Regular 5-polytopes
Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face. There are exactly three such regular polytopes, all convex:
- {3,3,3,3} - Hexateron (5-simplex)
- {4,3,3,3} - Penteract (5-hypercube)
- {3,3,3,4} - Pentacross (5-orthoplex)
There are no nonconvex regular polytopes in 5 or more dimensions.
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—Henry David Thoreau (1817–1862)