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Available in 16 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-16" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D8%A7_%D8%A7%D9%86%D8%B7%D8%A8%D8%A7%D9%82%D9%8A%D8%A9" title="لا انطباقية – Arabic" lang="ar" hreflang="ar" data-title="لا انطباقية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Хиралност (математика) – Bulgarian" lang="bg" hreflang="bg" data-title="Хиралност (математика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quiralitat_(geometria)" title="Quiralitat (geometria) – Catalan" lang="ca" hreflang="ca" data-title="Quiralitat (geometria)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Chiralit%C3%A4t_(Mathematik)" title="Chiralität (Mathematik) – German" lang="de" hreflang="de" data-title="Chiralität (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Quiralidad_(matem%C3%A1ticas)" title="Quiralidad (matemáticas) – Spanish" lang="es" hreflang="es" data-title="Quiralidad (matemáticas)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nememspegulsimetrieco" title="Nememspegulsimetrieco – Esperanto" lang="eo" hreflang="eo" data-title="Nememspegulsimetrieco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%E2%80%8C%D8%B3%D8%A7%D9%86%DB%8C_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="دست‌سانی (هندسه) – Persian" lang="fa" hreflang="fa" data-title="دست‌سانی (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Quiralidade_(xeometr%C3%ADa)" title="Quiralidade (xeometría) – Galician" lang="gl" hreflang="gl" data-title="Quiralidade (xeometría)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Chiralit%C3%A0_(matematica)" title="Chiralità (matematica) – Italian" lang="it" hreflang="it" data-title="Chiralità (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Chiraliteit_(wiskunde)" title="Chiraliteit (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Chiraliteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Symetria_chiralna" title="Symetria chiralna – Polish" lang="pl" hreflang="pl" data-title="Symetria chiralna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Quiralidade_(matem%C3%A1tica)" title="Quiralidade (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Quiralidade (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Chiralitate_(matematic%C4%83)" title="Chiralitate (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Chiralitate (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Хиральность (математика) – Russian" lang="ru" hreflang="ru" data-title="Хиральность (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kiralnost_(matematika)" title="Kiralnost (matematika) – Slovenian" lang="sl" hreflang="sl" data-title="Kiralnost (matematika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A5%D1%96%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%96%D1%81%D1%82%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Хіральність (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Хіральність (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2467946#sitelinks-wikipedia" 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title="Enantiomorphous">Enantiomorphous</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property of an object that is not congruent to its mirror image</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2_parallel_footprints.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/2_parallel_footprints.png/250px-2_parallel_footprints.png" decoding="async" width="220" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/eb/2_parallel_footprints.png 1.5x" data-file-width="265" data-file-height="280" /></a><figcaption>The footprint here demonstrates chirality. Individual left and right footprints are chiral <b>enantiomorphs</b> in a plane because they are mirror images while containing no mirror symmetry individually.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a figure is <b>chiral</b> (and said to have <b>chirality</b>) if it is not identical to its <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a>, or, more precisely, if it cannot be mapped to its mirror image by <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> and <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a> alone. An object that is not chiral is said to be <i>achiral</i>. </p><p>A chiral object and its mirror image are said to be <b>enantiomorphs</b>. The word <i>chirality</i> is derived from the Greek <span title="Ancient Greek (to 1453)-language text"><span lang="grc">χείρ</span></span> (cheir), the hand, the most familiar chiral object; the word <i>enantiomorph</i> stems from the Greek <span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἐναντίος</span></span> (enantios) 'opposite' + <span title="Ancient Greek (to 1453)-language text"><span lang="grc">μορφή</span></span> (morphe) 'form'. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:3D_Cartesian_Coodinate_Handedness.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/3D_Cartesian_Coodinate_Handedness.jpg/220px-3D_Cartesian_Coodinate_Handedness.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/3D_Cartesian_Coodinate_Handedness.jpg/330px-3D_Cartesian_Coodinate_Handedness.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/3D_Cartesian_Coodinate_Handedness.jpg/440px-3D_Cartesian_Coodinate_Handedness.jpg 2x" data-file-width="800" data-file-height="450" /></a><figcaption>Left and <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rules</a> in three dimensions</figcaption></figure> <table class="wikitable" align="right"> <caption>The <a href="/wiki/Tetromino" title="Tetromino">tetrominos</a> S and Z are enantiomorphs in 2-dimensions </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Tetromino_S.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Tetromino_S.svg/100px-Tetromino_S.svg.png" decoding="async" width="100" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Tetromino_S.svg/150px-Tetromino_S.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Tetromino_S.svg/200px-Tetromino_S.svg.png 2x" data-file-width="167" data-file-height="115" /></a></span><br />S </th> <th><span typeof="mw:File"><a href="/wiki/File:Tetromino_Z.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetromino_Z.svg/94px-Tetromino_Z.svg.png" decoding="async" width="94" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetromino_Z.svg/141px-Tetromino_Z.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetromino_Z.svg/188px-Tetromino_Z.svg.png 2x" data-file-width="159" data-file-height="124" /></a></span><br />Z </th></tr></tbody></table> <p>Some chiral three-dimensional objects, such as the <a href="/wiki/Helix" title="Helix">helix</a>, can be assigned a right or left <a href="/wiki/Handedness" title="Handedness">handedness</a>, according to the <a href="/wiki/Right-hand_rule" title="Right-hand rule">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 <a href="https://en.wiktionary.org/wiki/inside_out" class="extiw" title="wiktionary:inside out">inside-out</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The J-, L-, S- and Z-shaped <i><a href="/wiki/Tetromino" title="Tetromino">tetrominoes</a></i> of the popular video game <a href="/wiki/Tetris" title="Tetris">Tetris</a> also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane. </p> <div class="mw-heading mw-heading2"><h2 id="Chirality_and_symmetry_group">Chirality and symmetry group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=2" title="Edit section: Chirality and symmetry group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A figure is achiral if and only if its <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> contains at least one <i><a href="/wiki/Orientation-reversing" class="mw-redirect" title="Orientation-reversing">orientation-reversing</a></i> isometry. (In Euclidean geometry any <a href="/wiki/Isometry" title="Isometry">isometry</a> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\mapsto Av+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">↦</mo> <mi>A</mi> <mi>v</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\mapsto Av+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5a8d8d31bcfbe8bce3798f79bdb950a6e3c087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.45ex; height:2.343ex;" alt="{\displaystyle v\mapsto Av+b}" /></span> with an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> and a vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span>. The <a href="/wiki/Determinant" title="Determinant">determinant</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> 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.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It does not refer to any orientation concept: an <a href="/wiki/Isometry" title="Isometry">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.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Chirality_in_two_dimensions">Chirality in two dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=3" title="Edit section: Chirality in two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bracelets33.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Bracelets33.svg/300px-Bracelets33.svg.png" decoding="async" width="300" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Bracelets33.svg/450px-Bracelets33.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Bracelets33.svg/600px-Bracelets33.svg.png 2x" data-file-width="1545" data-file-height="482" /></a><figcaption>The colored <a href="/wiki/Necklace_(combinatorics)" title="Necklace (combinatorics)">necklace</a> in the middle is <b>chiral</b> in two dimensions; the two others are <b>achiral</b>.<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.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Triangle.Scalene.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/250px-Triangle.Scalene.svg.png" decoding="async" width="250" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/375px-Triangle.Scalene.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/500px-Triangle.Scalene.svg.png 2x" data-file-width="245" data-file-height="110" /></a><figcaption>A <a href="/wiki/Scalene_triangle" class="mw-redirect" title="Scalene triangle">scalene triangle</a> does not have mirror symmetries, and hence is a <a href="/wiki/Chiral_polytope" title="Chiral polytope">chiral polytope</a> in 2 dimensions.</figcaption></figure> <p>In two dimensions, every figure which possesses an <a href="/wiki/Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is a line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is invariant under the mapping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto (x,-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto (x,-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d51515da267a3ef35a0040901de49906824542fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.079ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto (x,-y)}" /></span>, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span> is chosen to be the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>-axis of the coordinate system.) For that reason, a <a href="/wiki/Triangle" title="Triangle">triangle</a> is achiral if it is <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral</a> or <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles</a>, and is chiral if it is <a href="/wiki/Triangle#By_lengths_of_sides" title="Triangle">scalene</a>. </p><p>Consider the following pattern: </p> <dl><dd><span typeof="mw:File"><a href="/wiki/File:Krok_6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Krok_6.svg/320px-Krok_6.svg.png" decoding="async" width="320" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Krok_6.svg/480px-Krok_6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Krok_6.svg/640px-Krok_6.svg.png 2x" data-file-width="2279" data-file-height="412" /></a></span></dd></dl> <p>This figure is chiral, as it is not identical to its mirror image: </p> <dl><dd><span typeof="mw:File"><a href="/wiki/File:Krok_6_mirrored.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Krok_6_mirrored.png/330px-Krok_6_mirrored.png" decoding="async" width="320" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Krok_6_mirrored.png/500px-Krok_6_mirrored.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Krok_6_mirrored.png/960px-Krok_6_mirrored.png 2x" data-file-width="1190" data-file-height="202" /></a></span></dd></dl> <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="/wiki/Frieze_group" title="Frieze group">frieze group</a> generated by a single <a href="/wiki/Glide_reflection" title="Glide reflection">glide reflection</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Chirality_in_three_dimensions">Chirality in three dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=4" title="Edit section: Chirality in three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg/250px-Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg.png" decoding="async" width="220" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg/330px-Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg/500px-Chiralit%C3%A4t_von_W%C3%BCrfeln_V.1.svg.png 2x" data-file-width="825" data-file-height="966" /></a><figcaption>Pair of chiral <a href="/wiki/Dice" title="Dice">dice</a> (enantiomorphs)</figcaption></figure> <p>In three dimensions, every figure that possesses a <a href="/wiki/Mirror_plane_of_symmetry" class="mw-redirect" title="Mirror plane of symmetry">mirror plane of symmetry</a> <i>S<sub>1</sub></i>, an inversion center of symmetry <i>S<sub>2</sub></i>, or a higher <a href="/wiki/Improper_rotation" title="Improper rotation">improper rotation</a> (rotoreflection) <i>S<sub>n</sub></i> axis of symmetry<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> is achiral. (A <i>plane of symmetry</i> of a figure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is a plane <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is invariant under the mapping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)\mapsto (x,y,-z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)\mapsto (x,y,-z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2f87652a0aa1bb5c8f7a4a304dbdcd75f9c7d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.323ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\mapsto (x,y,-z)}" /></span>, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> is chosen to be the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>-plane of the coordinate system. A <i>center of symmetry</i> of a figure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /></span> is invariant under the mapping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58621041fdf3a9a1f65c92e16495e596fe76dc38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.939ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}" /></span>, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> 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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fe9e4a8992203873eec62a8ae2397795714ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.591ex; height:2.843ex;" alt="{\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\}}" /></span></dd></dl> <p>which is invariant under the orientation reversing isometry <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)\mapsto (-y,x,-z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)\mapsto (-y,x,-z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcc600278c3ad7a86e39471537ad7324702caf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.131ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\mapsto (-y,x,-z)}" /></span> and thus achiral, but it has neither plane nor center of symmetry. The figure </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f95c21a79302afbe5a61407e942ab354e8f4206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.37ex; height:2.843ex;" alt="{\displaystyle F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}}" /></span></dd></dl> <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="/wiki/Point_groups_in_three_dimensions#Center_of_symmetry" title="Point groups in three dimensions">center axis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Knot_theory">Knot theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=5" title="Edit section: Knot theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> is called <a href="/wiki/Amphichiral_knot" class="mw-redirect" title="Amphichiral knot">achiral</a> if it can be continuously deformed into its mirror image, otherwise it is called a <a href="/wiki/Chiral_knot" title="Chiral knot">chiral knot</a>. For example, the <a href="/wiki/Unknot" title="Unknot">unknot</a> and the <a href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">figure-eight knot</a> are achiral, whereas the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a> is chiral. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Chiral_polytope" title="Chiral polytope">Chiral polytope</a></li> <li><a href="/wiki/Chirality_(physics)" title="Chirality (physics)">Chirality (physics)</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">Parity (physics)</a></li> <li><a href="/wiki/Chirality_(chemistry)" title="Chirality (chemistry)">Chirality (chemistry)</a></li> <li><a href="/wiki/Asymmetry" title="Asymmetry">Asymmetry</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li> <li><a href="/wiki/Vertex_algebra" class="mw-redirect" title="Vertex algebra">Vertex algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFToongWang1997" class="citation journal cs1">Toong, Yock Chai; Wang, Shih Yung (April 1997). "An example of a human topological rubber glove act". <i>Journal of Chemical Education</i>. <b>74</b> (4): 403. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997JChEd..74..403T">1997JChEd..74..403T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fed074p403">10.1021/ed074p403</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Chemical+Education&rft.atitle=An+example+of+a+human+topological+rubber+glove+act&rft.volume=74&rft.issue=4&rft.pages=403&rft.date=1997-04&rft_id=info%3Adoi%2F10.1021%2Fed074p403&rft_id=info%3Abibcode%2F1997JChEd..74..403T&rft.aulast=Toong&rft.aufirst=Yock+Chai&rft.au=Wang%2C+Shih+Yung&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPetitjean,_M.2020" class="citation journal cs1">Petitjean, M. 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Symmetry operations and symmetry elements"</a>. <i>chemwiki.ucdavis.edu</i>. 3 March 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">25 March</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=chemwiki.ucdavis.edu&rft.atitle=2.+Symmetry+operations+and+symmetry+elements&rft.date=2014-03-03&rft_id=http%3A%2F%2Fchemwiki.ucdavis.edu%2FTheoretical_Chemistry%2FSymmetry%2FSymmetry_operations_and_symmetry_elements&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28mathematics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFlapan2000" class="citation book cs1"><a href="/wiki/Erica_Flapan" title="Erica Flapan">Flapan, Erica</a> (2000). <a href="/wiki/When_Topology_Meets_Chemistry" title="When Topology Meets Chemistry"><i>When Topology Meets Chemistry</i></a>. Outlook. Cambridge University Press and Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66254-0" title="Special:BookSources/0-521-66254-0"><bdi>0-521-66254-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=When+Topology+Meets+Chemistry&rft.series=Outlook&rft.pub=Cambridge+University+Press+and+Mathematical+Association+of+America&rft.date=2000&rft.isbn=0-521-66254-0&rft.aulast=Flapan&rft.aufirst=Erica&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28mathematics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(mathematics)&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html">Symmetry, Chirality, Symmetry Measures and Chirality Measures:</a> General Definitions</li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ChiralPolyhedra/">Chiral Polyhedra</a> by <a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. 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