There are many relations among the uniform polyhedra.

Here they are grouped by the Wythoff symbol.

Key

Image Name Bowers pet name V Number of vertices,E Number of edges,F Number of faces=Face configuration ?=Euler characteristic, group=Symmetry group Wythoff symbol – Vertex figure W – Wenninger number, U – Uniform number, K- Kaleido number, C -Coxeter number alternative name second alternative name

Regular

All the faces are identical, each edge is identical and each vertex is identical. They all have a Wythoff symbol of the form p|q 2.

Convex The Platonic solids.

Non-convex The Kepler-Poinsot solids.

Quasi-regular Each edge is identical and each vertex is identical. There are two types of faces which appear in an alternating fashion around each vertex. The first row are semi-regular with 4 faces around each vertex. They have Wythoff symbol 2|p q. The second row are ditrigonal with 6 faces around each vertex. They have Wythoff symbol 3|p q or 3/2|p q.

Wythoff p q|r

Truncated regular forms Each vertex has three faces surrounding it, two of which are identical. These all have Wythoff symbols 2 p|q, some are constructed by truncating the regular solids.

Hemipolyhedra The hemipolyhedra all have faces which pass through the origin. Their Wythoff symbols are of the form p p/m|q or p/m p/n|q. With the exception of the tetrahemihexahedron they occur in pairs, and are closely related to the semi-regular polyhedra, like the cuboctohedron.

Rhombic quasi-regular Four faces around the vertex in the pattern p.q.r.q. The name rhombic stems from inserting a square in the cuboctahedron and icosidodecahedron. The Wythoff symbol is of the form p q|r.

Even-sided forms

Wythoff p q r| These have three different faces around each vertex, and the vertices do not lie on any plane of symmetry. They have Wythoff symbol p q r|, and vertex figures 2p.2q.2r.

Wythoff p q (r s)|

Vertex figure p.q.-p.-q. Wythoff p q (r s)|, mixing pqr| and pqs|.

Snub polyhedra

These have Wythoff symbol |p q r, and one non-Wythoffian construction is given |p q r s.

Wythoff |p q r

Symmetry groupOIhIII

Wythoff |p q r s Symmetry groupIh