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

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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."

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 [...]

Let me say at the outset, I was very excited when this book came out. It was awhile before I got around to reading it, and in the interim my expectations grew. Having just finished KoIS, I am sad to say, my expectations were greatly disappointed. SR may have written a fine magazine profile (I haven't read it; I can't say; but I don't doubt she did), but as (evidently) neither a mathematician nor a historian(/biographer) by training, she was not, in my view, up to the task of developing a book length treatment of her subject. The opening chapters of KOIS read as competent if uninspired biography. Coxeter strikes me as perhaps not the most scintillating of biographical subjects. But judging from his massive scholarly output, and the influence he has exerted on several generations of mathematicians and scientists in disparate fields, he is a patently fascinating intellectual subject. Clearly SR knows this too, and tries admirably to convey as much. But to do her subject true justice, she would have needed to delve far more deeply into his scholarship, and the fundamentals of his discipline, than she is able to. I can only hope that Conway or Hofstadter or Dyson or some other deeply knowledgeable -- and sympathetic -- Coxeterian will take up the challenge of writing a real scientific biography of HSM. He richly deserves one.

One additional criticism of KoIS: in its concluding several chapters, KoIS nearly abandons its subject altogether, and roves randomly over topics in the geometry of biochemistry and physics. These are fascinating topics, to be sure, but a) RS treats them only superficially and b) their connection to Coxeter is far from clear. The subtitle of the present book seems to suggest that but for Coxeter advances in biochemical and string theoretical thinking might never have occurred. Really? I don't doubt Coxeter's key role in keeping geometry alive in the mathematics of the 20th century. But he was not THE only geometrist of the era nor THE only geometry-minded mathematician or scientist. SR is practicing an annoying kind of hyperbole here that is all too common in popular-science treatments. These closing chapters, then, read almost entirely as a kind of filler, chock full of interesting details, but of only tangential relation to the topic at hand: Coxeter's life and work.

Having come down on KoIS rather negatively, I do credit SR with taking this project on. She clearly has great affection for Coxeter the geometer and the man, and the density of her endnotes (a bit excessive to be sure) attest the hard work she did in the library and on the phone. She is an engaging writer and will doubtless do fine work in the future. That said, both the biography and the intellectual biography of HSM Coxeter have yet to be written. I look forward eagerly to reading them one day.

As I very rarely read biographies and almost never books about math, this book, recommended to me my father, a geometry and coxeter enthusiast, came both as a pleasant surprise and a nice change in my reading habits. First of all, I must admit that it upsets some of the stereotypes we may have about mathematicians being rigid minded reasoners, inclined to reduce reality to barren numbers and calculations. This, as we are shown, is not the case for Coxeter.

In King of Infinite Space, Siobhan Roberts depicts Coxeter as a veritable artist: driven by esthetic impulses and indifferent to scientific utility, we are shown Coxeter's geometry as being inspired overall by a sense of beauty . Roberts also shows how Coxeter's work, in spite of all this, has a significant place in the history of mathematics, as well as its current developments, and how it has found practical applications in many scientific fields.

Avoiding tedious technical descriptions that could weigh down the book and providing adequate clarification when needed (usually indexed), the writing is accessible and engaging; it is written in an intelligent yet unacademic style which is inviting to all types readers, mathematicians and biography enthusiasts alike .

One thing I especially enjoyed about this book is that we are given an image of the artistic, imaginative side of mathematics much different from the pencil and paper problems and encumbering calculations that many of us are put off by. Coxeter's mathematical world is replete with dynamic, rotating shapes and multi-dimensional figures, kaleidoscopes and mirrors, honeycombs and reflections; all of these are of interest not for their scientific utility or practical purposes but rather because they satisfy Coxeter's penchant for symmetry, harmony, and beauty. No matter how indifferent you may be to the math, what comes across in this book is something undeniably appealing in Coxeter's romantic quest for higher spaces and heroic in his attempt to 'save geometry'. Really, what is there not to like about an aloof mathematician whose primary tools are folding mirrors, kaleidoscopes, and escher paintings?

In the end, we can make no claims of understanding Coxeter's math or individual work, a special privilege afforded only to geometers, career specialists, or die-hard enthusiasts (my father); however, what we can appreciate is the spirit and nature of his thought, its scope and importance, his esthetic approach to the subject and his humble outlook regarding his own work. This book, written for a popular audience, excels at informing us of just that.

Is geometry dead? For most of us, geometry is but a faint memory of school days, an old-fashioned toy collecting dust cast in a corner of our childhood memories. So old-fashioned indeed that geometry was indeed pronounced dead by a group of French mathematicians, Bourbaki. In fact we find in this first book by Siobhan Roberts that geometry was already falling out of fashion when the protagonist of her work, Dr. Donald Coxeter, chose the subject to be his lifelong pursuit. What he made out of it for his extremely long and fruitful academic career, therefore, is an indispensable lesson that one's passion and talent would eventually lead to extraordinary achievements regardless of the trendiness of the nature of one's target subject.

"King of Infinite Space" is about a mathematician who reinvented and reinvigorated geometry. It will be a delight to anyone with basic understanding of geometry, but readers' enjoyment does not depend on it. It can be read also as an arresting biography of a lonely scholar who sought solace in the beauty of mathematics while being emotionally alienated by his family. Instead of his wife, it seems as if geometry was his partner of life, bringing him many joys and even unexpected interactions with artists such as M.C. Escher or with George Odom, an admirer and a resident in the Hudson River Psychiatric Center in upstate New York who kept sending Dr. Coxeter numerous models.

For the rest of us less endowed with mathematical talent, this book offers a glimpse to life and mind of a genius. The unfashionableness of his research subject seems to have been compensated by the longevity of his academic career -- he was writing new papers well into his nineties. Offering also a digested and concise version of the rich history of geometry, Roberts' work is a wonderful choice to a wide range of readers.

Coxeter was a widely respected mathematician for whom many people had a deep fondness, and this is successfully captured in Roberts' book. The book contains little mathematics, but what it has, is well chosen.

Like many biographies of mathematicians, the books errs a little on the side of excessive hero worship, and as such it gives a somewhat simplistic and distorted view of Coxeter's contributions to mathematics and his status in the mathematical community. In particular, the supposed opposition between Coxeter and Bourbaki really misses the mark; and the portrayal of Jean Dieudonné, a truly inspirational intellectual, is very shabby.

While the book isn't a scholarly work, it is nevertheless very well documented. The book has lots of interesting information, particularly on Coxeter's early years, and his connections with Buckminister Fuller, and Escher. And it contains interesting interviews with numerous mathematicians. Above all, the book is well written and entertaining. Well worth the read.

I am very happy to have received this book as a gift from a former math professor when I began teaching high school geometry last year. Initially during the year, it was difficult to get the students excited; I found that having them read excerpts from this book really helped! The narrative way the book is written allowed students to follow along, and they began to understand some of the joy that can come from mathematics.
I hope this book inspires a new generation of geometers!

If you enjoy Mathematics and particularly Recreational Mathematics;you'll find this book a real treasure trove. This book touches on so many interests I've had over the years,that I was mesmerized how one thing after another kept popping up.I have been interested in Recreational Mathematics,Puzzles and a wide range of things these pursuits lead to. I have over 500 books in my library connected to these interests in one way or another,but this book really shines in that it covers so much and over such a long period.
Starting off with the author's name. It's pronounced "she-von" and is the Irish form of Joan,a name which was introduced into Ireland by the Anglo-Normans.It has been anglicized as Judith and Julia.
I have also developed an interest in the concept of "connections" and as one goes through the fascinating journey of Coxeter's life ,connections endlessly come up. I started getting interested in puzzles in school during the early 50's and have never ceased to be drawn to them.
It's amazing how one encounters things without really trying.Two years ago ,while Birding at Pointe Peele,another obsession;things were quiet one evening,and I was doing some Sudoku puzzles when a young lad came by. I got him interested,and his uncle also came by. He told me that if I liked puzzles,I should meet his friend who constructs amazing stick and ball models.I took him up on it and met a retired engineer ,like myself,and for a whole afternoon,he showed me a number of models he had constructed.They were fascinating polyhedra,similar to those being held by Coxeter on the cover of this book. One big difference though;he used wooden balls,about 3/4" diam. and sticks about 1/8" diam. to construct his models. He painted the balls different colors to emphasize their positions in the structure. This approach allows the internal structure of the model to be seen. They remind me of the way chemistry uses similar balls and sticks to display molecular structures.This man told me about Coxeter and that he discussed his models with him and how he actually built a computer to determine what stellations of models were or were not possible to construct.At a later visit to his home,he showed me dozens of other models he had constructed and even a machine he had built to drill the holes in the balls. By that time I had turned up "Polyhedron Models" by Magnus Wenninger and my friend told me he also discussed these models with him.He gave me one of his models and it is one of my prized possessions.
Later on I found Wenninger's website,and if you'd like to see his wonderful models,visit it.
So, naturally,when I got Robert,s book,I knew it would be of great interest.
Although I had purchased Rouse Ball's "Mathematical Recreations" in 1960,and had even constructed a Rotating Ring of tetrahedra,I hadn't noticed that Coxeter had done the revision.
I recently read "The Universal Book of Mathematics" by David Darling,(see my review on June 22,2007)and found it a very handy reference as I read this book.
There is so much covered in this book ,one hardly knows where to start in writing a review. It has us meeting so many people in Coxeter's world;Einstein ,Buckminister Fuller,M.C.Escher,Martin Gardner,Paul Erdos,and on and on.You will meet their families ,students,likes and passions. These great minds are the people who have been in the forefront of Mahematics during the 20th Century and all who took over and expanded the horizons of the great Mathematicians who paved the way.
Like ,in the world of music, there are the great composers and great players,but even though most of us can never really enter their worlds or minds,we can thankfully still enjoy and appreciate their work. Such is the same with mathematics.
I was glad to read where M.C.Echer couldn't begin to follow Coxeter when he tried to explain math to him.It just confirms that one does not necessarily have to understand something to enjoy it. Even Coxeter understood that, and felt likewise about many areas of math and science.
If there is anything that this book really brought home to me is that it shows why I have enjoyed my pursuits in Recreational Mathematics and Puzzles so much over the years.
On top of being a wonderful biography on the life of Donald Coxeter and his associates and friends;it is a goldmine of a reference. 63 pages of Endnotes provide endless resource information;coupled with 13 pages of Bibliography and an extremly well compiled Index;makes this a major reference source that I will turn to often.

Siobhan Roberts first book is a gem. This is up there with Hofstadter's books in terms of the content. I was very impressed with the depth the author went to in describing mathematical objects. I was introduced to interesting characters such as George Odom, a life long fan of Coxeter.

This is a book all fans of math beyond the textbook must read!