French researchers Vincent Borrelli, Saïd Jabrane, Francis Lazarus and Boris Thibert have described an isometric embedding of the flat torus in 3D space, using the convex integration theory developed by Gromov in the 1970s. That means they’ve produced a surface which is topologically a torus – it has a single hole — which preserves distances between points in the 4D flat torus. Interestingly, the tangent plane is defined everywhere — the surface is in a sense smooth — but the normal vector is not defined, so it’s also a fractal. This is impossible in higher dimensions
I’ve recorded a short video explaining in a handwavey fashion, with a few props made from things I had lying around, just what has been done.
In the 1950s, Nash and Kuiper proved that such an embedding existed, but couldn’t visualise it or give an explicit description of it. Now, using convex integration theory, the embedding described in the paper is the limit of a series of “corrugations” – wrinkles added in each of three directions along the surface. By beginning with an embedding which is strictly short, i.e. the distances between points are never greater than they were in the original shape, the corrugations can stretch the shortened distances while leaving others alone.
Since the process is defined recursively, the authors were able to write a program to generate a 3D mesh of an approximation to the final shape using only the first four corrugations. That program produced the eye-catching images above, which the authors hope demonstrates that convex integration theory “can produce computationally tractable solutions of partial differential relations.”
The work was done not really because the world was crying out for an isometric embedding of the flat torus, but because the authors wanted to promote convex integration theory as a widely applicable tool.
Source: Wrinkled doughnut solves geometrical mystery.
Original paper: Flat tori in three-dimensional space and convex integration.
This is actually somewhat related to turning a sphere inside out, which this Outside In video so clearly explains, that it almost seems patronising.
Both mappings use wiggles to avoid creating kinks (sphere eversion can also be achieved with a $C_1$-isometric embeddings).
And, according to Wikipedia, the convex integration this paper promotes, can be used to prove the homotopy principle, which both generalises sphere eversion (Smale’s theorem) and the Nash embedding theorem which this torus construction demonstrates.
There’s also a good explanation of this torus embedding on the Hevea project website.