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A flat torus can be isometrically embedded in 3D space with a continuously differentiable embedding. Visualising this embedding using computer graphics was the aim of a French project, completed in 2012. Turned out a flat torus in 3D has cool looking fractal structure. http://www2.cnrs.fr/en/2027.htm

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Nice result!

But, is this construction injective? I.E. do the little waves not produce any fake crossings in the surface?

IIRC from my Differential Geometry classes, immersions and embeddings are more general that I expected, and they include some strange cases. I don't remember the details, but looking at the Wikipedia page they include the embedding of the Klein bottle in R^3.

It's immersions that include some edge cases. Embeddings effectively identify a manifold with a submanifold of another manifold.

Klein bottles can't be embedded into R^3.

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