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 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:
40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
The uniform tilings (infinite polyhedra)
11 Euclidean uniform tessellations with convex faces;
14 Euclidean uniform tilings with nonconvex faces;
Infinite number of uniform tilings in hyperbolic plane.
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
[C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
[W] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
[K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetry, 6-9 with tetrahedral symmetry, 10-26 with Octahedral symmetry, 46-80 with icosahedral symmetry.
[U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.
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.
Column key
Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+)
Kaleido software indexing: K01-K80 (Kn = Un-5 for n = 6 to 80) (prisms 1-5, Tetrahedron etc. 6+)
Magnus Wenninger Polyhedron Models: W001-W119
1-18 - 5 convex regular and 13 convex semiregular
20-22, 41 - 4 non-convex regular
19-66 Special 48 stellations/compounds (Nonregulars not given on this list)
67-109 - 43 non-convex non-snub uniform
110-119 - 10 non-convex snub uniform
Chi: the Euler characteristic, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
Density: the Density (polytope) represents the number of windings of a polyhedron around its center. This is left blank for non-orientable polyhedra and hemipolyhedra (polyhedra with faces passing through their centers), for which the density is not well-defined.
Note on Vertex figure images:
The white polygon lines represent the "vertex figure" polygon. The colored faces are included on the vertex figure images help see their relations. Some of the intersecting faces are drawn visually incorrectly because they are not properly intersected visually to show which portions are in front.
