There are many relations among the

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

. Here they are grouped by the Wythoff symbol.

Key

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 In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

'' construction is given , p q r s.

Wythoff , p q r

Wythoff , p q r s

{, class="wikitable" , - !Symmetry group , - , Ih , {{Uniform polyhedra db, Polyhedra smallbox2, Gidrid Polyhedra Uniform polyhedra