List of Regular Polytopes - Four-dimensional Regular Polytopes

Four-dimensional Regular Polytopes

Regular 4-polytopes (called polychora) with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures .

• A vertex figure (of a polychoron) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
• An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.

The existence of a regular polychoron is constrained by the existence of the regular polyhedra .

Each will exist in a space dependent upon this expression:

: Hyperspherical 3-space honeycomb or 4-space polychoron

: Euclidean 3-space honeycomb

: Hyperbolic 3-space honeycomb

These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

The Euler characteristic for polychora is and is zero for all forms.