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href="#In_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>In geometry</span> </div> </a> <button aria-controls="toc-In_geometry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In geometry subsection</span> </button> <ul id="toc-In_geometry-sublist" class="vector-toc-list"> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalence_of_shapes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equivalence_of_shapes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equivalence of shapes</span> </div> </a> <button aria-controls="toc-Equivalence_of_shapes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Equivalence of shapes subsection</span> </button> <ul id="toc-Equivalence_of_shapes-sublist" class="vector-toc-list"> <li id="toc-Congruence_and_similarity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congruence_and_similarity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Congruence and similarity</span> </div> </a> <ul id="toc-Congruence_and_similarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homeomorphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homeomorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Homeomorphism</span> </div> </a> <ul id="toc-Homeomorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shape_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shape_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Shape analysis</span> </div> </a> <ul id="toc-Shape_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Similarity_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Similarity_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Similarity classes</span> </div> </a> <ul id="toc-Similarity_classes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Human_perception_of_shapes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Human_perception_of_shapes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Human perception of shapes</span> </div> </a> <ul id="toc-Human_perception_of_shapes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Shape</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 57 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-57" 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">57 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vorm" title="Vorm – Afrikaans" lang="af" hreflang="af" data-title="Vorm" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D9%83%D9%84" title="شكل – Arabic" lang="ar" hreflang="ar" data-title="شكل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Forma_(predmet)" title="Forma (predmet) – Azerbaijani" lang="az" hreflang="az" data-title="Forma (predmet)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%95%E0%A7%83%E0%A6%A4%E0%A6%BF" title="আকৃতি – Bangla" lang="bn" hreflang="bn" data-title="আকৃতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Korte" title="Korte – Central Bikol" lang="bcl" hreflang="bcl" data-title="Korte" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Forma_geom%C3%A8trica" title="Forma geomètrica – Catalan" lang="ca" hreflang="ca" data-title="Forma geomètrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Tvar" title="Tvar – Czech" lang="cs" hreflang="cs" data-title="Tvar" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Chiwumbiko" title="Chiwumbiko – Shona" lang="sn" hreflang="sn" data-title="Chiwumbiko" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Si%C3%A2p" title="Siâp – Welsh" lang="cy" hreflang="cy" data-title="Siâp" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kujund" title="Kujund – Estonian" lang="et" hreflang="et" data-title="Kujund" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Forma_(figura)" title="Forma (figura) – Spanish" lang="es" hreflang="es" data-title="Forma (figura)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%DA%A9%D9%84" title="شکل – Persian" lang="fa" hreflang="fa" data-title="شکل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Forme_(g%C3%A9om%C3%A9trie)" title="Forme (géométrie) – French" lang="fr" hreflang="fr" data-title="Forme (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cumadh" title="Cumadh – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Cumadh" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Forma_(xeometr%C3%ADa)" title="Forma (xeometría) – Galician" lang="gl" hreflang="gl" data-title="Forma (xeometría)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%BD%A2" title="形 – Gan" lang="gan" hreflang="gan" data-title="形" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%A8%EC%96%91" title="모양 – Korean" lang="ko" hreflang="ko" data-title="모양" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%88%D6%82%D6%80%D5%BE%D5%A1%D5%BA%D5%A1%D5%BF%D5%AF%D5%A5%D6%80" title="Ուրվապատկեր – Armenian" lang="hy" hreflang="hy" data-title="Ուրվապատկեր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Oblik" title="Oblik – Croatian" lang="hr" hreflang="hr" data-title="Oblik" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Formo" title="Formo – Ido" lang="io" hreflang="io" data-title="Formo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bentuk" title="Bentuk – Indonesian" lang="id" hreflang="id" data-title="Bentuk" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Forma" title="Forma – Interlingua" lang="ia" hreflang="ia" data-title="Forma" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Form" title="Form – Icelandic" lang="is" hreflang="is" data-title="Form" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%86%E0%B2%95%E0%B2%BE%E0%B2%B0" title="ಆಕಾರ – Kannada" lang="kn" hreflang="kn" data-title="ಆಕಾರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/%C5%9E%C3%AAwe" title="Şêwe – Kurdish" lang="ku" hreflang="ku" data-title="Şêwe" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Forma" title="Forma – Latvian" lang="lv" hreflang="lv" data-title="Forma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Geometrin%C4%97_fig%C5%ABra" title="Geometrinė figūra – Lithuanian" lang="lt" hreflang="lt" data-title="Geometrinė figūra" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Alakzat_(geometria)" title="Alakzat (geometria) – Hungarian" lang="hu" hreflang="hu" data-title="Alakzat (geometria)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D0%B1%D0%BB%D0%B8%D0%BA" title="Облик – Macedonian" lang="mk" hreflang="mk" data-title="Облик" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bentuk" title="Bentuk – Malay" lang="ms" hreflang="ms" data-title="Bentuk" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Bantuak" title="Bantuak – Minangkabau" lang="min" hreflang="min" data-title="Bantuak" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D1%8D%D0%BB%D0%B1%D1%8D%D1%80" title="Хэлбэр – Mongolian" lang="mn" hreflang="mn" data-title="Хэлбэр" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Bulia" title="Bulia – Fijian" lang="fj" hreflang="fj" data-title="Bulia" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vorm_(fysiek)" title="Vorm (fysiek) – Dutch" lang="nl" hreflang="nl" data-title="Vorm (fysiek)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Figur_i_matematikk" title="Figur i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Figur i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Boca" title="Boca – Oromo" lang="om" hreflang="om" data-title="Boca" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Shiep" title="Shiep – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Shiep" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Forma_(figura)" title="Forma (figura) – Portuguese" lang="pt" hreflang="pt" data-title="Forma (figura)" 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/Form%C4%83" title="Formă – Romanian" lang="ro" hreflang="ro" data-title="Formă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Panpa_suyu" title="Panpa suyu – Quechua" lang="qu" hreflang="qu" data-title="Panpa suyu" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0_%D0%BF%D1%80%D0%B5%D0%B4%D0%BC%D0%B5%D1%82%D0%B0" title="Форма предмета – Russian" lang="ru" hreflang="ru" data-title="Форма предмета" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Forma" title="Forma – Albanian" lang="sq" hreflang="sq" data-title="Forma" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Forma" title="Forma – Sicilian" lang="scn" hreflang="scn" data-title="Forma" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Shape" title="Shape – Simple English" lang="en-simple" hreflang="en-simple" data-title="Shape" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B4%DA%AA%D9%84" title="شڪل – Sindhi" lang="sd" hreflang="sd" data-title="شڪل" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B4%DB%8E%D9%88%DB%95" title="شێوە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="شێوە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Oblik" title="Oblik – Serbian" lang="sr" hreflang="sr" data-title="Oblik" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Oblik" title="Oblik – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Oblik" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tasokuvio" title="Tasokuvio – Finnish" lang="fi" hreflang="fi" data-title="Tasokuvio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Hugis" title="Hugis – Tagalog" lang="tl" hreflang="tl" data-title="Hugis" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a 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free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></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">Form of an object</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Shape_(disambiguation)" class="mw-disambig" title="Shape (disambiguation)">Shape (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Geometric shape" redirects here. For the Unicode symbols, see <a href="/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes (Unicode block)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D,_objectnr_83212.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D%2C_objectnr_83212.JPG/220px-Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D%2C_objectnr_83212.JPG" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D%2C_objectnr_83212.JPG/330px-Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D%2C_objectnr_83212.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D%2C_objectnr_83212.JPG/440px-Plastic_vormenstoof_of_puzzelbal_van_%E2%80%9CTupperware_Toy%E2%80%9D%2C_objectnr_83212.JPG 2x" data-file-width="3872" data-file-height="2592" /></a><figcaption>A children's toy called Shape-O made by <a href="/wiki/Tupperware_Brands" title="Tupperware Brands">Tupperware</a> used for learning various shapes.</figcaption></figure> <p>A <b>shape</b> is a <a href="/wiki/Graphics" title="Graphics">graphical representation</a> of an object's form or its external boundary, outline, or external <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a>. It is distinct from other object properties, such as <a href="/wiki/Color" title="Color">color</a>, <a href="/wiki/Surface_texture" class="mw-redirect" title="Surface texture">texture</a>, or <a href="/wiki/Material" title="Material">material</a> type. In <a href="/wiki/Geometry" title="Geometry">geometry</a>, <i>shape</i> excludes information about the object's <a href="/wiki/Position_(geometry)" title="Position (geometry)">position</a>, <a href="/wiki/Size" title="Size">size</a>, <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a> and <a href="/wiki/Chirality" title="Chirality">chirality</a>.<sup id="cite_ref-Kendall_1-0" class="reference"><a href="#cite_note-Kendall-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A <i><b>figure</b></i> is a representation including both shape and size (as in, e.g., <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">figure of the Earth</a>). </p><p>A <b>plane shape</b> or <b>plane figure</b> is constrained to lie on a <i><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></i>, in contrast to <i><a href="/wiki/Solid_figure" class="mw-redirect" title="Solid figure">solid</a></i> 3D shapes. A <b>two-dimensional shape</b> or <b>two-dimensional figure</b> (also: <b>2D shape</b> or <b>2D figure</b>) may lie on a more general curved <i><a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a></i> (a <a href="/wiki/Two-dimensional_space" title="Two-dimensional space">two-dimensional space</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Classification_of_simple_shapes">Classification of simple shapes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=1" title="Edit section: Classification of simple shapes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lists_of_shapes" title="Lists of shapes">Lists of shapes</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Polygon_types.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Polygon_types.svg/300px-Polygon_types.svg.png" decoding="async" width="300" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Polygon_types.svg/450px-Polygon_types.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Polygon_types.svg/600px-Polygon_types.svg.png 2x" data-file-width="750" data-file-height="630" /></a><figcaption>A variety of <a href="/wiki/Polygon" title="Polygon">polygonal</a> shapes.</figcaption></figure> <p>Some simple shapes can be put into broad categories. For instance, <a href="/wiki/Polygon" title="Polygon">polygons</a> are classified according to their number of edges as <a href="/wiki/Triangle" title="Triangle">triangles</a>, <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilaterals</a>, <a href="/wiki/Pentagon" title="Pentagon">pentagons</a>, etc. Each of these is divided into smaller categories; triangles can be <a href="/wiki/Equilateral" class="mw-redirect" title="Equilateral">equilateral</a>, <a href="/wiki/Isosceles" class="mw-redirect" title="Isosceles">isosceles</a>, <a href="/wiki/Obtuse_triangle" class="mw-redirect" title="Obtuse triangle">obtuse</a>, <a href="/wiki/Triangle#By_internal_angles" title="Triangle">acute</a>, <a href="/wiki/Triangle" title="Triangle">scalene</a>, etc. while quadrilaterals can be <a href="/wiki/Rectangle" title="Rectangle">rectangles</a>, <a href="/wiki/Rhombi" class="mw-redirect" title="Rhombi">rhombi</a>, <a href="/wiki/Trapezoids" class="mw-redirect" title="Trapezoids">trapezoids</a>, <a href="/wiki/Squares" class="mw-redirect" title="Squares">squares</a>, etc. </p><p>Other common shapes are <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planes</a>, and <a href="/wiki/Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a> such as <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>, <a href="/wiki/Circle" title="Circle">circles</a>, and <a href="/wiki/Parabola" title="Parabola">parabolas</a>. </p><p>Among the most common 3-dimensional shapes are <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a>, which are shapes with flat faces; <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoids</a>, which are egg-shaped or sphere-shaped objects; <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinders</a>; and <a href="/wiki/Cone" title="Cone">cones</a>. </p><p>If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a <a href="/wiki/Manhole_cover" title="Manhole cover">manhole cover</a> is a <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a>, because it is approximately the same geometric object as an actual geometric disk. </p> <div class="mw-heading mw-heading2"><h2 id="In_geometry">In geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=2" title="Edit section: In geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Area.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area.svg/250px-Area.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area.svg/375px-Area.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area.svg/500px-Area.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>A set of geometric shapes in 2 dimensions: <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>, <a href="/wiki/Triangle" title="Triangle">triangle</a> & <a href="/wiki/Circle" title="Circle">circle</a></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Basic_shapes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Basic_shapes.svg/250px-Basic_shapes.svg.png" decoding="async" width="250" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Basic_shapes.svg/375px-Basic_shapes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Basic_shapes.svg/500px-Basic_shapes.svg.png 2x" data-file-width="697" data-file-height="503" /></a><figcaption>A set of geometric shapes in 3 dimensions: <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a>, <a href="/wiki/Sphere" title="Sphere">sphere</a> & <a href="/wiki/Cube" title="Cube">cube</a></figcaption></figure> <p>A <b>geometric shape</b> consists of the <a href="/wiki/Geometry" title="Geometry">geometric</a> information which remains when <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">location</a>, <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scale</a>, <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a> and <a href="/wiki/Reflection_(geometry)" class="mw-redirect" title="Reflection (geometry)">reflection</a> are removed from the description of a <a href="/wiki/Geometric_object" class="mw-redirect" title="Geometric object">geometric object</a>.<sup id="cite_ref-Kendall_1-1" class="reference"><a href="#cite_note-Kendall-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. </p><p>Many two-dimensional geometric shapes can be defined by a set of <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> or <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> and <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called <a href="/wiki/Polygon" title="Polygon">polygons</a> and include <a href="/wiki/Triangle" title="Triangle">triangles</a>, <a href="/wiki/Square" title="Square">squares</a>, and <a href="/wiki/Pentagon" title="Pentagon">pentagons</a>. Other shapes may be bounded by <a href="/wiki/Curve" title="Curve">curves</a> such as the <a href="/wiki/Circle" title="Circle">circle</a> or the <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>. Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> enclosed by those lines, as well as the resulting interior points. Such shapes are called <a href="/wiki/Polyhedron" title="Polyhedron">polyhedrons</a> and include <a href="/wiki/Cube" title="Cube">cubes</a> as well as <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a> such as <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedrons</a>. Other three-dimensional shapes may be bounded by curved surfaces, such as the <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> and the <a href="/wiki/Sphere" title="Sphere">sphere</a>. </p><p>A shape is said to be <a href="/wiki/Convex_polytope" title="Convex polytope">convex</a> if all of the points on a line segment between any two of its points are also part of the shape. </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are multiple ways to compare the shapes of two objects: </p> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a>: Two objects are <i>congruent</i> if one can be transformed into the other by a sequence of rotations, translations, and/or reflections.</li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a>: Two objects are <i>similar</i> if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reflections.</li> <li><a href="/wiki/Homotopy#Isotopy" title="Homotopy">Isotopy</a>: Two objects are <i>isotopic</i> if one can be transformed into the other by a sequence of <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformations</a> that do not tear the object or put holes in it.</li></ul> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Similar-geometric-shapes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Similar-geometric-shapes.svg/300px-Similar-geometric-shapes.svg.png" decoding="async" width="300" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Similar-geometric-shapes.svg/450px-Similar-geometric-shapes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Similar-geometric-shapes.svg/600px-Similar-geometric-shapes.svg.png 2x" data-file-width="936" data-file-height="648" /></a><figcaption>Figures shown in the same color have the same shape as each other and are said to be similar.</figcaption></figure> <p>Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters "<b>b</b>" and "<b>d</b>" are a reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere. <a href="/wiki/Procrustes_analysis" title="Procrustes analysis">Procrustes analysis</a> is used in many sciences to determine whether or not two objects have the same shape, or to measure the difference between two shapes. In advanced mathematics, <a href="/wiki/Quasi-isometry" title="Quasi-isometry">quasi-isometry</a> can be used as a criterion to state that two shapes are approximately the same. </p><p>Simple shapes can often be classified into basic <a href="/wiki/Geometry" title="Geometry">geometric</a> objects such as a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a>, a <a href="/wiki/Curve" title="Curve">curve</a>, a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, a <a href="/wiki/Plane_figure" class="mw-redirect" title="Plane figure">plane figure</a> (e.g. <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> or <a href="/wiki/Circle" title="Circle">circle</a>), or a solid figure (e.g. <a href="/wiki/Cube" title="Cube">cube</a> or <a href="/wiki/Sphere" title="Sphere">sphere</a>). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, or as <a href="/wiki/Fractal" title="Fractal">fractals</a>. </p><p>Some common shapes include: <a href="/wiki/Circle" title="Circle">Circle</a>, <a href="/wiki/Square" title="Square">Square</a>, <a href="/wiki/Triangle" title="Triangle">Triangle</a>, <a href="/wiki/Rectangle" title="Rectangle">Rectangle</a>, <a href="/wiki/Oval" title="Oval">Oval</a>, <a href="/wiki/Star_(polygon)" class="mw-redirect" title="Star (polygon)">Star (polygon)</a>, <a href="/wiki/Rhombus" title="Rhombus">Rhombus</a>, <a href="/wiki/Semicircle" title="Semicircle">Semicircle</a>. Regular polygons starting at pentagon follow the naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See <a href="/wiki/Polygon" title="Polygon"> polygon</a> </p> <div class="mw-heading mw-heading2"><h2 id="Equivalence_of_shapes">Equivalence of shapes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=4" title="Edit section: Equivalence of shapes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In geometry, two subsets of a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> have the same shape if one can be transformed to the other by a combination of <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>, <a href="/wiki/Rotation" title="Rotation">rotations</a> (together also called <a href="/wiki/Rigid_transformation" title="Rigid transformation">rigid transformations</a>), and <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">uniform scalings</a>. In other words, the <i>shape</i> of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, and accordingly a precise mathematical definition of the notion of shape can be given as being an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of subsets of a Euclidean space having the same shape. </p><p>Mathematician and statistician <a href="/wiki/David_George_Kendall" title="David George Kendall">David George Kendall</a> writes:<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> </p> <blockquote><p>In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale<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> and rotational effects are filtered out from an object.’</p></blockquote> <p>Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on the size and placement in space of the object. For instance, a "<b><small>d</small></b>" and a "<b><big>p</big></b>" have the same shape, as they can be perfectly superimposed if the "<b><small>d</small></b>" is translated to the right by a given distance, rotated upside down and magnified by a given factor (see <a href="/wiki/Procrustes_superimposition" class="mw-redirect" title="Procrustes superimposition">Procrustes superimposition</a> for details). However, a <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a> could be called a different shape. For instance, a "<b><big>b</big></b>" and a "<b><big>p</big></b>" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if the object is scaled non-uniformly. For example, a <a href="/wiki/Sphere" title="Sphere">sphere</a> becomes an <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> when scaled differently in the vertical and horizontal directions. In other words, preserving axes of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object. </p> <div class="mw-heading mw-heading3"><h3 id="Congruence_and_similarity">Congruence and similarity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=5" title="Edit section: Congruence and similarity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence (geometry)</a> and <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity (geometry)</a></div> <p>Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a>. An object is therefore congruent to its <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a> (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size. </p><p>Objects that have the same shape or mirror image shapes are called <a href="/wiki/Geometrically_similar" class="mw-redirect" title="Geometrically similar">geometrically similar</a>, whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size. </p> <div class="mw-heading mw-heading3"><h3 id="Homeomorphism">Homeomorphism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=6" title="Edit section: Homeomorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Homeomorphism" title="Homeomorphism">Homeomorphism</a></div> <p>A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. </p><p>One way of modeling non-rigid movements is by <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> and a <a href="/wiki/Circle" title="Circle">circle</a> are homeomorphic to each other, but a <a href="/wiki/Sphere" title="Sphere">sphere</a> and a <a href="/wiki/Torus" title="Torus">donut</a> are not. An often-repeated <a href="/wiki/Mathematical_joke" title="Mathematical joke">mathematical joke</a> is that topologists cannot tell their coffee cup from their donut,<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> since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. </p><p>A described shape has external lines that you can see and make up the shape. If you were putting your coordinates on a coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has a outline and boundary so you can see it and is not just regular dots on a regular paper. </p> <div class="mw-heading mw-heading3"><h3 id="Shape_analysis">Shape analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=7" title="Edit section: Shape analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Statistical_shape_analysis" title="Statistical shape analysis">Statistical shape analysis</a></div> <p>The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of <a href="/wiki/Statistical_shape_analysis" title="Statistical shape analysis">statistical shape analysis</a>. In particular, <a href="/wiki/Procrustes_analysis" title="Procrustes analysis">Procrustes analysis</a> is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example <a href="/wiki/Spectral_shape_analysis" title="Spectral shape analysis">Spectral shape analysis</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Similarity_classes">Similarity classes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=8" title="Edit section: Similarity classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All <a href="/wiki/Similar_triangles" class="mw-redirect" title="Similar triangles">similar triangles</a> have the same shape. These shapes can be classified using <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <span class="texhtml mvar" style="font-style:italic;">u</span>, <span class="texhtml mvar" style="font-style:italic;">v</span>, <span class="texhtml mvar" style="font-style:italic;">w</span> for the vertices, in a method advanced by J.A. Lester<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> and <a href="/wiki/Rafael_Artzy" title="Rafael Artzy">Rafael Artzy</a>. For example, an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> can be expressed by the complex numbers 0, 1, <span class="nowrap">(1 + i√3)/2</span> representing its vertices. Lester and Artzy call the ratio <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <mi>u</mi> <mo>−</mo> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b5496e0625704562c1045998778191dbcc5a63" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.267ex; height:5.176ex;" alt="{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}"></span> the <b>shape</b> of triangle <span class="texhtml">(<i>u</i>, <i>v</i>, <i>w</i>)</span>. Then the shape of the equilateral triangle is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>0</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mn>0</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f140050e037b44d1dbe337b4f799420b2854986" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:54.058ex; height:7.176ex;" alt="{\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.}"></span> For any <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</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 z\mapsto az+b,\quad a\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mi>z</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto az+b,\quad a\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef6e2ccd29e2b67fbdecbc79665f418ca70719f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.352ex; height:2.676ex;" alt="{\displaystyle z\mapsto az+b,\quad a\neq 0,}"></span>   a triangle is transformed but does not change its shape. Hence shape is an <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> of <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>. The shape <span class="texhtml"><i>p</i> = S(<i>u</i>,<i>v</i>,<i>w</i>)</span> depends on the order of the arguments of function S, but <a href="/wiki/Permutation" title="Permutation">permutations</a> lead to related values. For instance, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <mi>u</mi> <mo>−</mo> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>w</mi> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mi>u</mi> <mo>−</mo> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <mi>v</mi> <mo>−</mo> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121fabf2f592f39e0173598057e17f31c50d5213" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:51.321ex; height:5.176ex;" alt="{\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).}"></span> Also <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}=S(u,w,v).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{-1}=S(u,w,v).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a880779d1a9ad32617fb73c263840b3004cac15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.835ex; height:3.176ex;" alt="{\displaystyle p^{-1}=S(u,w,v).}"></span> Combining these permutations gives <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 S(v,w,u)=(1-p)^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(v,w,u)=(1-p)^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3646c5f937736d2dd3e4b56ae453f36249e42113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.558ex; height:3.176ex;" alt="{\displaystyle S(v,w,u)=(1-p)^{-1}.}"></span> Furthermore, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <mi>v</mi> <mo>−</mo> <mi>w</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c24f602cd480562e7b0acb360f276c029c6548" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:55.679ex; height:5.176ex;" alt="{\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).}"></span> These relations are "conversion rules" for shape of a triangle. </p><p>The shape of a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> is associated with two complex numbers <span class="texhtml mvar" style="font-style:italic;">p</span>, <span class="texhtml mvar" style="font-style:italic;">q</span>. If the quadrilateral has vertices <span class="texhtml"><i>u</i></span>, <span class="texhtml"><i>v</i></span>, <span class="texhtml"><i>w</i></span>, <span class="texhtml"><i>x</i></span>, then <span class="texhtml"><i>p</i> = S(<i>u</i>,<i>v</i>,<i>w</i>)</span> and <span class="texhtml"><i>q</i> = S(<i>v</i>,<i>w</i>,<i>x</i>)</span>. Artzy proves these propositions about quadrilateral shapes: </p> <ol><li>If <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=(1-q)^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(1-q)^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07c1e92aeb2a2c5751606daa59aae18e8807da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.219ex; height:3.176ex;" alt="{\displaystyle p=(1-q)^{-1},}"></span> then the quadrilateral is a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>.</li> <li>If a parallelogram has <span class="texhtml">| arg <i>p</i> | = | arg <i>q</i> |</span>, then it is a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a>.</li> <li>When <span class="texhtml"><i>p</i> = 1 + i</span> and <span class="texhtml"><i>q</i> = (1 + i)/2</span>, then the quadrilateral is <a href="/wiki/Square" title="Square">square</a>.</li> <li>If <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=r(1-q^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=r(1-q^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5236dc8cc0aa29ec8f8f00ba1e3a0c3cc30471f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.63ex; height:3.176ex;" alt="{\displaystyle p=r(1-q^{-1})}"></span> and <span class="texhtml">sgn <i>r</i> = sgn(Im <i>p</i>)</span>, then the quadrilateral is a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a>.</li></ol> <p>A <a href="/wiki/Polygon" title="Polygon">polygon</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 (z_{1},z_{2},...z_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z_{1},z_{2},...z_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993c2db7b948707d48cab297d6a8af814f4c33f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.549ex; height:2.843ex;" alt="{\displaystyle (z_{1},z_{2},...z_{n})}"></span> has a shape defined by <i>n</i> − 2 complex numbers <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 S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ff30a824655ade8d1b4dfc6f91a941cf37a524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.598ex; height:3.009ex;" alt="{\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.}"></span> The polygon bounds a <a href="/wiki/Convex_set" title="Convex set">convex set</a> when all these shape components have imaginary components of the same sign.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Human_perception_of_shapes">Human perception of shapes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Shape&action=edit&section=9" title="Edit section: Human perception of shapes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Human vision relies on a wide range of shape representations.<sup id="cite_ref-ShapeComp_7-0" class="reference"><a href="#cite_note-ShapeComp-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called <a href="/wiki/Geon_(psychology)" title="Geon (psychology)">geons</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Meanwhile, others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their <i>segmentability</i>, <i>compactness</i> and <i>spikiness</i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> When comparing shape similarity, however, at least 22 independent dimensions are needed to account for the way natural shapes vary. <sup id="cite_ref-ShapeComp_7-1" class="reference"><a href="#cite_note-ShapeComp-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>There is also clear evidence that shapes guide human <a href="/wiki/Visual_spatial_attention" title="Visual spatial attention">attention</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </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=Shape&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Glossary_of_shapes_with_metaphorical_names" title="Glossary of shapes with metaphorical names">Glossary of shapes with metaphorical names</a></li> <li><a href="/wiki/Lists_of_shapes" title="Lists of shapes">Lists of shapes</a></li> <li><a href="/wiki/Shape_factor_(disambiguation)" class="mw-redirect mw-disambig" title="Shape factor (disambiguation)">Shape factor</a></li> <li><a href="/wiki/Size" title="Size">Size</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew polygon</a></li> <li><a href="/wiki/Solid_geometry" title="Solid geometry">Solid geometry</a></li> <li><a href="/wiki/Region_(mathematics)" class="mw-redirect" title="Region (mathematics)">Region (mathematics)</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=Shape&action=edit&section=11" 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 mw-references-columns"><ol class="references"> <li id="cite_note-Kendall-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kendall_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kendall_1-1"><sup><i><b>b</b></i></sup></a></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="CITEREFKendall,_D.G.1984" class="citation journal cs1">Kendall, D.G. (1984). "Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces". <i>Bulletin of the London Mathematical Society</i>. <b>16</b> (2): 81–121. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fblms%2F16.2.81">10.1112/blms/16.2.81</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+London+Mathematical+Society&rft.atitle=Shape+Manifolds%2C+Procrustean+Metrics%2C+and+Complex+Projective+Spaces&rft.volume=16&rft.issue=2&rft.pages=81-121&rft.date=1984&rft_id=info%3Adoi%2F10.1112%2Fblms%2F16.2.81&rft.au=Kendall%2C+D.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" 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="CITEREFKendall,_D.G.1984" class="citation journal cs1">Kendall, D.G. (1984). <a rel="nofollow" class="external text" href="http://image.diku.dk/imagecanon/material/kendall-shapes.pdf">"Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the London Mathematical Society</i>. <b>16</b> (2): 81–121. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fblms%2F16.2.81">10.1112/blms/16.2.81</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+London+Mathematical+Society&rft.atitle=Shape+Manifolds%2C+Procrustean+Metrics%2C+and+Complex+Projective+Spaces&rft.volume=16&rft.issue=2&rft.pages=81-121&rft.date=1984&rft_id=info%3Adoi%2F10.1112%2Fblms%2F16.2.81&rft.au=Kendall%2C+D.G.&rft_id=http%3A%2F%2Fimage.diku.dk%2Fimagecanon%2Fmaterial%2Fkendall-shapes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Here, scale means only <a href="/wiki/Uniform_scaling" class="mw-redirect" title="Uniform scaling">uniform scaling</a>, as non-uniform scaling would change the shape of the object (e.g., it would turn a square into a rectangle).</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHubbardWest1995" class="citation book cs1">Hubbard, John H.; West, Beverly H. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SHBj2oaSALoC&q=%22coffee+cup%22+topologist+joke&pg=PA204"><i>Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems</i></a>. Texts in Applied Mathematics. Vol. 18. Springer. p. 204. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94377-0" title="Special:BookSources/978-0-387-94377-0"><bdi>978-0-387-94377-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Equations%3A+A+Dynamical+Systems+Approach.+Part+II%3A+Higher-Dimensional+Systems&rft.series=Texts+in+Applied+Mathematics&rft.pages=204&rft.pub=Springer&rft.date=1995&rft.isbn=978-0-387-94377-0&rft.aulast=Hubbard&rft.aufirst=John+H.&rft.au=West%2C+Beverly+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSHBj2oaSALoC%26q%3D%2522coffee%2Bcup%2522%2Btopologist%2Bjoke%26pg%3DPA204&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">J.A. Lester (1996) "Triangles I: Shapes", <i><a href="/wiki/Aequationes_Mathematicae" title="Aequationes Mathematicae">Aequationes Mathematicae</a></i> 52:30–54</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Rafael_Artzy" title="Rafael Artzy">Rafael Artzy</a> (1994) "Shapes of Polygons", <i>Journal of Geometry</i> 50(1–2):11–15</span> </li> <li id="cite_note-ShapeComp-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-ShapeComp_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ShapeComp_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorgensternHartmannSchmidtTiedemann2021" class="citation journal cs1">Morgenstern, Yaniv; Hartmann, Frieder; Schmidt, Filipp; Tiedemann, Henning; Prokott, Eugen; Maiello, Guido; Fleming, Roland (2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8195351">"An image-computable model of visual shape similarity"</a>. <i>PLOS Computational Biology</i>. <b>17</b> (6): 34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pcbi.1008981">10.1371/journal.pcbi.1008981</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8195351">8195351</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34061825">34061825</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+Computational+Biology&rft.atitle=An+image-computable+model+of+visual+shape+similarity&rft.volume=17&rft.issue=6&rft.pages=34&rft.date=2021&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8195351%23id-name%3DPMC&rft_id=info%3Apmid%2F34061825&rft_id=info%3Adoi%2F10.1371%2Fjournal.pcbi.1008981&rft.aulast=Morgenstern&rft.aufirst=Yaniv&rft.au=Hartmann%2C+Frieder&rft.au=Schmidt%2C+Filipp&rft.au=Tiedemann%2C+Henning&rft.au=Prokott%2C+Eugen&rft.au=Maiello%2C+Guido&rft.au=Fleming%2C+Roland&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8195351&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndreopoulosTsotsos2013" class="citation journal cs1">Andreopoulos, Alexander; Tsotsos, John K. (2013). "50 Years of object recognition: Directions forward". <i>Computer Vision and Image Understanding</i>. <b>117</b> (8): 827–891. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cviu.2013.04.005">10.1016/j.cviu.2013.04.005</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Vision+and+Image+Understanding&rft.atitle=50+Years+of+object+recognition%3A+Directions+forward&rft.volume=117&rft.issue=8&rft.pages=827-891&rft.date=2013&rft_id=info%3Adoi%2F10.1016%2Fj.cviu.2013.04.005&rft.aulast=Andreopoulos&rft.aufirst=Alexander&rft.au=Tsotsos%2C+John+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Marr, D., & Nishihara, H. (1978). Representation and recognition of the spatial organization of three-dimensional shapes. Proceedings of the Royal Society of London, 200, 269–294.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuang2020" class="citation journal cs1">Huang, Liqiang (2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7405702">"Space of preattentive shape features"</a>. <i>Journal of Vision</i>. <b>20</b> (4): 10. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1167%2Fjov.20.4.10">10.1167/jov.20.4.10</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7405702">7405702</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32315405">32315405</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+Vision&rft.atitle=Space+of+preattentive+shape+features&rft.volume=20&rft.issue=4&rft.pages=10&rft.date=2020&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7405702%23id-name%3DPMC&rft_id=info%3Apmid%2F32315405&rft_id=info%3Adoi%2F10.1167%2Fjov.20.4.10&rft.aulast=Huang&rft.aufirst=Liqiang&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7405702&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexanderSchmidtZelinsky2014" class="citation journal cs1">Alexander, R. G.; Schmidt, J.; Zelinsky, G.Z. (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4500174">"Are summary statistics enough? Evidence for the importance of shape in guiding visual search"</a>. <i>Visual Cognition</i>. <b>22</b> (3–4): 595–609. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F13506285.2014.890989">10.1080/13506285.2014.890989</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4500174">4500174</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26180505">26180505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Visual+Cognition&rft.atitle=Are+summary+statistics+enough%3F+Evidence+for+the+importance+of+shape+in+guiding+visual+search.&rft.volume=22&rft.issue=3%E2%80%934&rft.pages=595-609&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4500174%23id-name%3DPMC&rft_id=info%3Apmid%2F26180505&rft_id=info%3Adoi%2F10.1080%2F13506285.2014.890989&rft.aulast=Alexander&rft.aufirst=R.+G.&rft.au=Schmidt%2C+J.&rft.au=Zelinsky%2C+G.Z.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4500174&rfr_id=info%3Asid%2Fen.wikipedia.org%3AShape" class="Z3988"></span></span> </li> </ol></div></div> <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=Shape&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/24px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/32px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> The dictionary definition of <a href="https://en.wiktionary.org/wiki/Special:Search/shape" class="extiw" title="wiktionary:Special:Search/shape"><i>shape</i></a> at Wiktionary</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Shape&oldid=1252373205">https://en.wikipedia.org/w/index.php?title=Shape&oldid=1252373205</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Elementary_geometry" title="Category:Elementary geometry">Elementary geometry</a></li><li><a href="/wiki/Category:Geometric_shapes" title="Category:Geometric shapes">Geometric shapes</a></li><li><a href="/wiki/Category:Morphology" title="Category:Morphology">Morphology</a></li><li><a href="/wiki/Category:Structure" title="Category:Structure">Structure</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 October 2024, at 01:43<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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