Three-torus
The three-dimensional torus, or three-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles,
- [math]\displaystyle{ \mathbb{T}^3 = S^1 \times S^1 \times S^1. }[/math]
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 the portion of space between two nested two-tori.
References
- Thurston, William P. (1997), Three-dimensional Geometry and Topology, Volume 1, Princeton University Press, p. 31, ISBN 9780691083049, https://books.google.com/books?id=9kkuP3lsEFQC&pg=PA31.
- Weeks, Jeffrey R. (2001), The Shape of Space (2nd ed.), CRC Press, p. 13, ISBN 9780824748371, https://books.google.com/books?id=A8WBiUWy3SgC&pg=PA13.
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