Regular Polytopes

Regular Polytopes is densely packed, with definitions coming rapid-fire and results following quickly, much like Stanley’s Enumerative Combinatorics. Years of results are elegantly summarized with just enough details for clarity, but not so many as to increase the length to a burdensome amount. Most of the chapters are definition-heavy, but still very readable. The key vocabulary is italicized with definitions given more casually within the narrative rather than set apart in a formal style. Similarly, theorems, propositions, and proof also occur naturally in the text, with section headings giving reference to the theorem or proof being addressed.

The readability is further enhanced by the consistent use of concrete examples with each new topic. For instance, reflection groups are illustrated with mirrors (Chapter 5) and illustrations or diagrams of the polytopes are given throughout the text when possible. These illustrations and figures are not flashy, but are good, clear, and effective.

The book was last revised in 1973, so it is occasionally out-of-date, although not frustratingly so. For example, in Chapter 5, Coxeter discusses the “novelty” of the use of Dynkin diagrams. In Chapter 1, we are reminded that the four-color problem is “unanswered.” Vocabulary has also experienced some changes over time, i.e. Coxeter’s use of “reciprocal” polytopes, which in more recent times are usually referred to as “dual” polytopes. Rather than detracting from the text, I found that these occasional differences give insight into the mathematical progress that has been made in the last half-century.

One of my favorite parts of this book are the historical remarks found at the end of each chapter. Coxeter carefully associates the results from the chapter with the major contributing mathematicians, but also adds a few interesting details setting the context for the mathematics. Personal details are also included, both about the mathematicians and occasionally about Coxeter himself; I particularly enjoyed reading about his acquaintance with Alicia Boole Stott in Chapter 13.

Overall, like the illustrations and diagrams, the book provides a well-written and comprehensive coverage of regular polytopes that is clear and effective without being elaborate or excessively detailed. Further, the historical perspective, found at the end of each chapter as well as in the treatment of the topics, gives this book a distinctly more entertaining style than a standard mathematical textbook.

Tricia Muldoon Brown (patricia.brown@armstrong.edu) is an Associate Professor at Armstrong State University with an interest in commutative algebra, combinatorics, and recreational mathematics.