Why a bagel equates with a coffee cup

15th December 2006 at 00:00
Marcus du Sautoy is professor of mathematics at Oxford university

This summer, the maths community handed out its most prestigious prize: the Fields Medals. These awards, for under-40s, have for years been regarded as the Nobel Prize of mathematics that the inventor of dynamite never bequeathed.

But it was the decision by one of the recipients to turn down his medal that hit world headlines. Russian mathematician Grigori Perelman was regarded as a worthy winner after cracking one of the greatest mysteries of mathematics: the Poincare conjecture. This conundrum is so fundamental to mathematics that it comes with a price tag of $1 million (pound;500.000) for whoever could crack it.

For Mr Perelman, the buzz of having understood one of the greatest enigmas of the 20th century is the ultimate reward. But what is this mystery, and is it possible, without a grounding in mathematics, to comprehend his achievement?

Although distinguished professors may find it difficult to comprehend the details of Mr Perelman's proof, I believe children can gain some appreciation of what he has achieved. We are often too scared to show pupils maths that go beyond arithmetic calculations, with their reassuring right or wrong answers. Much of maths is about finding new ways to look at the world.

The Poincare conjecture is part of a new sort of geometry called topology.

Often called "bendy geometry", it is concerned with how an object is connected. London commuters exploit a topological view of the world every day on their way to work. The London underground map is not a physical representation of the locations of stations. Rather, the physical map was morphed by Harry Beck into the iconic plan recognised throughout the world.

Commuters are mainly interested in the connection between stations rather than distances. Topologists at the start of the last century began classifying all the shapes possible in this bendy geometry. They saw that although a bagel was different from a coffee cup, from a topological perspective they could be viewed as the same. One can be moulded into the other without cuts.

The Poincare conjecture continues the attempt to understand this way of looking at shapes and provides an insight into what mathematical shape our universe might be: a big ball, perhaps, or something more exotic? A glimpse of these exciting worlds might provide inspiration for children to continue the hard graft in the classroom.

Marcus du Sautoy will present the Royal Institution Christmas Lectures: The Num8er My5teries (five broadcasts), starting December 25, 7.15pm on Channel Five

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