HAVE YOU EVER wondered what shape the Universe is? It’s a mind-bending problem. How can we possibly know the Universe’s shape unless we can get outside to take a look?

Thank goodness for mathematicians. They can come up with plausible answers to questions like this using a pencil and some imagination – and may even win a million dollars in the process. One mathematician who thought about the shape of the Universe, ingeniously as it turns out, was named Henri Poincaré. We’ll come back to him and the million dollars in a moment.

But I know what you’re thinking. What’s all the fuss? Isn’t it obvious that the Universe is either an extremely large (and possibly lumpy) ball, or that it goes on for ever?

Not so fast! Nothing is obvious. That’s the kind of thinking that led people to believe that the surface of the Earth was a vast flat expanse, possibly with some sort of edge off which sailors would drop if they ventured too far. Those flat-earth people got a bit of a shock when Magellan’s ship kept sailing west and ended up back where it started.

Misconceptions of the Universe can be similar to those of the flat Earth, but with an extra dimension. What if, like our planet, the Universe is also “curved” (in a fourth dimension) and doesn’t have an edge, so that if we travelled for long enough our path could sweep out every cubic inch of its content, and never come to an outside wall? In this Universe, a spaceman could head off in a “straight line” and still end up back where he started.

Hard though it is for most three-dimensional mortals to imagine, there are many physicists and mathematicians who believe that our Universe is indeed finite but without an edge. The trouble is that there is more than one higher-dimensional shape that could meet these conditions. Some are very exotic, such as two double-holed doughnuts glued together in hyperspace. The simplest – and I say that cautiously – is a “3sphere”.

Now, back to our spaceman, and to Poincaré. Suppose our spaceman has an extremely long piece of string, ties one end to his front doorknob, and then heads off to explore every corner of the universe, trailing the string behind him until he gets back home. He’ll have created an almighty loop of string. If he pulls the loose end, if he is lucky he will be able to wind all of his string back into a ball, without any of it getting looped around some warped hole in the universe.

I say if he is lucky because he will get all his string back only if the shape of the Universe is a 3-sphere. In 1904, Poincaré conjectured that this would be the case, but it was only last year that a Russian named Grigori Perelman finally proved that Poincaré was right. What is remarkable is that mathematicians regarded this proof as so important that they offered Perelman $1 million as a prize. Even more remarkably, he turned it down.

All of this knowledge I gleaned from The Poincaré Conjecture by Donal O’Shea. But it was hard work, mainly because some of the maths involved is, well, quite hard. In fact, while I have attempted to summarise the Poincaré Conjecture itself in a few hundred words, it takes O’Shea 45 pages to lead the reader clearly and gently to the point where he is able to explain what it actually is. On the way, you have to understand some precise mathematical terms, such as manifold and homeomorphism, because these return time and again.

O’Shea is nobly following a modern tradition of trying to make important but abstract mathematical ideas accessible to everyone. He succeeds only partially. Simon Singh hit the jackpot with Fermat’s Last Theorem, a problem that was familiar, easy to state, and had tons of human interest.

Poincaré’s problem is more abstract and far harder to explain – and if you don’t understand the question, you can’t hope to understand (or care about) the solution. For a general reader, the most intriguing part of the Poincaré story is the Perelman mystery. Why on earth would a mathematician turn down a prize of $1 million? The book largely ignores this, instead, particularly in the second half, exploring mathematical ideas beyond most people’s comfort zone.

Now that we know that Poincaré's conjecture is correct, what does this tell the man in the street about the shape of the universe? Directly, not a great deal. But apparently, whatever shape the universe turns out to be, they are now confident that they will be able to create an atlas of it. If that sounds opaque, it is because this is not a trivial topic. If it was, they wouldn’t have offered such a big prize.

The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O'Shea

Allen Lane, £17.99; 304pp

Buy the book here at the offer price of £16.19 (free p&p) timesonline.co.uk/booksfirst

Erica Wagner, literary editor of The Times, will discuss poetry, mathematics and beauty with Rob Eastaway at the Dulwich Picture Gallery, London SE21, on Friday at 7.30pm as part of the Dulwich Festival.

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