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The three-dimensional torus, or triple torus, is defined as the Cartesian product of three circles,

\( \mathbb{T}^3 = S^1 \times S^1 \times S^1. \)

In contrast, the usual torus is the Cartesian product of two circles only.

The triple torus is a three-dimensional compact manifold with no boundary. It can be obtained by "gluing" the three pairs of opposite faces of a cube, where being "glued" can be intuitively understood to mean that when a particle moving by inertia in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face, in the same direction. (After gluing the first pair of opposite faces the cube looks like a thick washer, after gluing the second pair — the flat faces of the washer — it looks like a hollow torus, the last gluing — the inner surface of the hollow torus to the outer surface — is physically impossible in three-dimensional space so it has to happen in four dimensions.)

Thurston, William P. (1997), Three-dimensional Geometry and Topology, Volume 1, Princeton University Press, p. 31, ISBN 9780691083049.
Weeks, Jeffrey R. (2001), The Shape of Space (2nd ed.), CRC Press, p. 13, ISBN 9780824748371.

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