Editing Torus (section)

=== 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>