June 2007

Elizabeth Holden


The Poincare Conjecture: In Search of the Shape of the Universe

The Poincare Conjecture author Donal O'Shea claims the book is intended for general readers, but I, with a Master’s in Physics, still found it rough going. Perhaps it’s because understanding the concepts involved doesn’t depend on an education in science so much as learning to think in a brand new way. Topology, an extension of geometry that considers the nature of space, contains concepts just too alien to easily absorb. Thinking in four spatial dimensions, as understanding The Poincare Conjecture requires, is something that just doesn’t come naturally to most of us, seeing as we only live in three.

The conjecture in question is a claim made by the mathematician Henri Poincare over 100 years ago. He said that all four dimensional spaces in which a loop in that space can be narrowed to a point are four-dimensional spheres -- there are no other possible shapes for the space. Until 2002, no one had been able to prove or disprove this claim. The framework for this book was the announcement by Russian mathematician Grigory Perelman that he had proven it. Besides the obvious momentousness of the proof of a previously unproved theory, it was also noteworthy for how he told the world -- by posting the proof on his website, forgoing the usual route of papers and conferences. The sensationalism of the move was enough to garner attention from media outlets, and it is easy to see why it seems like a topic that any members of the general public with an interest in science and math might find entertaining.

Unfortunately, getting the reader to the point where he or she is well versed enough in topology to understand the conjecture and its proof requires some pretty intense work. One major but inevitable weakness of the book is that it takes nine chapters before O’Shea even brings in the Poincare Conjecture, and another several before reaching its proof. Popular books on science and math are often enjoyable because the reader can meet the human beings behind the concepts, becoming privy to their habits and idiosyncrasies. Due to the concepts involved here, readers of The Poincare Conjecture have to muscle through over 100 pages of intense geometry before even reaching those stories.

That’s not to suggest the geometry involved is boring -- far from it. Even if I felt that after the third chapter I had the sort of mind powerful enough to bend spoons, I was still dazzled by what I was learning. An example: ever since I was a kid, the idea of a finite universe has bothered me. If our universe has an end, then, if we reached the boundary, what would be beyond it? But our universe can be finite, O’Shea says, yet have no boundary. The Poincare Conjecture addresses this idea as it builds up the reader’s understanding of topology. Imagine, O’Shea posits, that you live on the surface of a sphere. You can’t go below the surface of the sphere, nor can you jump. Your life is like an ant’s as it is crawling on a tennis ball. Your world is completely two-dimensional -- North, East, South, West, but no up or down. Now, the sphere that you live is clearly finite. You can walk all the way around it; it does not go on forever. Yet you never reach a boundary. The shape of our universe is like that, says O’Shea, only scaled up one dimension. We live on the three-dimensional skin of a four-dimensional shape. I love it.

If you are looking only to read engrossing tales of the scientists that change humanity’s understanding of our world, this book may be a bit intense. But if you want an introductory course in topology, including, in my opinion, some mind-blowing concepts, then you just might love The Poincare Conjecture.

The Poincare Conjecture: In Search of the Shape of the Universe by Donal O'Shea
Walker & Company
ISBN: 080271532X
304 Pages