2 - Abstract Regular Polytopes

Summary

The monograph ‘Abstract Regular Polytopes’ described the rich abstract theory, of which some basics are needed here. A new recursive definition is given, which corresponds more closely than that of the monograph to one’s intuitive idea of what a polytope should be. Regularity of abstract polytopes and the central idea of string C-groups are then introduced, and it is shown that the two concepts are equivalent. The intersection property defines a C-group; various conditions on a group are established that ensure it, in particular some quotient criteria. Presentations of the groups of regular polytopes are treated next, including the circuit criterion, and some related general concepts and notation are introduced. Maps or polyhedra, the polytopes of rank three, form an important class of regular polytopes; some of their properties and some useful examples are described. There is a brief discussion of amalgamation (constructing polytopes with given facets and vertex-figures) and universality. Finally, there is a treatment of certain special properties of regular polytopes, such as central symmetry, flatness and collapsibility.