Egon Schulte

Regulre Inzidenzkomplexe I

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.

Chirality and projective linear groups

In recent years the term ‘chiral’ has been used for geometric and combinatorial figures which are symmetrical by rotation but not by reflection. The correspondence of groups and polytopes is used to construct infinite series of chiral and regular polytopes whose facets or vertex-figures are chiral or regular toroidal maps. In particular, the groups PSL2(Zm) are used to construct chiral polytopes, while PSL2(Zm[i]) and PSL2(Zm[ω]) are used to construct regular polytopes.

Regular Polytopes in Ordinary Space

Abstract. The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grünbaum—Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E3 in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in E3. The paper gives a complete classification of the discrete regular polytopes in ordinary space.

Constructions for regular polytopes

Abstract The paper discusses a general method for constructing regular incidence-polytopes P from certain operations on the generators of a group W, which is generated by involutions. When W is the group of a regular incidence-polytope L , this amounts to constructing the regular skew polytopes (or skew polyhedra) P associated with L .

Groups of type L 2 (q) acting on polytopes

We prove that if G is a string C-group of rank 4 and G ∼ L2(q) with q a prime power, then q must be 11 or 19. The polytopes arising are

Hermitian Forms and Locally Toroidal Regular Polytopes

In the classical theory of regular polytopes the structure of a finite or infinite polytope (honeycomb) is governed by a real quadratic form. This form determines a geometry into which the polytope is embedded in such a way that all its symmetries are realized by isometries. The symmetry group is the Coxeter group associated with the quadratic form (cf. Coxeter [8]). These facts have important consequences for other fields in mathematics; see, for example, Tits [29, 303. The purpose of this paper is to show that much of the correspondence between polytopes and forms remains true for the theory of abstract regular polytopes, that is, regular incidence-polytopes. The concept of regular incidence-polytopes provides a suitable setting for combinatorial structures resembling the classical regular polytopes (cf. Danzer-Schulte [ 131); for related notions see also McMullen [ 171, Griinbaum [ 161, Dress [14], Buekenhout [2], and Tits [29]. In [I163 Griinbaum suggested studying abstract regular polytopes whose faces and vertex-figures are not necessarily of spherical type. He posed the problem of classifying all finite universal (or, in his notation, naturally generated) abstract regular polytopes {Y,, p see also Coxeter-Shephard [ll], Weiss [31, 321, and [24,25]. In [20,21] this problem was attacked by twisting operations on Coxeter groups and unitary reflexion groups, leading to the explicit recognition of the universal I

Regular polytopes from twisted Coxeter groups and unitary reflexion groups

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Reguläre Inzidenzkomplexe II

p,) for many choices of 9, and P>. In this paper we completely classify all the finite universal polytopes of

On Coxete's regular skew polyhedra

Abstract In this paper we discuss the construction of regular incidence-polytopes p by twisting operations on Coxeter groups and quotients of Coxeter groups. Particular attention is paid to the polytopes obtained from the unitary complex groups generated by reflexions of period 2. In particular this leads to the explicit recognition of the universal regular incidence-polytopes { p 1, p 2} in a number of interesting cases of regular incidence-polytopes p 1 and p 2.

Regular Incidence- Polytopes with Euclidean or Toroidal Faces and Vertex- Figures

The concept of regular incidence complexes generalizes the notion of regular polyhedra in a combinatorial and grouptheoretical sense. A regular (incidence) complex K is a special type of partially ordered structure with regularity defined by the flag-transitivity of its group A(K) of automorphisms.The structure of a regular complex K can be characterized by certain sets of generators and ‘relations’ of its group. The barycentric subdivision of K leads to a simplicial complex, from which K can be rebuilt by fitting together faces.Moreover, we characterize the groups that act flag-transitively on regular complexes. Thus we have a correspondence between regular complexes on the one hand and certain groups on the other hand.Especially, this principle is used to give a geometric representation for an important class of regular complexes, the so-called regular incidence polytopes. There are certain universal incidence polytopes associated to Coxeter groups with linear diagram, from which each regular incidence polytope can be deduced by identifying faces. These incidence polytopes admit a geometric representation in the real space by convex cones.