Concept

Clifford torus

Summary
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S1 and S1 (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 and S1 each exists in its own independent embedding space R2 and R2, 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 simi
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