Uniform polyhedron

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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 face- and edge-transitive), quasi-regular (if also 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.

There are two infinite classes of uniform polyhedra, together with 75 other polyhedra:

• Infinite classes:
  • prisms,
  • antiprisms.
• Convex exceptional:
  • 5 Platonic solids: regular convex polyhedra,
  • 13 Archimedean solids: 2 quasiregular and 11 semiregular convex polyhedra.
• Star (nonconvex) exceptional:
  • 4 Kepler–Poinsot polyhedra: regular nonconvex polyhedra,
  • 53 uniform star polyhedra: 14 quasiregular and 39 semiregular.

Hence 5 + 13 + 4 + 53 = 75.

There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron (Skilling's figure).

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.

The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space.