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I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem, a claim repeated in this Wikipedia entry. I have been unsuccessful in finding a description of such a mapping, or an image of what the embedding looks like. I'd be grateful to any pointers on this topic. Thanks!

Addendum. It seems Benoît Kloeckner's answer below is definitive. What I asked for apparently does not yet exist, but is "in process" and will soon be available through the work of the Hévéa project.

[23Apr2012] This is taken from the link in DamienC's comment and Benoît's update in the latter's answer below:
   corrugations

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    $\begingroup$ Is there any reason to expect there to be a closed form expression for this object? Nash-Kuiper embedding theorem is through a limiting process where the embedding converges in $C^1$ and not higher. So in general I am somewhat doubtful of a formula. Now, on the other hand, from the statement of the Nash-Kuiper theorem there should be a $C^1$ embedding that is $\epsilon$-close to the standard picture of a torus in $\mathbb{R}^3$. So that sort of tells you what one possible embedding looks like. $\endgroup$
    – Willie Wong
    Commented Jul 9, 2010 at 18:00
  • $\begingroup$ jstor.org/stable/1969840 is also very, very readable. $\endgroup$
    – Willie Wong
    Commented Jul 9, 2010 at 18:03
  • $\begingroup$ Great question, I've wondered this myself. The closest thing I've ever seen for this kind of thing is here, though the goal is the hyperbolic plane, and the method of construction seems ad-hoc and is not that of the Nash-Kuiper theorem: xs4all.nl/~westy31/Geometry/Geometry.html#Embed $\endgroup$
    – j.c.
    Commented Jul 9, 2010 at 18:03
  • $\begingroup$ Would it be feasible to generate by computer an image of an approximate embedding under a few iterations of the Nash-Kuiper technique? $\endgroup$
    – j.c.
    Commented Jul 9, 2010 at 18:06
  • $\begingroup$ @jc: I was hoping someone did this. Even searched Google images, unsuccessfully. $\endgroup$
    – Joseph O'Rourke
    Commented Jul 9, 2010 at 18:09

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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 http://hevea-project.fr/ENPageToreDossierDePresse.html

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 accordion), with large amplitude on the inside and small amplitude on the outside. If you design this perturbation well, you can manage so that the parallels now all have the same length. Of course, the perturbed meridian have now varying lengths! So you do the same thing by adding small waves in another direction, getting all meridians to have the same length again. You can iterate this procedure in a way so that the embedding converges in the $C^1$ topology to a flat embedded torus. But to prove that the precise perturbation you chose in order to get a nice image does converge, and that your maps are embeddings needs work (getting an immersion is easier if I remember well).

Also, the Hévéa project plans to draw images of Nash spheres, that is $C^1$ isometric embeddings of spheres of radius $>1$ inside a ball of unit radius.

Edit: Roman Kogan mentions in an answer below the following relevant, more recent links:

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  • $\begingroup$ Cool! I'll track the project, and look forward to the release of images. $\endgroup$
    – Joseph O'Rourke
    Commented Jul 9, 2010 at 18:42
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    $\begingroup$ The h-principle technique is really cool, and has some strange applications. One of my favourites is DeLellis and Szekelyhidi's demonstration that weak solutions to the Euler equation is non-unique arxiv.org/abs/math/0702079 Essentially you start with a compactly supported (in space and in time) function and alternatingly add horizontal and vertical ripples to make it converge to a solution in the weak sense. $\endgroup$
    – Willie Wong
    Commented Jul 10, 2010 at 11:36
  • $\begingroup$ Follow-up: there are some news from the Hevea project. A paper in PNAS (pnas.org/content/early/2012/04/18/1118478109.abstract) and a broad-audience presentation (math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html), with beautiful pictures!!!! $\endgroup$
    – DamienC
    Commented Apr 23, 2012 at 7:22
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    $\begingroup$ @BenoîtKloeckner I have added an answer with more links to the Hevea project results - not sure whether it's better to keep them in a separate response, or integrate into this answer, which reflects the state of affairs before the Hevea project was completed. $\endgroup$
    – Roman Kogan
    Commented Mar 19, 2018 at 20:43
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    $\begingroup$ They have now also successfully drawn pictures of a $C^1$ isometric embedding of the hyperbolic plane. $\endgroup$
    – Deane Yang
    Commented Mar 16, 2022 at 16:41
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On the other hand, if you are willing to settle for conformally flat, there is a beautiful theory of these. (The idea is to consider flat embeddings in the three-sphere, and then "project them into $R^3$ using stereographic projection.) The classification of flat embeddings of the torus in the three-sphere goes back to Bianchi in the 1800's, and Ulrich Pinkall recently found some particularly nice ones (the so-called "Bianchi-Pinkall Tori) by taking inverse images of a simple-closed curve under the Hopf fibration (so one set of circles are Hopf fibers). If you would like to see some example images, some applets to play with and morph them, and an explanatory pdf file, have a look here:

http://virtualmathmuseum.org/Surface/bianchi-pinkall_tori/bianchi-pinkall_tori.html

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  • $\begingroup$ Wow, what beautiful tori at that link! I will investigate this notion of a conformally flat metric, of which I was only dimly aware. Thanks! $\endgroup$
    – Joseph O'Rourke
    Commented Jul 9, 2010 at 20:43
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I would like to mention something I learned from Igor Pak subsequent to posing this question: there is a piecewise-linear embedding of the flat torus! In the paper by V. A. Zalgaller, "Some bendings of a long cylinder," Journal of Mathematical Sciences, 100(3):2228--2238, 2000 (translated from a 1997 article in the Russian journal Zapiski Nauchnykh Seminarov POMI), he proves this theorem:

"Theorem 1. A direct flat torus can be isometrically embedded in $\mathbb{R}^3$ 'in the origami style' if its development is a rectangle sufficiently large compared to its altitude."

He defines a direct flat torus as the result of identifying the opposite sides of a rectangle. (I have never seen the term "direct flat torus," and I don't know what role the modifier "direct" plays.) "In the origami style" describes how he bends a triangular prism such that "its middle part is broken to a complicated form." The embedding is a triangular prism bent into the shape of a regular hexagon. The "bendings" are piecewise-linear crinklings of the surface.

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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

The paper (considerably longer than the PNAS one) is available here.

There is also a fantastic video showcasing the 3D model, also made by the Hevea project.

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