# Three-torus

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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.

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In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1
a
and S1
b
. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
b
and S1
b
each exists in its own independent embedding space R2
a
and R2
b
, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.

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## 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 .