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In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles Sa and Sb. It is named after William Kingdon Clifford. It resides in R, as opposed to in R. To see why R is necessary, note that if Sa and Sb each exists in its own independent embedding space Ra and Rb, the resulting product space will be R rather than R. The historically popular view that the cartesian product of two circles is an R torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
Stated another way, a torus embedded in R is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
If Sa and Sb each has a radius of 1 / 2 {\displaystyle \textstyle {\sqrt {1/2}}} , their Clifford torus product will fit perfectly within the unit 3-sphere S, which is a 3-dimensional submanifold of R. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C, since C is topologically equivalent to R.
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 ; 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.