Abstract Regular Polytopes
Regular polytopes are described in detail in classical monographs by H. S. M. Coxeter. The main theme of the book – abstract regular polytopes – forms a generalization of classical regular polytopes. The automorphism groups of abstract regular polytopes can be characterized as the so called string C-groups. After a description of basic notions, the authors discuss Coxeter groups, amalgamation of abstract regular polytopes, their realizations and regular polytopes on space-forms. Various techniques for a construction of new regular polytopes (mixing or twisting) are then studied. There is a strong relation between regular polytopes and real quadratic forms. Similar relations also hold for an important class of abstract regular polytopes (real forms are replaced by hermitian forms). The next three chapters contain a description of locally toroidal regular polytopes. The classification problem for locally toroidal regular polytopes was an inspiration for the theory of abstract polytopes. The book ends with a description of certain families of regular polytopes to linear groups. There is a rich bibliography containing a half thousand items.