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nDIMENSIONAL TORUS
The notion of ndimensional torus (or hypertorus) refers to any topological space homeomorphic to the Cartesian product of a circle n times by itself, written , equivalent to the quotient ; it is therefore an ndimensional manifold.
For n = 1 we get the circle , for n = 2, the usual torus, and for n = 3, a 3dimensional manifold called in general hypertorus.
A model of the ndimensional torus embedded in
is Clifford's torus of dimension n.
problem: does there exist a model embedded in R^(n+1)?
Much as the usual torus can be seen as a full square the opposite sides of which have been identified, the ndimensional torus can be seen as a full ndimensional hypercube the n–1 opposite cells of which have been identified (identification by symmetry with respect to a hyperplane); the hypertorus is therefore a cube the opposite faces of which have been identified by a plane symmetry.
A way of imagining the 3dimensional hypertorus is to
imagine a rectangular room, the ceiling and floor of which are upholstered
with mirrors of a special kind: an observer, instead of seeing their face
reflected into the mirror, will see their back, with the right hand on
the right, the left hand on the left.
The video
game Portal provides such representations; in this game, a shot opens
in a first wall a yellow portal and a second shot opens a blue portal that
happens to be exactly the other face of the yellow portal that was just
opened. Therefore, for example, if the two walls are face to face, we get
the wanted identification.
To make a hypertorus, we would need to open 3 yellow portals and 3 blue portals on the 3 pairs of opposite faces of the room. Download this software that enables to visualize the various 3dimensional
spaces, including the hypertorus:

Do not mistake the ndimensional torus
for the nholed torus;
in particular, .
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© Robert FERRÉOL 2017