Determining the shape of the surface on which we live is, as Donal O’Shea observes in this historically minded little book, a delicate matter. Columbus’s idea was not (at least, not only) the lascivious fantasy of a hoary sea dog. He believed that he had reached India, not America. But he also knew that he had completed the journey much more quickly than the accepted size of the world allowed: the well-travelled southern route suggested Asia was thousands of miles further away. A mammary planet, God-seeking nipple northward, was the only explanation. Even after Ferdinand Magellan returned from his circumnavigation in 1522, it wasn’t (as O’Shea, who is pernickety as well as entertaining, remarks) clear that Earth was a sphere. There were just so many potential complications that he might have missed. It could have been an American doughnut: he could have sailed through the chocolate icing, and returned to Spain without even noticing the hole that he’d looped in the middle. Worse still, it might have been a pretzel.
O’Shea’s The Poincaré Conjecture concerns the next level up: the shape of our universe in the fourth dimension. Personally, my heart freezes to a one-dimensional dot when scientists start talking about more than three dimensions. How do you picture them? The best way is by analogy. Imagine a piece of paper (a two-dimensional object) gliding along. It moves straight ahead, never reversing, but one day finds itself back where it started. How did that happen? It doesn’t know that it has been slipping over the surface of a sphere, because it can’t see in three dimensions. It just thinks the laws of physics have gone mad. You can make a similar analogy for three-dimensional creatures. Fly off into space for a trillion light years: if you end up back where you started, then you know something dodgy is going on in the fourth dimension. As Professor O’Shea explains himself, Poincaré’s conjecture is only indirectly about the shape of universe; it is more to do with shapes in general. But talking cosmic gives The Poincaré Conjecture a dramatic tremble.
Four years ago, Grigory Perelman, still under 40, all alone in the mathematical world, peculiar, very bearded and living with his mother, using some of the most astounding theoretical leaps ever seen, finally proved that Poincaré’s ideas were correct. He won $1 million in prize money.
Mathematicians treat all numbers of dimension, even if there are ridiculous thousands of them, with a careless spirit. Ordinary humans need help. I’m working on a book about a mathematician at the moment (almost as odd a man as Perelman, but not quite as hairy), and my desk and chairs are piled high with papers trying to make sense of the simplest elements of this stuff. This morning I found a note on ‘homo-
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