Regular Polytopes (book)

Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries. The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations. It has 14 chapters, along with multiple appendices, providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers. It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject. The first chapter discusses regular polygons, regular polyhedra, basic concepts of graph theory, and the Euler characteristic. Using the Euler characteristic, Coxeter derives a Diophantine equation whose integer solutions describe and classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their duals to generate related polyhedra, including the semiregular polyhedra, and discusses zonohedra and Petrie polygons. Here and throughout the book, the shapes it discusses are identified and classified by their Schläfli symbols. Chapters 3 through 5 describe the symmetries of polyhedra, first as permutation groups and later, in the most innovative part of the book, as the Coxeter groups, groups generated by reflections and described by the angles between their reflection planes. This part of the book also describes the regular tessellations of the Euclidean plane and the sphere, and the regular honeycombs of Euclidean space. Chapter 6 discusses the star polyhedra including the Kepler–Poinsot polyhedra.