Chirality (mathematics)

<h2 id="examples">Examples</h2>

The <a href="/facts/Tetromino/QR8rjoaw">tetrominos</a> S and Z are enantiomorphs in 2-dimensions<table><tbody><tr><th>S</th><th>Z</th></tr></tbody></table>
<p>Some chiral three-dimensional objects, such as the <a href="/facts/Helix/BqC4AKwL">helix</a>, can be assigned a right or left <a href="/facts/Handedness/wb7Urd7u">handedness</a>, according to the <a href="/facts/Right-hand_rule/LJe0Qqh4">right-hand rule</a>.
</p><p>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 inside-out.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>The J-, L-, S- and Z-shaped <i><a href="/facts/Tetromino/QR8rjoaw">tetrominoes</a></i> of the popular video game <a href="/facts/Tetris/djLfTK0n">Tetris</a> also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.
</p>
<h2 id="chirality-and-symmetry-group">Chirality and symmetry group</h2>
<p>A figure is achiral if and only if its <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> contains at least one <i><a href="/facts/Orientation-reversing/lGXs42Nw">orientation-reversing</a></i> isometry. (In Euclidean geometry any <a href="/facts/Isometry/K5wX8ah6">isometry</a> can be written as 
  
    
      
        v
        ↦
        A
        v
        +
        b
      
    
    {\displaystyle v\mapsto Av+b}
  
 with an <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrix</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and a vector 
  
    
      
        b
      
    
    {\displaystyle b}
  
. The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.
</p><p>A general definition of chirality based on group theory exists.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> It does not refer to any orientation concept: an <a href="/facts/Isometry/K5wX8ah6">isometry</a> 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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="chirality-in-two-dimensions">Chirality in two dimensions</h2>

<p>In two dimensions, every figure which possesses an <a href="/facts/Axis_of_symmetry/bpvJqcnS">axis of symmetry</a> is achiral, and it can be shown that every <i>bounded</i> achiral figure must have an axis of symmetry. (An <i>axis of symmetry</i> of a figure 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is a line 
  
    
      
        L
      
    
    {\displaystyle L}
  
, such that 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is invariant under the mapping 
  
    
      
        (
        x
        ,
        y
        )
        ↦
        (
        x
        ,
        −
        y
        )
      
    
    {\displaystyle (x,y)\mapsto (x,-y)}
  
, when 
  
    
      
        L
      
    
    {\displaystyle L}
  
 is chosen to be the 
  
    
      
        x
      
    
    {\displaystyle x}
  
-axis of the coordinate system.) For that reason, a <a href="/facts/Triangle/cRXUwsMn">triangle</a> is achiral if it is <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral</a> or <a href="/facts/Isosceles_triangle/rvLe3yqK">isosceles</a>, and is chiral if it is <a href="/facts/Triangle/cRXUwsMn">scalene</a>.
</p><p>Consider the following pattern:
</p>

<p>This figure is chiral, as it is not identical to its mirror image:
</p>

<p>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 <a href="/facts/Frieze_group/LVknTDnt">frieze group</a> generated by a single <a href="/facts/Glide_reflection/BfDylbEM">glide reflection</a>.
</p>
<h2 id="chirality-in-three-dimensions">Chirality in three dimensions</h2>

<p>In three dimensions, every figure that possesses a <a href="/facts/Mirror_plane_of_symmetry/xkbUhBzI">mirror plane of symmetry</a> <i>S1</i>, an inversion center of symmetry <i>S2</i>, or a higher <a href="/facts/Improper_rotation/kcLdCD9r">improper rotation</a> (rotoreflection) <i>Sn</i> axis of symmetry<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> is achiral. (A <i>plane of symmetry</i> of a figure 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is a plane 
  
    
      
        P
      
    
    {\displaystyle P}
  
, such that 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is invariant under the mapping 
  
    
      
        (
        x
        ,
        y
        ,
        z
        )
        ↦
        (
        x
        ,
        y
        ,
        −
        z
        )
      
    
    {\displaystyle (x,y,z)\mapsto (x,y,-z)}
  
, when 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is chosen to be the 
  
    
      
        x
      
    
    {\displaystyle x}
  
-
  
    
      
        y
      
    
    {\displaystyle y}
  
-plane of the coordinate system. A <i>center of symmetry</i> of a figure 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is a point 
  
    
      
        C
      
    
    {\displaystyle C}
  
, such that 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is invariant under the mapping 
  
    
      
        (
        x
        ,
        y
        ,
        z
        )
        ↦
        (
        −
        x
        ,
        −
        y
        ,
        −
        z
        )
      
    
    {\displaystyle (x,y,z)\mapsto (-x,-y,-z)}
  
, when 
  
    
      
        C
      
    
    {\displaystyle C}
  
 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
</p>

F
          
            0
          
        
        =
        
          {
          
            (
            1
            ,
            0
            ,
            0
            )
            ,
            (
            0
            ,
            1
            ,
            0
            )
            ,
            (
            −
            1
            ,
            0
            ,
            0
            )
            ,
            (
            0
            ,
            −
            1
            ,
            0
            )
            ,
            (
            2
            ,
            1
            ,
            1
            )
            ,
            (
            −
            1
            ,
            2
            ,
            −
            1
            )
            ,
            (
            −
            2
            ,
            −
            1
            ,
            1
            )
            ,
            (
            1
            ,
            −
            2
            ,
            −
            1
            )
          
          }
        
      
    
    {\displaystyle 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\}}

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

F
          
            1
          
        
        =
        
          {
          
            (
            1
            ,
            0
            ,
            0
            )
            ,
            (
            −
            1
            ,
            0
            ,
            0
            )
            ,
            (
            0
            ,
            2
            ,
            0
            )
            ,
            (
            0
            ,
            −
            2
            ,
            0
            )
            ,
            (
            1
            ,
            1
            ,
            1
            )
            ,
            (
            −
            1
            ,
            −
            1
            ,
            −
            1
            )
          
          }
        
      
    
    {\displaystyle F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}}

<p>also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.
</p><p>Achiral figures can have a <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">center axis</a>.
</p>
<h2 id="knot-theory">Knot theory</h2>
<p>A <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a> is called <a href="/facts/Amphichiral_knot/e0KtTEtn">achiral</a> if it can be continuously deformed into its mirror image, otherwise it is called a <a href="/facts/Chiral_knot/e0KtTEtn">chiral knot</a>. For example, the <a href="/facts/Unknot/VvgWgJEr">unknot</a> and the <a href="/facts/Figure-eight_knot_(mathematics)/AAmcDoGu">figure-eight knot</a> are achiral, whereas the <a href="/facts/Trefoil_knot/c3K4nUu9">trefoil knot</a> is chiral.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Chiral_polytope/U0AvlB7p">Chiral polytope</a></li>
<li><a href="/facts/Chirality_(physics)/H8Xbcvzt">Chirality (physics)</a></li>
<li><a href="/facts/Parity_(physics)/emSbJ4hD">Parity (physics)</a></li>
<li><a href="/facts/Chirality_(chemistry)/p5a9pftB">Chirality (chemistry)</a></li>
<li><a href="/facts/Asymmetry/U009nepR">Asymmetry</a></li>
<li><a href="/facts/Skewness/rdS8luFa">Skewness</a></li>
<li><a href="/facts/Vertex_algebra/npzGRlsD">Vertex algebra</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Erica_Flapan/BoHLG3hQ">Flapan, Erica</a> (2000). <a href="/facts/When_Topology_Meets_Chemistry/LW5Kqb3k"><i>When Topology Meets Chemistry</i></a>. Outlook. Cambridge University Press and Mathematical Association of America. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-66254-0.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html">Symmetry, Chirality, Symmetry Measures and Chirality Measures:</a> General Definitions</li>
<li><a href="http://demonstrations.wolfram.com/ChiralPolyhedra/">Chiral Polyhedra</a> by <a href="/facts/Eric_W._Weisstein/FbMuVDep">Eric W. Weisstein</a>, <a href="/facts/The_Wolfram_Demonstrations_Project/l0sh4gln">The Wolfram Demonstrations Project</a>.</li>
<li><a href="http://www.map.mpim-bonn.mpg.de/Chiral_manifold">Chiral manifold</a> at the Manifold Atlas.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Toong, Yock Chai; Wang, Shih Yung (April 1997). "An example of a human topological rubber glove act". Journal of Chemical Education. 74 (4): 403. Bibcode:1997JChEd..74..403T. doi:10.1021/ed074p403. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Petitjean, M. (2020). "Chirality in metric spaces. In memoriam Michel Deza". Optimization Letters. 14 (2): 329–338. doi:10.1007/s11590-017-1189-7. <a href="https://doi.org/10.1007%2Fs11590-017-1189-7" target="_blank">https://doi.org/10.1007%2Fs11590-017-1189-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Petitjean, M. (2021). "Chirality in geometric algebra". Mathematics. 9 (13). 1521. doi:10.3390/math9131521. <a href="https://doi.org/10.3390%2Fmath9131521" target="_blank">https://doi.org/10.3390%2Fmath9131521</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Petitjean, M. (2022). "Chirality in affine spaces and in spacetime". arXiv:2203.04066 [math-ph]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"2. Symmetry operations and symmetry elements". chemwiki.ucdavis.edu. 3 March 2014. Retrieved 25 March 2016. <a href="http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Symmetry_operations_and_symmetry_elements" target="_blank">http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Symmetry_operations_and_symmetry_elements</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Chirality (mathematics) open-in-new

Chirality (mathematics)