The notion of n-dimensional 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 n-dimensional manifold.

For n = 1 we get the circle , for n = 2, the usual torus, and for n = 3, a 3-dimensional manifold called in general hypertorus.

A model of the n-dimensional 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 n-dimensional torus can be seen as a full n-dimensional 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 3-dimensional 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 3-dimensional spaces, including the hypertorus: http://www.geometrygames.org/CurvedSpaces/index.html

Do not mistake the n-dimensional torus  for the n-holed torus; in particular, .
 

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© Robert FERRÉOL 2017