Abstract Regular Polytopes [Hardcover]

A modern, comprehensive review of abstract regular polytopes.Abstract regular polytopes are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties. This comprehensive up-to-date account of the subject meets a critical need for a text in this area; no book has been published in this topic since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974).Abstract regular polytopes are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties. This comprehensive up-to-date account of the subject meets a critical need for a text in this area; no book has been published in this topic since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974).Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. This is the first comprehensive, up-to-date account of the subject and its ramifications. It meets a critical need for such a text, because no book has been published in this area 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 shol;