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:
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- : 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.
Read more about this topic: List Of Regular Polytopes
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