Abstract. This memoir is concerned with isometric embeddings of a square flat torus in the three dimensional Euclidean space. The existence of such embeddings was proved by John Nash and Nicolaas Kuiper in the mid 50s. However, the geometry of these embeddings could barely be conceived from their original papers. Here we provide an explicit construction based on the convex integration theory introduced by Mikhail Gromov in the 70s. Wethen turn this construction into a computer implementation leading us to the visualisation of an isometrically embedded flat torus. The pictures reveal a geometric object in-between fractals and ordinary surfaces. We call this object a C1 fractal.
Just the other day I've had a discussion with my companion about how one would determine they live on a toroidal[1] planet (assuming the atmosphere is too cloudy, and one can only see a small neighborhood).
She suggested measuring circumferences of circles at various points, as a means of getting curvature; the torus is negatively curved at some places.
"But what if we are on a flat torus?", I asked. "We are talking about a potentially realistic scenario here, not a topological abstraction", she objected. I didn't have anything to say.
But now I do.
On page 66, they even show a 3D-printed model!
I knew of Whitney embedding theorem, but somehow, not of the Nash embedding theorem which proves this is possible - and certainly not of this result, which is delightfully amazing.
[1]...which, according to articles like this one, are possible: https://io9.gizmodo.com/what-would-the-earth-be-like-if-it-w...
https://www.emis.de/journals/em/images/pdf/em_24.pdf