Clifford's torus

CLIFFORD'S TORUS

William Clifford (1845 - 1879): British mathematician.

System of Cartesian equations: .
Cartesian parametrization: .
Algebraic translation surface of degree 4 of R⁴.

Like the Bohemian dome, Clifford's torus is the surface generated by the translation of a circle along another circle, but here, the two circles are in directly orthogonal planes in .
It can also be seen as the Cartesian product of two circles; it is therefore one of the representations of the topological torus.
It is a Riemannian manifold of dimension 2 the Gauss curvature of which is zero, which is why it is also called "flat torus".
It is included in a 3-dimensional sphere of with radius .
Its affine projections in R³ are homeomorphic to the Bohemian dome (and therefore have a self-intersection curve), while its stereographic projections in R³ are the Dupin cyclides (including the usual tori)??? (cf. banchoff)

It can be generalized to the n-dimensional Clifford's torus, embedded in parametrized by ., which is a representation of the n-dimensional torus.

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