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The Sensual (Quadratic) Form

John Horton Conway
Mathematical Association of America
Publication Date: 
Number of Pages: 
Carus Mathematical Monographs 26
[Reviewed by
Michael Berg
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A book with such a title cannot fail to capture the attention of any mathematician, independent of specialty. On top of this, the author is John Horton Conway, of monster, moonshine, and game of life fame, to name but a few things cementing this unique mathematician’s reputation. Indeed, Conway is very well known, not just for his eclecticism and breadth (and depth, of course: just look at what monstrous moonshine hath wrought), but also for his playfulness and humor. And these qualities are richly represented in The Sensual (Quadratic) Form. For one thing, it is surprising, right off the bat, to encounter so huge a number of pictures and diagrams in a book devoted to a quintessentially arithmetical topic.

Along these lines, Conway presents in the book’s preface the following appraisal of the “topographs” that pepper his first chapter: “The ‘topograph’ of the First Lecture makes the entire theory of binary quadratic forms so easy that we no longer need to think or prove theorems about these forms — just look! In some sense the experts already knew something like this picture — but why did they only use it in the analytic theory, rather than right from the start?” Tantalizing stuff, and more than a raison d’être for the artwork.

The book is a carefully crafted (Conway thanks Francis Fung for “turning the bundles I had into the book you hold”) collection of four Hedrick Lectures dating to 1991; their titles are: “Can you see the shape of 3x2 + 6xy – 5y2 ?,” “Can you hear the shape of a lattice?,” “…And can you feel its form?,” and “The primary fragrances.” Each lecture is followed by an “afterthought,” including two that are particularly attractive on hard-core arithmetical grounds: “PSL2Z and Farey fractions” and “More about the invariants: p-adic numbers.”

And then there’s even more: Conway appends a postscript titled, “A taste of number theory” (Yeah!), where we encounter three famous theorems: quadratic reciprocity (for the Jacobi symbol), the fact that an even unimodular quadratic form’s signature is divisible by 8, and Legendre’s three-squares theorem. We also find here the strong Hasse-Minkowski principle and Gauss’ theorem on triangular numbers (done at the speed of light), and more besides.

The book overflows with good stuff, done in Conway’s unique style. It first came out as a hardcover Carus Monograph in 1997; the present book is the 2005 softcover version. This should boost its circulation considerably. And deservedly so.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.