A modern, comprehensive review of abstract regular polytopes.

Abstract 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.

Foreword
1. Classical regular polytopes
2. Regular polytopes
3. Coxeter groups
4. Amalgamation
5. Realizations
6. Regular polytopes on space-forms
7. Mixing
8. Twisting
9. Unitary groups and hermitian forms
10. Locally toroidal 4-polytopes: I
11. Locally toroidal 4-polytopes: II
12. Higher toroidal polytopes
13. Regular polytopes related to linear groups
14. Miscellaneous classes of regular polytopes
Bibliography
Indices.