When constructing a "flat" torus in 3-dimensional space there is significant distortion of distance along one axis. By "flat", mathematicians mean it has no Gaussian curvature, the way a cylinder is considered "flat", but not a sphere (which has positive curvature) or hyperbolic paraboloid (which has negative curvature).

The distortion of distances can be avoided by folding the torus in 4-dimensional space instead, as a Clifford Torus. (Or, equivalently, in a 2-dimensional complex space.)

It has recently been discovered (2012) that there is an alternate way to construct a flat torus in 3-dimensioal space, which will not distort distances. The construction involves a weird corrugated fractal surface.

http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html

http://docmadhattan.fieldofscience.com/2012/04/flat-torus-in-three-dimensional-space.html

http://www2.cnrs.fr/en/2027.htm

http://www.science4all.org/le-nguyen-hoang/flat-torus/

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