- Advanced Studies in Pure Mathematics
- Semiregular polytope
- Semi-regular polyhedra - Encyclopedia of Mathematics

The series D n does not yield polytopes because they would have digonal faces; similarly all the other non-linear diagrams are excluded.

Exceptional regular polytopes only exist in dimensions 3 and 4: the regular dodecahedron and icosahedron, the cell, the cell and the cell. Specializing to dimension 3, a polytope is regular if and only if the facets are pairwise congruent regular m-gons and all vertex figures are pairwise congruent regular n-gons. A 3-polytope is called Archimedean if its automorphism group acts transitively on the vertices and each facet is a regular polygon.

Up to linear isomorphy there are 13 Archimedean polytopes in addition to the 5 regular ones. Many of the regular and Archimedean polytopes do not have a rational coordinate representation.

### Advanced Studies in Pure Mathematics

There are forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Twelve cases are shown below: ten single-ring rectified forms, and two truncations. There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space:.

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There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

From Wikipedia, the free encyclopedia. Graphs of three regular and related uniform polytopes. Fundamental convex regular and uniform polytopes in dimensions 2— Categories : polytopes.

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## Semiregular polytope

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## Semi-regular polyhedra - Encyclopedia of Mathematics

Truncated simplex. Rectified simplex. Cantellated simplex. Runcinated simplex.

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Stericated simplex. Pentellated simplex.