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 his the latter's answer below:
