*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 ErdÃ¶s, 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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