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Publisher:

Walker & Company

Publication Date:

2006

Number of Pages:

416

Format:

Hardcover

Price:

27.95

ISBN:

0802714994

Category:

General

[Reviewed by , on ]

Branko Grünbaum

08/28/2006

H. S. M. (Donald) Coxeter was unquestionably the best known and most successful practitioners of "intuitive geometry" during much of the twentieth century; some writers call this he "classical geometry". "Intuitive geometry" is used here to denote the part of geometry that starts with the study of simple figures (points, lines, polygons, circles, polyhedra, and such) and proceeds by generalization to more complicated objects and to higher dimensions. Tools from combinatorics, algebra, analysis, topology and other fields are often used, but an intuitively accessible interpretation remains the core of the subject. During the first third of the twentieth century this aspect of geometry became overshadowed by disciplines oriented towards the exploitation of heavier mathematical tools. This gave rise to flourishing of topology as well as algebraic and differential geometry but made preoccupation with intuitive geometry all but disreputable.

Starting in the 1930s, Coxeter showed that this attitude is not justified, that intuitive geometry is not only challenging but also leads to a better understanding of problems in many other fields. With great consistency and extraordinary efforts Coxeter produced many books (at various levels of technical difficulty) as well as well over 200 papers reviewed in *Mathematical Reviews* (and many other works not listed there). A possibly even better measure of his involvement in explaining and promoting geometry is the fact that he authored close to 900 reviews in *Mathematical Reviews* ; this is surely a record.

*King of Infinite Space* is a biography of Coxeter as well as an account of his efforts to resist and counteract the Bourbakist tendencies and mindset. The book describes in detail both Coxeter's life and his interaction with people and institutions, mathematical, artistic and others. It is written in a lively and easy-to-read style, and provides documentation and references for all the claims. In pursuing the goal of presenting a complete account of Coxeter' life and influence, the author conducted interviews with an astonishingly large number of individuals, from Nobel prize winners to a resident of a psychiatric facility. All the facts are presented with great tact and sensibility.

The book is not intended to be a compendium of Coxeter's work. Instead, it is meant to convey a feeling for the obstacles Coxeter had to overcome, and the directions in which he advanced geometry and mathematics in general. The author is not a mathematician, but even so managed to convey the essentials of many of Coxeter's ideas.

Although its title seems rather strange, I can recommend the book in warmest terms to all those who wish to get detailed information about Coxeter's life, and an overall image of his achievements and influence. It is very unfortunate that whatever visibility accrued to intuitive geometry through Coxeter's life-long efforts appears to be disappearing in much of the academic world. As Siobhan Roberts puts it (page 363) in discussing Coxeter's disagreements with the University of Toronto during the last years of his life:

Another disappointment was the void left upon his retirement. The greatest living classical geometer, who had put the University of Toronto on the international mathematical map, was not replaced; there is now no classical geometer, not even a classically inclined geometer, on staff. Mathematical traditions usually do not reside in isolated individuals. Certainly, one of Coxeter's legacies is the community of his students who still come together in celebration of their work, and share his classical spirit with up and coming generations. And, quite conspicuously, Coxeter's students are the sort of mathematicians who take more pleasure in their work than most; they live and breathe and truly love the art. But for many, the decision taken by the University of Toronto, and other institutions internationally, not to invest in the future of classical geometry is alarming, if not tragic.

Branko Grünbaum is Professor Emeritus at the University of Washington. He lists his current interests as follows:

- Polyhedra in a general sense (not necessarily convex).
- Configurations of points and lines in the Euclidean or real projective planes.
- Graphs, in particular their imbeddings in manifolds.
- Tilings and patterns in the plane and in space.

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