The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations.[1][2] It has 14 chapters, along with multiple appendices,[3] providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers.[1] It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject.[1][2]
The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the regular polytopes, two chapters on higher-dimensional Euler characteristics and background on quadratic forms, two chapters on higher-dimensional Coxeter groups, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and polytope compounds.[3]
Later editions
The second edition was published in paperback;[9][11] it adds some more recent research of Robert Steinberg on Petrie polygons and the order of Coxeter groups,[9][12] appends a new definition of polytopes at the end of the book, and makes minor corrections throughout.[9] The photographic plates were also enlarged for this printing,[10][12] and some figures were redrawn.[12] The nomenclature of these editions was occasionally cumbersome,[2] and was modernized in the third edition. The third edition also included a new preface with added material on polyhedra in nature, found by the electron microscope.[13][14]
Reception
The book only assumes a high-school understanding of algebra, geometry, and trigonometry,[2][3] but it is primarily aimed at professionals in this area,[2] and some steps in the book's reasoning which a professional could take for granted might be too much for less-advanced readers.[3] Nevertheless, reviewer J. C. P. Miller recommends it to "anyone interested in the subject, whether from recreational, educational, or other aspects",[4] and (despite complaining about the omission of regular skew polyhedra) reviewer H. E. Wolfe suggests more strongly that every mathematician should own a copy.[7] Geologist A. J. Frueh Jr., describing the book as a textbook rather than a monograph, suggests that the parts of the book on the symmetries of space would likely be of great interest to crystallographers; however, Frueh complains of the lack of rigor in its proofs and the lack of clarity in its descriptions.[6]
Already in its first edition the book was described as "long awaited",[3] and "what is, and what will probably be for many years, the only organized treatment of the subject".[7] In a review of the second edition, Michael Goldberg (who also reviewed the first edition)[1] called it "the most extensive and authoritative summary" of its area of mathematics.[10] By the time of Tricia Muldoon Brown's 2016 review, she described it as "occasionally out-of-date, although not frustratingly so", for instance in its discussion of the four color theorem, proved after its last update. However, she still evaluated it as "well-written and comprehensive".[15]
