Uniform 5-polytope

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

The complete set of convex uniform 5-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.

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:

There are no nonconvex regular polytopes in 5 or more dimensions.

There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.

The 5-simplex is the regular form in the A₅ family. The 5-cube and 5-orthoplex are the regular forms in the B₅ family. The bifurcating graph of the D₆ family contains the pentacross, as well as a 5-demicube which is an alternated 5-cube.

Fundamental families

Uniform prisms There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes:

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