Torus
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dab toroid
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[[Image:Torus.png|right|thumb|250px|A torus]]
{{about|the surface and mathematical concept of a torus}}
In [[geometry]], a '''torus''' (pl. '''tori''') is a [[surface of revolution]] generated by revolving a [[circle]] in three dimensional space about an axis [[coplanar]] with the circle, which does not touch the circle. Examples of tori include the surfaces of [[doughnut]]s and [[inner tube]]s. The solid contained by the surface is known as a [[toroid (geometry)|toroid]]. A circle rotated about a [[chord (geometry)|chord]] of the circle is called a torus in some contexts, but this is not a common usage in mathematics. The shape produced when a circle is rotated about a chord resembles a round cushion. ''Torus'' was the [[Latin]] word for a [[cushion]] of this shape.
==Geometry==
A torus can be defined parametrically by:
:<math>x(u, v) = (R + r \cos{v}) \cos{u} \, </math>
:<math>y(u, v) = (R + r \cos{v}) \sin{u} \, </math>
:<math>z(u, v) = r \sin{v} \, </math>
where
:''u'', ''v'' are in the interval [0, 2π],
:''R'' is the distance from the center of the tube to the center of the torus,
:''r'' is the radius of the tube.
An equation in [[Cartesian coordinates]] for a torus radially symmetric about the ''z-''[[Coordinate_axis|axis]] is
:<math>\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2, \,\!</math>
and clearing the square root produces a quartic:
:<math> (x^2+y^2+z^2 + R^2 - r^2)^2 = 4R^2(x^2+y^2) . \,\!</math>
The [[surface area]] and interior [[volume]] of this torus are given by
:<math>A = 4 \pi^2 R r = \left( 2\pi r \right) \left( 2 \pi R \right) \,</math>
:<math>V = 2 \pi^2 R r^2 = \left( \pi r^2 \right) \left( 2\pi R \right). \,</math>
These formulas are the same as for a cylinder of length 2π''R'' and radius ''r'', created by cutting the tube and unrolling it by straightening out the line running around the centre of the tube. The losses in surface area and volume on the inner side of the tube happen to exactly cancel out the gains on the outer side.
According to a broader definition, the [[Generator (mathematics)| generator]] of a torus need not be a circle but could also be an [[ellipse]] or any other [[conic section]].
==Topology==
[[Image:torus_cycles.png|thumb|right|A torus is the product of two circles.]]
[[Topology|Topologically]], a '''torus''' is a closed [[surface]] defined as the [[product topology|product]] of two [[circle]]s: ''S''<sup>1</sup> × ''S''<sup>1</sup>. This can be viewed as lying in '''C'''<sup>2</sup> and is a subset of the 3-sphere ''S''<sup>3</sup> of radius <math>\sqrt{2}</math>. This topological torus is also often called the [[Clifford torus]]. In fact, ''S''<sup>3</sup> is [[Foliation|filled out]] by a family of nested tori in this manner (with two degenerate cases, a circle and a straight line), a fact which is important in the study of ''S''<sup>3</sup> as a [[fiber bundle]] over ''S''<sup>2</sup> (the [[Hopf bundle]]).
The surface described above, given the [[relative topology]] from '''R'''<sup>3</sup>, is [[homeomorphic]] to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by [[Stereographic projection|stereographically projecting]] the topological torus into '''R'''<sup>3</sup> from the north pole of ''S''<sup>3</sup>.
The torus can also be described as a [[quotient space|quotient]] of the [[Cartesian plane]] under the identifications
:(''x'',''y'') ~ (''x''+1,''y'') ~ (''x'',''y''+1).
Or, equivalently, as the quotient of the [[unit square]] by pasting the opposite edges together, described as a [[fundamental polygon]] <math>ABA^{-1}B^{-1}</math>.
[[Image:Inside-out torus (halfway).gif|thumb|right|170px|Turning a torus inside-out ([[:image:Inside-out torus (animated, small).gif|animated version]])]]
The [[fundamental group]] of the torus is just the [[direct product]] of the fundamental group of the circle with itself:
:<math>\pi_1(\mathbb{T}^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathbb{Z} \times \mathbb{Z}.</math>
Intuitively speaking, this means that a closed [[path (topology)|path]] that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding.
If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged.
The first [[homology group]] of the torus is [[isomorphic]] to the fundamental group (this follows from [[Hurewicz theorem]] since the fundamental group is [[abelian group|abelian]]).
== The ''n''-dimensional torus ==
The torus has a generalization to higher dimensions, the ''n''-'''dimensional torus''', often called the ''n''-'''torus''' for short. (This is one of two different meanings of the term "''n''-torus".)
Recalling that the torus is the product space of two circles, the ''n''-dimensional torus is the product of ''n'' circles.
That is:
:<math>\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n</math>
The torus discussed above is the 2-dimensional torus. The 1-dimensional torus is just the circle. The 3-dimensional torus is rather difficult to visualize. Just as for the 2-torus, the ''n''-torus can be described as a quotient of '''R'''<sup>''n''</sup> under integral shifts in any coordinate. That is, the ''n''-torus is '''R'''<sup>''n''</sup> modulo the [[group action|action]] of the integer [[lattice (group)|lattice]] '''Z'''<sup>''n''</sup> (with the action being taken as vector addition). Equivalently, the ''n''-torus is obtained from the ''n''-dimensional [[hypercube]] by gluing the opposite faces together.
An ''n''-torus in this sense is an example of an ''n-''dimensional [[Compact space|compact]] [[manifold]]. It is also an example of a compact [[abelian group|abelian]] [[Lie group]]. This follows from the fact that the [[unit circle]] is a compact abelian Lie group (when identified with the unit [[complex number]]s with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
Toroidal groups play an important part in the theory of [[compact Lie group]]s. This is due in part to the fact that in any compact Lie group ''G'' one can always find a [[maximal torus]]; that is, a closed [[subgroup]] which is a torus of the largest possible dimension. Such maximal tori ''T'' have a controlling role to play in theory of connected ''G''.
Automorphisms of ''T'' are easily constructed from automorphisms of the lattice '''Z'''<sup>''n''</sup>, which are classified by [[integral matrices]] ''M'' of size ''n''×''n'' which are [[invertible matrix|invertible]] with integral inverse; these are just the integral ''M'' of determinant +1 or −1. Making ''M'' act on '''R'''<sup>''n''</sup> in the usual way, one has the typical '''toral automorphism''' on the quotient.
The [[fundamental group]] of an ''n''-torus is a [[free abelian group]] of rank ''n''. The ''k''-th [[homology group]] of an ''n''-torus is a free abelian group of rank ''n'' [[binomial coefficient|choose]] ''k''. It follows that the [[Euler characteristic]] of the ''n''-torus is 0 for all ''n''. The [[cohomology ring]] ''H''<sup>•</sup>('''T'''<sup>''n''</sup>,'''Z''') can be identified with the [[exterior algebra]] over the '''Z'''-[[module (mathematics)|module]] '''Z'''<sup>''n''</sup> whose generators are the duals of the ''n'' nontrivial cycles.
==The ''n''-fold torus==
[[Image:Triple torus illustration.png|right|thumb|A triple torus]]
In the theory of [[surface]]s the term ''n-''torus has a different meaning. Instead of the product of ''n'' circles, they use the phrase to mean the [[connected sum]] of ''n'' 2-dimensional tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the disks' boundary circles. To form the connected sum of more than two surfaces, sum two of them at a time until they are all connected together. In this sense, an ''n''-torus resembles the surface of ''n'' doughnuts stuck together side by side, or a 2-dimensional [[sphere]] with ''n'' handles attached.
An ordinary torus is a 1-torus, a 2-torus is called a [[double torus]], a 3-torus a triple torus, and so on. The ''n''-torus is said to be an "[[orientability|orientable surface]]" of "[[Genus (mathematics)|genus]]" ''n'', the genus being the number of handles. The 0-torus is the 2-dimensional [[sphere]].
The [[classification theorem]] for surfaces states that every [[compact space|compact]] [[connected space|connected]] surface is either a sphere, an ''n''-torus with ''n'' > 0, or the connected sum of ''n'' [[projective plane]]s (that is, projective planes over the [[real numbers]]) with ''n'' > 0.
== Coloring a torus ==
<!--Chromatic number-->
If a torus is divided into regions, then it is always possible to color the regions with no more than seven colors so that neighboring regions have different colors. (Contrast with the [[four color theorem]] for the [[plane (mathematics)|plane]].)
[[Image:Projection color torus.png|480px|thumb|center|This construction shows the torus divided into the maximum of seven regions, every one of which touches every other.]]
==See also==
<div style="-moz-column-count:2; column-count:2;">
*[[Dupin cyclide]]
*[[Standard torus]]
*[[Algebraic torus]]
*[[Villarceau circles]]
*[[Annulus (mathematics)|Annulus]]
*[[Doughnut]]
*[[Elliptic curve]]
*[[Loewner's torus inequality]]
*[[Maximal torus]]
*[[Period lattice]]
*[[Sphere]]
<!-- *[[spiric sections]] what's this? -->
*[[Surface]]
*[[Torus (nuclear physics)]]
*[[Torus mandibularis]]
*[[Torus palatinus]]
*[[Umbilic Torus]]
</div>
==External links==
{{commons|Torus|Torus}}
* [http://www.cut-the-knot.org/shortcut.shtml#torus Creation of a torus] at [[cut-the-knot]]
* {{mathworld|Torus|Torus}}
* [http://www.dr-mikes-maths.com/4d-torus.html "4D torus"] Fly-through cross-sections of a four dimensional torus.
* [http://www.visumap.net/index.aspx?p=Resources/RpmOverview "Relational Perspective Map"] An algorithm that uses flat torus to visualize high dimensional data.
* [http://www.geometrygames.org/TorusGames/ "Torus Games"] Several games that highlight the topology of a torus.
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