Clifford torus

In geometric topology, the Clifford torus is the simplest and most symmetric Euclidean space embedding of the cartesian product of two circles S_a¹ and S_b¹. It resides in R⁴, as opposed to R³. To see why R⁴ is required, note that if S_a¹ and S_b¹ each exist in their own independent embedding spaces R_a² and R_b², the resulting product space will be R⁴ rather than R³. The historically popular view that the cartesian product of two circles is an R³torus 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, the R³ torus is an asymmetric reduced-dimension projection of the maximally symmetric R⁴ Clifford torus. 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 S_a¹ and S_b¹ each has a radius of ${\displaystyle {\sqrt {1/2}}}$ their Clifford torus product will fit perfectly within the unit 3-sphere S³, which is a 3-dimensional submanifold of R⁴. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C², since C² is topologically equivalent to R⁴.

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