Uniform Polyhedra - Convex Forms By Wythoff Construction

Convex Forms By Wythoff Construction

The convex uniform polyhedra can be named by Wythoff construction operations and can be named in relation to the regular form.

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

Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The octahedron is a regular polyhedron, and a triangular antiprism. The octahedron is also a rectified tetrahedron. Many polyhedra are repeated from different construction sources and are colored differently.

The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of hosohedrons and dihedrons which are degenerate polyhedra.

These symmetry groups are formed from the reflectional point groups in three dimensions, each represented by a fundamental triangle (p q r), where p>1, q>1, r>1 and 1/p+1/q+1/r<1.

• Tetrahedral symmetry (3 3 2) - order 24
• Octahedral symmetry (4 3 2) - order 48
• Icosahedral symmetry (5 3 2) - order 120
• Dihedral symmetry (n 2 2), for all n=3,4,5,... - order 4n

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

Along with the prisms and their dihedral symmetry, the spherical Wythoff construction process adds two regular classes which become degenerate as polyhedra - the dihedra and hosohedra, the first having only two faces, and the second only two vertices. The truncation of the regular hosohedra creates the prisms.

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

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

1. Hosohedrons H_2... (Only as spherical tilings)
2. Dihedrons D_2... (Only as spherical tilings)
3. Prisms P_3... (Truncated hosohedrons)
4. Antiprisms A_3... (Snub prisms)