A sphere is a sphere, right? Yes, if you mean a globe or a beach ball – what mathematicians call a two-dimensional sphere – but not if you are talking about a sphere in seven dimensions.
Now the mathematician who discovered that spheres start to behave differently in higher dimensional space – an insight that seeded a whole new field of mathematics – has been awarded the $1 million dollar Abel prize by the Norwegian Academy of Science and Letters.
John Milnor of the Institute for Mathematical Sciences at Stony Brook University in New York, was recognised for his “pioneering discoveries in topology, geometry and algebra”.
“It feels very good,” Milnor told New Scientist, though he says the award was somewhat unexpected: “One is always surprised by a call at 6 o’clock in the morning.”
Inflated cube
Topologists like Milnor study shapes whose mathematical properties aren’t changed by stretching or twisting, but they aren’t concerned with the exact geometrical properties of a particular shape, like lengths or angles. For example, you can turn a cube into a sphere by inflating it, so the two shapes are topologically identical. But you can’t turn a sphere into a doughnut without tearing a hole, so they are topologically different.
It is also possible to apply stricter rules to these transformations by making them much “smoother” – what mathematicians call differentiable. For shapes in three dimensions or less, those that share a topological geometry – for example a sphere and a cube – also have the same differentiable structure.
But mathematicians also study shapes in higher dimensions – even if they’re difficult to imagine. “You can often think of analogous things that are small enough to visualise,” explains Milnor. “The human brain is amazingly able to tackle all sorts of things.”
Tangled
sphere
Milnor did just that in 1956 when he discovered a seven-dimensional mathematical object that is identical to a seven-dimensional sphere under the rules of topology, but has a different differentiable structure. He called this shape an “exotic sphere”.
This was the first time a shape had been found sharing the topological properties – but not the differentiable structure – of its lower-dimensional counterpart. It led to the field that is now known as “differential topology”.
What does an exotic sphere look like? It’s difficult to imagine but bear in mind that it’s possible to tangle up a higher-dimensional sphere in a way that isn’t possible in two.
Imagine splitting an ordinary sphere into two halves along the middle, so that each half has a copy of every point on the equator. Now rejoin the two halves so that the southern copy of a point doesn’t join its northern counterpoint. In two dimensions, there’s only one way to do this: by twisting the sphere. But in seven dimensions the points can be mixed up with respect to each other in multiple different ways.
Smooth Poincaré
It turns out there are a total of 28 exotic spheres in seven dimensions, and they also exist in other dimensions. Dimension 15 has as many as 16,256, while others like dimensions five and six only have the ordinary sphere. Mathematicians don’t yet know whether exotic spheres exist in four dimensions – a problem known as the smooth Poincaré conjecture, and related to the generalised Poincaré conjecture, which was solved in 2003.
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