Torus - Flat Torus

Flat Torus

The flat torus is a torus with the metric inherited from its representation as the quotient, ℝ2/ℤ2, of the Cartesian plane under the identifications (x,y) ~ (x+1,y) ~ (x,y+1). This gives it the structure of a Riemannian manifold.

This metric can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is "flat" in the same sense that the surface of a cylinder is "flat". In 3 dimensions one can bend a flat sheet of paper into a cylinder without stretching the paper, but you cannot then bend this cylinder into a torus without stretching the paper (unless you give up some regularity and differentiability conditions, see below). In 4 dimensions one can (mathematically).

A simple 4-d Euclidean embedding is as follows: <x,y,z,w> = <R cos u, R sin u, P cos v, P sin v> where R and P are constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be isometrically embedded into Euclidean 3-space. Mapping it into 3-space requires you to stretch it, in which case it looks like a regular torus, for example, the following map <x,y,z> = <(R + P sin v)cos u, (R + P sin v)sin u, P cos v>.

A flat torus partitions the 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary.

Recently (April 2012), an embedding of a flat torus into three dimensions was found. It is similar in structure to a fractal as it is constructed by repeatedly corrugating a normal torus. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals.