Uniform polytope

A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.

Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram.

An extended Schläfli symbol can be used for representing rectified forms, with a single subscript: Truncation operations that can be applied to regular n-polytopes in any combination.

The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them.

In addition combinations of truncations can be performed which also generate new uniform polytopes.

Uniform polytopes can be constructed from their vertex figure, the arrangement of edges, faces, cells, etc.

A smaller number of nonreflectional uniform polytopes have a single vertex figure but are not repeated by simple reflections.

An omnitruncated polytope (all nodes ringed) will always have an irregular simplex as its vertex figure.

Also the cuboctahedron divides into 8 regular tetrahedra and 6 square pyramids (half octahedron), and it is the vertex figure for the alternated cubic honeycomb.

In two dimensions, there is an infinite family of convex uniform polytopes, the regular polygons, the simplest being the equilateral triangle.

There are five convex regular polyhedra, known as the Platonic solids: In addition to these, there are also 13 semiregular polyhedra, or Archimedean solids, which can be obtained via Wythoff constructions, or by performing operations such as truncation on the Platonic solids, as demonstrated in the following table: There is also the infinite set of prisms, one for each regular polygon, and a corresponding set of antiprisms.

The Wythoffian uniform polyhedra and tilings can be defined by their Wythoff symbol, which specifies the fundamental region of the object.

An extension of Schläfli notation, also used by Coxeter, applies to all dimensions; it consists of the letter 't', followed by a series of subscripted numbers corresponding to the ringed nodes of the Coxeter diagram, and followed by the Schläfli symbol of the regular seed polytope.

In four dimensions, there are 6 convex regular 4-polytopes, 17 prisms on the Platonic and Archimedean solids (excluding the cube-prism, which has already been counted as the tesseract), and two infinite sets: the prisms on the convex antiprisms, and the duoprisms.

Every regular polytope can be seen as the images of a fundamental region in a small number of mirrors.

For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume.

This includes all nonprismatic uniform 4-polytopes, except for the non-Wythoffian grand antiprism, which has no Coxeter family.

By placing rings on a nonzero number of nodes of the Coxeter diagrams, one can obtain 39 new 6-polytopes, 127 new 7-polytopes and 255 new 8-polytopes.

Uniform honeycombs generated by compact groups have finite facets and vertex figures, and exist in 2 through 4 dimensions.

Paracompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and exist in 2 through 10 dimensions.