Regular and Archimedean Polytopes

A convex polytope is regular if it admits a flag transitive group of (linear) automorphisms. This group turns out to be a finite reflection group (or Coxeter group).

The regular n-gons are eaxctly the regular 2-polytopes.

The Coxeter groups (of rank at least 3) come in three infinite series An, Bn=Cn, Dn plus finitely many so-called exceptional Coxeter groups.

The series An corresponds to the simplices. Bn=Cn gives cross polytopes and cubes (which form dual pairs). The series Dn 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 24-cell, the 120-cell and the 600-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. Therefore, one can associate to a regular polytope its so-called Schläfli symbol (m,n).

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. In these cases we give rational coordinates nonetheless: but, these coordinates are not naively obtained from rounding real numbers to arbitrary rational numbers. Instead, the corresponding polymake file contains the correct description of some polytope with rational coordinates which has the same combinatorial type as the corresponding regular one and which is close to the corresponding regular one. This works out since all combinatorial types of 3-dimensional polytopes do have a rational representation. It is known that there are (combinatorial types of) higher dimensional polytopes without any rational representation.

References

1. H.S.M. Coxeter: Regular Polytopes, Dover (1973).
2. Peter R. Cromwell: Polyhedra, Cambridge University Press (1999).
3. George Hart: Virtual Polyhedra - The Encyclopedia of Polyhedra.
4. James E. Humphreys: Reflection groups and Coxeter groups, Cambridge University Press (1992).