“Thanks, I needed that!” I hasten to ask forgiveness for setting a seemingly frivolous tone in this review of a biography that far transcends the cradle-to-grave story of a great mathematician--as well told as that story is in this book. My fellow old-timers, along with younger classic movie aficionados, will recognize the quotation from The High and the Mighty, wherein the John Wayne character prevents his friend (the slap’s recipient, played by Robert Stack) from proceeding with a terribly ill-advised action. My thanks (and perhaps yours will too) addresses the book’s detailed elaboration of how Donald Coxeter restored a powerfully valid system of mathematical thinking (and proving). The story, in fact, captures the 20th century’s fascinating sociology of mathematics and of groups of mathematicians to a tee.
Again, what the world “needed” was and continues to be a redress of balance among the mathematical modes of thinking--the visual and intuitive being a time-honored one of them--which Coxeter treated as rigorously as the rest of us treat more algebraic or formalist-style mathematics. Coxeter’s and his students’ and acolytes’ work contributed singly, as the book shows, in the case of geometry. Visual and intuitive are merely points of departure here, as Coxeter’s work encompasses n-dimensional Euclidean and non-Euclidean geometries, with n sometimes being much larger than three.
Douglas Hofstadter, world-famous author of Gödel, Escher, Bach [1], provides a substantive foreword that hits several bull’s-eyes in characterizing his friend Coxeter’s style and substance: Coxeter’s “strict adherence to ... proving geometric theorems using geometric methods, not using algebra ... builds an intuition that is totally unlike intuition that comes through formulas.” His synthetic style served the more abstract projective geometry as well--to the extent that Hofstadter found it to be “among the most gratifying mathematical experiences [he has] ever had.”
Multi-chapter Part 1 is titled “Pure Coxeter.” Having read the book, I realize the play on words, as “pure” describes both Coxeter’s mathematics and his normative/ethical philosophy, which included pacifism. As with his colleague G. H. Hardy, he had neither interest nor investment in applications--but the hidden reality of pure mathematics foiled them both with “inadvertent applications” or facts of nature that were being discovered. For example, I was interested to learn in chapter 0 that spearmint and caraway molecules are chiral twins, that is, mirror images. Coxeter was to use kaleidoscopic, multi-mirror reflections in generating geometric figures and their (Coxeter) groups of transformations. Chapter 1, “Mr Polytope Goes to Budapest,” has a vivid narrative about young Donald, plus a general overview of geometry’s ancient history; the Platonic solids were apparently discovered quite early and at far-apart places. The famous parallel postulate, and young Coxeter’s related “forays in[to] the non-Euclidean realm,” comprise a very interesting discussion.
Chapter 2’s description of Coxeter’s family and childhood holds one’s interest by being a mixture of significant but mundane facts, such as his parents’ divorce, and what I would call Donald’s personal epistemology (which includes his mathematical style). His Aunt Alice was none other than the great George Boole’s granddaughter, Alicia Boole Stott. Coxeter saw ever more clearly the contrast between the increasingly dominant Descartes/Hilbert approach and his own continuation and elaboration of synthetic geometry.
Coxeter’s illustrious career included Cambridge (UK), Princeton, and Toronto universities and brief to extensive interactions with some who were better known than he, for example, Weyl and Wittgenstein. Regarding Weyl’s discovery of continuous groups, “Coxeter had hit upon a primordial and indispensable tool [Coxeter groups] that permeates that field of mathematics.” As regards such non-mathematicians as the flighty (my characterization) Buckminster Fuller (of difficult-to-pin-down profession) and the robust quasi-mathematical artist M. C. Escher, their interactions with Coxeter were substantive (especially that with Escher). One need not have been in either Fuller’s or Escher’s cult to find that part of Coxeter’s story interesting. Not surprisingly, Coxeter’s quasi-austere personality remained invariant throughout.
Although all the chapters are of the highest quality in terms of style and substance, I found chapter 6, “Death to Triangles!,” the most interesting for a personal reason. Dumb luck during my college days resulted in my interaction and study with at least three members of the secretive Bourbaki group, one of whom was “one of [its] two brains” at the time. Bourbaki’s (translated) full battle cry was “Down with Euclid! Death to Triangles!” Bourbaki’s goal was to build all of mathematics from the ground up whilst avoiding diagrams to the maximum extent. I make the point here that the book’s narrative comports one hundred percent with this personal experience. That several math majors would yell “Classiciste! Classiciste!” (in French) at any transgressor (albeit with humor) evidences the spirit of those times. Coxeter, “the antithesis of Bourbaki,” managed eventually and heroically to win Bourbaki over.
The eight appendices are certainly in the right place, and are first-class components of this excellent book. John Horton Conway’s perspicuous proof of Morley’s theorem (also known as Miracle, Appendix 5) regarding the angle trisectors of any triangle is wonderful to behold.
This outstanding book has my highest recommendation.
