Editing Torus (section)

== ''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>