Mathematicians Produce First-Ever Image of Flat Torus in 3D

Apr 25, 2012 by Enrico de Lazaro

A team of mathematicians and computer scientists led by Dr. Vincent Borrelli of the Université Lyon I in France has succeeded for the first time in constructing and visually representing an image of a flat torus in three-dimensional space.

Isometric embedding of a square flat torus in 3D space, seen from the outside (Vincent Borrelli et al)

This is a smooth fractal, halfway between fractals and ordinary surfaces. The findings are published in the Proceedings of the National Academy of Sciences.

In the 1950s, Nicolaas Kuiper and the Nobel laureate John Nash demonstrated the existence of a representation of an abstract mathematical object called flat torus, without being able to visualize it. Since then, constructing a representation of this surface has remained a challenge that has finally been met by the team.

On the basis of the Convex Integration Theory developed by Mikhail Gromov in the 1970s, the team used the corrugation technique. This abstract mathematical method helps to determine atypical solutions to partial differential equations. This enabled the researchers to obtain images of a flat torus in 3D for the first time.

Halfway between fractals and ordinary surfaces, the image shows a smooth fractal.

These findings open up new avenues in applied mathematics, especially in the visualization of the differential equations encountered in physics and biology. The astounding properties of smooth fractals could also play a central role in the analysis of the geometry of shapes.