Uniform polyhedron

There are two infinite classes of uniform polyhedra, together with 75 other polyhedra:

There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron (Skilling's figure).

The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space.

The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term “regular polyhedra” was, and is, contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call “polyhedra”, with those special ones that deserve to be called “regular”. But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,…—the writers failed to define what are the “polyhedra” among which they are finding the “regular” ones.

Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and to intersect each other.

The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron

The convex uniform polyhedra can be named by Wythoff construction operations on the regular form.

In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group.

The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides.

Below the convex uniform polyhedra are indexed 1–18 for the nonprismatic forms as they are presented in the tables by symmetry form.

For the infinite set of prismatic forms, they are indexed in four families:

(The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.)

The tetrahedral symmetry of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation.

There are 24 triangles, visible in the faces of the tetrakis hexahedron, and in the alternately colored triangles on a sphere:

The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six of these forms are repeated from the tetrahedral symmetry table above.

There are 48 triangles, visible in the faces of the disdyakis dodecahedron, and in the alternately colored triangles on a sphere:

The icosahedral symmetry of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above.

The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere.

There are 8 fundamental triangles, visible in the faces of the square bipyramid (Octahedron) and alternately colored triangles on a sphere:

There are 12 fundamental triangles, visible in the faces of the hexagonal bipyramid and alternately colored triangles on a sphere:

There are 16 fundamental triangles, visible in the faces of the octagonal bipyramid and alternately colored triangles on a sphere:

There are 20 fundamental triangles, visible in the faces of the decagonal bipyramid and alternately colored triangles on a sphere:

There are 24 fundamental triangles, visible in the faces of the dodecagonal bipyramid and alternately colored triangles on a sphere.