TITLE: $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
QUESTION [66 upvotes]: 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:

REPLY [7 votes]: 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.