Vincent Borrelli, Roland Denis, Francis Lazarus, Boris Thibert and Damien Rohmer

Researcher in Computer Science
CNRS, Grenoble
Grenoble, France

It follows from a simple curvature obstruction that a flat torus cannot be isometrically embedded in Euclidean 3-space. From the same obstruction, neither can a sphere be isometrically reduced to fit in the interior of a ball of smaller radius. However, sixty years ago, John Nash and Nicolas Kuiper demonstrated the existence of such isometric embedding or reduction if we just require the isometric maps to be C1. Misha Gromov later developed the Convex Integration theory as a powerful tool to construct such maps. Nonetheless, until a few years ago no one had been able to visualize those paradoxical surfaces. Here, we present images and 3D printings of the Hevea Project computed thanks to the convex integration theory of Gromov.

3D printing of a reduced sphere
3D printing of a reduced sphere
25 x 25 x 25 cm
plastic powder for 3d printing

3D printing of an isometrically reduced sphere.

Flying over a 3D fractal flat torus
Flying over a 3D fractal flat torus

This short video shows the embedding of an (abstract) flat torus in 3D ambient space. The fractal aspect of the embedding results from the Convex Integration process, which accumulates corrugations on a given initial non-isometric torus. More information can be found on the Hevea project site: