Simple set: Difference between revisions
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Fixed the definition. Clearly the definition given would include N itself. |
Deltahedron (talk | contribs) →Formal definitions and some properties: hyperimmune and hypersimple |
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The '''three-dimensional torus''', or '''triple torus''', is defined as 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 two circles only. | |||
The triple 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]]. (After gluing the first pair of opposite faces the cube looks like a thick [[Washer (hardware)|washer]], after gluing the second pair — the flat faces of the washer — it looks like a hollow torus, the last gluing — the inner surface of the hollow torus to the outer surface — is physically impossible in three-dimensional space so it has to happen in four dimensions.) | |||
[[Category:Topology]] | |||
[[Category:Differential topology]] | |||
[[Category:Differential geometry]] | |||
[[Category:Geometric topology]] | |||
[[Category:Manifolds]] | |||
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Revision as of 20:03, 14 April 2013
Template:Multiple issues The three-dimensional torus, or triple torus, is defined as the Cartesian product of three circles,
In contrast, the usual torus is the Cartesian product of two circles only.
The triple torus is a three-dimensional compact manifold with no boundary. It can be obtained by gluing the three pairs of opposite faces of a cube. (After gluing the first pair of opposite faces the cube looks like a thick washer, after gluing the second pair — the flat faces of the washer — it looks like a hollow torus, the last gluing — the inner surface of the hollow torus to the outer surface — is physically impossible in three-dimensional space so it has to happen in four dimensions.)