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

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of three circles,

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

In contrast, the usual

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

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

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

, 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 mod ...

. 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 visual ...

while gluing two of these pairs produces the solid space between two nested tori. In 1984,

Alexei Starobinsky Alexei Alexandrovich Starobinsky (; 19 April 1948 – 21 December 2023) was a Soviet and Russian-Jewish theoretical physicist and cosmologist. He was a pioneer of the theory of cosmic inflation, for which he received the 2014 Kavli Prize in ...

and

Yakov Zeldovich Yakov Borisovich Zeldovich (, ; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet people, Soviet Physics, physicist of Belarusians, Belarusian origin, who is known for his prolific contributions in physical Physical c ...

at the Landau Institute in

Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...

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 fu ...

where the

shape of the universe In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curv ...

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

Sources

* . * . {{topology-stub 3-manifolds