Editing Chirality (mathematics)

{{short description|Property of an object that is not congruent to its mirror image}}
[[File:2 parallel footprints.png|thumb|The footprint here demonstrates chirality. Individual left and right footprints are chiral '''enantiomorphs''' in a plane because they are mirror images while containing no mirror symmetry individually.]]
In [[geometry]], a figure is '''chiral''' (and said to have '''chirality''') if it is not identical to its [[mirror image]], or, more precisely, if it cannot be mapped to its mirror image by [[Rotation (mathematics)|rotation]]s and [[Translation (geometry)|translation]]s alone. An object that is not chiral is said to be ''achiral''.

A chiral object and its mirror image are said to be '''enantiomorphs'''. The word ''chirality'' is derived from the Greek {{lang|grc|χείρ}} (cheir), the hand, the most familiar chiral object; the word ''enantiomorph'' stems from the Greek {{lang|grc|ἐναντίος}} (enantios) 'opposite' + {{lang|grc|μορφή}} (morphe) 'form'.

== Examples==
[[File:3D Cartesian Coodinate Handedness.jpg|thumb|left|Left and [[right-hand rule]]s in three dimensions]]
{| class=wikitable align=right
|+ The [[tetromino]]s S and Z are enantiomorphs in 2-dimensions
![[File:Tetromino S.svg|100px]]<BR>S
![[File:Tetromino Z.svg|94px]]<BR>Z
|}
Some chiral three-dimensional objects, such as the [[helix]], can be assigned a right or left [[handedness]], according to the [[right-hand rule]].

Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them [[wiktionary:inside out|inside-out]].<ref>{{cite journal
 | last1 = Toong | first1 = Yock Chai
 | last2 = Wang | first2 = Shih Yung
 | date = April 1997
 | doi = 10.1021/ed074p403
 | issue = 4
 | journal = Journal of Chemical Education
 | page = 403
 | title = An example of a human topological rubber glove act
 | volume = 74| bibcode = 1997JChEd..74..403T
 }}</ref>

The J-, L-, S- and Z-shaped ''[[tetromino]]es'' of the popular video game [[Tetris]] also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.

==Chirality and symmetry group==
A figure is achiral<!--sic!--> if and only if its [[symmetry group]] contains at least one ''[[orientation-reversing]]'' isometry. (In Euclidean geometry any [[isometry]] can be written as <math>v\mapsto Av+b</math> with an [[orthogonal matrix]] <math>A</math> and a vector <math>b</math>. The [[determinant]] of <math>A</math> is either 1 or &minus;1 then. If it is &minus;1 the isometry is orientation-reversing, otherwise it is orientation-preserving.

A general definition of chirality based on group theory exists.<ref>{{cite journal | author = Petitjean, M. | title = Chirality in metric spaces. In memoriam Michel Deza | journal = Optimization Letters | year=2020 | volume=14 | issue=2 | pages=329–338 | doi = 10.1007/s11590-017-1189-7 | doi-access=free }}</ref> It does not refer to any orientation concept: an [[isometry]] is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.<ref>{{cite journal | author = Petitjean, M. | title = Chirality in geometric algebra | journal=Mathematics | year=2021 | volume=9 | issue=13 | at=1521 | no-pp=yes | doi = 10.3390/math9131521 | doi-access=free }}</ref><ref>{{cite arXiv |last=Petitjean |first=M. |date=2022 |title=Chirality in affine spaces and in spacetime |eprint=2203.04066 |class=math-ph }}</ref>

==Chirality in two dimensions==
[[File:Bracelets33.svg|thumb|300px|The colored [[Necklace (combinatorics)|necklace]] in the middle is '''chiral''' in two dimensions; the two others are '''achiral'''.<br>This means that as physical necklaces on a table the left and right ones can be rotated into their mirror image while remaining on the table. The one in the middle, however, would have to be picked up and turned in three dimensions.]]
[[File:Triangle.Scalene.svg|thumb|250px|A [[scalene triangle]] does not have mirror symmetries, and hence is a [[chiral polytope]] in 2 dimensions.]]
In two dimensions, every figure which possesses an [[axis of symmetry]] is achiral, and it can be shown that every ''bounded'' achiral figure must have an axis of symmetry. (An ''axis of symmetry'' of a figure <math>F</math> is a line <math>L</math>, such that <math>F</math> is invariant under the mapping <math>(x,y)\mapsto(x,-y)</math>, when <math>L</math> is chosen to be the <math>x</math>-axis of the coordinate system.) For that reason, a [[triangle]] is achiral if it is [[equilateral triangle|equilateral]] or [[isosceles triangle|isosceles]], and is chiral if it is [[Triangle#By_lengths_of_sides|scalene]].

Consider the following pattern:
:[[File:Krok 6.svg|320px]]

This figure is chiral, as it is not identical to its mirror image:
:[[File:Krok 6 mirrored.png|320px]]

But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a [[frieze group]] generated by a single [[glide reflection]].

==Chirality in three dimensions==
[[File:Chiralität von Würfeln V.1.svg|thumb|Pair of chiral [[dice]] (enantiomorphs)]]
In three dimensions, every figure that possesses a [[mirror plane of symmetry]] ''S<sub>1</sub>'', an inversion center of symmetry ''S<sub>2</sub>'', or a higher [[improper rotation]] (rotoreflection) ''S<sub>n</sub>'' axis of symmetry<ref>{{cite web|title=2. Symmetry operations and symmetry elements|url=http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Symmetry_operations_and_symmetry_elements|website=chemwiki.ucdavis.edu|date=3 March 2014 |access-date=25 March 2016}}</ref> is achiral. (A ''plane of symmetry'' of a figure <math>F</math> is a plane <math>P</math>, such that <math>F</math> is invariant under the mapping <math>(x,y,z)\mapsto(x,y,-z)</math>, when <math>P</math> is chosen to be the <math>x</math>-<math>y</math>-plane of the coordinate system. A ''center of symmetry'' of a figure <math>F</math> is a point <math>C</math>, such that <math>F</math> is invariant under the mapping <math>(x,y,z)\mapsto(-x,-y,-z)</math>, when <math>C</math> is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure

:<math>F_0=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\right\}</math>

which is invariant under the orientation reversing isometry <math>(x,y,z)\mapsto(-y,x,-z)</math> and thus achiral, but it has neither plane nor center of symmetry. The figure

:<math>F_1=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}</math>

also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.

Achiral figures can have a [[Point groups in three dimensions#Center of symmetry|center axis]].

==Knot theory==<!-- This section is linked from [[Figure-eight knot (mathematics)]] -->

A [[knot (mathematics)|knot]] is called [[amphichiral knot|achiral]] if it can be continuously deformed into its mirror image, otherwise it is called a [[chiral knot]]. For example, the [[unknot]] and the [[figure-eight knot (mathematics)|figure-eight knot]] are achiral, whereas the [[trefoil knot]] is chiral.

==See also==
* [[Chiral polytope]]
* [[Chirality (physics)]]
* [[Parity (physics)]]
* [[Chirality (chemistry)]]
* [[Asymmetry]]
* [[Skewness]]
* [[Vertex algebra]]

==References==
{{Reflist}}

==Further reading==
*{{cite book|title=When Topology Meets Chemistry|title-link=When Topology Meets Chemistry|first=Erica|last=Flapan|author-link=Erica Flapan|year=2000|publisher=Cambridge University Press and Mathematical Association of America|series=Outlook|isbn=0-521-66254-0}}

== External links ==
*[http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html Symmetry, Chirality, Symmetry Measures and Chirality Measures:] General Definitions
* [http://demonstrations.wolfram.com/ChiralPolyhedra/ Chiral Polyhedra] by [[Eric W. Weisstein]], [[The Wolfram Demonstrations Project]].
*[http://www.map.mpim-bonn.mpg.de/Chiral_manifold Chiral manifold] at the Manifold Atlas.

[[Category:Knot theory]]
[[Category:Polyhedra]]
[[Category:Chirality]]
[[Category:Topology]]