tessellation of the surface of a sphere with spherical triangles. The symbol appears as a set of three numbers and a vertical bar, in one of the following arrangements:

p q r |,   p q | r ,   p | q r ,   | p q r .

In all cases, the three numbers denote a spherical triangle with angles π/p, π/q, π/r, whose corresponding vertices we label P, Q, and R. There are restrictions on which triads of numbers are allowed, in that a set of such triangles must form a tessellation of the sphere by repeated reflections. In the case of star polyhedra the tessellation is allowed to cover the surface multiple times.

The position of the vertical bar determines how the vertices of the polyhedron are found from the resulting tessellation:

• p q r | — We take the set of points that are the incentres of the spherical triangles (i.e. points equidistant from the nearest point on each edge).
• p q | r — We take the set of points lying on the side PQ that bisect the angle at R.
• p | q r — We takes the set of points corresponding to the vertices P of the spherical triangles.
• | p q r — Here we take a special point inside the triangle, but only on even reflections. This is used for the snub polyhedra.