Chiral extensions of chiral polytopes

Given a chiral d-polytope K with regular facets, we describe a construction for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is finite whenever K is finite. We provide explicit examples of chiral 4-polytopes constructed in this way from chiral toroidal maps.Comment: 21 pages. [v2] includes several minor revisions for clarit

Hereditary Polytopes

Every regular polytope has the remarkable property that it inherits all symmetries of each of its facets. This property distinguishes a natural class of polytopes which are called hereditary. Regular polytopes are by definition hereditary, but the other polytopes in this class are interesting, have possible applications in modeling of structures, and have not been previously investigated. This paper establishes the basic theory of hereditary polytopes, focussing on the analysis and construction of hereditary polytopes with highly symmetric faces.Comment: Discrete Geometry and Applications (eds. R.Connelly and A.Ivic Weiss), Fields Institute Communications, (23 pp, to appear

Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear

Polygonal Complexes and Graphs for Crystallographic Groups

The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the classification of regular polygonal complexes, chiral polyhedra, and more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss and W.Whiteley), Fields Institute Communications, to appea

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder, A. Deza, and A. Ivic Weiss (eds), Springe