(These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics. Genus g surface
In the theory of surfaces there is another object, the "genus" g surface. Instead of the product of n circles, a genus g surface is the connected sum of g two-tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. To form the connected sum of more than two surfaces, sum two of them at a time until they are all connected. In this sense, a genus g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles attached. As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or, rarely, n-fold tori). The terms double torus and triple torus are also occasionally used. The classification theorem for surfaces states that every compact connected surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real projective planes. Toroidal polyhedra
Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0.
It is diffeomorphic to a regular torus but not isometric. It can not be analytically embedded (smooth of class Ck, 2 ≤ k ≤ ∞) into Euclidean 3-space. Mapping it into 3-space requires one to stretch it, in which case it looks like a regular torus.