Winner of the 2009 Euler Book Prize, awarded by the Mathematical Association of America for an outstanding book about mathematics. The Euler Prize citation reads:
"This book by Siobhan Roberts gives an intimate and engaging portrait of one of the most influential mathematicians of the last century. It also provides a mathematical history of those years, including the currents set in motion by Hilbert's 23 problems, the influence of Bourbaki, and the unexpected applications of mathematics to computer science, communications, information, crystallography, medical research, environmental studies, as well as art -- Coxeter's work directly inspired Circle Limit III by M.C. Escher. Above all, it gives a superbly readable account, in personal terms, of the search for beauty that sets mathematics in motion, and of the synergy of individual and group efforts that make it happen. It's an engaging page-turner, even for nonmathematically trained readers, and it will offer them an insider's look at the world of mathematics and the people who create it ... [The book] will also strike a special chord with mathematicians, because it honors the spirit of wonder and openness that Coxeter embodied in his approach to mathematics." ... Read more
Customer Reviews (16)
Behold a Fearless Symmetry
An enjoyable biography of one of the 20th century's great mathematicians. It shines is in its description of Coxeter's interaction and influence with other luminaries including Veblen,Wittgenstein, Lefshetz,Conway, Escher, Bourbaki (the author holds back for a while, playing into the deception that Bourbaki is a group, not an individual), Buckminister Fuller, Hofstatder and others. The time at Princeton has its moments but at best Coxeter was at Fine Hall at the same time as Einstein and Von Neuman, but that's about it.
The book does dragwhen it delves into his marriage to Rien Brouwerand his politics.Coxeter was a pacifist but an armchair one.His most dramatic act was to walk out on a talk at Hart House by George H. Bush along with about half thefaculty.It picks up again when discussing the relevance and importance of his work to such diverse areas as logistics, immunology and molecular structures, telecommunications, string theory and ultimately the shape of the universe. Geometry is incredibly important to physics as parallel to every symmetry that you find in n-dimensions there is believed to be a corresponding conservation law.They become one and the same.
As I grew up in Toronto and was a student who took several courses in the Math department at UofT I found it interesting how Coxeter had in various ways indirectly affected my own life. As a teenager I spent many happy hours figuring out magic squares and other puzzles in his edited version of H. Rousse Ball's Mathematical Recreations). I still have my copy of Father Magnus Wenninger's Polyhedron Models which I used to create numerous stellated polyhedra and hang them in my room and give them as gifts. Father Wenninger corresponded with Coxeter and is mentioned in the book, however you need to delve into the End Notes to get more of the details. Coxeter had already retired when I entered the University of Toronto. Apparently he was glad that the department had hived off Computer Science.I wound up with a minor in Applied Math and was not required to take a computer course, though I did learn SPSS in order to chug out statistics and that proved to be fairly useful.
There's some good understandable coverage of some mathematical concepts such as Coxeter numbers and diagrams. Siobhan Roberts communicates the sense that Coxeter kept the field of geometry alive, but the lack of successors is troubling.
Coxeter's life message would be about the importance of diagrams and visual geometry as a foundation of Western intellectual thought. He apparently disliked computers and never learned how to use one though there is one short excerpt that says that he set a problem for his calculator- either it was a TI programmable, or he did a combinatorial calculation and had a great deal of faith that it hadn't died on him as it worked out the answer overnight. What I find paradoxical about this is that towards the end of his life (and he was quite active) some of the most interesting descriptive work in geometry was in computer generated graphics for GIS, Engineering and Film.
In classic tradition the Quadrivium - Arithmetic, Geometry, Astronomy and Music along with the Trivium - Grammar, Logic and Rhetoric, formed the basis for a liberal education. As the book notes, we've somehow lost the idea that Mathematical Thinking is part of that.Looking over my eldest's options for electives in University I was appalled to find that other than an Astronomy course there wasn't a single course in Math or Science offered. More Coxeter, please.
Difficult topic handled womanfully
Biog of Coxeter, a Canadian professor who flourished in the 1960s and was famous for having 'saved geometry'. Famous amongst mathematicians, at least. His books were and I expect still are on university maths reading lists. The authoress is Canadian, and it's her first book. It has to be said she hasn't been bold in her comments. For example it's not entirely clear what Coxeter's life work really amounted to. Nor does she try to tackle the highly disputable elements which seem taken for granted by these people, on such things as multiple dimensions and groups (defined in ways not in conformity with the usual meanings, I'd guess deliberately), geometry of curved surfaces - which has been given a sort of fantasy ethereal feel, and the extremely simple and therefore implausible derivation of e=mc squared.
Coxeter seems to have been happiest with extensions of the Platonic solids (in 'Regular Polytopes'), and related activities, such as the standard system for describing crystals (by their nodes etc), geodesic domes, possible implications for molecular structure, and so on. There's quite a bit on domes (which he didn't invent) and buckminsterfullerene, a form of carbon generated by evaporating carbon in a vacuum whereupon the atoms because of carbon's valency of 4 tend to like as sheets, but by default curl round. Whether anyone's found a use for this, seems not to be known by Roberts. Another thing Coxeter didn't find was Penrose's simple-seeming slightly asymmetrical tiles.
Coxeter used algebra rather heavily and extended into regions where he then tailed off and others' expertise ruled. His book of 'twelve essays' hardly sold any copies. There are quite striking accounts, if you read between the lines, of inter-departmental and inter-university rivalries.
Coxeter was a pacifist and appears to have had little to do with WW2; and was also a vegetarian. Although highly praised (e.g. there's an intro by Douglas Hofstadter) his own university (Toronto) seems to have eased him out - though this may have simply been standard practice.
Various other figures - Newton, Kepler, G H Hardy, Escher, Martin Gardner, Buckminster Fuller, Polya, Einstein, several Penroses etc naturally enough appear.
Recommended as a general biography though it's light on both explanations and sceptical analyses.
Bound in a nutshell, for the moment
My namesake BrianO Nualláin aka Flann O'Brien/Myles na Gopaleen (we are not related, but share Oct 5 as a birthday) wrotea science fiction comic novel called "the third policeman" in which the footnotes competefor space with the main narrative. In this reverential account of a great geometer, Siobhán (I have reinstated the accent she eschews; it is pronounced Shove-awn) Roberts is so keen to get the details right that the end matter (notes, appendices, indices) occupies Pp259-399. Indeed, whether to focus on the copious notes or the main narrative is a real issue.
Coxeter first came to my attention in the beautiful "hypergami" work described by the Eisenbergs in my "spatial cognition" collection. Spatial Cognition: Foundations and Applocations : Selected Papers from Mind Iii, Annual Conference of the Cognitive Science Society of Ireland, 1998 (Advances in Consciousness Research)He had three adversaries; the anti-diagram Bourbaki school
(150-170),anybody who wanted practical applications like Buckminster Fuller (180-181) and , finally, non-classical geometry like Riemann. The book is extremely informative, and should perhaps have received more attention. It is still not too late to combine with the likes of the Eisenbergs to revive this project; pace another reviewer here, another book on this subject may take a long time.
Seán O Nualláin Ph.D. 5u Meitheamh2009
Good bed-time read
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.
Coxeter deserves better...
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.
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