Regul�re Inzidenzkomplexe I

Abstract

The concept of regular incidence-complexes generalizes the notion of regular polyhedra in a combinatorial sense. A regular incidence-complex is a partially ordered set with regularity defined by certain transitivity properties of its automorphism group. The concept includes all regular d-polytopes and all regular complex d-polytopes as well as many geometries and well-known configurations.

Twisted Groups and Locally Toroidal Regular Polytopes

“…We shall construct a diagram D by adjoining (8) to G as follows. We identify the node m − 1 of (8) We can now define the (m + n)-polytope L K,G by its group W A(K) (the semi-direct product induced by the action of A(K) on W), and the distinguished generators ρ 0 , .…”

“…In recent years, the classical notion of a regular polytope has been generalized to abstract regular polytopes; these are combinatorial structures with a distinctive geometric or topological flavour, which resemble the classical regular polytopes (Danzer-Schulte [8], McMullen-Schulte [23]). For related notions in geometric or group theoretic contexts, see also McMullen [12], Grünbaum [10], Dress [9], Buekenhout [1] and Tits [29].…”

Generating combinatorial complexes of polyhedral type

“…It is somewhat analogous to Danzer's notion of an incidence-complex (cf. [10]) and Griinbaum's notion of a polystroma (cf. [14]).…”

“…Note that the word 'complex' was used in a slightly different sense in [10,21], where it abbreviates the word 'incidence-complex'.…”

“…For notation and basic results the reader is referred to Griinbaum [13] and McMullenShephard [18] for convex polytopes, to Griinbaum-Shephard [15] for tilings, and to Danzer-Schulte [10] for regular incidence-complexes.…”