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CLIFFORD'S TORUS

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

System of Cartesian equations: .
Cartesian parametrization: . Algebraictranslation 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