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.