# 3-torus

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Cartesian product of 3 circles

This article is about the three-dimensional space. For the two-dimensional surface with three holes, see [triple torus](/source/Triple_torus).

A view from inside a 3-torus. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops.

The **three-dimensional torus**, or **3-torus**, is defined as any topological space that is [homeomorphic](/source/Homeomorphism) to the [Cartesian product](/source/Cartesian_product) of three circles, T 3 = S 1 × S 1 × S 1 . {\displaystyle \mathbb {T} ^{3}=S^{1}\times S^{1}\times S^{1}.} In contrast, the usual [torus](/source/Torus) is the Cartesian product of only two circles.

## Description

The 3-torus is a three-dimensional [compact](/source/Compact_space) [manifold](/source/Manifold) with no [boundary](/source/Manifold#Manifold_with_boundary). It can be obtained by "gluing" the three pairs of opposite faces of a [cube](/source/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](/source/Periodic_boundary_conditions). Gluing only one pair of opposite faces produces a [solid torus](/source/Solid_torus) while gluing two of these pairs produces the solid space between two nested tori.

## Usage

In 1984, [Alexei Starobinsky](/source/Alexei_Starobinsky) and [Yakov Zeldovich](/source/Yakov_Zeldovich) at the [Landau Institute](/source/Landau_Institute_for_Theoretical_Physics) in [Moscow](/source/Moscow) proposed a [cosmological model](/source/Cosmological_model) where the [shape of the universe](/source/Shape_of_the_universe) is a 3-torus.[1]

## References

1. **[^](#cite_ref-NYT_1-0)** Overbye, Dennis. *New York Times* 11 March 2003: Web. 16 January 2011. [“Universe as Doughnut: New Data, New Debate”](https://www.nytimes.com/2003/03/11/science/universe-as-doughnut-new-data-new-debate.html)

### Sources

- [Thurston, William P.](/source/William_Thurston) (1997), [*Three-dimensional Geometry and Topology, Volume 1*](https://books.google.com/books?id=9kkuP3lsEFQC&pg=PA31), Princeton University Press, p. 31, [ISBN](/source/ISBN_(identifier)) [9780691083049](https://en.wikipedia.org/wiki/Special:BookSources/9780691083049).

- [Weeks, Jeffrey R.](/source/Jeffrey_Weeks_(mathematician)) (2001), [*The Shape of Space*](https://books.google.com/books?id=A8WBiUWy3SgC&pg=PA13) (2nd ed.), CRC Press, p. 13, [ISBN](/source/ISBN_(identifier)) [9780824748371](https://en.wikipedia.org/wiki/Special:BookSources/9780824748371).

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