Roberts, Siobhan. King of infinite space: Donald Coxeter, the man who saved geometry. Toronto: Anansi, 2006. 399pp.

I'm reading a lot of science auto/biography these days, and generally enjoying it a lot. While generally not much of a fan of the "great man" theory of science history, I also tend to like a really good story. Siobhan Roberts's biography of mathematician Harold Scott MacDonald "Donald" Coxeter is a little more heavily weighted on the "great man" side of the equation a perhaps a little light on the "good story" but I enjoyed it tremendously nevertheless. Not least because I have a rather interesting personal connection to this book, but more on that later.

So, who was Donald Coxeter? In a sense, he was the Einstein of geometry in the 20th century, in the sense that Einstein demonstrated that science wasn't exhausted in the 19th century, Coxeter demonstrated that classical geometry wasn't exhausted either, that there was a lot of interesting avenues for important research and applications, both in science and the arts.

A nice quote from page 4, giving a sense of the man and his life's passion.

Coxeter was also known to be both instructive and entertaining in revealing the hidden symmetry of an apple. Around the dinner table with colleagues gathered for the American Math Society conference in 1981, he asked: " Did you know t hat apples do not have cores?" They thought he was pulling their legs, until the hostess, Marjorie Senechal, a mathematics professor at Smith College, procured an apple and placed it before him with a knife, as requested. He filleted the fruit into thin horizontal sections, demonstrating that there was no stem-to-stern core, but rather elongated pods of seeds within. The piece de resistence occurred when he reached the center of the apple and sliced through the equator. There lay it's secret symmetry -- not nature's sloppy attempt at spherical symmetry, as suggested by an apple's exterior, but rather perfect fivefold symmetry, hidden at the apple's heart: the apple seeds were arranged in a five-point star. Everyone around the table gasped when they saw it. "It just shows," said Senechal, "that he was looking everywhere, and looking deeply. Coxeter delighted in the geometry of everyday objects, and, because he was so curious and astute, he found symmetries and regularities in these objects that the rest of us never suspected."

And this is what Benoit Mandelbrot had to say about Coxeter style and place in history (p127):

"He was viewed as a throwback...He was a bit marginal...I remember feeling the strength of his style. The enjoyment Coxeter always had handling shapes, models, and letting models help him dream, is something I find very attractive and very important -- the spirit of loving shapes and the role of the eye and the hand, that what I dound so marvelous in Coxeter."

"Most people are not strong enough to have a well-defined personal style...The should bend according to fashion or circumstance and he clearly did not bend. He kept with his classical tradition of geometry, wich had been totally flattened -- pulverized would be even closer -- by Bourbaki. to learn mathematics without pictures is criminal, a ridiculous enterprise."

So, what is it about Roberts' book that makes it worth reading? First of all, it's quite a good outline of Coxeter's life, if a little shallow on his non-mathematical life. We hit the high points, like birth, death & marriage, but we see his children on stage only peripherally until the end when his daughter Susan starts taking care of him. To me this is a bit of a weakness of the book, the lack of color and emotion in the tale of Coxeter's life. Maybe there wasn't much, perhaps the life of a mathematician is like the life of a novelist, where all the good stuff happens between their ears rather than on a grander stage. And there are hints that Coxeter was a bit cold and distant. But still, remarkably little seemed to have happened in a life of over 90 years.

On the other hand, the story of Coxeter's intellectual life is absolutely griping, entwined as it was with the history of mathematics in the 20th century, especially the place of geometry in that history. In a way, you can almost see the story of Coxeter's life as the intellectual history of geometry in the last century, rising from the doldrums to take it's place in as a driving force in physics, ecommerce, databases and even bioinformatics. The overriding theme of the book is the interrelationship of geometry and life, the visual elements that we interact with in film, in art, in science, in architecture, in virtually every aspect of our lives. Coxeter's lifelong battle was to bring the visual sense back into math and math education, rejecting the more algebra-based ideas of the French Bourbaki collective, making math more understandable and accessible. And it was very clear that Coxeter was passionately concerned with the teaching of math and geometry and that cared a great deal about this students. There's a great story about how one of his students, Asia Ivic Weiss, now of York Unversity, was a bit leary of telling him about an error he'd made. When she did, he was actually delighted that she'd found the error and even gave her a wedding gift that commemorated the occasion (p129-130).

Yes, the more math, especially geometry, you know, the more you will enjoy this book. It doesn't shy away from challenging the reader to grasp subtle concepts, to make connections, to understand and enjoy geometry for it's own sake. But, we are rewarded by our efforts. We see how Coxeter's life intersects those of various notable personalities, both scientific and artistic and how Coxeter always takes something away to improve his own work. The artist M.C. Escher is a perfect example. He and Coxeter had a wonderfully odd, mutually beneficial relationship, a relationship well explored in the book as was Coxeter's relationship with Buckminster Fuller.

So, yes, this book is not perfect. I would have appreciated a bit more about Coxeter the man It's also a bit strange how much information was packed into 80+ pages of endnotes, almost like a parallel book actually more about Coxeter's life. I would have appreciated it if a lot of the material in the notes was expanded and pulled into the main narrative. Also, a couple of times a glossary would have been helpful.

But, overall this is a great book that tells a very important story. At the beginning I said I wasn't too fond of the "great man" theory of science history. That's certainly true, but at the same time I realize that so much of the intellectual history of an age can be seen through the works of individual scholars, that to ignore their stories is as great a error as to glorify them. And this book does strike a balance between the man, his work and the intellectual currents that surrounded him.

(A note to the math librarians out there reading this -- the bibliography is a wonderful source for collection development in the roots of classical geometry, in particular I guess we should all make sure our libraries are full of Coxeter!)

I guess I should elaborate on my personal connection to this fine book. In my capacity as Mathematics librarian at York University, I've obviously gotten to know many of the math profs at York and one of them is Asia Ivic Weiss, who happens to have been Coxeter's last grad student at the University of Toronto. Prof. Weiss was the graduate program director in the math department for a few years, including around when Coxeter passed away. As it happens, before he died Coxeter donated a significant amount of his mathematical papers to the Math Department at York, to be housed in their Coxeter Reading Room. Well, a few years ago, after Coxeter had passed away, Asia approached me for some advice on what to do with these papers. They were nice to have, and scholars certainly took advantage of them, but she was not sure if this was the best place. So, we met and discussed the situation. We quickly saw that the issue was larger in scope, that really we should come to terms with all of Coxeter's papers, most of whom were at his house in the Rosedale neighbourhood in Toronto. So, Asia, Coxeter's daughter Susan Thomas, Coxeter's mathematical executor Arthur Sherk and I all met at the Coxeter house where Susan was still living. I also consulted with one of York's archivists, Susanne Dubeau, to get her advice on the situation. After that meeting and subsequent conversations we all agreed that Coxeter's papers belonged at the University of Toronto Archives, including the ones that were currently at York. And so, that's where they are now. It was gratifying that everytime I checked a footnote in Robert's book and saw that the source was from the UofT archives that I played a small part in making sure that Coxeter's papers are accessible to scholars and journalists. It's also great to see the names of so many of the profs know from my work mentioned, like Lee Lorch, John Andraos from Chemistry and Walter Whiteley. Walter was probably the most interviewed person in the book, with many insights on the role of Coxeter's work in other areas of math and science and on the "geometry gap" -- the idea that if we don't teach geometry to scientists, they'll miss out spacial or geometric connections in their work.

Tomorrow, December 12, 2006, there's a book launch at the Fields Institute in Toronto. I hope to be there and report back my experiences.

Update 2006.12.16: I went to the book launch at the Fields institute on the 12th and it was a very nice event. Not a reading, more of a cocktail party with a short talk/film clips in the middle. The relaxed atmosphere was very congenial; it was nice to see my York colleagues Asia Ivic Weiss and Walter Whiteley as well as to meet Susan Thomas, Coxeter's daughter, again after a couple of years. As usual at such events, it's always a treat to shake the author's hand and say how much you enjoyed the book.