List of uniform polyhedra
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. 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.
Not included are:
- 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges ;
- The uniform tilings
- * 11 Euclidean uniform tessellations with convex faces;
- * 14 Euclidean uniform tilings with nonconvex faces;
- * Infinite number of uniform tilings in hyperbolic plane.
- Any polygons or 4-polytopes
Indexing
- 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.
- 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.
- 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.
- Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.
Names of polyhedra by number of sides
Table of polyhedra
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.Convex uniform polyhedra
Uniform star polyhedra
| Name | Image | Wyth sym | Vert. fig | Sym. | C# | W# | U# | K# | Vert. | Edges | Faces | Chi | Orient able? | Dens. | Faces by type |
| Great disnub dirhombidodecahedron* | | 5/3 5/2 | /2 | Ih | -- | -- | -- | -- | 60 | 360 | 204 | -96 | No | 120+60+24 |
: The great disnub dirhombidodecahedron has 240 of its 360 edges coinciding in space 120 pairs. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron.
Column key
- Uniform indexing: U01-U80
- Kaleido software indexing: K01-K80
- 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
- * 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 represents the number of windings of a polyhedron around its center. This is left blank for non-orientable polyhedra and hemipolyhedra, 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.