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.
• First comprehensive account of the modern theory of regular polytopes and their groups • Attractive interplay of several mathematical areas including geometry, combinatorics, group theory and topology • First book in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974)
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.
'The book gives a comprehensive, complete overview of recent developments in a n important area of discrete geometry. it really fills an existing gap … and it shows in an impressive manner the interplay between the different methods that are important in this field. It can be strongly recommended to researchers and graduate students working in geometry, combinatorics and group theory.' Bulletin of the London Mathematical Society
'This book should be properly seen as the primary reference for the theory of abstract polytopes, especially of abstract regular polytopes … The book is very comprehensive and deep in its coverage of the topic. Almost everything known about abstract regular polytopes until the date of publication may be found somewhere within its 551 pages.' Zentralblatt MATH