Flying over a 3D fractal flat torus
Why MathOverflow recommends it
MathOverflow answerers point people to it as the visual payoff for questions about C^1 isometric embeddings of the flat torus, showing what the Nash-Kuiper theorem's abstract surface actually looks like.
Recommended by 1 people across 1 different months, 2010 to 2019.
What it is
A 78-second flyover animation of the Hevea project's rendering of a C^1 isometric embedding of the flat torus in 3D space, a surface whose existence follows from the Nash-Kuiper theorem. The images were computed using Gromov's convex integration, and it is presented by the French Hevea research team.
What people said
Strong rec Roman Kogan · 2018-03 · MathOverflow · on “$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$”
I just wanted to add that there is also a very detailed paper on the isometric embedding of a flat torus by the four authors of the Hevea project (namely, Borrelli, Jabrane, Lazarus, and Thibert), which also includes photographs of a 3D-printed model on p.67. 3D-printed model of flat torus [LINK] The paper (considerably longer than the PNAS one) is available here [LINK] . There is also a fantastic video showcasing the 3D model [this video] , also made by the Hevea project.
Cites Benoît Kloeckner · 2010-07 · MathOverflow · on “$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$”
A group of french mathematicians and computer scientists are currently working on this. The project is named Hévéa, and has already produced a few images. Edit: a few images and the PNAS paper have been released, see [LINK] Just a few word to explain what I understood of their method (which is by using h-principle) from the few image I saw in preview. Start with a revolution torus. The meridians are cool, because they all have the same length, as expected from those of a flat torus. But the parallels are totally uncool, because their lengths differ greatly: they witness the non-flatness of the revolution torus. Now perturb your torus by adding waves in the direction of the meridians (like an
Cites Piotr Hajlasz · 2019-02 · MathOverflow · on “Applications of flat submanifolds to other fields of mathematics”
Crystallographic groups define flat compact manifolds and they are used to describe symmetries of crystals. Flat tori are used in computational physics and chemistry: if you want to investigate dynamics, say of a gas and and for computational reasons you can only consider 1000 particles, you cannot place the particles in $\mathbb{R}^3$ because they would escape. The trick is to place the particles in $\mathbb{S}^1\times \mathbb{S}^1\times \mathbb{S}^1$ which is represented as a "periodic" cube: if a particle leaves a cube through one side, in enters the cube on the opposite side. Math and art: By the famous Nash-Kuiper theorem, a flat torus $\mathbb{S}^1\times \mathbb{S}^1$ does admit a $C^1
Cites Sean Lawton · 2019-03 · MathOverflow · on “Isometric embedding of a genus g surface”
The Nash-Kuiper Theorem [LINK] implies the answer is yes if you only require the embedding to be of class $C^1$ . However, I believe the actual visualization problem for $g\geq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$ -embedding of the flat torus [this video] , which has a $C^\infty$ -isometric embedding into $\mathbb{R}^4$ ). Since the smallest known [LINK] $C^\infty$ -embedding for the hyperbolic plane is $\mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open). Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$ -embedded into $\mathbb{R}^3$ for $r\g