🔗 Clifford torus

🔗 Mathematics

In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S¹_a and S¹_b (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R⁴, as opposed to in R³. To see why R⁴ is necessary, note that if S¹_a and S¹_b each exist in their own independent embedding spaces R²_a and R²_b, the resulting product space will be R⁴ rather than R³. The historically popular view that the cartesian product of two circles is an R³ 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 R³ is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R⁴. 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 S¹_a and S¹_b each has a radius of ${\displaystyle \textstyle {\sqrt {1/2}}}$, 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.

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.