Doesn’t look much like a square, but a square can be transformed into this torus (Image: V. Borrelli et al/PNAS) Twisted at a fine scale, no wonder it has been a challenge to visualise (Image: V. Borrelli et al/PNAS)

This may be the weirdest doughnut you have ever seen, but it solves a long-standing geometrical puzzle that evaded mathematicians including Nobel laureate John Nash, who inspired the film A Beautiful Mind.


Topology is the branch of mathematics concerned with the geometric deformations of objects. According to its rules, a certain type of flat square – in which opposite edges have been mathematically linked – is equivalent to a holed-doughnut, or torus, because one can easily be turned into the other. First, form a cylinder by joining the top edge of the square to the bottom edge, then bend that cylinder into a circle and join its two open ends.

There is just one problem: for the two ends to meet, the torus must be stretched in a way that distorts the original shape of the square. Any horizontal lines on the original square will be stretched on the torus, while vertical lines will remain the same. (Cartographers encounter a similar problem when unwrapping a globe of the Earth to create flat maps. They are forced then to make compromises such as inflating the size of Greenland, which can appear similar in size to Africa on standard maps but is actually one-fourteenth as big.)

Molecular doughnut

But could there be an alternative torus that leaves both horizontal and vertical line lengths unchanged? In the 1950s, game theorist and economist John Nash, together with mathematician Nicolaas Kuiper, proved that such a torus could exist.

However, their methods only worked at a tiny scale, making it too difficult to actually visualise the shape. As a result, no one knew what it would look like. “It’s like describing a cooking recipe at the molecular level,” says Francis Lazarus at the University of Grenoble in France.

Now, Lazarus and a team of mathematicians from Grenoble and the University of Lyon have managed to visualise the shape of this torus. Starting with a shrunken version of the regular, smooth torus, they wrinkle the surface in the horizontal direction, increasing the length of just the vertical lines.

3D printout

They then apply further wrinkles in other directions until the lengths of both vertical and horizontal lines are equal to the lengths of these lines on the square. The result is the bizarre-looking torus that is pictured above-right.

The method of wrinkling is known as convex integration theory. “Until now people though it was a very complicated technique,” says Lazarus. “This work proves you can actually use it.”

He says the theory could now be applied to solving complicated systems of equations that arise from problems in physics and biology. Meanwhile, he and his colleagues plan to bring their strange new shape into the real world using a 3D printer.

Journal reference: Proceedings of the National Academy of Sciences, DOI: 10.1073/pnas.1118478109