My meeting with this book fell considerably short of love at first sight. Not saw it on sale yesterday at a Melbourne bookstore and asked if I thought it might be interesting. I picked it up, glanced at the less-than-brilliant cover and leafed through it for a minute or two; the writing seemed lackluster and the first anecdote I found was one Id seen before. I was about to put it back when I reconsidered. It cost $10 and was evidently an easy read. Id always wondered what the deal was with the mysterious Poincaré conjecture. Why not find out?Well, I couldnt have more wrong: this is a truly excellent book. The bare bones of the story are easy to summarize. The Poincaré conjecture, formulated in 1900 by Henri Poincaré, states cryptically that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. It remained an important unsolved problem for about a century, until it was proved correct by the reclusive Russian mathematician Grigori Perelman. Perelman was awarded two of the most prestigious prizes in mathematics, but turned them down.On that description it doesnt sound very interesting, but the author makes it come alive; hes done a huge amount of background reading on both the mathematics and the history, and when he puts it in its historical context you see how fascinating it is. Well over half the book is a history of geometry, starting from its foundations in antiquity with the Babylonians, Pythagoras and Euclid. OShea, a cultured mathematician with an intense interest in the history of his subject, gives you plenty of material on the Greeks (did you know theres a mistake in the proof of Euclids Proposition 1?), then traces how their work was passed through the Arabs to Renaissance Europe. En route, he finds a delightful way to explain to the non-mathematicians what a 3-sphere is: it turns out to be the shape of the universe as described in Dantes Divine Comedy, two sets of concentric spheres mystically joined at their common surface. He illustrates with a famous picture from Doré:As he progresses towards the present day, he finds opportunities to introduce the other terms that will eventually be used in the Conjecture, and the narrative starts to focus in on the key concepts: manifolds, connectedness, topology and, above all, non-Euclidean geometry. This is the clearest overview of the subject Ive ever seen, and he has a whole bunch of stories and observations I hadnt come across before. One thing I found particularly remarkable was the long guerilla war waged by the 19th century German mathematicians against Kants conceptions of geometry. I have had several discussions with philosophically knowledgeable people on this site about Einsteins claim to have refuted Kant. What I didnt realize was that it was just the final battle in a campaign that had gone on for a century. Gauss laid the groundwork, but thought it was so controversial that he couldnt publish: at least in Germany, it wasnt possible to openly say that Kant was wrong, and non-Euclidean geometries made perfectly good sense. But other great mathematicians - Riemann, Lobachevsky and Bolyai - found the same ideas, and they gradually came out in the open. Einstein finished it off: not only is it logically possible that the space we live in might be non-Euclidean, it actually happens to be true! Another remarkable story from the end of this period is the intense rivalry between the German Klein (who, I learned, married Hegels granddaughter) and the French Poincaré, a professional duel which so exhausted them that they both suffered nervous breakdowns as a result. OShea, who knows both French and German, includes lovely quotations from their correspondence. By the time we reach 1900 and the formulation of the Conjecture, it all makes perfect sense, and its obvious why the problem captivated several generations of top mathematicians. I was worried that the last third would be anticlimactic, but my fears again turned out to be groundless. OShea hardly loses momentum at all as he goes into the finishing stretch, which involves explaining some horribly difficult mathematics; once again, he finds clever visual analogies to make the esoteric technique of Ricci flow seem reasonable and intuitive. Its obviously impossible to give us the details of Perelmans proof, but he successfully conveys both its general outline and the process which led to its acceptance by the world mathematical community.At the end, there is the tantalizing mystery: why did Perelman turn down the huge prizes hed won, and what was the even larger discovery he hinted at, which would make the Poincaré conjecture no more than a stepping stone? If this had been a novel, I would have groaned at the authors unsubtle attempt to set up a sequel, but oddly enough it happens to be real life. Stranger than fiction, you know.