Clifford torus

In geometric topology, the Clifford torus is a special kind of torus sitting inside the unit 3-sphere S³ in R⁴, the Euclidean space of four dimensions. Or equivalently, it can be seen as a torus sitting inside C² since C² is topologically equivalent to R⁴. It is specifically the torus in S³ that is geometrically the cartesian product of two circles, each of radius sqrt(1/2).

The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.