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

Siobhan Roberts

Walker and Company 2006
A book review by Danny Yee © 2008
Living from 1907 to 2003, Harold Scott MacDonald "Donald" Coxeter's career as a mathematician nearly spanned the 20th century. In King of Infinite Space, Siobhan Roberts tells the story of his life, but also uses it as a framework to explore topics in the broader history of mathematics, its applications and its teaching.

Mathematicians pose obvious challenges for a biographer. Geometry is probably the most accessible area of mathematics, but Coxeter's work is still too technical to present to a general reader in any depth. And his core work was on polytopes, the equivalents of polygons and polyhedra in higher dimensions, which are hard to depict in three dimensions, let alone two. Roberts presents just enough to give some feel for Coxeter's mathematics, and throughout tries to explain why it is important.

Coxeter's personal life isn't the stuff of drama. A childhood interest in geometry led to study at Cambridge and a career as a mathematician, at Princeton and for most of his life at Toronto. The publication of Regular Polytopes and An Introduction to Geometry brought him broader recognition. He remained active right to the end, giving a conference lecture in Budapest the year before his death. Outside his work, Roberts says little about his marriage and family, but describes his pacifism during World War II and support for civil liberties during the McCarthy era.

Much of the interest in King of Infinite Space comes from the notable figures whom Coxeter worked with or knew, including Albert Einstein, Paul Erdös, Martin Gardener, Freeman Dyson, Buckminster Fuller, John Conway, and Douglas Hofstadter (who contributes a foreword). The artist M.C. Escher became a kind of collaborator, despite not understanding any mathematics, and a lesser known correspondent was George Odom, a kind of idiot-savant who chose to live in an asylum.

Roberts also sets Coxeter in the context of 20th century mathematics, in particular as a leading exception to a broad move towards abstraction and neglect of geometry. So she gives a brief biography of Bourbaki, the French "group mathematician" notorious for not using any diagrams and pronouncing "Death to Triangles", and an account of the trend which led to the "New Math" and the downgrading of geometry in mathematics teaching. Coxeter opposed this through his work and by encouraging geometry in schools.

Coxeter's mathematics is also set in its broader context. Towards the end of King of Infinite Space Roberts describes some of the applications of Coxeter-style geometry with which Coxeter wasn't himself involved: computer animation, the Geometer's Sketchpad software, protein folding, Buckminsterfullerene, speculations about the shape of the universe and the structure of space-time, and string theory.

All of this makes 250 pages go by fast, and feel quite full. King of Infinite Space is a fun read which should appeal to non-mathematicians without putting off those who do have a mathematics background. Eight appendices provide slightly more technical material, the most difficult being Conway's 1995 proof of Morley's Miracle. And there are endnotes and a bibliography for those who want to delve deeper either into the mathematics or into other aspects of Coxeter's life.

June 2008

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%T King of Infinite Space
%S Donald Coxeter, the Man Who Saved Geometry
%A Roberts, Siobhan
%I Walker and Company
%D 2006
%O hardcover, notes, bibliography, index
%G ISBN 0802714994
%P 399pp