King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry

King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, Siobhan Roberts,  2006. 320 pp., illustrations, $25.00 Hardcover. ISBN : 0802714994. Walker & Company, 104 Fifth Avenue, New York, NY 10011, (212) 727-8300, http://www.walkerbooks.com/

Siobhan Roberts’ King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry is by far one of the best math related books I have read in years. Admittedly I am a geometer at heart, but it was far more than the mathematical content of the book which excited me. First of all, the author provides a detailed and very human look at  the life of a world class mathematician. We follow Coxeter’s career from Trinity College to the University of Toronto with stops along the way at Berkeley and the Institute for Advanced Studies at Princeton. Secondly, we get a good look at how and why geometry fell out of favor in the twentieth century, thanks in good part to efforts of Bourbaki, the French mathematics critical montoring group. Thirdly, we see how Coxeter developed many of his important results in a way that is accessible to anyone with a decent secondary mathematical background. The book contains seven appendices and an extensive set of endnotes all of which I found to be both very readable and very helpful. The author does a nice job of showing how the concept of symmetry was central to Coxeter’s work in second, third, fourth, and higher dimensions. I found particularly interesting Coxeter’s admiration for the work of M. C. Escher but as hard as Coxeter tried he could not get Escher to understand the mathematical significance of his own art work. During the later part of the book, the author shows how Coxeter’s work has been used in a variety of fields both inside and outside of mathematics:  Buckminster Fuller, in his work with geodesic structures, was inspired by Coxeter’s polyhedral theories; Macarthur Fellow Jeff Weeks employed Coxeter’s work with higher dimensional polytopes in developing his theories on the shape of the universe.

One theme that occurs again and again throughout the book is that Coxeter’s work was always characterized by his excellent taste, his sense of beauty, and the exquisite simplicity of his mathematics. I hope that anyone who reads this book will run out and get copies of Coxeter’s three wonderful books:  Introduction to Geometry (2nd edition 1989), Regular Polytopes( 1973 ),  and Geometry Revisited ( 1967 ) (co-authored with Samuel Grietzer ) . Finally, the author has shown how Coxeter’s efforts have helped rekindle people’s interest in geometry. Many prominent people in the mathematics community, such as Douglas Hofstadter and John Conway, have been inspired by Coxeter’s work and are helping to revive interest in this beautiful subject. This is a book that should be read by everyone who teaches geometry and by anyone who has any interest in the way in which some of the most elegant work in mathematics during the last century evolved.

Siobhan Roberts has been working with members of the York University Mathematics Department to put together a web site for the book. The site is still under construction but an early version  can be found at  www.donaldcoxeter.com.

Jon Choate, Mathematics Depart., Groton School, Groton, MA

See also the MAA Review by Branko Grünbaum.