Clifford Torus - Formal Definition

Formal Definition

The unit circle S1 in R2 can be parameterized by an angle coordinate:

In another copy of R2, take another copy of the unit circle

Then the Clifford torus is

Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in R2 × R2 = R4.

If R4 is given by coordinates (x1, y1, x2, y2), then the Clifford torus is given by

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