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.

It was proven in 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 by Schwarz triangle|uniform polyhedra with degenerate] vertex figures which have overlapping edges ;
The uniform tilings
* 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

Indexing

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

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, 27–80 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

There are generic geometric names for the most common polyhedra. The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube.

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.
There are infinitely many prisms and antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.

Uniform star polyhedra

The forms containing only convex faces are listed first, followed by the forms with star faces. Again infinitely many prisms and antiprisms exist; they are listed here up to the 8-sided ones.
The uniform polyhedra 3 3,, 3, 3, and have some faces occurring as coplanar pairs.

Name

Image

Wyth sym

Vert. fig

Sym.

Vert.

Edges

Faces

Chi

Orient- able?

Dens.

Faces by type