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:

• all 75 nonprismatic uniform polyhedra;
• a few representatives of the infinite sets of prisms and antiprisms;
• one degenerate polyhedron, Skilling's figure with overlapping edges.

It was proven in Sopov (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:

• The uniform polyhedron compounds.
• 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
• The uniform tilings (infinite polyhedra)
  • 11 Euclidean convex uniform tilings;
  • 28 Euclidean nonconvex or apeirogonal uniform tilings;
  • Infinite number of uniform tilings in hyperbolic plane.
• Any polygons or 4-polytopes