Editing Torus

{{short description|Doughnut-shaped surface of revolution}}
{{Distinguish|Taurus (disambiguation){{!}}Taurus}}
{{About|the mathematical surface|the volume|Solid torus|other uses}}
{{For|the meteorological research project|TORUS Project}}
{{Use dmy dates|date=August 2021}}

[[File:Tesseract torus.png|thumb|A ring torus with a selection of circles on its surface]]
[[File:Ring Torus to Degenerate Torus (Short).gif|thumb|As the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally [[degeneracy (mathematics)|degenerates]] into a double-covered [[sphere]].]]
[[File:torus cycles.svg|thumb|right|A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle.]]

In [[geometry]], a '''torus''' ({{plural form}}: '''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 include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a '''donut''' or '''doughnut'''.

If the [[axis of revolution]] does not touch the circle, the surface has a ring shape and is called a '''torus of revolution''', also known as a '''ring torus'''. If the axis of revolution is [[tangent]] to the circle, the surface is a '''horn torus'''. If the axis of revolution passes twice through the circle, the surface is a '''[[Lemon (geometry)|spindle torus]]''' (or ''self-crossing torus'' or ''self-intersecting torus''). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered [[sphere]]. If the revolved curve is not a circle, the surface is called a ''[[toroid]]'', as in a square toroid.

Real-world objects that approximate a torus of revolution include [[swim ring]]s, [[inner tube]]s and [[ringette ring]]s. 

A torus should not be confused with a ''[[solid torus]]'', which is formed by rotating a [[disk (geometry)|disk]], rather than a circle, around an axis. A solid torus is a torus plus the [[volume]] inside the torus. Real-world objects that approximate a ''solid torus'' include [[O-ring]]s, non-inflatable [[lifebuoy]]s, ring [[doughnut]]s, and [[bagel]]s.

In [[topology]], a ring torus is [[homeomorphism|homeomorphic]] to the [[product topology|Cartesian product]] of two [[circle]]s: {{math|''S''<sup>1</sup> × ''S''<sup>1</sup>}}, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into [[Euclidean space]], but another way to do this is the Cartesian product of the [[embedding]] of {{math|''S''<sup>1</sup>}} in the plane with itself. This produces a geometric object called the [[Clifford torus]], a surface in [[four-dimensional space|4-space]].

In the field of [[topology]], a torus is any topological space that is homeomorphic to a torus.<ref>
{{cite book
 |last1=Gallier |first1=Jean |author1-link=Jean Gallier
 |last2=Xu |first2=Dianna |author2-link=Dianna Xu
 |doi=10.1007/978-3-642-34364-3
 |isbn=978-3-642-34363-6
 |mr=3026641
 |publisher=Springer, Heidelberg
 |series=Geometry and Computing
 |title=A Guide to the Classification Theorem for Compact Surfaces
 |title-link=A Guide to the Classification Theorem for Compact Surfaces
 |volume=9
 |year=2013
}}</ref> The surface of a coffee cup and a doughnut are both topological tori with [[Genus (mathematics)|genus]] one.

An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare [[Klein bottle]]).

== Etymology ==
''[[wikt:torus|Torus]]'' is a Latin word denoting something round, a swelling, an elevation, a protuberance.

== Geometry ==

{{Multiple image
|align =
|direction=vertical
|total_width=230
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|header=Bottom-halves and<br />vertical cross-sections
|image1=Standard_torus-ring.png
|alt1=ring
|caption1={{math|''R'' > ''r''}}: ring torus or anchor ring
|image2=Standard_torus-horn.png
|alt2=horn
|caption2={{math|1=''R''=''r''}}: horn torus
|image3=Standard_torus-spindle.png
|alt3=spindle
|caption3={{math|''R'' < ''r''}}: self-intersecting spindle torus
|footer =
}}
[[File:Toroidal coord.png|thumb|Poloidal direction (red arrow) and toroidal direction (blue arrow)]]

A torus of revolution in 3-space can be [[parametric equation|parametrized]] as:<ref>{{cite web |url=http://www.geom.uiuc.edu/zoo/toptype/torus/standard/eqns.html |title=Equations for the Standard Torus |publisher=Geom.uiuc.edu |date=6 July 1995 |access-date=21 July 2012 |url-status=live |archive-url=https://web.archive.org/web/20120429011957/http://www.geom.uiuc.edu/zoo/toptype/torus/standard/eqns.html |archive-date=29 April 2012}}</ref>
<math display="block">\begin{align}
x(\theta, \varphi) &= (R + r \sin \theta) \cos{\varphi}\\
y(\theta, \varphi) &= (R + r \sin \theta) \sin{\varphi}\\
z(\theta, \varphi) &= r \cos \theta\\
\end{align}</math>
using angular coordinates {{math|''θ''}}, {{math|''φ'' ∈ [0, 2π)}}, representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the ''major radius'' {{math|''R''}} is the distance from the center of the tube to the center of the torus and the ''minor radius'' {{math|''r''}} is the radius of the tube.<ref>{{cite web| title=Torus | url=http://doc.spatial.com/index.php/Torus|publisher=Spatial Corp. | access-date=16 November 2014|url-status=live | archive-url=https://web.archive.org/web/20141213210422/http://doc.spatial.com/index.php/Torus |archive-date=13 December 2014}}</ref>

The ratio {{math|''R''/''r''}} is called the ''[[aspect ratio]]'' of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2.

An [[implicit function|implicit]] equation in [[Cartesian coordinates]] for a torus radially symmetric about the z-[[coordinate axis|axis]] is
<math display="block">{\textstyle \bigl(\sqrt{x^2 + y^2} - R\bigr)^2} + z^2 = r^2.</math>

Algebraically eliminating the [[square root]] gives a [[quartic equation]],
<math display="block">\left(x^2 + y^2 + z^2 + R^2 - r^2\right)^2 = 4R^2\left(x^2+y^2\right).</math>

The three classes of standard tori correspond to the three possible aspect ratios between {{mvar|R}} and {{mvar|r}}:
* When {{math|''R'' > ''r''}}, the surface will be the familiar ring torus or anchor ring.
* {{math|1=''R'' = ''r''}} corresponds to the horn torus, which in effect is a torus with no "hole".
* {{math|''R'' < ''r''}} describes the self-intersecting spindle torus; its inner shell is a ''[[lemon (geometry)|lemon]]'' and its outer shell is an ''[[apple (geometry)|apple]]''.
* When {{math|1=''R'' = 0}}, the torus degenerates to the sphere radius {{math|''r''}}.
* When {{math|1=''r'' = 0}}, the torus degenerates to the circle radius {{math|''R''}}.

When {{math|''R'' ≥ ''r''}}, the [[interior (topology)|interior]]
<math display="block">{\textstyle \bigl(\sqrt{x^2 + y^2} - R\bigr)^2} + z^2 < r^2</math>
of this torus is [[diffeomorphism|diffeomorphic]] (and, hence, homeomorphic) to a [[Cartesian product|product]] of a [[disk (geometry)|Euclidean open disk]] and a circle. The [[volume]] of this solid torus and the [[surface area]] of its torus are easily computed using [[Pappus's centroid theorem]], giving:<ref>{{MathWorld|Torus|Torus}}</ref>
<math display="block">\begin{align}
A &= \left( 2\pi r \right) \left(2 \pi R \right) = 4 \pi^2 R r, \\[5mu]
V &= \left ( \pi r^2 \right ) \left( 2 \pi R \right) = 2 \pi^2 R r^2.
\end{align}</math>

These formulae are the same as for a cylinder of length {{math|2π''R''}} and radius {{mvar|r}}, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.

Expressing the surface area and the volume by the distance {{mvar|p}} of an outermost point on the surface of the torus to the center, and the distance {{mvar|q}} of an innermost point to the center (so that {{math|1=''R'' = {{sfrac|''p'' + ''q''|2}}}} and {{math|1=''r'' = {{sfrac|''p'' − ''q''|2}}}}), yields
<math display="block">\begin{align}
A &= 4 \pi^2 \left(\frac{p+q}{2}\right) \left(\frac{p-q}{2}\right) = \pi^2 (p+q) (p-q), \\[5mu]
V &= 2 \pi^2 \left(\frac{p+q}{2}\right) \left(\frac{p-q}{2}\right)^2 = \tfrac14 \pi^2 (p+q) (p-q)^2.
\end{align}</math>

As a torus is the product of two circles, a modified version of the [[spherical coordinate system]] is sometimes used.
In traditional spherical coordinates there are three measures, {{mvar|R}}, the distance from the center of the coordinate system, and {{mvar|θ}} and {{mvar|φ}}, angles measured from the center point.

As a torus has, effectively, two center points, the centerpoints of the angles are moved; {{mvar|φ}} measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of {{mvar|θ}} is moved to the center of {{mvar|r}}, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".<ref>{{cite web |url=http://dictionary.oed.com/cgi/entry/50183023?single=1&query_type=word&queryword=poloidal&first=1&max_to_show=10 |work=Oxford English Dictionary Online |access-date=10 August 2007 |title=poloidal |publisher=Oxford University Press}}</ref>

In modern use, [[toroidal and poloidal]] are more commonly used to discuss [[magnetic confinement fusion]] devices.

== Topology ==
{{No footnotes|section|date=November 2015}}
[[Topology|Topologically]], a torus is a [[closed surface]] defined as the [[product topology|product]] of two [[circle]]s: {{math|''S''<sup>1</sup> × ''S''<sup>1</sup>}}. This can be viewed as lying in [[complex coordinate space|{{math|'''C'''<sup>2</sup>}}]] and is a subset of the [[3-sphere]] {{math|''S''<sup>3</sup>}} of radius {{math|√2}}. This topological torus is also often called the [[Clifford torus]].<ref>{{Cite journal |last=De Graef |first=Marc |date=March 7, 2024 |title=Applications of the Clifford torus to material textures |url=https://journals.iucr.org/j/issues/2024/03/00/iu5046/iu5046.pdf |journal=Journal of Applied Crystallography |volume=57 |issue=3 |pages=638–648|doi=10.1107/S160057672400219X |pmid=38846769 |pmc=11151663 |bibcode=2024JApCr..57..638D }}</ref> In fact, {{math|''S''<sup>3</sup>}} is [[foliation|filled out]] by a family of nested tori in this manner (with two degenerate circles), a fact that is important in the study of {{math|''S''<sup>3</sup>}} as a [[fiber bundle]] over {{math|''S''<sup>2</sup>}} (the [[Hopf bundle]]).

The surface described above, given the [[relative topology]] from [[real coordinate space|{{math|'''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  {{math|'''R'''<sup>3</sup>}} from the north pole of {{math|''S''<sup>3</sup>}}.

The torus can also be described as a [[quotient space (topology)|quotient]] of the [[Cartesian plane]] under the identifications
: <math>(x,y) \sim (x+1,y) \sim (x,y+1), \,</math>
or, equivalently, as the quotient of the [[unit square]] by pasting the opposite edges together, described as a [[fundamental polygon]] {{math|''ABA''<sup>−1</sup>''B''<sup>−1</sup>}}.

[[File:Inside-out torus (animated, small).gif|thumb|Turning a punctured torus inside-out]]
The [[fundamental group]] of the torus is just the [[direct product of groups|direct product]] of the fundamental group of the circle with itself:
: <math>\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathrm{Z} \times \mathrm{Z}.</math><ref>Padgett, Adele (2014). "Fundamental groups: motivation, computation methods, and applications", REA Program, Uchicago. https://math.uchicago.edu/~may/REU2014/REUPapers/Padgett.pdf</ref>

Intuitively speaking, this means that a [[loop (topology)|closed path]] that circles the torus's "hole" (say, a circle that traces out a particular latitude) and then circles the torus's "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. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding.

The fundamental group can also be derived from taking the torus as the quotient <math>T^2\cong \mathbb{R}^2/\mathbb{Z}^2</math> (see below), so that <math>\mathbb{R}^2</math> may be taken as its [[universal cover]], with [[deck transformation]] group <math>\mathbb{Z}^2=\pi_1(T^2)</math>.  

Its higher [[Homotopy group|homotopy groups]] are all trivial, since a universal cover projection <math>p:\widetilde{X}\rightarrow X</math> always induces isomorphisms between the groups <math>\pi_n(\widetilde{X})</math> and <math>\pi_n(X)</math> for <math>n>1</math>, and <math>\mathbb{R}^2</math> is [[Contractible space|contractible]].

The torus has [[homology groups]]

<math>H_n(T^2)=\begin{cases}\mathbb{Z},& n=0,2\\

\mathbb{Z}\oplus \mathbb{Z},& n=1\\

0&\text{else.}\end{cases}</math>

Thus, the first homology group of the torus is [[isomorphism|isomorphic]] to its fundamental group-- which in particular can be deduced from [[Hurewicz theorem]] since <math>\pi_1(T^2)</math> is [[abelian group|abelian]].

The cohomology groups with integer coefficients are isomorphic to the homology ones-- which can be seen either by direct computation, [[Universal coefficient theorem|the universal coefficient theorem]] or even [[Poincaré duality]].

If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.

== Two-sheeted cover ==
The 2-torus is a twofold branched cover of the 2-sphere, with four [[ramification point]]s. Every [[conformal structure]] on the 2-torus can be represented as such a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the [[Weierstrass point]]s. In fact, the conformal type of the torus is determined by the [[cross-ratio]] of the four points.

== ''n''-dimensional torus ==
[[File:Clifford-torus.gif|thumb|A stereographic projection of a [[Clifford torus]] in four dimensions performing a simple rotation through the ''xz''-plane]]

The torus has a generalization to higher dimensions, the {{em|{{visible anchor|''n''-dimensional torus|Finite dimensional torus}}}}, often called the {{em|{{math|''n''}}-torus}} or {{em|hypertorus}} for short. (This is the more typical meaning of the term "{{math|''n''}}-torus", the other referring to {{math|''n''}} holes or of genus {{math|''n''}}.<ref>{{cite web |last=Weisstein |first=Eric W. |title=Torus |url=https://mathworld.wolfram.com/Torus.html |access-date=2021-07-27 |website=mathworld.wolfram.com}}</ref>) Just as the ordinary torus is topologically the product space of two circles, the {{math|''n''}}-dimensional torus is ''topologically equivalent to'' the product of {{math|''n''}} circles. That is:
: <math>T^n = \underbrace{S^1 \times \cdots \times S^1}_n.</math>

The standard 1-torus is just the circle: {{math|1=''T''<sup>1</sup> = ''S''<sup>1</sup>}}. The torus discussed above is the standard 2-torus, {{math|1=''T''<sup>2</sup>}}. And similar to the 2-torus, the {{math|''n''}}-torus, {{math|1=''T''<sup>''n''</sup>}} can be described as a quotient of {{math|'''R'''<sup>''n''</sup>}} under integral shifts in any coordinate. That is, the ''n''-torus is {{math|'''R'''<sup>''n''</sup>}} modulo the [[group action (mathematics)|action]] of the integer [[lattice (group)|lattice]] {{math|'''Z'''<sup>''n''</sup>}} (with the action being taken as vector addition). Equivalently, the {{math|''n''}}-torus is obtained from the {{math|''n''}}-dimensional [[hypercube]] by gluing the opposite faces together.

An {{math|''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 {{math|''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 {{math|''T''}} have a controlling role to play in theory of connected {{math|''G''}}. Toroidal groups are examples of [[protorus|protori]], which (like tori) are compact connected abelian groups, which are not required to be [[manifold]]s.

[[Automorphism]]s of {{math|''T''}} are easily constructed from automorphisms of the lattice {{math|'''Z'''<sup>''n''</sup>}}, which are classified by [[invertible matrix|invertible]] [[integral matrices]] of size {{math|''n''}} with an integral inverse; these are just the integral matrices with determinant {{math|±1}}. Making them act on {{math|'''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&nbsp;{{math|''n''}}. The {{math|''k''}}th [[homology group]] of an {{math|''n''}}-torus is a free abelian group of rank ''n'' [[binomial coefficient|choose]]&nbsp;{{math|''k''}}. It follows that the [[Euler characteristic]] of the {{math|''n''}}-torus is {{math|0}} for all&nbsp;{{math|''n''}}. The [[cohomology ring]] ''H''<sup>•</sup>(<math>T^{n}</math>,&nbsp;'''Z''') can be identified with the [[exterior algebra]] over the {{math|'''Z'''}}-[[module (mathematics)|module]] {{math|'''Z'''<sup>''n''</sup>}} whose generators are the duals of the {{math|''n''}} nontrivial cycles.

{{see also|Quasitoric manifold}}
=== Configuration space ===
[[File:Moebius Surface 1 Display Small.png|thumb|The configuration space of 2 not necessarily distinct points on the circle is the [[orbifold]] quotient of the 2-torus, {{math|''T''<sup>2</sup> / ''S''<sub>2</sub>}}, which is the [[Möbius strip]].]]
[[File:Neo-Riemannian Tonnetz.svg|thumb|left|The ''[[Tonnetz]]'' is an example of a torus in music theory.{{br}}<small>The Tonnetz is only truly a torus if [[enharmonic equivalence]] is assumed, so that the {{nowrap|(F♯-A♯)}} segment of the right edge of the repeated parallelogram is identified with the {{nowrap|(G♭-B♭)}} segment of the left edge.</small>]]
As the {{math|''n''}}-torus is the {{math|''n''}}-fold product of the circle, the {{math|''n''}}-torus is the [[configuration space (physics)|configuration space]] of {{math|''n''}} ordered, not necessarily distinct points on the circle. Symbolically, {{math|1=''T''<sup>''n''</sup> = (''S''<sup>1</sup>)<sup>''n''</sup>}}. The configuration space of ''unordered'', not necessarily distinct points is accordingly the [[orbifold]] {{math|''T''<sup>''n''</sup> / ''S''<sup>''n''</sup>}}, which is the quotient of the torus by the [[symmetric group]] on {{math|''n''}} letters (by permuting the coordinates).

For {{math|1=''n'' = 2}}, the quotient is the [[Möbius strip]], the edge corresponding to the orbifold points where the two coordinates coincide. For {{math|1=''n'' = 3}} this quotient may be described as a solid torus with cross-section an [[equilateral triangle]], with a [[Dehn twist|twist]]; equivalently, as a [[triangular prism]] whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.

These orbifolds have found significant [[orbifold#Music theory|applications to music theory]] in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model [[triad (music)|musical triad]]s.<ref>{{Cite journal |last=Tymoczko |first=Dmitri |url=http://www.brainmusic.org/EducationalActivitiesFolder/Tymoczko_chords2006.pdf |title=The Geometry of Musical Chords |date=7 July 2006 |journal=[[Science (journal)|Science]] |volume=313 |pages=72–74 |bibcode=2006Sci...313...72T |citeseerx=10.1.1.215.7449 |doi=10.1126/science.1126287 |pmid=16825563 |archive-url=https://web.archive.org/web/20110725100537/http://www.brainmusic.org/EducationalActivitiesFolder/Tymoczko_chords2006.pdf |archive-date=25 July 2011 |url-status=live |issue=5783 |s2cid=2877171}}</ref><ref>{{Cite web |last=Phillips |first=Tony |date=October 2006 |title=Take on Math in the Media |url=http://www.ams.org/mathmedia/archive/10-2006-media.html |publisher=[[American Mathematical Society]]|archive-url=https://web.archive.org/web/20081005194933/http://www.ams.org/mathmedia/archive/10-2006-media.html |archive-date=2008-10-05 }}</ref>

== Flat torus ==

[[File:Torus from rectangle.gif|thumb|In three dimensions, one can bend a rectangle into a torus, but doing this necessarily affects the distances measured along the surface, as seen by the distortion of the checkered pattern.]]
[[File:Duocylinder ridge animated.gif|thumb|Seen in [[stereographic projection]], a 4D ''flat torus'' can be projected into 3-dimensions and rotated on a fixed axis.]]
[[File:Toroidal monohedron.png|thumb|The simplest tiling of a flat torus is [[regular map (graph theory)#Toroidal polyhedra|{4,4}<sub>1,0</sub>]], constructed on the surface of a [[duocylinder]] with 1 vertex, 2 orthogonal edges, and one square face. It is seen here stereographically projected into 3-space as a torus.]]
A flat torus is a torus with the metric inherited from its representation as the [[quotient space (topology)|quotient]], {{math|1='''R'''<sup>2</sup> / '''L'''}}, where {{math|'''L'''}} is a discrete subgroup of {{math|1='''R'''<sup>2</sup>}} isomorphic to {{math|1='''Z'''<sup>2</sup>}}. This gives the quotient the structure of a [[Riemannian manifold]], as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when {{math|1='''L''' = '''Z'''<sup>2</sup>}}: {{math|'''R'''<sup>2</sup> / '''Z'''<sup>2</sup>}}, which can also be described as the [[Cartesian plane]] under the identifications {{math|(''x'', ''y'') ~ (''x'' + 1, ''y'') ~ (''x'', ''y'' + 1)}}. This particular flat torus (and any uniformly scaled version of it) is known as the "square" flat torus.

This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero [[Gaussian curvature]] everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below).

A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:
: <math>(x,y,z,w) = (R\cos u, R\sin u, P\cos v, P\sin v)</math>
where ''R'' and ''P'' are positive constants determining the aspect ratio. It is [[diffeomorphism|diffeomorphic]] to a regular torus but not [[isometry|isometric]]. It can not be [[analytic function|analytically]] embedded ([[smooth function|smooth]] of class {{math|''C<sup>k</sup>'', 2 ≤ ''k'' ≤ ∞}}) into Euclidean 3-space. [[Map (mathematics)|Mapping]] it into ''3''-space requires one to stretch it, in which case it looks like a regular torus. For example, in the following map:
: <math>(x,y,z) = ((R+P\sin v)\cos u, (R+P\sin v)\sin u, P\cos v).</math>

If {{math|''R''}} and {{math|''P''}} in the above flat torus parametrization form a unit vector {{math|1=(''R'', ''P'') = (cos(''η''), sin(''η''))}} then ''u'', ''v'', and {{math|0 < ''η'' < π/2}} parameterize the unit 3-sphere as [[3-sphere#Hopf coordinates|Hopf coordinates]]. In particular, for certain very specific choices of a square flat torus in the [[3-sphere]] ''S''<sup>3</sup>, where {{math|1=''η'' = π/4}} above, the torus will partition the 3-sphere into two [[congruence (geometry)|congruent]] solid tori subsets with the aforesaid flat torus surface as their common [[boundary (topology)|boundary]]. One example is the torus {{math|''T''}} defined by
: <math>T = \left\{ (x,y,z,w) \in S^3 \mid x^2+y^2 = \frac 1 2, \ z^2+w^2 = \frac 1 2 \right\}.</math>

Other tori in {{math|''S''<sup>3</sup>}} having this partitioning property include the square tori of the form {{math|''Q'' ⋅ ''T''}}, where {{math|''Q''}} is a rotation of 4-dimensional space {{math|'''R'''<sup>4</sup>}}, or in other words {{math|''Q''}} is a member of the Lie group {{math|SO(4)}}.

It is known that there exists no {{math|''C''<sup>2</sup>}} (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the [[Nash embedding theorem|Nash-Kuiper theorem]], which was proven in the 1950s, an isometric ''C''<sup>1</sup> embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding.

[[File:Flat torus Havea embedding.png|thumb|right|<math>C^1</math> isometric embedding of a flat torus in {{math|'''R'''<sup>3</sup>}}, with corrugations]]
In April 2012, an explicit ''C''<sup>1</sup> (continuously differentiable) isometric embedding of a flat torus into 3-dimensional Euclidean space {{math|'''R'''<sup>3</sup>}} was found.<ref>{{cite journal|last=Filippelli|first=Gianluigi|date=27 April 2012|title=Doc Madhattan: A flat torus in three dimensional space|url=http://docmadhattan.fieldofscience.com/2012/04/flat-torus-in-three-dimensional-space.html|url-status=live|journal=Proceedings of the National Academy of Sciences|volume=109|issue=19|pages=7218–7223|doi=10.1073/pnas.1118478109|pmc=3358891|pmid=22523238|archive-url=https://web.archive.org/web/20120625222341/http://docmadhattan.fieldofscience.com/2012/04/flat-torus-in-three-dimensional-space.html|archive-date=25 June 2012|access-date=21 July 2012|doi-access=free}}</ref><ref>{{cite web|url=http://www.sci-news.com/othersciences/mathematics/article00279.html|title=Mathematicians Produce First-Ever Image of Flat Torus in 3D {{pipe}} Mathematics|last=Enrico de Lazaro|date=18 April 2012|website=Sci-News.com|url-status=live|archive-url=https://web.archive.org/web/20120601021059/http://www.sci-news.com/othersciences/mathematics/article00279.html|archive-date=1 June 2012|access-date=21 July 2012}}</ref><ref>{{cite web|url=http://www2.cnrs.fr/en/2027.htm|title=Mathematics: first-ever image of a flat torus in 3D – CNRS Web site – CNRS|url-status=dead|archive-url=https://web.archive.org/web/20120705120058/http://www2.cnrs.fr/en/2027.htm|archive-date=5 July 2012|access-date=21 July 2012}}</ref><ref>{{cite web |url=http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html |title=Flat tori finally visualized! |publisher=Math.univ-lyon1.fr |date=18 April 2012 |access-date=21 July 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120618084643/http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html |archive-date=18 June 2012}}</ref> It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a [[fractal]] as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined [[normal (geometry)|surface normals]], yielding a so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths".<ref>{{cite web |url=http://www.science4all.org/article/flat-torus/ |title=The Tortuous Geometry of the Flat Torus |last=Hoang |first=Lê Nguyên |date=2016 |website=Science4All |access-date=November 1, 2022}}</ref> (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics.

=== Conformal classification of flat tori ===

In the study of [[Riemann surface|Riemann surfaces]], one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The [[Uniformization theorem]] guarantees that every Riemann surface is [[Conformal map|conformally equivalent]] to one that has constant [[Gaussian curvature]]. In the case of a torus, the constant curvature must be zero. Then one defines the "[[moduli space]]" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space ''M'' may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle π and the other has total angle 2π/3. 

''M'' may be turned into a compact space ''M*'' – topologically equivalent to a sphere – by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space  is a sphere with ''three'' points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, ''M*'' may be constructed by glueing together two congruent [[geodesic triangle]]s in the [[hyperbolic plane]] along their (identical) boundaries, where each triangle has angles of {{math|π/2}}, {{math|π/3}}, and {{math|0}}. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the [[Gauss–Bonnet theorem]] shows that the area of each triangle can be calculated as {{math|1=π − (π/2 + π/3 + 0) = π/6}}, so it follows that the compactified moduli space ''M*'' has area equal to {{math|π/3}}.

The other two cusps occur at the points corresponding in ''M*'' to (a) the square torus (total angle {{math|π}}) and (b) the hexagonal torus (total angle {{math|2π/3}}). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation.

== Genus ''g'' surface ==
{{Main|Genus g surface}}
In the theory of [[surface (topology)|surface]]s there is a more general family of objects, the "[[genus (mathematics)|genus]]" {{math|''g''}} surfaces. A genus {{math|''g''}} surface is the [[connected sum]] of {{math|''g''}} two-tori. (And so the torus itself is the surface of genus 1.) To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. (That is, merge the two boundary circles so they become just one circle.) To form the connected sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus {{math|''g''}} surface resembles the surface of {{math|''g''}} doughnuts stuck together side by side, or a [[sphere|2-sphere]] with {{math|''g''}} handles attached.

As examples, a genus zero surface (without boundary) is the [[sphere|two-sphere]] while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called {{math|''n''}}-holed tori (or, rarely, {{math|''n''}}-fold tori). The terms [[double torus]] and [[triple torus]] are also occasionally used.

The [[classification theorem]] for surfaces states that every [[compact space|compact]] [[connected space|connected]] surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real [[projective plane]]s.

{| class=wikitable
|- align=center
|[[File:Double torus illustration.png|160px]]{{br}}[[double torus|genus two]]
|[[File:Triple torus illustration.png|240px]]{{br}}[[triple torus|genus three]]
|}

== Toroidal polyhedra ==
{{Further|Toroidal polyhedron}}
[[File:Hexagonal torus.svg|thumb|A [[toroidal polyhedron]] with {{nowrap|6 × 4 {{=}} 24}} [[quadrilateral]] faces]]
[[Polyhedron|Polyhedra]] with the topological type of a torus are called toroidal polyhedra, and have [[Euler characteristic]] {{math|1=''V'' − ''E'' + ''F'' = 0}}. For any number of holes, the formula generalizes to {{math|1=''V'' − ''E'' + ''F'' = 2 − 2''N''}}, where {{math|''N''}} is the number of holes.

The term "toroidal polyhedron" is also used for higher-genus polyhedra and for [[Immersion (mathematics)|immersions]] of toroidal polyhedra.

{{Expand section|date=April 2010}}

== Automorphisms ==
The [[homeomorphism group]] (or the subgroup of diffeomorphisms) of the torus is studied in [[geometric topology]]. Its [[mapping class group]] (the connected components of the homeomorphism group) is surjective onto the group <math>\operatorname{GL}(n,\mathbf{Z})</math> of invertible integer matrices, which can be realized as linear maps on the universal covering space <math>\mathbf{R}^{n}</math> that preserve the standard lattice <math>\mathbf{Z}^{n}</math> (this corresponds to integer coefficients) and thus descend to the quotient.

At the level of [[homotopy]] and [[homology (mathematics)|homology]], the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the [[fundamental group]], as these are all naturally isomorphic; also the first [[cohomology group]] generates the [[cohomology]] algebra:
: <math>\operatorname{MCG}_{\operatorname{Ho}}(T^n) = \operatorname{Aut}(\pi_1(X)) = \operatorname{Aut}(\mathbf{Z}^n) = \operatorname{GL}(n,\mathbf{Z}).</math>

Since the torus is an [[Eilenberg–MacLane space]] {{math|''K''(''G'', 1)}}, its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.

Thus the [[short exact sequence]] of the mapping class group splits (an identification of the torus as the quotient of <math>\mathbf{R}^{n}</math> gives a splitting, via the linear maps, as above):
: <math>1 \to \operatorname{Homeo}_0(T^n) \to \operatorname{Homeo}(T^n) \to \operatorname{MCG}_{\operatorname{TOP}}(T^n) \to 1.</math>

The mapping class group of higher genus surfaces is much more complicated, and an area of active research.

== Coloring a torus==
<!--Chromatic number-->
The torus's [[Heawood number]] is seven, meaning every graph that can be [[toroidal graph|embedded on the torus]] has a [[chromatic number]] of at most seven. (Since the [[complete graph]] <math>\mathsf{K_7}</math> can be embedded on the torus, and <math>\chi (\mathsf{K_7}) = 7</math>, the upper bound is tight.) Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color. (Contrast with the [[four color theorem]] for the [[plane (mathematics)|plane]].)
[[File:Projection color torus.png|480px|thumb|center|This construction shows the torus divided into seven regions, every one of which touches every other, meaning each must be assigned a unique color.]]

== de Bruijn torus ==
{{main|de Bruijn torus}}
[[File:de_bruijn_torus_3x3.stl|thumb|250px|link=http://viewstl.com/classic/?embedded&url=http://upload.wikimedia.org/wikipedia/commons/1/1e/De_bruijn_torus_3x3.stl&bgcolor=black|[[STL (file format)|STL]] model of de Bruijn torus {{nowrap|(16,32;3,3)<sub>2</sub>}} with 1s as panels and 0s as holes in the mesh &ndash; with consistent orientation, every 3×3 matrix appears exactly once]]
In [[combinatorics|combinatorial]] mathematics, a ''de Bruijn torus'' is an [[matrix (mathematics)|array]] of symbols from an alphabet (often just 0 and 1) that contains every {{math|''m''}}-by-{{math|''n''}} [[matrix (mathematics)|matrix]] exactly once.  It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the [[De Bruijn sequence]], which can be considered a special case where {{math|''n''}} is&nbsp;1 (one dimension).

== Cutting a torus ==
A solid torus of revolution can be cut by ''n'' (> 0) planes into at most
: <math>\begin{pmatrix}n+2 \\ n-1\end{pmatrix} +\begin{pmatrix}n \\ n-1\end{pmatrix} = \tfrac{1}{6}(n^3 + 3n^2 + 8n)</math>
parts.<ref>{{MathWorld|urlname=TorusCutting|title=Torus Cutting}}</ref> (This assumes the pieces may not be rearranged but must remain in place for all cuts.)

The first 11 numbers of parts, for {{math|0 ≤ ''n'' ≤ 10}} (including the case of {{math|1=''n'' = 0}}, not covered by the above formulas), are as follows:
: 1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... {{OEIS|id=A003600}}.

== See also ==
{{Portal|Mathematics}}
{{columns-list|colwidth=22em|
* [[3-torus]]
* [[Algebraic torus]]
* [[Angenent torus]]
* [[Annulus (geometry)]]
* [[Clifford torus]]
* [[Complex torus]]
* [[Dupin cyclide]]
* [[Elliptic curve]]
* [[Irrational winding of a torus]]
* [[Joint European Torus]]
* [[Klein bottle]]
* [[Loewner's torus inequality]]
* [[Maximal torus]]
* [[Period lattice]]
* [[Real projective plane]]
* [[Sphere]]
* [[Spiric section]]
* [[Surface (topology)]]
* [[Toric lens]]
* [[Toric section]]
* [[Toric variety]]
* [[Toroid]]
* [[Toroidal and poloidal]]
* [[Torus-based cryptography]]
* [[Torus knot]]
* [[Umbilic torus]]
* [[Villarceau circles]]
}}

== Notes ==
* ''Nociones de Geometría Analítica y Álgebra Lineal'', {{ISBN|978-970-10-6596-9}}, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish
* Allen Hatcher. [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic Topology'']. Cambridge University Press, 2002. {{ISBN|0-521-79540-0}}.
* V. V. Nikulin, I. R. Shafarevich. ''Geometries and Groups''. Springer, 1987. {{ISBN|3-540-15281-4}}, {{ISBN|978-3-540-15281-1}}.
* [http://www.mathcurve.com/surfaces/tore/tore.shtml "Tore (notion géométrique)" at ''Encyclopédie des Formes Mathématiques Remarquables'']

== References ==
{{reflist}}

== External links ==
{{Wiktionary}}
{{Commons and category|Torus}}
* [http://www.cut-the-knot.org/shortcut.shtml Creation of a torus] at [[cut-the-knot]]
* [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"] Visualizing high dimensional data with flat torus
* [http://tofique.fatehi.us/Mathematics/Polydoes/polydoes.html Polydoes, doughnut-shaped polygons]
* Archived at [https://ghostarchive.org/varchive/youtube/20211211/3_VydFQmtZ8 Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20140128170125/http://www.youtube.com/watch?v=3_VydFQmtZ8&gl=US&hl=en Wayback Machine]{{cbignore}}: {{cite web |last=Séquin |first=Carlo H |author-link=Carlo H. Séquin |title=Topology of a Twisted Torus – Numberphile |date=27 January 2014 |url=https://www.youtube.com/watch?v=3_VydFQmtZ8 |publisher=[[Brady Haran]] |format=video}}{{cbignore}}
* {{cite web |author=Anders Sandberg |title=Torus Earth |date=4 February 2014 |url=http://www.aleph.se/andart/archives/2014/02/torusearth.html |access-date=24 July 2019}}

{{Compact topological surfaces}}

{{Authority control}}

[[Category:Surfaces]]