3-torus

{{Short description|Cartesian product of 3 circles}}

{{about|the three-dimensional space|the two-dimensional surface with three holes|triple torus}}

Image:3-Manifold 3-Torus.png

The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, $\backslash mathbb\{T\}^3\; =\; S^1\; \backslash times\; S^1\; \backslash times\; S^1.$ In contrast, the usual torus is the Cartesian product of only two circles.

The 3-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, producing periodic boundary conditions. Gluing only one pair of opposite faces produces a solid torus while gluing two of these pairs produces the solid space between two nested tori.

In 1984, Alexei Starobinsky and Yakov Zeldovich at the Landau Institute in Moscow proposed a cosmological model where the shape of the universe is a 3-torus.Overbeye, Dennis. New York Times 11 March 2003: Web. 16 January 2011. [https://www.nytimes.com/2003/03/11/science/universe-as-doughnut-new-data-new-debate.html “Universe as Doughnut: New Data, New Debate”]