List of uniform polyhedra

In geometry, 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 can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both.

This list includes these:

It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide.

Not included are:

Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:

The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.

(*) : The great disnubdirhombidodecahedron has 240 of its 360 edges coinciding in 120 pairs of edges with the same image in space. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron. If these 120 pairs are considered to be single edges with 4 faces meeting, then the number of edges drops to 240 and the Euler characteristic becomes 24.

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