This article was taken from the July 2012 issue of Wired magazine. Be the first to read Wired's articles in print before they're posted online, and get your hands on loads of additional content by subscribing online.
A flat torus is a theoretical square with "pairwise identified" sides -- like the wraparound screen in Pac-Man where characters exit through one side of the square and reappear on the opposite side. "[Mathematician] John Nash proved it was theoretically possible in the 50s, but no one had been able to show its shape," says mathematician Francis Lazarus, one of the four-person French team who produced the images.
In the past, it's been visualised as the inner tube of a bicycle wheel but this shape is not a mathematically correct representation. "The distance between two points becomes distorted," says Lazarus. "There is no correspondence between two points on the flat torus and the same on the 3D surface." To correct this defect, mathematicians in three labs -- the Institut Camille Jordan in Lyon, the Laboratoire Jean Kuntzmann and the Gipsa-Lab, both in Grenoble -- built an algorithm that warped the inner-tube shape by piling an infinite number of "corrugations" or ripples on to it, until the distance between points became accurate.
The resulting object (pictured) is a completely new type of shape: a smooth fractal. "People thought the theory was too out-of-reach and abstract," says Lazarus. "But we have converted it into an algorithm -- one that can be implemented by mathematicians in real life." Coming to a theme park near you: the flat torus water-tube rapids ride.
This article was originally published by WIRED UK
