There are many relations among the uniform polyhedra.
Here they are grouped by the Wythoff symbol.
| 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 |
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.
The Platonic solids.
The Kepler-Poinsot solids.
Each edge is identical and each vertex is identical. There are two types of faceswhich 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.
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.
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.
Four faces around the vertex in the pattern p.q.r.q. The name rhombic stems from insertinga square in the cuboctahedron and icosidodecahedron. The Wythoff symbol is of the form 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.
Vertex figure p.q.-p.-q. Wythoff p q (r s)|, mixing pqr| and pqs|.
These have Wythoff symbol |p q r, and one non-Wythoffian construction is given |p q r s.
| Symmetry group | ||||
|---|---|---|---|---|
| O | ||||
| Ih | ||||
| I | ||||
| I | ||||
| I |