The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

of three circles,

\mathbb^3 = S^1 \times S^1 \times S^1.

In contrast, the usual

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

is the Cartesian product of only two circles. The 3-torus is a three-dimensional

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

with no

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...

. It can be obtained by "gluing" the three pairs of opposite faces of a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

, 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 Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical mode ...

. 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 Borisovich Zel'dovich Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ков Бори́сович Зельдо́вич; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Bel ...

at the Landau Institute in Moscow proposed a cosmological model where the

shape of the universe The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes gen ...

is a 3-torus.Overbeye, Dennis. ''New York Times'' 11 March 2003: Web. 16 January 2011
“Universe as Doughnut: New Data, New Debate”
/ref>

References

*. *. Topology Differential topology Differential geometry Geometric topology 3-manifolds {{topology-stub