Uniform Polyhedra

Uniform Polyhedra

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.

Uniform polyhedra may be regular (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.

Excluding the infinite sets, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).

• Convex
  • 5 Platonic solids – regular convex polyhedra
  • 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra
• Star
  • 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra
  • 53 uniform star polyhedra – 5 quasiregular and 48 semiregular
  • 1 star polyhedron found by John Skilling with pairs of edges that coincide, called the great disnub dirhombidodecahedron (Skilling's figure).

There are also two infinite sets of uniform prisms and antiprisms, including convex and star forms.

Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.