regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic, or topological properties, in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory, and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics, and group theory.

Chiral polyhedra in ordinary euclidean space E 3 are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinct orbits. This paper completely enumerates the discrete infinite chiral polyhedra in E 3 with finite skew faces and finite skew vertex-figures. There are several families of such polyhedra of types {4, 6}, {6, 4} and {6, 6}. Their geometry and combinatorics are discussed in detail. It is also proved that a chiral polyhedron in E 3 cannot be finite. Part II of the paper will complete the classification of all chiral polyhedra in E 3 . All chiral polyhedra not described in Part I have infinite, helical faces and again occur in families. So, in effect, Part I enumerates all chiral polyhedra in E 3 with finite faces.

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.

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 E 3 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 E 3 . The paper gives a complete classification of the discrete regular polytopes in ordinary space.

No abstract