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CLIFFORD'S TORUS

William Clifford (1845 - 1879): British mathematician. |

System of Cartesian equations: .
Cartesian parametrization: . Algebraic translation surface of degree 4 of R ^{4}. |

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 R^{3} are homeomorphic to the Bohemian dome (and therefore have a self-intersection curve), while its stereographic projections in R^{3} 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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© Robert FERRÉOL 2017