Abstract Polytope - Abstract Regular Polytopes

Abstract Regular Polytopes

Formally, an abstract polytope is defined to be "regular" if its automorphism group acts transitively on the set of its flags. In particular, any two k-faces F, G of an n-polytope are "the same", i.e. that there is an automorphism which maps F to G. When an abstract polytope is regular, its automorphism group is isomorphic to a quotient of a Coxeter group.

All polytopes of rank ≤ 2 are regular. The most famous regular polyhedra are the five Platonic solids. The hemicube (shown) is also regular.

Informally, for each rank k, this means that there is no way to distinguish any k-face from any other - the faces must be identical, and must have identical neighbors, and so forth. For example, a cube is regular because all the faces are squares, each square's vertices are attached to three squares, and each of these squares is attached to identical arrangements of other faces, edges and vertices, and so on.

This condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (n−1)-faces and isomorphic regular vertex figures.

This is a weaker condition than regularity for traditional polytopes, in that it refers to the (combinatorial) automorphism group, not the (geometric) symmetry group. For example, any abstract polygon is regular, since angles, edge-lengths, edge curvature, skewness etc. don't exist for abstract polytopes.

There are several other weaker concepts, some not yet fully standardised, such as semi-regular, quasi-regular, uniform, chiral, and Archimedean that apply to polytopes that have some, but not all of their faces equivalent in each rank.