{{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|upright=1.25|thumb|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 [[Homeomorphism|homeomorphic]] to the [[Cartesian product]] of three circles, <math>\mathbb{T}^3 = S^1 \times S^1 \times S^1.</math> In contrast, the usual [[torus]] is the Cartesian product of only two circles.
== Description == The 3-torus is a three-dimensional [[compact space|compact]] [[manifold]] with no [[manifold#Manifold with boundary|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.
== Usage == In 1984, [[Alexei Starobinsky]] and [[Yakov Zeldovich]] at the [[Landau Institute for Theoretical Physics|Landau Institute]] in [[Moscow]] proposed a [[cosmological model]] where the [[shape of the universe]] is a 3-torus.<ref name="NYT">Overbye, 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”]</ref>
==References== {{Reflist}}
===Sources=== * {{citation|title=Three-dimensional Geometry and Topology, Volume 1|first=William P.|last=Thurston|authorlink=William Thurston|publisher=Princeton University Press|year=1997|isbn=9780691083049|page=31|url=https://books.google.com/books?id=9kkuP3lsEFQC&pg=PA31}}. * {{citation|title=The Shape of Space|first=Jeffrey R.|last=Weeks|authorlink=Jeffrey Weeks (mathematician)|edition=2nd|publisher=CRC Press|year=2001|isbn=9780824748371|page=13|url=https://books.google.com/books?id=A8WBiUWy3SgC&pg=PA13}}.
{{Manifolds}} {{topology-stub}}
[[Category:3-manifolds]]