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\ ...
of three circles,
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 ...
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 ...
with no
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
. 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 on ...
, 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
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology.
A standard way to visu ...
while gluing two of these pairs produces the solid space between two nested tori.
In 1984,
Alexei Starobinsky
Alexei Alexandrovich Starobinsky (russian: Алексе́й Алекса́ндрович Староби́нский; born 19 April 1948) is a Soviet and Russian astrophysicist and cosmologist. He received the Kavli Prize in Astrophysics "for ...
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
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
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