List of Regular Polytopes - Five-dimensional Regular Polytopes and Higher

Five-dimensional Regular Polytopes and Higher

In five dimensions, a regular polytope can be named as where is the hypercell (or teron) type, is the cell type, is the face type, and is the face figure, is the edge figure, and is the vertex figure.

A 5-polytope has been called a polyteron, and if infinite (i.e. a honeycomb) a tetracomb.

A vertex figure (of a 5-polytope) is a polychoron, seen by the arrangement of neighboring vertices to each vertex.

An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.

A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.

A regular polytope exists only if and are regular polychora.

The space it fits in is based on the expression:

: Spherical 4-space tessellation or 5-space polytope

: Euclidean 4-space tessellation

: hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations. There are no non-convex regular polytopes in five dimensions or higher.

Higher-dimensional polytopes have sometimes received names. 6-polytopes have sometimes been called polypeta, 7-polytopes polyexa, 8-polytopes polyzetta, and 9-polytopes polyyotta.