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

In the interest of full disclosure, I must admit from the start that I started to read this book because of the tireless promotion of Glenn Smith, my father-in-law, friend of Coxeter and passionate geometer (who also happens to be quoted in the book). I even went to U of T, even if my studies kept me far from the mathematics department. But although geometry was my least favorite of all the maths in school, with Glenn's passion for this book there was no way I could continue to ignore my husband's copy lying in the living room. I had to pick it up and see for myself just what his excitement was all about.

I expected to be in over my head, possibly bored, reading it more out of curiosity than intrinsic interest. But after the first few pages I was hooked. This book, while delineating the history of geometrical inquiry, is also a captivating narrative of Coxeter's life. This is the story of a man who pursued his passion with his own quirks and habits, told in a way that rendered him human to me even as it allowed me to fully understand why he is considered a genius in his field.

Yes, there were certain paragraphs full of mathematical explanations where I had to simply breath deep and hope that whatever on earth that meant had no direct impact on the unfolding of the narrative at large. The abundance of footnotes were also awkward at first, a sign that the author wasn't sure if her audience would be academic or popular, but after a while they faded from my attention as I became engrossed in Coxeter's story.

By the end, I was ready to pull out the Zome tools and mirrors so that I could start building models and see if I, too, could see in four dimensions or more. So far I'm still stuck in the regular three, but with the inspiration of Coxeter to guide me there's always hope.

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 [...]

Here is an unlikely candidate for real-life hero: a geometry professor. H. S. M. "Donald" Coxeter, a classical geometer interested in shapes, lines, vertices, polygons and the visualization of such geometric entities, saved the world from being overtaken by formalists who wanted to algebra-ize everything in geometry. Coxeter is not well known by most people; his geometry encompassed higher dimensions than most of us can think about. But he was truly a hero for the mathematicians who knew him and worked with him, and he did make differences in their discipline that have proved to have surprisingly widespread and even practical results. He has had the good fortune, four years after the end of his long life, to be the subject of a full and admiring biography _King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry_ (Walker) by Siobhan Roberts. Roberts is a journalist who interviewed Coxeter himself over a period of two years toward the end of his life, and has interviewed many leading mathematicians and scholars who were the best ones to explain the exalted status in which Coxeter is held. Roberts's book is not a geometry text; she give analogies about Coxeter's work and hints at its themes rather than going into any mathematical detail, so even if you are intimidated by mathematics, you can get an idea of Coxeter's thoughts and just why he was such a revolutionary. This makes not only for an interesting biography, but an agreeable tour of just how mathematics has gotten done in the past few decades.

Coxeter, born in London in 1907, was one of the mathematicians that broke the rule that doing math is a young man's game. He did make his first discoveries when he was thirteen, but was active until his death in 2003, still writing, proving, and presenting. He was a student at Cambridge, and in 1936 he immigrated to Toronto and took a teaching post at the university there, where he remained for the rest of his life. There are many descriptions of cranky mathematicians in these pages, but Coxeter was never like that. A fellow mathematician said, "He was almost courtly. He was very gentle, even when he managed to show you that you were thinking like an idiot." He had the archetypal lack of interest in any practical applications of his ideas, appalled that his lovely theories could be sullied by practical utility. He firmly believed in pictures, visualization, and intuition, putting himself successfully at odds with the formalists who had inspired the New Math that was taught in grade schools forty years ago. His insistence on visual appeal linked him to M. C. Escher who incorporated mathematical ideas into his art. Coxeter even wrote explications of certain Escher prints; that he did so gratified the artist, but privately the non-mathematician Escher said that the "hocus-pocus text is no use to me at all." Coxeter also had a close, not always collegial, relationship with Buckminster Fuller, whose geodesic domes he admired.

Coxeter's professional life was without reproach. His family life was much less than perfect. Part of the problem was that he had all the bumbling of an absent-minded professor, causing his wife Rien to screech names at him in her native Dutch. Coxeter conceded, "I was not able to love Rien as fully and completely as one should his wife," but when she developed Alzheimer's, he took uncharacteristically close care of her bathing, dressing, and feeding. Neither of their two children had interest in mathematics. His daughter "ran hot and cold on his status as a mathematical legend," but escorted the elderly Coxeter to his last conferences. She said, "Dad would hate to be equated with Elvis Presley, but Elvis gave people some moments of joy, happiness, inspiration. And if that's what Dad's work does for these people, that's wonderful. Personally, I get more from Elvis Presley." She isn't the only one, of course, but Roberts's delightful biography can't help but show even non-mathematicians just how important a figure Coxeter was. Do not fret that you don't understand all the math here. Coxeter once admitted that even in the geometry that he loved, "There are so many branches of the subject in which I am almost as ignorant as the proverbial man in the street."