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*The Poincaré Conjecture: In Search of the Shape of the Universe*, by Donal O’Shea, 2007, ix + 293 pp., $26.95, ISBN-10 0-8027-1532-X, ISBN-13 978-0-8027-1532-6, Walker and Co., 104 Fifth Avenue, New York, New York, 10011, www.walkerbooks.com

Donal O’Shea’s book The Poincare Conjecture is a very readable popular account of the history of the Poincare Conjecturé, recently proved by Grigori Perelman. It is clear that the author tried hard to make the book accessible to everyone from the curious high school student to the research mathematician. I think he succeeded rather well. For those not already familiar with the history of mathematics and its Dramatis Personae there is a Glossary of Terms, a Glossary of Names and a Timeline. Naturally the more mathematical maturity you can bring to bear on the topic the richer the text will appear, but there is something here for everyone. The higher level discussions along with some nonessential background material are relegated to the extensive, 40 page notes section at the end.

The book is bracketed by two modern stories. The first chapter shows Perelman presenting his work at MIT in April, 2003 and the final chapter describes the events at the 25th meeting of the International Congress of Mathematicians in August, 2006 in Madrid, where Perelman was awarded, but declined, the Fields Medal. In between is the tale of Poincare’s Conjecture and its many lustrous players.

Beginning with the Pythagoreans and Euclid, O’Shea gives a brief overview of the history of Mathematics up to the late nineteenth century, as seen through the lens of geometry, Euclidean and otherwise. The story is richly embroidered with unexpected but illuminative details (Did you know that at the end of his life Columbus seriously entertained the idea that the surface of the earth was more pear-shaped than round?) and editorial asides (“One wonders how much fear of mathematics stems from the disjuncture between the assertion that Euclid is perfect and some students’ intuitive, but difficult to articulate, sense that some things in it are not quite right.”). These keep the reader engaged even through those passages where the story is already familiar.

In the modern era O’Shea focuses on three mathematicians, Riemann, Klein, and of course, Poincaré. He paints a brief biographical sketch of each against the backdrop of his time and place in the history of mathematics. Along the way we briefly meet such luminaries as Gauss, Hilbert, Dehn, Smale and others, although all of these appear only briefly as their part in the story is minimal. O’Shea keeps the focus always on the Poincaré Conjecture itself. Individual mathematicians appear only inasmuch as they help move that story along.

I have two complaints, both minor.

First: O’Shea comments repeatedly that our universe is three-dimensional. This is not strictly true. Current physical theories place the dimensionality of our universe considerably higher than three and it seems pointless to pretend otherwise.

Second: The text could have benefited from one more pass under the eyes of a good editor. There were several instances of missing words, mismatched subject and verb and the like. I was inclined to disregard them at first as some such errors are essentially unavoidable. However by page 95 it had become irritating and I began keeping track. Thereafter I found at least 5 instances of this kind of error. The bulk of them appeared in the Notes section.

Despite these peccadillos, I found *The Poincaré Conjecture* to be very informative and entertaining. I recommend it to all.

Eugene Boman, Associate Professor of Mathematics, The Pennsylvania State University, Capital College, Middletown, PA.

Eugene Boman, reviewer, "The Poincare Conjecture: In Search of the Shape of the Universe," *Convergence* (August 2007)