Three-torus

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

${\displaystyle \mathbb {T} ^{3}=S^{1}\times S^{1}\times S^{1}.}$

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

The three-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 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. After gluing the first pair of opposite faces, the cube looks like a thick washer (annular cylinder), after gluing the second pair — the flat faces of the washer — it looks like two nested two-tori, the last gluing — the inner nested torus to the outer nested torus — is physically impossible in three-dimensional space so it has to happen in four dimensions.

References

• 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.