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class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Measurement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Measurement</span> </div> </a> <ul id="toc-Measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Symmetry</span> </div> </a> <ul id="toc-Symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inscribed_and_circumscribed_circles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inscribed_and_circumscribed_circles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Inscribed and circumscribed circles</span> </div> </a> <ul id="toc-Inscribed_and_circumscribed_circles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Constructions</span> </div> </a> <button aria-controls="toc-Constructions-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 Constructions subsection</span> </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Coordinates_and_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coordinates_and_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Coordinates and equations</span> </div> </a> <ul id="toc-Coordinates_and_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compass_and_straightedge" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compass_and_straightedge"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Compass and straightedge</span> </div> </a> <ul id="toc-Compass_and_straightedge-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_topics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Related topics</span> </div> </a> <button aria-controls="toc-Related_topics-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 Related topics subsection</span> </button> <ul id="toc-Related_topics-sublist" class="vector-toc-list"> <li id="toc-Inscribed_squares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inscribed_squares"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Inscribed squares</span> </div> </a> <ul id="toc-Inscribed_squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area_and_quadrature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Area_and_quadrature"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Area and quadrature</span> </div> </a> <ul id="toc-Area_and_quadrature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tiling_and_packing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tiling_and_packing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Tiling and packing</span> </div> </a> <ul id="toc-Tiling_and_packing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Counting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Counting"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Counting</span> </div> </a> <ul id="toc-Counting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Euclidean_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-Euclidean_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Non-Euclidean geometry</span> </div> </a> <ul id="toc-Non-Euclidean_geometry-sublist" class="vector-toc-list"> </ul> </li> </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">6</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">7</span> <span>References</span> </div> </a> <ul id="toc-References-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" title="Table of Contents" > <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">Square</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 145 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-145" 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">145 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%97%D1%8D%D0%B1%D0%B3%D1%8A%D1%83%D0%B7%D1%8D%D0%BD%D0%B0%D1%82%D3%80%D1%8D" title="ЗэбгъузэнатӀэ – Kabardian" lang="kbd" hreflang="kbd" data-title="ЗэбгъузэнатӀэ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target"><span>Адыгэбзэ</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vierkant" title="Vierkant – Afrikaans" lang="af" hreflang="af" data-title="Vierkant" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Quadrat_(Geometrie)" title="Quadrat (Geometrie) – Alemannic" lang="gsw" hreflang="gsw" data-title="Quadrat (Geometrie)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9" title="مربع – Arabic" lang="ar" hreflang="ar" data-title="مربع" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Cuadrau" title="Cuadrau – Aragonese" lang="an" hreflang="an" data-title="Cuadrau" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A7%B0%E0%A7%8D%E0%A6%97%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="বৰ্গক্ষেত্ৰ – Assamese" lang="as" hreflang="as" data-title="বৰ্গক্ষেত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Cuadr%C3%A1u" title="Cuadráu – Asturian" lang="ast" hreflang="ast" data-title="Cuadráu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Pusi_k%27uchuni" title="Pusi k'uchuni – Aymara" lang="ay" hreflang="ay" data-title="Pusi k'uchuni" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Azerbaijani" lang="az" hreflang="az" data-title="Kvadrat" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D9%88%D8%B1%D8%A8%D8%B9" title="موربع – South Azerbaijani" lang="azb" hreflang="azb" data-title="موربع" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="বর্গক্ষেত্র – Bangla" lang="bn" hreflang="bn" data-title="বর্গক্ষেত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%A8ng-hong-h%C3%AAng" title="Chèng-hong-hêng – Minnan" lang="nan" hreflang="nan" data-title="Chèng-hong-hêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Bashkir" lang="ba" hreflang="ba" data-title="Квадрат" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Belarusian" lang="be" hreflang="be" data-title="Квадрат" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Квадрат" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Kwadrado" title="Kwadrado – Central Bikol" lang="bcl" hreflang="bcl" data-title="Kwadrado" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Bulgarian" lang="bg" hreflang="bg" data-title="Квадрат" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%98%E0%BD%82%E0%BE%B2%E0%BD%B4%E0%BC%8B%E0%BD%96%E0%BD%9E%E0%BD%B2%E0%BC%8B%E0%BD%81%E0%BC%8B%E0%BD%82%E0%BD%84%E0%BC%8B%E0%BC%8D" title="མགྲུ་བཞི་ཁ་གང་། – Tibetan" lang="bo" hreflang="bo" data-title="མགྲུ་བཞི་ཁ་གང་།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Bosnian" lang="bs" hreflang="bs" data-title="Kvadrat" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Karrez" title="Karrez – Breton" lang="br" hreflang="br" data-title="Karrez" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quadrat_(pol%C3%ADgon)" title="Quadrat (polígon) – Catalan" lang="ca" hreflang="ca" data-title="Quadrat (polígon)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%C4%83%D0%B2%D0%B0%D1%82%D0%BA%D0%B0%D0%BB" title="Тăваткал – Chuvash" lang="cv" hreflang="cv" data-title="Тăваткал" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C4%8Ctverec" title="Čtverec – Czech" lang="cs" hreflang="cs" data-title="Čtverec" 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/Skweya" title="Skweya – Shona" lang="sn" hreflang="sn" data-title="Skweya" 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/Sgw%C3%A2r" title="Sgwâr – Welsh" lang="cy" hreflang="cy" data-title="Sgwâr" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Danish" lang="da" hreflang="da" data-title="Kvadrat" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%83%D8%A7%D8%B1%D9%88" title="كارو – Moroccan Arabic" lang="ary" hreflang="ary" data-title="كارو" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Kvadr%C3%A1hta" title="Kvadráhta – Northern Sami" lang="se" hreflang="se" data-title="Kvadráhta" data-language-autonym="Davvisámegiella" data-language-local-name="Northern Sami" class="interlanguage-link-target"><span>Davvisámegiella</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quadrat" title="Quadrat – German" lang="de" hreflang="de" data-title="Quadrat" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Kwadrat" title="Kwadrat – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Kwadrat" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ruut" title="Ruut – Estonian" lang="et" hreflang="et" data-title="Ruut" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%AC%CE%B3%CF%89%CE%BD%CE%BF" title="Τετράγωνο – Greek" lang="el" hreflang="el" data-title="Τετράγωνο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Quadr%C3%AA_(giometr%C3%ACa)" title="Quadrê (giometrìa) – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Quadrê (giometrìa)" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuadrado" title="Cuadrado – Spanish" lang="es" hreflang="es" data-title="Cuadrado" 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/Kvadrato_(geometrio)" title="Kvadrato (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Kvadrato (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Karratu" title="Karratu – Basque" lang="eu" hreflang="eu" data-title="Karratu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9" title="مربع – Persian" lang="fa" hreflang="fa" data-title="مربع" 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/Carr%C3%A9" title="Carré – French" lang="fr" hreflang="fr" data-title="Carré" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cearn%C3%B3g" title="Cearnóg – Irish" lang="ga" hreflang="ga" data-title="Cearnóg" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cadrado" title="Cadrado – Galician" lang="gl" hreflang="gl" data-title="Cadrado" 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/%E6%96%B9%E5%BD%A2" title="方形 – Gan" lang="gan" hreflang="gan" data-title="方形" 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-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%9A%E0%AB%8B%E0%AA%B0%E0%AA%B8" title="ચોરસ – Gujarati" lang="gu" hreflang="gu" data-title="ચોરસ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Chang-f%C3%B4ng-h%C3%ACn" title="Chang-fông-hìn – Hakka Chinese" lang="hak" hreflang="hak" data-title="Chang-fông-hìn" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%82%AC%EA%B0%81%ED%98%95" title="정사각형 – Korean" lang="ko" hreflang="ko" data-title="정사각형" 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%94%D5%A1%D5%BC%D5%A1%D5%AF%D5%B8%D6%82%D5%BD%D5%AB" title="Քառակուսի – Armenian" lang="hy" hreflang="hy" data-title="Քառակուսի" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%BE%E0%A4%95%E0%A4%BE%E0%A4%B0" title="वर्गाकार – Hindi" lang="hi" hreflang="hi" data-title="वर्गाकार" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Kwadrat" title="Kwadrat – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Kwadrat" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Croatian" lang="hr" hreflang="hr" data-title="Kvadrat" 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/Quadrato_(geometrio)" title="Quadrato (geometrio) – Ido" lang="io" hreflang="io" data-title="Quadrato (geometrio)" 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/Persegi" title="Persegi – Indonesian" lang="id" hreflang="id" data-title="Persegi" 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/Quadrato" title="Quadrato – Interlingua" lang="ia" hreflang="ia" data-title="Quadrato" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-ik mw-list-item"><a href="https://ik.wikipedia.org/wiki/Ka%C5%8Bi%C4%A1allulik" title="Kaŋiġallulik – Inupiaq" lang="ik" hreflang="ik" data-title="Kaŋiġallulik" data-language-autonym="Iñupiatun" data-language-local-name="Inupiaq" class="interlanguage-link-target"><span>Iñupiatun</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Ossetic" lang="os" hreflang="os" data-title="Квадрат" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-square" title="I-square – Xhosa" lang="xh" hreflang="xh" data-title="I-square" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/IGcagcane" title="IGcagcane – Zulu" lang="zu" hreflang="zu" data-title="IGcagcane" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferningur" title="Ferningur – Icelandic" lang="is" hreflang="is" data-title="Ferningur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quadrato" title="Quadrato – Italian" lang="it" hreflang="it" data-title="Quadrato" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A8%D7%99%D7%91%D7%95%D7%A2" title="ריבוע – Hebrew" lang="he" hreflang="he" data-title="ריבוע" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pesagi" title="Pesagi – Javanese" lang="jv" hreflang="jv" data-title="Pesagi" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%93%E1%83%A0%E1%83%90%E1%83%A2%E1%83%98" title="კვადრატი – Georgian" lang="ka" hreflang="ka" data-title="კვადრატი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B0%D1%80%D1%88%D1%8B" title="Шаршы – Kazakh" lang="kk" hreflang="kk" data-title="Шаршы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Pedrek" title="Pedrek – Cornish" lang="kw" hreflang="kw" data-title="Pedrek" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mraba" title="Mraba – Swahili" lang="sw" hreflang="sw" data-title="Mraba" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Kare" title="Kare – Haitian Creole" lang="ht" hreflang="ht" data-title="Kare" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/%C3%87ar%C3%A7ik" title="Çarçik – Kurdish" lang="ku" hreflang="ku" data-title="Çarçik" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Kyrgyz" lang="ky" hreflang="ky" data-title="Квадрат" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%AE%E0%BA%B9%E0%BA%9A%E0%BA%88%E0%BA%B1%E0%BA%94%E0%BA%95%E0%BA%B8%E0%BA%A5%E0%BA%B1%E0%BA%94" title="ຮູບຈັດຕຸລັດ – Lao" lang="lo" hreflang="lo" data-title="ຮູບຈັດຕຸລັດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Quadrum" title="Quadrum – Latin" lang="la" hreflang="la" data-title="Quadrum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kvadr%C4%81ts" title="Kvadrāts – Latvian" lang="lv" hreflang="lv" data-title="Kvadrāts" 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/Kvadratas" title="Kvadratas – Lithuanian" lang="lt" hreflang="lt" data-title="Kvadratas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Veerkantj" title="Veerkantj – Limburgish" lang="li" hreflang="li" data-title="Veerkantj" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Cuadro" title="Cuadro – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Cuadro" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Quadr%C3%A0t_(geometr%C3%ACa)" title="Quadràt (geometrìa) – Lombard" lang="lmo" hreflang="lmo" data-title="Quadràt (geometrìa)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gyzet" title="Négyzet – Hungarian" lang="hu" hreflang="hu" data-title="Négyzet" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Macedonian" lang="mk" hreflang="mk" data-title="Квадрат" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Efamira" title="Efamira – Malagasy" lang="mg" hreflang="mg" data-title="Efamira" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%AE%E0%B4%9A%E0%B4%A4%E0%B5%81%E0%B4%B0%E0%B4%82" title="സമചതുരം – Malayalam" lang="ml" hreflang="ml" data-title="സമചതുരം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%9A%E0%A5%8C%E0%A4%B0%E0%A4%B8" title="चौरस – Marathi" lang="mr" hreflang="mr" data-title="चौरस" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%93%E1%83%A0%E1%83%90%E1%83%A2%E1%83%98" title="კვადრატი – Mingrelian" lang="xmf" hreflang="xmf" data-title="კვადრატი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9" title="مربع – Egyptian Arabic" lang="arz" hreflang="arz" data-title="مربع" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Segi_empat_sama" title="Segi empat sama – Malay" lang="ms" hreflang="ms" data-title="Segi empat sama" 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-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%81%D1%8C" title="Квадратсь – Moksha" lang="mdf" hreflang="mdf" data-title="Квадратсь" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Mongolian" lang="mn" hreflang="mn" data-title="Квадрат" 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/Sikuea" title="Sikuea – Fijian" lang="fj" hreflang="fj" data-title="Sikuea" 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/Vierkant" title="Vierkant – Dutch" lang="nl" hreflang="nl" data-title="Vierkant" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97_(%E0%A4%86%E0%A4%95%E0%A4%BE%E0%A4%B0)" title="वर्ग (आकार) – Nepali" lang="ne" hreflang="ne" data-title="वर्ग (आकार)" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形 – Japanese" lang="ja" hreflang="ja" data-title="正方形" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kwadroot" title="Kwadroot – Northern Frisian" lang="frr" hreflang="frr" data-title="Kwadroot" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvadrat" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvadrat" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Cairat_(geometria)" title="Cairat (geometria) – Occitan" lang="oc" hreflang="oc" data-title="Cairat (geometria)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B9)" title="Квадрат (геометрий) – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Квадрат (геометрий)" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Arfakkuu" title="Arfakkuu – Oromo" lang="om" hreflang="om" data-title="Arfakkuu" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Uzbek" lang="uz" hreflang="uz" data-title="Kvadrat" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%B0%E0%A8%97" title="ਵਰਗ – Punjabi" lang="pa" hreflang="pa" data-title="ਵਰਗ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Quadrat_(Geometrie)" title="Quadrat (Geometrie) – Palatine German" lang="pfl" hreflang="pfl" data-title="Quadrat (Geometrie)" data-language-autonym="Pälzisch" data-language-local-name="Palatine German" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9" title="مربع – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مربع" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%85%D9%84%D9%88%D8%B1%D9%8A%DA%81_(%D9%85%DB%90%DA%86%D9%BE%D9%88%D9%87%D9%86%D9%87)" title="څلوريځ (مېچپوهنه) – Pashto" lang="ps" hreflang="ps" data-title="څلوريځ (مېچپوهنه)" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Skwier_(jaamichri)" title="Skwier (jaamichri) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Skwier (jaamichri)" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9E%B6%E1%9E%9A%E1%9F%89%E1%9F%81" title="ការ៉េ – Khmer" lang="km" hreflang="km" data-title="ការ៉េ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Quadr%C3%A0" title="Quadrà – Piedmontese" lang="pms" hreflang="pms" data-title="Quadrà" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kwadrat" title="Kwadrat – Polish" lang="pl" hreflang="pl" data-title="Kwadrat" 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/Quadrado" title="Quadrado – Portuguese" lang="pt" hreflang="pt" data-title="Quadrado" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Murabba" title="Murabba – Crimean Tatar" lang="crh" hreflang="crh" data-title="Murabba" data-language-autonym="Qırımtatarca" data-language-local-name="Crimean Tatar" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/P%C4%83trat" title="Pătrat – Romanian" lang="ro" hreflang="ro" data-title="Pătrat" 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/T%27asra" title="T'asra – Quechua" lang="qu" hreflang="qu" data-title="T'asra" 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-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82_(%D2%91%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F)" title="Квадрат (ґеометрія) – Rusyn" lang="rue" hreflang="rue" data-title="Квадрат (ґеометрія)" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Russian" lang="ru" hreflang="ru" data-title="Квадрат" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Square" title="Square – Scots" lang="sco" hreflang="sco" data-title="Square" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Katrori" title="Katrori – Albanian" lang="sq" hreflang="sq" data-title="Katrori" 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/Quatratu" title="Quatratu – Sicilian" lang="scn" hreflang="scn" data-title="Quatratu" 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/Square" title="Square – Simple English" lang="en-simple" hreflang="en-simple" data-title="Square" 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/%DA%86%D9%88%D8%B1%D8%B3" title="چورس – Sindhi" lang="sd" hreflang="sd" data-title="چورس" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C5%A0tvorec" title="Štvorec – Slovak" lang="sk" hreflang="sk" data-title="Štvorec" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvadrat_(geometrija)" title="Kvadrat (geometrija) – Slovenian" lang="sl" hreflang="sl" data-title="Kvadrat (geometrija)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Kwadrat" title="Kwadrat – Silesian" lang="szl" hreflang="szl" data-title="Kwadrat" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Labajibaarane" title="Labajibaarane – Somali" lang="so" hreflang="so" data-title="Labajibaarane" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%86%D9%88%D8%A7%D8%B1%DA%AF%DB%86%D8%B4%DB%95" title="چوارگۆشە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="چوارگۆشە" 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/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Serbian" lang="sr" hreflang="sr" data-title="Квадрат" 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/Kvadrat" title="Kvadrat – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kvadrat" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Pasagi_bener" title="Pasagi bener – Sundanese" lang="su" hreflang="su" data-title="Pasagi bener" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neli%C3%B6_(geometria)" title="Neliö (geometria) – Finnish" lang="fi" hreflang="fi" data-title="Neliö (geometria)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kvadrat" title="Kvadrat – Swedish" lang="sv" hreflang="sv" data-title="Kvadrat" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Parisukat" title="Parisukat – Tagalog" lang="tl" hreflang="tl" data-title="Parisukat" 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 href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%A4%E0%AF%81%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="சதுரம் – Tamil" lang="ta" hreflang="ta" data-title="சதுரம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Tatar" lang="tt" hreflang="tt" data-title="Квадрат" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%9A%E0%B0%A4%E0%B1%81%E0%B0%B0%E0%B0%B8%E0%B1%8D%E0%B0%B0%E0%B0%82" title="చతురస్రం – Telugu" lang="te" hreflang="te" data-title="చతురస్రం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B5%E0%B9%88%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1%E0%B8%88%E0%B8%B1%E0%B8%95%E0%B8%B8%E0%B8%A3%E0%B8%B1%E0%B8%AA" title="รูปสี่เหลี่ยมจัตุรัส – Thai" lang="th" hreflang="th" data-title="รูปสี่เหลี่ยมจัตุรัส" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Tajik" lang="tg" hreflang="tg" data-title="Квадрат" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kare" title="Kare – Turkish" lang="tr" hreflang="tr" data-title="Kare" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Квадрат – Ukrainian" lang="uk" hreflang="uk" data-title="Квадрат" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9_(%DB%81%D9%86%D8%AF%D8%B3%DB%81)" title="مربع (ہندسہ) – Urdu" lang="ur" hreflang="ur" data-title="مربع (ہندسہ)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Quadrato" title="Quadrato – Venetian" lang="vec" hreflang="vec" data-title="Quadrato" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_vu%C3%B4ng" title="Hình vuông – Vietnamese" lang="vi" hreflang="vi" data-title="Hình vuông" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Cw%C3%A5r%C3%A9" title="Cwåré – Walloon" lang="wa" hreflang="wa" data-title="Cwåré" data-language-autonym="Walon" data-language-local-name="Walloon" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="正方形" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Vierkant" title="Vierkant – West Flemish" lang="vls" hreflang="vls" data-title="Vierkant" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Kwatro_kantos" title="Kwatro kantos – Waray" lang="war" hreflang="war" data-title="Kwatro kantos" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形 – Wu" lang="wuu" hreflang="wuu" data-title="正方形" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%95%D7%95%D7%90%D7%93%D7%A8%D7%90%D7%98" title="קוואדראט – Yiddish" lang="yi" hreflang="yi" data-title="קוואדראט" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/Al%E1%BB%8D%CC%81pom%C3%A9j%C3%AC" title="Alọ́poméjì – Yoruba" lang="yo" hreflang="yo" data-title="Alọ́poméjì" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形 – Cantonese" lang="yue" hreflang="yue" data-title="正方形" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Kvadrots" title="Kvadrots – Samogitian" lang="sgs" hreflang="sgs" data-title="Kvadrots" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形 – Chinese" lang="zh" hreflang="zh" data-title="正方形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B4%B0%E2%B5%8E%E2%B4%BD%E2%B4%BD%E2%B5%93%E2%B5%A5" title="ⴰⵎⴽⴽⵓⵥ – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⴰⵎⴽⴽⵓⵥ" 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							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">This is the <a href="/wiki/Wikipedia:Pending_changes" title="Wikipedia:Pending changes">latest accepted revision</a>, <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:Log&type=review&page=Square">reviewed</a> on <i>31 March 2025</i>.</div></div><div tabindex="0"></div></div></div></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">Shape with four equal sides and angles</div> <p class="mw-empty-elt"> </p> <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">This article is about the shape. For other uses, see <a href="/wiki/Square_(disambiguation)" class="mw-disambig" title="Square (disambiguation)">Square (disambiguation)</a>.</div> <div class="skin-invert-image"><style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Square</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Regular_polygon_4_annotated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Regular_polygon_4_annotated.svg/250px-Regular_polygon_4_annotated.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Regular_polygon_4_annotated.svg/330px-Regular_polygon_4_annotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Regular_polygon_4_annotated.svg/500px-Regular_polygon_4_annotated.svg.png 2x" data-file-width="515" data-file-height="515" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li><a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a></li> <li><a href="/wiki/Hypercube" title="Hypercube">hypercube</a></li> <li><a href="/wiki/Cross-polytope" title="Cross-polytope">cross-polytope</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a> and <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Dihedral_group_of_order_8" title="Dihedral group of order 8">order-8 dihedral</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Area" title="Area">Area</a></th><td class="infobox-data">side<sup>2</sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Internal_angle" class="mw-redirect" title="Internal angle">Internal angle</a> (<a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>)</th><td class="infobox-data"><span class="texhtml mvar" style="font-style:italic;">π</span>/2 (90°)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Perimeter" title="Perimeter">Perimeter</a></th><td class="infobox-data">4 · side</td></tr></tbody></table></div> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>square</b> is a <a href="/wiki/Regular_polygon" title="Regular polygon">regular</a> <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a>. It has four straight sides of equal length and four equal <a href="/wiki/Angle" title="Angle">angles</a>. Squares are special cases of <a href="/wiki/Rectangle" title="Rectangle">rectangles</a>, which have four equal angles, and of <a href="/wiki/Rhombus" title="Rhombus">rhombuses</a>, which have four equal sides. As with all rectangles, a square's angles are <a href="/wiki/Right_angle" title="Right angle">right angles</a> (90 <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>, or <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>/2 <a href="/wiki/Radian" title="Radian">radians</a>), making adjacent sides <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>. The <a href="/wiki/Area" title="Area">area</a> of a square is the side length multiplied by itself, and so in <a href="/wiki/Algebra" title="Algebra">algebra</a>, multiplying a number by itself is called <a href="/wiki/Square_(algebra)" title="Square (algebra)">squaring</a>. </p><p>Equal squares can tile the plane edge-to-edge in the <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a>. Square tilings are ubiquitous in <a href="/wiki/Tile" title="Tile">tiled</a> floors and walls, <a href="/wiki/Graph_paper" title="Graph paper">graph paper</a>, image <a href="/wiki/Pixel" title="Pixel">pixels</a>, and <a href="/wiki/Game_board" title="Game board">game boards</a>. Square shapes are also often seen in building <a href="/wiki/Floor_plan" title="Floor plan">floor plans</a>, <a href="/wiki/Origami_paper" title="Origami paper">origami paper</a>, food servings, in <a href="/wiki/Graphic_design" title="Graphic design">graphic design</a> and <a href="/wiki/Heraldry" title="Heraldry">heraldry</a>, and in instant photos and fine art. </p><p>The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a> by <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>, now known to be impossible. Squares can be inscribed in any smooth or convex curve such as a circle or triangle, but it remains unsolved <a href="/wiki/Inscribed_square_problem" title="Inscribed square problem">whether a square can be inscribed in every simple closed curve</a>. Several problems of <a href="/wiki/Squaring_the_square" title="Squaring the square">squaring the square</a> involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes. </p><p>Squares can be constructed by <a href="/wiki/Straightedge_and_compass" class="mw-redirect" title="Straightedge and compass">straightedge and compass</a>, through their <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, or by repeated multiplication by <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. They form the <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">metric balls</a> for <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab geometry</a> and <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a>, two forms of non-Euclidean geometry. Although <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a> and <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> both lack polygons with four equal sides and right angles, they have square-like regular polygons with four sides and other angles, or with right angles and different numbers of sides. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions_and_characterizations">Definitions and characterizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=1" title="Edit section: Definitions and characterizations"><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:The_square_among_a_family_of_rectangles_or_rhombuses.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/The_square_among_a_family_of_rectangles_or_rhombuses.png/280px-The_square_among_a_family_of_rectangles_or_rhombuses.png" decoding="async" width="280" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/The_square_among_a_family_of_rectangles_or_rhombuses.png/420px-The_square_among_a_family_of_rectangles_or_rhombuses.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/The_square_among_a_family_of_rectangles_or_rhombuses.png/560px-The_square_among_a_family_of_rectangles_or_rhombuses.png 2x" data-file-width="1716" data-file-height="956" /></a><figcaption>Among rectangles (top row), the square is the shape with equal sides (blue, middle). Among rhombuses (bottom row), the square is the shape with right angles (blue, middle).</figcaption></figure> <p>Squares can be defined or characterized in many equivalent ways. If a <a href="/wiki/Polygon" title="Polygon">polygon</a> in the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> satisfies any one of the following criteria, it satisfies all of them: </p> <ul><li>A square is a polygon with four equal sides and four <a href="/wiki/Right_angle" title="Right angle">right angles</a>; that is, it is a quadrilateral that is both a rhombus and a rectangle<sup id="cite_ref-zalgri_1-0" class="reference"><a href="#cite_note-zalgri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>A square is a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> with four equal sides.<sup id="cite_ref-zalgri_1-1" class="reference"><a href="#cite_note-zalgri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>A square is a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a> with a right angle between a pair of adjacent sides.<sup id="cite_ref-zalgri_1-2" class="reference"><a href="#cite_note-zalgri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>A square is a rhombus with all angles equal.<sup id="cite_ref-zalgri_1-3" class="reference"><a href="#cite_note-zalgri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>A square is a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> with one right angle and two adjacent equal sides.<sup id="cite_ref-zalgri_1-4" class="reference"><a href="#cite_note-zalgri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>A square is a quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other. That is, it is a rhombus with equal diagonals.<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></li> <li>A square is a quadrilateral with successive sides <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></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 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>, <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/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> whose area is<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> <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 A={\frac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7888d4e7ff6047da0e9ee2a6b2e160b3c3d34f3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.487ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2}).}" /></span></li></ul> <p>Squares are the only <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> whose <a href="/wiki/Internal_and_external_angle" class="mw-redirect" title="Internal and external angle">internal angle</a>, <a href="/wiki/Central_angle#Central_angle_of_a_regular_polygon" title="Central angle">central angle</a>, and <a href="/wiki/Internal_and_external_angle" class="mw-redirect" title="Internal and external angle">external angle</a> are all equal (they are all right angles).<sup id="cite_ref-rich_4-0" class="reference"><a href="#cite_note-rich-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A square is a special case of a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a> (equal sides, opposite equal angles), a <a href="/wiki/Kite_(geometry)" title="Kite (geometry)">kite</a> (two pairs of adjacent equal sides), a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> (one pair of opposite sides parallel), a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> (all opposite sides parallel), a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> or tetragon (four-sided polygon), and a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> (opposite sides equal, right-angles),<sup id="cite_ref-zalgri_1-5" class="reference"><a href="#cite_note-zalgri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and therefore has all the properties of all these shapes, namely: </p> <ul><li>All four internal angles of a square are equal (each being 90°, a right angle).<sup id="cite_ref-rich_4-1" class="reference"><a href="#cite_note-rich-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-godsid_5-0" class="reference"><a href="#cite_note-godsid-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>The central angle of a square is equal to 90°.<sup id="cite_ref-rich_4-2" class="reference"><a href="#cite_note-rich-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>The external angle of a square is equal to 90°.<sup id="cite_ref-rich_4-3" class="reference"><a href="#cite_note-rich-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>The diagonals of a square are equal and <a href="/wiki/Bisection" title="Bisection">bisect</a> each other, meeting at 90°.<sup id="cite_ref-godsid_5-1" class="reference"><a href="#cite_note-godsid-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>The diagonals of a square bisect its internal angles, forming <a href="/wiki/Angle#adjacent" title="Angle">adjacent angles</a> of 45°.<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></li> <li>All four sides of a square are equal.<sup id="cite_ref-FOOTNOTEGodfreySiddons1919135_7-0" class="reference"><a href="#cite_note-FOOTNOTEGodfreySiddons1919135-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>Opposite sides of a square are <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a>.<sup id="cite_ref-FOOTNOTESchorlingClarkCarter1935101_8-0" class="reference"><a href="#cite_note-FOOTNOTESchorlingClarkCarter1935101-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <p>All squares are <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> to each other, meaning they have the same shape.<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> One parameter (typically the length of a side or diagonal)<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> suffices to specify a square's size. Squares of the same size are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</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-heading3"><h3 id="Measurement">Measurement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=3" title="Edit section: Measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Five_Squared.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Five_Squared.svg/150px-Five_Squared.svg.png" decoding="async" width="150" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Five_Squared.svg/225px-Five_Squared.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Five_Squared.svg/300px-Five_Squared.svg.png 2x" data-file-width="600" data-file-height="770" /></a><figcaption>The area of a square is the product of the lengths of its sides.</figcaption></figure> <p>A square whose four sides have length <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> has <a href="/wiki/Perimeter" title="Perimeter">perimeter</a><sup id="cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen147_131]_12-0" class="reference"><a href="#cite_note-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen147_131]-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <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=4\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>4</mn> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=4\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d009f88863d47239430002ef235d8b3d8e603f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.976ex; height:2.176ex;" alt="{\displaystyle P=4\ell }" /></span> and <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> length <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 d={\sqrt {2}}\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {2}}\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f40925a7920f169927c5c417f05ca550cce3dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.382ex; height:3.009ex;" alt="{\displaystyle d={\sqrt {2}}\ell }" /></span>.<sup id="cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]_13-0" class="reference"><a href="#cite_note-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> (The <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>, appearing in this formula, is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>, meaning that it is not the ratio of any two <a href="/wiki/Integer" title="Integer">integers</a>. It is approximately equal to 1.414.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>) A square's <a href="/wiki/Area" title="Area">area</a> is<sup id="cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]_13-1" class="reference"><a href="#cite_note-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <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 A=\ell ^{2}={\tfrac {1}{2}}d^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\ell ^{2}={\tfrac {1}{2}}d^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b2898b5ce48dff04a24144f997c5e0d9fe0c0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.541ex; height:3.509ex;" alt="{\displaystyle A=\ell ^{2}={\tfrac {1}{2}}d^{2}.}" /></span> This formula for the area of a square as the second power of its side length led to the use of the term <i><a href="/wiki/Square_(algebra)" title="Square (algebra)">squaring</a></i> to mean raising any number to the second power.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Reversing this relation, the side length of a square of a given area is the <a href="/wiki/Square_root" title="Square root">square root</a> of the area. Squaring an integer, or taking the area of a square with integer sides, results in a <a href="/wiki/Square_number" title="Square number">square number</a>; these are <a href="/wiki/Figurate_number" title="Figurate number">figurate numbers</a> representing the numbers of points that can be arranged into a square grid.<sup id="cite_ref-FOOTNOTEConwayGuy199630–33,_38–40_16-0" class="reference"><a href="#cite_note-FOOTNOTEConwayGuy199630–33,_38–40-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an <a href="/wiki/Equable_shape" title="Equable shape">equable shape</a>. The only other equable integer rectangle is a three by six rectangle.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Because it is a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Indeed, if <i>A</i> and <i>P</i> are the area and perimeter enclosed by a quadrilateral, then the following <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a> holds: <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 16A\leq P^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mi>A</mi> <mo>≤</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16A\leq P^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd38306aee3d076f8ac44eaa889716b73d242b28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.042ex; height:2.843ex;" alt="{\displaystyle 16A\leq P^{2}}" /></span> with equality if and only if the quadrilateral is a square.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEAlsinaNelsen2020187Theorem_9.2.2_20-0" class="reference"><a href="#cite_note-FOOTNOTEAlsinaNelsen2020187Theorem_9.2.2-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=4" title="Edit section: Symmetry"><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/Symmetry_group_of_a_square" class="mw-redirect" title="Symmetry group of a square">Symmetry group of a square</a></div> <p>The square is the most symmetrical of the quadrilaterals.<sup id="cite_ref-berger_21-0" class="reference"><a href="#cite_note-berger-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Eight <a href="/wiki/Rigid_transformation" title="Rigid transformation">rigid transformations</a> of the plane take the square to itself:<sup id="cite_ref-miller_22-0" class="reference"><a href="#cite_note-miller-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.3emem;"><style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tnone"><div class="thumbinner multiimageinner" style="width:736px;max-width:736px"><div class="trow"><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_I.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Square_symmetry_%E2%80%93_I.png/180px-Square_symmetry_%E2%80%93_I.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Square_symmetry_%E2%80%93_I.png/270px-Square_symmetry_%E2%80%93_I.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Square_symmetry_%E2%80%93_I.png/360px-Square_symmetry_%E2%80%93_I.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">The square's initial position <br /> (the <a href="/wiki/Identity_transformation" class="mw-redirect" title="Identity transformation">identity transformation</a>)</div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_R1.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Square_symmetry_%E2%80%93_R1.png/180px-Square_symmetry_%E2%80%93_R1.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Square_symmetry_%E2%80%93_R1.png/270px-Square_symmetry_%E2%80%93_R1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Square_symmetry_%E2%80%93_R1.png/360px-Square_symmetry_%E2%80%93_R1.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption"><a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">Rotation</a> by 90° anticlockwise</div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_R2.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Square_symmetry_%E2%80%93_R2.png/180px-Square_symmetry_%E2%80%93_R2.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Square_symmetry_%E2%80%93_R2.png/270px-Square_symmetry_%E2%80%93_R2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Square_symmetry_%E2%80%93_R2.png/360px-Square_symmetry_%E2%80%93_R2.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">Rotation by 180°</div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_R3.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Square_symmetry_%E2%80%93_R3.png/250px-Square_symmetry_%E2%80%93_R3.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Square_symmetry_%E2%80%93_R3.png/330px-Square_symmetry_%E2%80%93_R3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Square_symmetry_%E2%80%93_R3.png/500px-Square_symmetry_%E2%80%93_R3.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">Rotation by 270°</div></div></div><div class="trow"><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_D1.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Square_symmetry_%E2%80%93_D1.png/180px-Square_symmetry_%E2%80%93_D1.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Square_symmetry_%E2%80%93_D1.png/270px-Square_symmetry_%E2%80%93_D1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Square_symmetry_%E2%80%93_D1.png/360px-Square_symmetry_%E2%80%93_D1.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">Diagonal NW–SE <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a></div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_H.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Square_symmetry_%E2%80%93_H.png/250px-Square_symmetry_%E2%80%93_H.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Square_symmetry_%E2%80%93_H.png/330px-Square_symmetry_%E2%80%93_H.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Square_symmetry_%E2%80%93_H.png/500px-Square_symmetry_%E2%80%93_H.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">Horizontal reflection</div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_D2.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Square_symmetry_%E2%80%93_D2.png/180px-Square_symmetry_%E2%80%93_D2.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Square_symmetry_%E2%80%93_D2.png/270px-Square_symmetry_%E2%80%93_D2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Square_symmetry_%E2%80%93_D2.png/360px-Square_symmetry_%E2%80%93_D2.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">Diagonal NE–SW reflection</div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="border:none"><span typeof="mw:File"><a href="/wiki/File:Square_symmetry_%E2%80%93_V.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Square_symmetry_%E2%80%93_V.png/250px-Square_symmetry_%E2%80%93_V.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Square_symmetry_%E2%80%93_V.png/330px-Square_symmetry_%E2%80%93_V.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Square_symmetry_%E2%80%93_V.png/500px-Square_symmetry_%E2%80%93_V.png 2x" data-file-width="1538" data-file-height="1538" /></a></span></div><div class="thumbcaption">Vertical reflection</div></div></div></div></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Quadrilateral_symmetries.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Quadrilateral_symmetries.svg/250px-Quadrilateral_symmetries.svg.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Quadrilateral_symmetries.svg/330px-Quadrilateral_symmetries.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Quadrilateral_symmetries.svg/500px-Quadrilateral_symmetries.svg.png 2x" data-file-width="684" data-file-height="855" /></a><figcaption>The axes of reflection symmetry and centers of rotation symmetry of a square (top), rectangle and rhombus (center), <a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoid</a>, kite, and parallelogram (bottom)</figcaption></figure> <p>For an axis-parallel square centered at the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>, each of these symmetries acts by a combination of negating and swapping the <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinates</a> of points.<sup id="cite_ref-ers_23-0" class="reference"><a href="#cite_note-ers-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> These symmetries permute the eight isosceles triangles between the half-edges and the square's center (which stays in place); any of these triangles can be taken as the <a href="/wiki/Fundamental_region" class="mw-redirect" title="Fundamental region">fundamental region</a> of the transformations.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Each two vertices, each two edges, and each two half-edges are mapped one to the other by at least one of the symmetries. Each pair of half-edges has exactly one symmetry that maps one to the other.<sup id="cite_ref-berger_21-1" class="reference"><a href="#cite_note-berger-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> All <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> also have these properties, which are expressed by saying that symmetries of a square and, more generally, a regular polygon act <a href="/wiki/Transitive_action" class="mw-redirect" title="Transitive action">transitively</a> on vertices and edges, and <a href="/wiki/Simply_transitively" class="mw-redirect" title="Simply transitively">simply transitively</a> on half-edges. </p><p>Combining any two of these transformations by performing one after the other continues to take the square to itself, and therefore produces another symmetry. Repeated rotation produces another rotation with the summed rotation angle. Two reflections with the same axis return to the identity transformation, while two reflections with different axes rotate the square. A rotation followed by a reflection, or vice versa, produces a different reflection. This <a href="/wiki/Function_composition" title="Function composition">composition operation</a> gives the eight symmetries of a square the mathematical structure of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, called the <i>group of the square</i> or the <i><a href="/wiki/Dihedral_group_of_order_8" title="Dihedral group of order 8">dihedral group of order eight</a></i>.<sup id="cite_ref-miller_22-1" class="reference"><a href="#cite_note-miller-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Other quadrilaterals, like the rectangle and rhombus, have only a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of these symmetries.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Perspective-3point.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Perspective-3point.svg/130px-Perspective-3point.svg.png" decoding="async" width="130" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Perspective-3point.svg/195px-Perspective-3point.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Perspective-3point.svg/260px-Perspective-3point.svg.png 2x" data-file-width="402" data-file-height="425" /></a><figcaption>Three-point perspective of a cube, showing perspective transformations of its six square faces into six different quadrilaterals</figcaption></figure> <p>The shape of a square, but not its size, is preserved by <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarities</a> of the plane. Other kinds of transformations of the plane can take squares to other kinds of quadrilateral. An <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> can take a square to any parallelogram, or vice versa; a <a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformation</a> can take a square to any <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a>, or vice versa. This implies that, when <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">viewed in perspective</a>, a square can look like any quadrilateral. A <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a> can take the vertices of a square (but not its edges) to the vertices of a <a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">harmonic quadrilateral</a>. </p><p>The <a href="/wiki/Wallpaper_group" title="Wallpaper group">wallpaper groups</a> are symmetry groups of two-dimensional repeating patterns. For many of these groups the basic unit of repetition (the unit cell of its <a href="/wiki/Period_lattice" class="mw-redirect" title="Period lattice">period lattice</a>) can be a square, and for three of these groups, p4, p4m, and p4g, it must be a square.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.3emem;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tnone"><div class="thumbinner multiimageinner" style="width:472px;max-width:472px"><div class="trow"><div class="tsingle" style="width:155px;max-width:155px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4-1.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Wallpaper_group-p4-1.jpg/153px-Wallpaper_group-p4-1.jpg" decoding="async" width="153" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Wallpaper_group-p4-1.jpg/230px-Wallpaper_group-p4-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Wallpaper_group-p4-1.jpg/306px-Wallpaper_group-p4-1.jpg 2x" data-file-width="1064" data-file-height="1070" /></a></span></div><div class="thumbcaption text-align-center">p4, Egyptian tomb ceiling</div></div><div class="tsingle" style="width:157px;max-width:157px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-1.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/155px-Wallpaper_group-p4m-1.jpg" decoding="async" width="155" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/233px-Wallpaper_group-p4m-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/310px-Wallpaper_group-p4m-1.jpg 2x" data-file-width="1065" data-file-height="1057" /></a></span></div><div class="thumbcaption text-align-center">p4m, Nineveh & Persia</div></div><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4g-2.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/152px-Wallpaper_group-p4g-2.jpg" decoding="async" width="152" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/228px-Wallpaper_group-p4g-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/304px-Wallpaper_group-p4g-2.jpg 2x" data-file-width="1054" data-file-height="1069" /></a></span></div><div class="thumbcaption text-align-center">p4g, China</div></div></div><div class="trow" style="display:flow-root"><div class="thumbcaption" style="text-align:center"><a href="/wiki/Wallpaper_group" title="Wallpaper group">Wallpaper groups</a> of tilings from <i><a href="/wiki/The_Grammar_of_Ornament" class="mw-redirect" title="The Grammar of Ornament">The Grammar of Ornament</a></i></div></div></div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Inscribed_and_circumscribed_circles">Inscribed and circumscribed circles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=5" title="Edit section: Inscribed and circumscribed circles"><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:Incircle_and_circumcircle_of_a_square.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Incircle_and_circumcircle_of_a_square.png/250px-Incircle_and_circumcircle_of_a_square.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Incircle_and_circumcircle_of_a_square.png/330px-Incircle_and_circumcircle_of_a_square.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Incircle_and_circumcircle_of_a_square.png/500px-Incircle_and_circumcircle_of_a_square.png 2x" data-file-width="1442" data-file-height="1448" /></a><figcaption>The <a href="/wiki/Inscribed_circle" class="mw-redirect" title="Inscribed circle">inscribed circle</a> (purple) and <a href="/wiki/Circumscribed_circle" title="Circumscribed circle">circumscribed circle</a> (red) of a square (black)</figcaption></figure> <p>The <a href="/wiki/Inscribed_circle" class="mw-redirect" title="Inscribed circle">inscribed circle</a> of a square is the largest circle that can fit inside that square. Its center is the center point of the square, and its radius (the <a href="/wiki/Inradius" class="mw-redirect" title="Inradius">inradius</a> of the square) is <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 r=\ell /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\ell /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1331e88b832075ac13224353f2270026726d58cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.442ex; height:2.843ex;" alt="{\displaystyle r=\ell /2}" /></span>. Because this circle touches all four sides of the square (at their midpoints), the square is a <a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a>. The <a href="/wiki/Circumscribed_circle" title="Circumscribed circle">circumscribed circle</a> of a square passes through all four vertices, making the square a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>. Its radius, the <a href="/wiki/Circumradius" class="mw-redirect" title="Circumradius">circumradius</a>, is <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 R=\ell /{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\ell /{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeccd53c48b9cd3cb48b7ead6a7301df92936103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.093ex; height:3.176ex;" alt="{\displaystyle R=\ell /{\sqrt {2}}}" /></span>.<sup id="cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen149_133]_28-0" class="reference"><a href="#cite_note-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen149_133]-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> If the inscribed circle of a square <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 ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}" /></span> has tangency points <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span> on <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 AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}" /></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 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> on <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 BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e0f24a49061dcd63874f7d81f395b5f38800f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle BC}" /></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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /></span> on <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 CD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9669379a3a9b8c55e7876c2371ccbc6e21b654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.691ex; height:2.176ex;" alt="{\displaystyle CD}" /></span>, and <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> on <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 DA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd3701b8cd2a436cf1beac3ea362a217e90911c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.667ex; height:2.176ex;" alt="{\displaystyle DA}" /></span>, then for any 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 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> on the inscribed circle,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><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 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>P</mi> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>P</mi> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>P</mi> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0062829ef063bf1eb190d32c0daff5b1386078c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.182ex; height:3.176ex;" alt="{\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}" /></span> 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 d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}" /></span> is the distance from an arbitrary point in the plane to the <span class="nowrap"><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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span>th</span> vertex of a square and <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is the <a href="/wiki/Circumradius" class="mw-redirect" title="Circumradius">circumradius</a> of the square, then<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><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 {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mn>3</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b679caaf0ba4e25df849f29246a0695183362837" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.854ex; height:8.009ex;" alt="{\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}" /></span> 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 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> and <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 d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}" /></span> are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then <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 d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605aef5c91fb204800787c51fd98e5859032aa43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.234ex; height:3.343ex;" alt="{\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}" /></span> and <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 d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab426f647805773fe69076743c8334eb2394231" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.942ex; height:3.343ex;" alt="{\displaystyle d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}" /></span> where <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is the circumradius of the square.<sup id="cite_ref-Mamuka_31-0" class="reference"><a href="#cite_note-Mamuka-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=6" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:232px;max-width:232px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Artesanal_tile_industry_(15323096610).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Artesanal_tile_industry_%2815323096610%29.jpg/250px-Artesanal_tile_industry_%2815323096610%29.jpg" decoding="async" width="230" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Artesanal_tile_industry_%2815323096610%29.jpg/500px-Artesanal_tile_industry_%2815323096610%29.jpg 1.5x" data-file-width="4592" data-file-height="3056" /></a></span></div><div class="thumbcaption">Square <a href="/wiki/Tile" title="Tile">tiles</a></div></div><div class="tsingle" style="width:156px;max-width:156px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Girl_with_a_Pearl_Earring_(pixelated).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Girl_with_a_Pearl_Earring_%28pixelated%29.jpg/250px-Girl_with_a_Pearl_Earring_%28pixelated%29.jpg" decoding="async" width="154" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Girl_with_a_Pearl_Earring_%28pixelated%29.jpg/330px-Girl_with_a_Pearl_Earring_%28pixelated%29.jpg 2x" data-file-width="4080" data-file-height="4061" /></a></span></div><div class="thumbcaption"><a href="/wiki/Pixelated" class="mw-redirect" title="Pixelated">Pixelated</a> <i><a href="/wiki/Girl_with_a_Pearl_Earring" title="Girl with a Pearl Earring">Girl with a Pearl Earring</a></i></div></div></div></div></div> <p>Squares are so well-established as the shape of <a href="/wiki/Tiles" class="mw-redirect" title="Tiles">tiles</a> that the <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a> word <a href="/wiki/Tessera" title="Tessera">tessera</a>, for a small tile as used in <a href="/wiki/Mosaic" title="Mosaic">mosaics</a>, comes from an ancient Greek word for the number four, referring to the four corners of a square tile.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Graph_paper" title="Graph paper">Graph paper</a>, preprinted with a <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a>, is widely used for <a href="/wiki/Data_visualization" class="mw-redirect" title="Data visualization">data visualization</a> using <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Pixel" title="Pixel">pixels</a> of <a href="/wiki/Bitmap_image" class="mw-redirect" title="Bitmap image">bitmap images</a>, as recorded by <a href="/wiki/Image_scanner" title="Image scanner">image scanners</a> and <a href="/wiki/Digital_camera" title="Digital camera">digital cameras</a> or displayed on <a href="/wiki/Electronic_visual_display" title="Electronic visual display">electronic visual displays</a>, conventionally lie at the intersections of a square grid, and are often considered as small squares, arranged in a square tiling.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Standard techniques for <a href="/wiki/Image_compression" title="Image compression">image compression</a> and <a href="/wiki/Video_compression" class="mw-redirect" title="Video compression">video compression</a>, including the <a href="/wiki/JPEG" title="JPEG">JPEG</a> format, are based on the subdivision of images into larger square blocks of pixels.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Quadtree" title="Quadtree">quadtree</a> data structure used in data compression and <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a> is based on the <a href="/wiki/Recursive" class="mw-redirect" title="Recursive">recursive</a> subdivision of squares into smaller squares.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg/220px-20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg/330px-20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg/440px-20240815_Site_of_Luoyang_City_from_Han_to_Wei_Dynasty_-_Site_of_the_Pagoda_of_Yongning_Temple_04.jpg 2x" data-file-width="1800" data-file-height="1200" /></a><figcaption>Site of the <a href="/wiki/Yongning_Pagoda" title="Yongning Pagoda">Yongning Pagoda</a></figcaption></figure> <p>Architectural structures from both ancient and modern cultures have featured a square floor plan, base, or footprint. Ancient examples include the <a href="/wiki/Egyptian_pyramids" title="Egyptian pyramids">Egyptian pyramids</a>,<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Mesoamerican_pyramids" title="Mesoamerican pyramids">Mesoamerican pyramids</a> such as those at <a href="/wiki/Teotihuacan" title="Teotihuacan">Teotihuacan</a>,<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Chogha_Zanbil" title="Chogha Zanbil">Chogha Zanbil</a> ziggurat in Iran,<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> the four-fold design of Persian walled gardens, said to model the four rivers of Paradise, and later structures inspired by their design such as the <a href="/wiki/Taj_Mahal" title="Taj Mahal">Taj Mahal</a> in India,<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> the square bases of Buddhist <a href="/wiki/Stupa" title="Stupa">stupas</a>,<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> and East Asian <a href="/wiki/Pagoda" title="Pagoda">pagodas</a>, buildings that symbolically face to the four points of the compass and reach to the heavens.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Norman <a href="/wiki/Keep" title="Keep">keeps</a> such as the <a href="/wiki/Tower_of_London" title="Tower of London">Tower of London</a> often take the form of a low square tower.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> In modern architecture, a majority of <a href="/wiki/Skyscraper" title="Skyscraper">skyscrapers</a> feature a square plan for pragmatic rather than aesthetic or symbolic reasons.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:352px;max-width:352px"><div class="trow"><div class="tsingle" style="width:173px;max-width:173px"><div class="thumbimage" style="height:172px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Mandalatibet_(cropped).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Mandalatibet_%28cropped%29.jpg/250px-Mandalatibet_%28cropped%29.jpg" decoding="async" width="171" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Mandalatibet_%28cropped%29.jpg/330px-Mandalatibet_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Mandalatibet_%28cropped%29.jpg/500px-Mandalatibet_%28cropped%29.jpg 2x" data-file-width="1470" data-file-height="1483" /></a></span></div><div class="thumbcaption">A Tibetan <a href="/wiki/Mandala" title="Mandala">mandala</a></div></div><div class="tsingle" style="width:175px;max-width:175px"><div class="thumbimage" style="height:172px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Piet_Mondrian,_1942_-_Broadway_Boogie_Woogie.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Piet_Mondrian%2C_1942_-_Broadway_Boogie_Woogie.jpg/173px-Piet_Mondrian%2C_1942_-_Broadway_Boogie_Woogie.jpg" decoding="async" width="173" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Piet_Mondrian%2C_1942_-_Broadway_Boogie_Woogie.jpg/260px-Piet_Mondrian%2C_1942_-_Broadway_Boogie_Woogie.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Piet_Mondrian%2C_1942_-_Broadway_Boogie_Woogie.jpg/346px-Piet_Mondrian%2C_1942_-_Broadway_Boogie_Woogie.jpg 2x" data-file-width="4104" data-file-height="4111" /></a></span></div><div class="thumbcaption"><i><a href="/wiki/Broadway_Boogie_Woogie" title="Broadway Boogie Woogie">Broadway Boogie Woogie</a></i>, <a href="/wiki/Piet_Mondrian" title="Piet Mondrian">Piet Mondrian</a></div></div></div></div></div> <p>The stylized nested squares of a Tibetan <a href="/wiki/Mandala" title="Mandala">mandala</a>, like the design of a stupa, function as a miniature model of the cosmos.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Some formats for film photography use a square <a href="/wiki/Aspect_ratio_(image)" title="Aspect ratio (image)">aspect ratio</a>, notably <a href="/wiki/Polaroid_camera" class="mw-redirect" title="Polaroid camera">Polaroid cameras</a>, <a href="/wiki/Medium_format" title="Medium format">medium format</a> cameras, and <a href="/wiki/Instamatic" title="Instamatic">Instamatic</a> cameras.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> Painters known for their frequent use of square frames and forms include <a href="/wiki/Josef_Albers" title="Josef Albers">Josef Albers</a>,<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Kazimir_Malevich" title="Kazimir Malevich">Kazimir Malevich</a><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Piet_Mondrian" title="Piet Mondrian">Piet Mondrian</a>.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Baseball_diamond" class="mw-redirect" title="Baseball diamond">Baseball diamonds</a><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Boxing_ring" title="Boxing ring">boxing rings</a> are square despite being named for other shapes.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> In the <a href="/wiki/Quadrille" title="Quadrille">quadrille</a> and <a href="/wiki/Square_dance" title="Square dance">square dance</a>, four couples form the sides of a square.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Samuel_Beckett" title="Samuel Beckett">Samuel Beckett</a>'s minimalist television play <i><a href="/wiki/Quad_(play)" title="Quad (play)">Quad</a></i>, four actors walk along the sides and diagonals of a square.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:473px;max-width:473px"><div class="trow"><div class="tsingle" style="width:156px;max-width:156px"><div class="thumbimage" style="height:150px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Chess_and_goose_game_board_MET_ES4614.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Chess_and_goose_game_board_MET_ES4614.jpg/154px-Chess_and_goose_game_board_MET_ES4614.jpg" decoding="async" width="154" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Chess_and_goose_game_board_MET_ES4614.jpg/231px-Chess_and_goose_game_board_MET_ES4614.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Chess_and_goose_game_board_MET_ES4614.jpg/308px-Chess_and_goose_game_board_MET_ES4614.jpg 2x" data-file-width="1748" data-file-height="1712" /></a></span></div><div class="thumbcaption">16th-century Indian <a href="/wiki/Chessboard" title="Chessboard">chessboard</a></div></div><div class="tsingle" style="width:157px;max-width:157px"><div class="thumbimage" style="height:150px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Stomachion.JPG" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Stomachion.JPG/155px-Stomachion.JPG" decoding="async" width="155" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Stomachion.JPG/233px-Stomachion.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/2/21/Stomachion.JPG 2x" data-file-width="270" data-file-height="263" /></a></span></div><div class="thumbcaption"><a href="/wiki/Ostomachion" title="Ostomachion">Ostomachion</a></div></div><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:150px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg/152px-Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg" decoding="async" width="152" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg/228px-Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg/304px-Horoskop_Johannette_Maria_zu_Wied_1615_img01.jpg 2x" data-file-width="704" data-file-height="700" /></a></span></div><div class="thumbcaption">1615 <a href="/wiki/Horoscope" title="Horoscope">horoscope</a></div></div></div></div></div> <p>The square <a href="/wiki/Go_board" class="mw-redirect" title="Go board">go board</a> is said to represent the earth, with the 361 crossings of its lines representing days of the year.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Chessboard" title="Chessboard">chessboard</a> inherited its square shape from a <a href="/wiki/Pachisi" title="Pachisi">pachisi</a>-like Indian race game and in turn passed it on to <a href="/wiki/Checkers" title="Checkers">checkers</a>.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> In two ancient games from <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a> and <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Ancient Egypt</a>, the <a href="/wiki/Royal_Game_of_Ur" title="Royal Game of Ur">Royal Game of Ur</a> and <a href="/wiki/Senet" title="Senet">Senet</a>, the game board itself is not square, but rectangular, subdivided into a grid of squares.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> The ancient Greek <a href="/wiki/Ostomachion" title="Ostomachion">Ostomachion</a> puzzle (according to some interpretations) involves rearranging the pieces of a square cut into smaller polygons, as does the Chinese <a href="/wiki/Tangram" title="Tangram">tangram</a>.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> Another set of puzzle pieces, the <a href="/wiki/Polyomino" title="Polyomino">polyominos</a>, are formed from squares glued edge-to-edge.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> Medieval and Renaissance <a href="/wiki/Horoscope" title="Horoscope">horoscopes</a> were arranged in a square format, across Europe, the Middle East, and China.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Other recreational uses of squares include the shape of <a href="/wiki/Origami" title="Origami">origami</a> paper,<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> and a common style of <a href="/wiki/Quilting" title="Quilting">quilting</a> involving the use of square quilt blocks.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:471px;max-width:471px"><div class="trow"><div class="tsingle" style="width:133px;max-width:133px"><div class="thumbimage" style="height:131px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:CHE_Vuadens_Flag.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/CHE_Vuadens_Flag.svg/250px-CHE_Vuadens_Flag.svg.png" decoding="async" width="131" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/CHE_Vuadens_Flag.svg/330px-CHE_Vuadens_Flag.svg.png 2x" data-file-width="750" data-file-height="750" /></a></span></div><div class="thumbcaption">Square flag of the municipality of <a href="/wiki/Vuadens" title="Vuadens">Vuadens</a>, based on the Swiss flag</div></div><div class="tsingle" style="width:133px;max-width:133px"><div class="thumbimage" style="height:131px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:QR_code_for_mobile_English_Wikipedia.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/QR_code_for_mobile_English_Wikipedia.svg/131px-QR_code_for_mobile_English_Wikipedia.svg.png" decoding="async" width="131" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/QR_code_for_mobile_English_Wikipedia.svg/197px-QR_code_for_mobile_English_Wikipedia.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/QR_code_for_mobile_English_Wikipedia.svg/262px-QR_code_for_mobile_English_Wikipedia.svg.png 2x" data-file-width="296" data-file-height="296" /></a></span></div><div class="thumbcaption"><a href="/wiki/QR_code" title="QR code">QR code</a> for the mobile English Wikipedia</div></div><div class="tsingle" style="width:199px;max-width:199px"><div class="thumbimage" style="height:131px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Waffles_(1).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Waffles_%281%29.jpg/250px-Waffles_%281%29.jpg" decoding="async" width="197" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Waffles_%281%29.jpg/330px-Waffles_%281%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Waffles_%281%29.jpg/500px-Waffles_%281%29.jpg 2x" data-file-width="2700" data-file-height="1800" /></a></span></div><div class="thumbcaption">Square <a href="/wiki/Waffle" title="Waffle">waffles</a></div></div></div></div></div> <p>Squares are a common element of <a href="/wiki/Graphic_design" title="Graphic design">graphic design</a>, used to give a sense of stability, symmetry, and order.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Heraldry" title="Heraldry">heraldry</a>, a <a href="/wiki/Canton_(heraldry)" title="Canton (heraldry)">canton</a> (a design element in the top left of a shield) is normally square, and a square flag is called a banner.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Flag_of_Switzerland" title="Flag of Switzerland">flag of Switzerland</a> is square, as are the <a href="/wiki/Flags_and_arms_of_cantons_of_Switzerland" title="Flags and arms of cantons of Switzerland">flags of the Swiss cantons</a>.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> <a href="/wiki/QR_code" title="QR code">QR codes</a> are square and feature prominent nested square alignment marks in three corners.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Robertson_screw" title="Robertson screw">Robertson screws</a> have a square drive socket.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Cracker_(food)" title="Cracker (food)">Crackers</a> and sliced <a href="/wiki/Cheese" title="Cheese">cheese</a> are often square,<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> as are <a href="/wiki/Waffle" title="Waffle">waffles</a>.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> Square foods named for their square shapes include <a href="/wiki/Caramel_square" class="mw-redirect" title="Caramel square">caramel squares</a>, <a href="/wiki/Date_square" title="Date square">date squares</a>, <a href="/wiki/Lemon_square" class="mw-redirect" title="Lemon square">lemon squares</a>,<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Square_sausage" class="mw-redirect" title="Square sausage">square sausage</a>,<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Carr%C3%A9_de_l%27Est" title="Carré de l'Est">Carré de l'Est</a> cheese.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=7" title="Edit section: Constructions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Coordinates_and_equations">Coordinates and equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=8" title="Edit section: Coordinates and equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Square_equation_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Square_equation_plot.svg/220px-Square_equation_plot.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Square_equation_plot.svg/330px-Square_equation_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Square_equation_plot.svg/440px-Square_equation_plot.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption><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|=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|+|y|=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bbde6c572f1f7f2ba72ec57c43abe593a7bd78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.174ex; height:2.843ex;" alt="{\displaystyle |x|+|y|=2}" /></span> plotted on <i><a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a></i>.</figcaption></figure> <p>A <a href="/wiki/Unit_square" title="Unit square">unit square</a> is a square of side length one. Often it is represented in <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinates</a> as the square enclosing the points <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></span> that have <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 0\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30810e06ad49f3a837bd2193d4392eda1f74e7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.852ex; height:2.343ex;" alt="{\displaystyle 0\leq x\leq 1}" /></span> and <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 0\leq y\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq y\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877a791a23880968c5e285359bbf496b626a4e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.677ex; height:2.509ex;" alt="{\displaystyle 0\leq y\leq 1}" /></span>. Its vertices are the four points that have 0 or 1 in each of their coordinates.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p><p>An axis-parallel square with its center at the 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 (x_{c},y_{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{c},y_{c})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16480a7cb48365325f8ebc21f15ee26287f11eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.201ex; height:2.843ex;" alt="{\displaystyle (x_{c},y_{c})}" /></span> and sides of length <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 2r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb337de6fbd1ab48176084f9c4534b8c55847042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle 2r}" /></span> (where <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> is the inradius, half the side length) has vertices at the four points <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_{c}\pm r,y_{c}\pm r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>±</mo> <mi>r</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>±</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{c}\pm r,y_{c}\pm r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3f5253f9a9827841f6ae03b09e40208e785c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.979ex; height:2.843ex;" alt="{\displaystyle (x_{c}\pm r,y_{c}\pm r)}" /></span>. Its interior consists of the points <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></span> with <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 \max(|x-x_{c}|,|y-y_{c}|)<r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(|x-x_{c}|,|y-y_{c}|)<r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d53609cf0aae34908d9bf8805a6040444cfe925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.427ex; height:2.843ex;" alt="{\displaystyle \max(|x-x_{c}|,|y-y_{c}|)<r}" /></span>, and its boundary consists of the points with <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 \max(|x-x_{c}|,|y-y_{c}|)=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(|x-x_{c}|,|y-y_{c}|)=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b96bfdfa3d650bf9afb583799c52134524ff9425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.427ex; height:2.843ex;" alt="{\displaystyle \max(|x-x_{c}|,|y-y_{c}|)=r}" /></span>.<sup id="cite_ref-iobst_76-0" class="reference"><a href="#cite_note-iobst-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> </p><p>A diagonal square with its center at the 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 (x_{c},y_{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{c},y_{c})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16480a7cb48365325f8ebc21f15ee26287f11eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.201ex; height:2.843ex;" alt="{\displaystyle (x_{c},y_{c})}" /></span> and diagonal of length <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 2R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5e012ed1fbb85fd15e40e08a4f375e37650c4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.926ex; height:2.176ex;" alt="{\displaystyle 2R}" /></span> (where <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is the circumradius, half the diagonal) has vertices at the four points <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_{c}\pm R,y_{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>±</mo> <mi>R</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{c}\pm R,y_{c})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bf97effbce5920c74781ab54b02ba5a9131c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.805ex; height:2.843ex;" alt="{\displaystyle (x_{c}\pm R,y_{c})}" /></span> and <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_{c},y_{c}\pm R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>±</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{c},y_{c}\pm R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24f9255fdc53b8f513a5ff685fba414bf2df4f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.805ex; height:2.843ex;" alt="{\displaystyle (x_{c},y_{c}\pm R)}" /></span>. Its interior consists of the points <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></span> with <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-x_{c}|+|y-y_{c}|<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x_{c}|+|y-y_{c}|<R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0df17b7eb6a5fcead42169d14b2b2d69926da6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.814ex; height:2.843ex;" alt="{\displaystyle |x-x_{c}|+|y-y_{c}|<R}" /></span>, and its boundary consists of the points with <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-x_{c}|+|y-y_{c}|=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x_{c}|+|y-y_{c}|=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf78437cb2166327b7e8775b9faf927f2c5997a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.814ex; height:2.843ex;" alt="{\displaystyle |x-x_{c}|+|y-y_{c}|=R}" /></span>.<sup id="cite_ref-iobst_76-1" class="reference"><a href="#cite_note-iobst-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> For instance the illustration shows a diagonal square centered at the origin <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 (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" /></span> with circumradius 2, given by the equation <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|=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|+|y|=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bbde6c572f1f7f2ba72ec57c43abe593a7bd78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.174ex; height:2.843ex;" alt="{\displaystyle |x|+|y|=2}" /></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:A_square_with_Gaussian_integer_vertices.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/A_square_with_Gaussian_integer_vertices.png/250px-A_square_with_Gaussian_integer_vertices.png" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/A_square_with_Gaussian_integer_vertices.png/330px-A_square_with_Gaussian_integer_vertices.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/A_square_with_Gaussian_integer_vertices.png/500px-A_square_with_Gaussian_integer_vertices.png 2x" data-file-width="1628" data-file-height="1502" /></a><figcaption>A square formed by multiplying the complex number <span class="texhtml mvar" style="font-style:italic;">p</span> by powers of <span class="texhtml mvar" style="font-style:italic;">i</span>, and its translation obtained by adding another complex number <span class="texhtml mvar" style="font-style:italic;">c</span>. The background grid shows the <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>.</figcaption></figure> <p>In the <a href="/wiki/Complex_plane" title="Complex plane">plane of complex numbers</a>, multiplication by the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> rotates the other term in the product by 90° around the origin (the number zero). Therefore, if any nonzero complex number <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> is repeatedly multiplied by <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span>, giving the four 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 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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></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 ip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe5f0b723b5de9e52ea19d164ea01758181b5d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.972ex; height:2.509ex;" alt="{\displaystyle ip}" /></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 -p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <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/233ea0d764a4823bcf8b9a31b2f25f3966e77845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.978ex; height:2.343ex;" alt="{\displaystyle -p}" /></span>, and <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 -ip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -ip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e779a4ca194fea0ea4859b9f406f8e0a94804c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.78ex; height:2.509ex;" alt="{\displaystyle -ip}" /></span>, these numbers will form the vertices of a square centered at the origin.<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> If one interprets the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> and <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a> of these four complex numbers as Cartesian coordinates, with <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=x+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7751a31aed9d051df15ee28e7587cced2266cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.485ex; height:2.509ex;" alt="{\displaystyle p=x+iy}" /></span>, then these four numbers have the coordinates <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></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,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d79850138ad0820a91339857596015da20fe0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.137ex; height:2.843ex;" alt="{\displaystyle (-y,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 (-x,-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</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)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25dfc8e5539521f54d9d8b5914073368f7068a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.945ex; height:2.843ex;" alt="{\displaystyle (-x,-y)}" /></span>, and <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,-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-y,-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ac575fde921f3a075329cd4d359d43da349714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.945ex; height:2.843ex;" alt="{\displaystyle (-y,-x)}" /></span>.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> This square can be translated to have any other complex number <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/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></span> is center, using the fact that the <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> from the origin to <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/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></span> is represented in complex number arithmetic as addition with <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/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></span>.<sup id="cite_ref-numberverse_79-0" class="reference"><a href="#cite_note-numberverse-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>, complex numbers with integer real and imaginary parts, form a <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a> in the complex plane.<sup id="cite_ref-numberverse_79-1" class="reference"><a href="#cite_note-numberverse-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Compass_and_straightedge">Compass and straightedge</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=9" title="Edit section: Compass and straightedge"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The construction of a square with a given side, using a <a href="/wiki/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">compass and straightedge</a>, is given in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a> I.46.<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> The existence of this construction means that squares are <a href="/wiki/Constructible_polygon" title="Constructible polygon">constructible polygons</a>. A regular <span class="nowrap"><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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>-gon</span> is constructible exactly when the odd <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> are distinct <a href="/wiki/Fermat_prime" class="mw-redirect" title="Fermat prime">Fermat primes</a>,<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> and in the case of a square <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 n=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d928ec15aeef83aade867992ee473933adb6139d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=4}" /></span> has no odd prime factors so this condition is <a href="/wiki/Vacuously_true" class="mw-redirect" title="Vacuously true">vacuously true</a>. </p><p><i>Elements</i> IV.6–7 also give constructions for a square inscribed in a circle and circumscribed about a circle, respectively.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Straight_Square_Inscribed_in_a_Circle_240px.gif" class="mw-file-description" title="Square with a given circumcircle"><img alt="Square with a given circumcircle" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Straight_Square_Inscribed_in_a_Circle_240px.gif/180px-Straight_Square_Inscribed_in_a_Circle_240px.gif" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2b/Straight_Square_Inscribed_in_a_Circle_240px.gif 1.5x" data-file-width="240" data-file-height="240" /></a></span></div> <div class="gallerytext">Square with a given circumcircle</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:01-Quadrat-Seite-gegeben.gif" class="mw-file-description" title="Square with a given side length, using Thales' theorem"><img alt="Square with a given side length, using Thales' theorem" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/01-Quadrat-Seite-gegeben.gif/250px-01-Quadrat-Seite-gegeben.gif" decoding="async" width="180" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/01-Quadrat-Seite-gegeben.gif/330px-01-Quadrat-Seite-gegeben.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/01-Quadrat-Seite-gegeben.gif/500px-01-Quadrat-Seite-gegeben.gif 2x" data-file-width="528" data-file-height="473" /></a></span></div> <div class="gallerytext">Square with a given side length, using <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:01-Quadrat-Diagonale-gegeben.gif" class="mw-file-description" title="Square with a given diagonal"><img alt="Square with a given diagonal" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/01-Quadrat-Diagonale-gegeben.gif/180px-01-Quadrat-Diagonale-gegeben.gif" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/01-Quadrat-Diagonale-gegeben.gif/270px-01-Quadrat-Diagonale-gegeben.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/01-Quadrat-Diagonale-gegeben.gif/360px-01-Quadrat-Diagonale-gegeben.gif 2x" data-file-width="650" data-file-height="650" /></a></span></div> <div class="gallerytext">Square with a given diagonal</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Related_topics">Related topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=10" title="Edit section: Related topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:473px;max-width:473px"><div class="trow"><div class="tsingle" style="width:155px;max-width:155px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Dual_Cube-Octahedron.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/250px-Dual_Cube-Octahedron.svg.png" decoding="async" width="153" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/330px-Dual_Cube-Octahedron.svg.png 2x" data-file-width="744" data-file-height="749" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Cube" title="Cube">cube</a> and <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>, next steps in sequences of <a href="/wiki/Regular_polytope" title="Regular polytope">regular polytopes</a> starting with squares</div></div><div class="tsingle" style="width:156px;max-width:156px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Sierpinski_carpet_6,_white_on_black.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Sierpinski_carpet_6%2C_white_on_black.svg/250px-Sierpinski_carpet_6%2C_white_on_black.svg.png" decoding="async" width="154" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Sierpinski_carpet_6%2C_white_on_black.svg/330px-Sierpinski_carpet_6%2C_white_on_black.svg.png 2x" data-file-width="729" data-file-height="729" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Sierpi%C5%84ski_carpet" title="Sierpiński carpet">Sierpiński carpet</a>, a square <a href="/wiki/Fractal" title="Fractal">fractal</a> with square holes</div></div><div class="tsingle" style="width:156px;max-width:156px"><div class="thumbimage" style="height:153px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Ising-tartan.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Ising-tartan.png/154px-Ising-tartan.png" decoding="async" width="154" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Ising-tartan.png/231px-Ising-tartan.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Ising-tartan.png/308px-Ising-tartan.png 2x" data-file-width="1024" data-file-height="1024" /></a></span></div><div class="thumbcaption">An invariant measure for the <a href="/wiki/Baker%27s_map" title="Baker's map">baker's map</a></div></div></div></div></div> <p>The <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> of a square is {4}.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">truncated</a> square is an <a href="/wiki/Octagon" title="Octagon">octagon</a>.<sup id="cite_ref-FOOTNOTECoxeter1948148_84-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1948148-84"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> The square belongs to a family of <a href="/wiki/Regular_polytope" title="Regular polytope">regular polytopes</a> that includes the <a href="/wiki/Cube" title="Cube">cube</a> in three dimensions and the <a href="/wiki/Hypercube" title="Hypercube">hypercubes</a> in higher dimensions,<sup id="cite_ref-FOOTNOTECoxeter1948122–123_85-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1948122–123-85"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> and to another family that includes the <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a> in three dimensions and the <a href="/wiki/Cross-polytope" title="Cross-polytope">cross-polytopes</a> in higher dimensions.<sup id="cite_ref-FOOTNOTECoxeter1948121–122_86-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1948121–122-86"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> The cube and hypercubes can be given vertex coordinates that are all <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 \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}" /></span>, giving an axis-parallel square in two dimensions, while the octahedron and cross-polytopes have one coordinate <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 \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}" /></span> and the rest zero, giving a diagonal square in two dimensions.<sup id="cite_ref-FOOTNOTECoxeter1948122,_126_87-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1948122,_126-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> As with squares, the <a href="/wiki/Hyperoctahedral_group" title="Hyperoctahedral group">symmetries of these shapes</a> can be obtained by applying a <a href="/wiki/Signed_permutation" class="mw-redirect" title="Signed permutation">signed permutation</a> to their coordinates.<sup id="cite_ref-ers_23-1" class="reference"><a href="#cite_note-ers-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Sierpi%C5%84ski_carpet" title="Sierpiński carpet">Sierpiński carpet</a> is a square <a href="/wiki/Fractal" title="Fractal">fractal</a>, with square holes.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curves</a> including the <a href="/wiki/Hilbert_curve" title="Hilbert curve">Hilbert curve</a>, <a href="/wiki/Peano_curve" title="Peano curve">Peano curve</a>, and <a href="/wiki/Sierpi%C5%84ski_curve" title="Sierpiński curve">Sierpiński curve</a> cover a square as the continuous image of a line segment.<sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Z-order_curve" title="Z-order curve">Z-order curve</a> is analogous but not continuous.<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> Other mathematical functions associated with squares include <a href="/wiki/Arnold%27s_cat_map" title="Arnold's cat map">Arnold's cat map</a> and the <a href="/wiki/Baker%27s_map" title="Baker's map">baker's map</a>, which generate chaotic <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a> on a square,<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Lemniscate_elliptic_functions" title="Lemniscate elliptic functions">lemniscate elliptic functions</a>, complex functions periodic on a square grid.<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Inscribed_squares">Inscribed squares</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=11" title="Edit section: Inscribed squares"><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:Calabi_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Calabi_triangle.svg/220px-Calabi_triangle.svg.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Calabi_triangle.svg/330px-Calabi_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Calabi_triangle.svg/440px-Calabi_triangle.svg.png 2x" data-file-width="400" data-file-height="266" /></a><figcaption>The <a href="/wiki/Calabi_triangle" title="Calabi triangle">Calabi triangle</a> and the three placements of its largest square.<sup id="cite_ref-FOOTNOTEConwayGuy1996[httpsbooksgooglecombooksid0--3rcO7dMYCpgPA206_206]_93-0" class="reference"><a href="#cite_note-FOOTNOTEConwayGuy1996[httpsbooksgooglecombooksid0--3rcO7dMYCpgPA206_206]-93"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> The placement on the long side of the triangle is inscribed; the other two are not.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Inscribed_square_problem" title="Inscribed square problem">Inscribed square problem</a> and <a href="/wiki/Inscribed_square_in_a_triangle" title="Inscribed square in a triangle">Inscribed square in a triangle</a></div> <p>A square is <a href="/wiki/Inscribed_figure" title="Inscribed figure">inscribed</a> in a curve when all four vertices of the square lie on the curve. The unsolved <a href="/wiki/Inscribed_square_problem" title="Inscribed square problem">inscribed square problem</a> asks whether every <a href="/wiki/Simple_closed_curve" class="mw-redirect" title="Simple closed curve">simple closed curve</a> has an inscribed square. It is true for every <a href="/wiki/Smooth_curve" class="mw-redirect" title="Smooth curve">smooth curve</a>,<sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> and for the boundary of any <a href="/wiki/Convex_set" title="Convex set">convex set</a>. The only other regular polygon that can always be inscribed in every convex set is the <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>, as there exists a convex set on which no other regular polygon can be inscribed.<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> </p><p>For an <a href="/wiki/Inscribed_square_in_a_triangle" title="Inscribed square in a triangle">inscribed square in a triangle</a>, at least one side of the square lies on a side of the triangle. Every <a href="/wiki/Acute_triangle" class="mw-redirect" title="Acute triangle">acute triangle</a> has three inscribed squares, one for each of its three sides. A <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> has two inscribed squares, one touching its right angle and the other lying on its hypotenuse. An <a href="/wiki/Obtuse_triangle" class="mw-redirect" title="Obtuse triangle">obtuse triangle</a> has only one inscribed square, on its longest. A square inscribed in a triangle can cover at most half the triangle's area.<sup id="cite_ref-gardner_96-0" class="reference"><a href="#cite_note-gardner-96"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Area_and_quadrature">Area and quadrature <span class="anchor" id="Squaring_the_circle"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=12" title="Edit section: Area and quadrature"><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">See also: <a href="/wiki/Area" title="Area">Area</a>, <a href="/wiki/Quadrature_(geometry)" title="Quadrature (geometry)">Quadrature (geometry)</a>, and <a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pythagorean.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/250px-Pythagorean.svg.png" decoding="async" width="180" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/330px-Pythagorean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/500px-Pythagorean.svg.png 2x" data-file-width="512" data-file-height="466" /></a><figcaption>The <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Squaring_the_Circle_J.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Squaring_the_Circle_J.svg/130px-Squaring_the_Circle_J.svg.png" decoding="async" width="130" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Squaring_the_Circle_J.svg/195px-Squaring_the_Circle_J.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Squaring_the_Circle_J.svg/260px-Squaring_the_Circle_J.svg.png 2x" data-file-width="149" data-file-height="149" /></a><figcaption>A circle and square with the same area</figcaption></figure> <p>Conventionally, since ancient times, most units of <a href="/wiki/Area" title="Area">area</a> have been defined in terms of various squares, typically a square with a standard unit of <a href="/wiki/Length" title="Length">length</a> as its side, for example a <a href="/wiki/Square_meter" class="mw-redirect" title="Square meter">square meter</a> or <a href="/wiki/Square_inch" title="Square inch">square inch</a>.<sup id="cite_ref-Treese_97-0" class="reference"><a href="#cite_note-Treese-97"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> The area of an arbitrary rectangle can then be simply computed as the product of its length and its width, and more complicated shapes can be measured by conceptually breaking them up into unit squares or into arbitrary triangles.<sup id="cite_ref-Treese_97-1" class="reference"><a href="#cite_note-Treese-97"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">ancient Greek deductive geometry</a>, the area of a planar shape was measured and compared by constructing a square with the same area by using only a finite number of steps with <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>, a process called <i><a href="/wiki/Quadrature_(geometry)" title="Quadrature (geometry)">quadrature</a></i> or <i>squaring</i>. <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a> shows how to do this for rectangles, parallelograms, triangles, and then more generally for <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygons</a> by breaking them into triangular pieces.<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> Some shapes with curved sides could also be squared, such as the <a href="/wiki/Lune_of_Hippocrates" title="Lune of Hippocrates">lune of Hippocrates</a><sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Quadrature_of_the_Parabola" title="Quadrature of the Parabola">parabola</a>.<sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>100<span class="cite-bracket">]</span></a></sup> </p><p>This use of a square as the defining shape for area measurement also occurs in the Greek formulation of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: squares constructed on the two sides of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> have equal total area to a square constructed on the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a>.<sup id="cite_ref-101" class="reference"><a href="#cite_note-101"><span class="cite-bracket">[</span>101<span class="cite-bracket">]</span></a></sup> Stated in this form, the theorem would be equally valid for other shapes on the sides of the triangle, such as equilateral triangles or semicircles,<sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>102<span class="cite-bracket">]</span></a></sup> but the Greeks used squares. In modern mathematics, this formulation of the theorem using areas of squares has been replaced by an algebraic formulation involving <a href="/wiki/Square_(algebra)" title="Square (algebra)">squaring numbers</a>: the lengths of the sides and hypotenuse of the right triangle obey the equation <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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}" /></span>.<sup id="cite_ref-103" class="reference"><a href="#cite_note-103"><span class="cite-bracket">[</span>103<span class="cite-bracket">]</span></a></sup> </p><p>Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully to <a href="/wiki/Squaring_the_circle" title="Squaring the circle">square the circle</a>, constructing a square with the same area as a given circle, again using finitely many steps with a compass and straightedge. In 1882, the task was proven to be impossible as a consequence of the <a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a>. This theorem proves that <a href="/wiki/Pi" title="Pi">pi</a> (<span class="texhtml mvar" style="font-style:italic;">π</span>) is a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a> rather than an <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic irrational number</a>; that is, it is not the <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> of any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients. A construction for squaring the circle could be translated into a polynomial formula for <span class="texhtml mvar" style="font-style:italic;">π</span>, which does not exist.<sup id="cite_ref-104" class="reference"><a href="#cite_note-104"><span class="cite-bracket">[</span>104<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Tiling_and_packing">Tiling and packing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=13" title="Edit section: Tiling and packing"><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/Square_tiling" title="Square tiling">Square tiling</a>, <a href="/wiki/Square_packing" title="Square packing">Square packing</a>, <a href="/wiki/Circle_packing_in_a_square" title="Circle packing in a square">Circle packing in a square</a>, and <a href="/wiki/Squaring_the_square" title="Squaring the square">Squaring the square</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Tiling_Regular_4-4_Square.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Tiling_Regular_4-4_Square.svg/200px-Tiling_Regular_4-4_Square.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Tiling_Regular_4-4_Square.svg/300px-Tiling_Regular_4-4_Square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Tiling_Regular_4-4_Square.svg/400px-Tiling_Regular_4-4_Square.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div><div class="thumbcaption"><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a></div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg/200px-Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg/300px-Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg/400px-Academ_Periodic_tiling_by_squares_of_two_different_sizes.svg.png 2x" data-file-width="750" data-file-height="750" /></a></span></div><div class="thumbcaption"><a href="/wiki/Pythagorean_tiling" title="Pythagorean tiling">Pythagorean tiling</a></div></div></div></div></div> <p>The <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a>, familiar from flooring and game boards, is one of three <a href="/wiki/Tiling_by_regular_polygons" class="mw-redirect" title="Tiling by regular polygons">regular tilings</a> of the plane. The other two use the <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> and the <a href="/wiki/Regular_hexagon" class="mw-redirect" title="Regular hexagon">regular hexagon</a>.<sup id="cite_ref-105" class="reference"><a href="#cite_note-105"><span class="cite-bracket">[</span>105<span class="cite-bracket">]</span></a></sup> The vertices of a square tiling form a <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a>.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198729_106-0" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198729-106"><span class="cite-bracket">[</span>106<span class="cite-bracket">]</span></a></sup> Squares of more than one size can also tile the plane,<sup id="cite_ref-FOOTNOTEGrünbaumShephard198776–78_107-0" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198776–78-107"><span class="cite-bracket">[</span>107<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-108" class="reference"><a href="#cite_note-108"><span class="cite-bracket">[</span>108<span class="cite-bracket">]</span></a></sup> for instance in the <a href="/wiki/Pythagorean_tiling" title="Pythagorean tiling">Pythagorean tiling</a>, named for its connection to proofs of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<sup id="cite_ref-109" class="reference"><a href="#cite_note-109"><span class="cite-bracket">[</span>109<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg/250px-Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg/330px-Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg/500px-Packing_11_unit_squares_in_a_square_with_side_length_3.87708359....svg.png 2x" data-file-width="1774" data-file-height="1777" /></a><figcaption>The smallest known square that can contain 11 unit squares has side length approximately 3.877084.<sup id="cite_ref-friedman_110-0" class="reference"><a href="#cite_note-friedman-110"><span class="cite-bracket">[</span>110<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><a href="/wiki/Square_packing" title="Square packing">Square packing</a> problems seek the smallest square or circle into which a given number of <a href="/wiki/Unit_square" title="Unit square">unit squares</a> can fit. A chessboard optimally packs a square number of unit squares into a larger square, but beyond a few special cases such as this, the optimal solutions to these problems remain unsolved;<sup id="cite_ref-friedman_110-1" class="reference"><a href="#cite_note-friedman-110"><span class="cite-bracket">[</span>110<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-111" class="reference"><a href="#cite_note-111"><span class="cite-bracket">[</span>111<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-112" class="reference"><a href="#cite_note-112"><span class="cite-bracket">[</span>112<span class="cite-bracket">]</span></a></sup> the same is true for <a href="/wiki/Circle_packing_in_a_square" title="Circle packing in a square">circle packing in a square</a>.<sup id="cite_ref-113" class="reference"><a href="#cite_note-113"><span class="cite-bracket">[</span>113<span class="cite-bracket">]</span></a></sup> Packing squares into other shapes can have high <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a>: testing whether a given number of unit squares can fit into an <a href="/wiki/Orthogonal_convexity" class="mw-redirect" title="Orthogonal convexity">orthogonally convex</a> <a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">rectilinear polygon</a> with <a href="/wiki/Half-integer" title="Half-integer">half-integer</a> vertex coordinates is <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>.<sup id="cite_ref-114" class="reference"><a href="#cite_note-114"><span class="cite-bracket">[</span>114<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Squaring_the_square" title="Squaring the square">Squaring the square</a> involves subdividing a given square into smaller squares, all having integer side lengths. A subdivision with distinct smaller squares is called a perfect squared square.<sup id="cite_ref-115" class="reference"><a href="#cite_note-115"><span class="cite-bracket">[</span>115<span class="cite-bracket">]</span></a></sup> Another variant of squaring the square called "Mrs. Perkins's quilt" allows repetitions, but uses as few smaller squares as possible in order to make the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of the side lengths be 1.<sup id="cite_ref-116" class="reference"><a href="#cite_note-116"><span class="cite-bracket">[</span>116<span class="cite-bracket">]</span></a></sup> The entire plane can be tiled by squares, with exactly one square of each integer side length.<sup id="cite_ref-117" class="reference"><a href="#cite_note-117"><span class="cite-bracket">[</span>117<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:441px;max-width:441px"><div class="trow"><div class="tsingle" style="width:145px;max-width:145px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Clifford-torus.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Clifford-torus.gif/250px-Clifford-torus.gif" decoding="async" width="143" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6f/Clifford-torus.gif 2x" data-file-width="255" data-file-height="255" /></a></span></div><div class="thumbcaption"><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> into 3d of a rotating <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a></div></div><div class="tsingle" style="width:145px;max-width:145px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Petrie-1.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Petrie-1.gif/143px-Petrie-1.gif" decoding="async" width="143" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Petrie-1.gif/215px-Petrie-1.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8c/Petrie-1.gif 2x" data-file-width="240" data-file-height="240" /></a></span></div><div class="thumbcaption"><a href="/wiki/Regular_skew_apeirohedron" title="Regular skew apeirohedron">Regular skew apeirohedron</a> with six squares per vertex</div></div><div class="tsingle" style="width:145px;max-width:145px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Teabag.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Teabag.jpg/143px-Teabag.jpg" decoding="async" width="143" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Teabag.jpg/215px-Teabag.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Teabag.jpg/286px-Teabag.jpg 2x" data-file-width="432" data-file-height="432" /></a></span></div><div class="thumbcaption">Numerical simulation of an inflated square pillow</div></div></div></div></div> <p>In higher dimensions, other surfaces than the plane can be tiled by equal squares, meeting edge-to-edge. One of these surfaces is the <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a>, the four-dimensional <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of two congruent circles; it has the same intrinsic geometry as a single square with each pair of opposite edges glued together.<sup id="cite_ref-118" class="reference"><a href="#cite_note-118"><span class="cite-bracket">[</span>118<span class="cite-bracket">]</span></a></sup> Another square-tiled surface, a <a href="/wiki/Regular_skew_apeirohedron" title="Regular skew apeirohedron">regular skew apeirohedron</a> in three dimensions, has six squares meeting at each vertex.<sup id="cite_ref-119" class="reference"><a href="#cite_note-119"><span class="cite-bracket">[</span>119<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Paper_bag_problem" title="Paper bag problem">paper bag problem</a> seeks the maximum volume that can be enclosed by a surface tiled with two squares glued edge to edge; its exact answer is unknown.<sup id="cite_ref-120" class="reference"><a href="#cite_note-120"><span class="cite-bracket">[</span>120<span class="cite-bracket">]</span></a></sup> Gluing two squares in a different pattern, with the vertex of each square attached to the midpoint of an edge of the other square (or alternatively subdividing these two squares into eight squares glued edge-to-edge) produces a pincushion shape called a <a href="/wiki/Biscornu" title="Biscornu">biscornu</a>.<sup id="cite_ref-121" class="reference"><a href="#cite_note-121"><span class="cite-bracket">[</span>121<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Counting">Counting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=14" title="Edit section: Counting"><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/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal number</a> and <a href="/wiki/Dividing_a_square_into_similar_rectangles" title="Dividing a square into similar rectangles">Dividing a square into similar rectangles</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Two_square_counting_puzzles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Two_square_counting_puzzles.svg/330px-Two_square_counting_puzzles.svg.png" decoding="async" width="260" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Two_square_counting_puzzles.svg/500px-Two_square_counting_puzzles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Two_square_counting_puzzles.svg/520px-Two_square_counting_puzzles.svg.png 2x" data-file-width="504" data-file-height="324" /></a><figcaption>Two square-counting puzzles: There are 14 squares in a <span class="texhtml">3 × 3</span> grid of squares (top), but as a <span class="texhtml">4 × 4</span> grid of points it has six more off-axis squares (bottom) for a total of 20.</figcaption></figure> <p>A common <a href="/wiki/Mathematical_puzzle" title="Mathematical puzzle">mathematical puzzle</a> involves counting the squares of all sizes in a square grid 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 n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /></span> squares. For instance, a square grid of nine squares has 14 squares: the nine squares that form the grid, four more <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 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}" /></span> squares, and one <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 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0d4d6106875f8006be1d898512ca5843bad8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" /></span> square. The answer to the puzzle is <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 n(n+1)(2n+1)/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(n+1)(2n+1)/6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801bfabe9a78e34cb154636452e814c12f637689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.296ex; height:2.843ex;" alt="{\displaystyle n(n+1)(2n+1)/6}" /></span>, a <a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">square pyramidal number</a>.<sup id="cite_ref-122" class="reference"><a href="#cite_note-122"><span class="cite-bracket">[</span>122<span class="cite-bracket">]</span></a></sup> For <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 n=1,2,3,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,3,\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b207567215497887a3250644b8876c23dc3ebf10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=1,2,3,\dots }" /></span> these numbers are:<sup id="cite_ref-123" class="reference"><a href="#cite_note-123"><span class="cite-bracket">[</span>123<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.6em;">1, 5, 14, 30, 55, 91, 140, 204, 285, ...</div> <p>A variant of the same puzzle asks for the number of squares formed by a grid 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 n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /></span> points, allowing squares that are not axis-parallel. For instance, a grid of nine points has five axis-parallel squares as described above, but it also contains one more diagonal square for a total of six.<sup id="cite_ref-124" class="reference"><a href="#cite_note-124"><span class="cite-bracket">[</span>124<span class="cite-bracket">]</span></a></sup> In this case, the answer is given by the <i>4-dimensional pyramidal numbers</i> <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 n^{2}(n^{2}-1)/12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}(n^{2}-1)/12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f56d5254b4d9474b9427ff9bd3f7c5aa402e68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.197ex; height:3.176ex;" alt="{\displaystyle n^{2}(n^{2}-1)/12}" /></span>. For <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 n=1,2,3,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,3,\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b207567215497887a3250644b8876c23dc3ebf10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=1,2,3,\dots }" /></span> these numbers are:<sup id="cite_ref-125" class="reference"><a href="#cite_note-125"><span class="cite-bracket">[</span>125<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.6em;">0, 1, 6, 20, 50, 105, 196, 336, 540, ...</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_square_partitions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Plastic_square_partitions.svg/220px-Plastic_square_partitions.svg.png" decoding="async" width="220" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Plastic_square_partitions.svg/330px-Plastic_square_partitions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Plastic_square_partitions.svg/440px-Plastic_square_partitions.svg.png 2x" data-file-width="414" data-file-height="126" /></a><figcaption>Partitions of a square into three similar rectangles</figcaption></figure> <p>Another counting problem involving squares asks for the number of different shapes of rectangle that can be used when <a href="/wiki/Dividing_a_square_into_similar_rectangles" title="Dividing a square into similar rectangles">dividing a square into similar rectangles</a>.<sup id="cite_ref-126" class="reference"><a href="#cite_note-126"><span class="cite-bracket">[</span>126<span class="cite-bracket">]</span></a></sup> A square can be divided into two similar rectangles only in one way, by bisecting it, but when dividing a square into three similar rectangles there are three possible <a href="/wiki/Aspect_ratio" title="Aspect ratio">aspect ratios</a> of the rectangles, 3:1, 3:2, and the square of the <a href="/wiki/Plastic_ratio" title="Plastic ratio">plastic ratio</a>. The number of proportions that are possible when dividing into <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> rectangles is known for small values 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, but not as a general formula. For <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 n=1,2,3,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,3,\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b207567215497887a3250644b8876c23dc3ebf10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=1,2,3,\dots }" /></span> these numbers are:<sup id="cite_ref-127" class="reference"><a href="#cite_note-127"><span class="cite-bracket">[</span>127<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.6em;">1, 1, 3, 11, 51, 245, 1372, ...</div> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Non-Euclidean_geometry">Non-Euclidean geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square&action=edit&section=15" title="Edit section: Non-Euclidean geometry"><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:TaxicabGeometryCircle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/TaxicabGeometryCircle.svg/130px-TaxicabGeometryCircle.svg.png" decoding="async" width="130" height="358" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/TaxicabGeometryCircle.svg/195px-TaxicabGeometryCircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/TaxicabGeometryCircle.svg/260px-TaxicabGeometryCircle.svg.png 2x" data-file-width="214" data-file-height="590" /></a><figcaption>Points (red) at equal distance from a central point (blue) according to <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab geometry</a></figcaption></figure> <p>Squares tilted at 45° to the coordinate axes are the <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">metric balls</a> for <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab geometry</a>, 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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" /></span> distance metric in the <a href="/wiki/Real_coordinate_plane" class="mw-redirect" title="Real coordinate plane">real coordinate plane</a>. According to this metric, the distance between any two points <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_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}" /></span> and <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_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}" /></span> is <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_{1}-x_{2}|+|y_{1}-y_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{1}-x_{2}|+|y_{1}-y_{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fbb2f18e5ce70da5f0afbf20ecedf0b461f466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.263ex; height:2.843ex;" alt="{\displaystyle |x_{1}-x_{2}|+|y_{1}-y_{2}|}" /></span> instead of the <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</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 {\sqrt {|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802e0e51e0eb05888576abc386b97ef8d556b2ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.696ex; height:4.843ex;" alt="{\displaystyle {\sqrt {|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2}}}}" /></span>. The points with taxicab distance <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> from any given point form a diagonal square, centered at the given point, with diagonal length <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 2d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8106478cb4da6af49992eeb3a3b8690d27797ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.176ex;" alt="{\displaystyle 2d}" /></span>. In the same way, axis-parallel squares form the metric balls for 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 L_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ecf05fae003abbd91bc8c749d5a8a807d6efd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.458ex; height:2.509ex;" alt="{\displaystyle L_{\infty }}" /></span> distance metric (called the <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a>), for which the distance is given by the formula <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 \max(|x_{1}-x_{2}|,|y_{1}-y_{2}|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(|x_{1}-x_{2}|,|y_{1}-y_{2}|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff16a809baefbe732a2c5dd3f0a92e49e298fe62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.592ex; height:2.843ex;" alt="{\displaystyle \max(|x_{1}-x_{2}|,|y_{1}-y_{2}|)}" /></span>. In this metric, the points with distance <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> from some point form an axis-parallel square, centered at the given point, with side length <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 2d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8106478cb4da6af49992eeb3a3b8690d27797ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.176ex;" alt="{\displaystyle 2d}" /></span>.<sup id="cite_ref-128" class="reference"><a href="#cite_note-128"><span class="cite-bracket">[</span>128<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-129" class="reference"><a href="#cite_note-129"><span class="cite-bracket">[</span>129<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-130" class="reference"><a href="#cite_note-130"><span class="cite-bracket">[</span>130<span class="cite-bracket">]</span></a></sup> </p><p>Other forms of non-Euclidean geometry, including <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a> and <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, also have polygons with four equal sides and equal angles. These are often called squares,<sup id="cite_ref-maraner_131-0" class="reference"><a href="#cite_note-maraner-131"><span class="cite-bracket">[</span>131<span class="cite-bracket">]</span></a></sup> but some authors avoid calling them that, instead calling them regular quadrilaterals, because unlike Euclidean squares they cannot have right angles. These geometries also have regular polygons with right angles, but with numbers of sides different from four.<sup id="cite_ref-singer_132-0" class="reference"><a href="#cite_note-singer-132"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a>, a <i>square</i> is a <a href="/wiki/Spherical_polygon" class="mw-redirect" title="Spherical polygon">polygon</a> whose edges are <a href="/wiki/Great_circle" title="Great circle">great-circle</a> arcs of equal length, which meet at equal angles. Unlike the square of Euclidean geometry, spherical squares have <a href="/wiki/Obtuse_angle" class="mw-redirect" title="Obtuse angle">obtuse angles</a>, larger than a right angle. Larger spherical squares have larger angles.<sup id="cite_ref-maraner_131-1" class="reference"><a href="#cite_note-maraner-131"><span class="cite-bracket">[</span>131<span class="cite-bracket">]</span></a></sup> An <a href="/wiki/Octant_of_a_sphere" title="Octant of a sphere">octant of a sphere</a> is a regular <a href="/wiki/Spherical_triangle" class="mw-redirect" title="Spherical triangle">spherical triangle</a> consisting of three straight sides and three right angles. The sphere can be tiled by eight such octants to make a spherical <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, with four octants meeting at each vertex.<sup id="cite_ref-133" class="reference"><a href="#cite_note-133"><span class="cite-bracket">[</span>133<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, squares with right angles do not exist. Rather, squares in hyperbolic geometry have <a href="/wiki/Acute_angle" class="mw-redirect" title="Acute angle">acute angles</a>, less than right angles. Larger hyperbolic squares have smaller angles. It is also possible to make a regular hyperbolic polygon with right angles at every vertex and any number of sides greater than four; such polygons can <a href="/wiki/Uniform_tilings_in_hyperbolic_plane" title="Uniform tilings in hyperbolic plane">uniformly tile the hyperbolic plane</a>.<sup id="cite_ref-singer_132-1" class="reference"><a href="#cite_note-singer-132"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Tetragonal_dihedron.png" class="mw-file-description" title="A dihedron with spherical square faces and 180° angles. The Peirce quincuncial projection for world maps conformally maps these faces to Euclidean squares.[134]"><img alt="A dihedron with spherical square faces and 180° angles. The Peirce quincuncial projection for world maps conformally maps these faces to Euclidean squares.[134]" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Tetragonal_dihedron.png/250px-Tetragonal_dihedron.png" decoding="async" width="178" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Tetragonal_dihedron.png/330px-Tetragonal_dihedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Tetragonal_dihedron.png/500px-Tetragonal_dihedron.png 2x" data-file-width="594" data-file-height="600" /></a></span></div> <div class="gallerytext">A <a href="/wiki/Dihedron" title="Dihedron">dihedron</a> with spherical square faces and 180° angles. The <a href="/wiki/Peirce_quincuncial_projection" title="Peirce quincuncial projection">Peirce quincuncial projection</a> for <a href="/wiki/World_map" title="World map">world maps</a> <a href="/wiki/Conformal_mapping" class="mw-redirect" title="Conformal mapping">conformally maps</a> these faces to Euclidean squares.<sup id="cite_ref-134" class="reference"><a href="#cite_note-134"><span class="cite-bracket">[</span>134<span class="cite-bracket">]</span></a></sup></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Square_on_sphere.svg" class="mw-file-description" title="Six squares tile the sphere with 3 squares around each vertex and 120° internal angles, forming a spherical cube.[135]"><img alt="Six squares tile the sphere with 3 squares around each vertex and 120° internal angles, forming a spherical cube.[135]" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/180px-Square_on_sphere.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/270px-Square_on_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/360px-Square_on_sphere.svg.png 2x" data-file-width="816" data-file-height="815" /></a></span></div> <div class="gallerytext">Six squares tile the sphere with 3 squares around each vertex and 120° internal angles, forming a spherical cube.<sup id="cite_ref-135" class="reference"><a href="#cite_note-135"><span class="cite-bracket">[</span>135<span class="cite-bracket">]</span></a></sup></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Square_on_hyperbolic_plane.png" class="mw-file-description" title="Squares can tile the hyperbolic plane with five around each vertex, each square having 72° internal angles, giving the order-5 square tiling. For every n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.[132]"><img alt="Squares can tile the hyperbolic plane with five around each vertex, each square having 72° internal angles, giving the order-5 square tiling. For every n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.[132]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Square_on_hyperbolic_plane.png/250px-Square_on_hyperbolic_plane.png" decoding="async" width="177" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Square_on_hyperbolic_plane.png/330px-Square_on_hyperbolic_plane.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Square_on_hyperbolic_plane.png/500px-Square_on_hyperbolic_plane.png 2x" data-file-width="613" data-file-height="624" /></a></span></div> <div class="gallerytext">Squares can tile the <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic plane</a> with five around each vertex, each square having 72° internal angles, giving the <a href="/wiki/Order-5_square_tiling" title="Order-5 square tiling">order-5 square tiling</a>. For every <span class="texhtml"><i>n</i> ≥ 5</span> there is a hyperbolic tiling with <span class="texhtml mvar" style="font-style:italic;">n</span> squares about each vertex.<sup id="cite_ref-singer_132-2" class="reference"><a href="#cite_note-singer-132"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:H2_tiling_246-1.png" class="mw-file-description" title="Regular hexagons with right angles can tile the hyperbolic plane with four hexagons meeting at each vertex, giving the order-4 hexagonal tiling. For every n ≥ 5 there is a hyperbolic tiling by right-angled regular n-gons, dual to the tiling with n squares about each vertex.[132]"><img alt="Regular hexagons with right angles can tile the hyperbolic plane with four hexagons meeting at each vertex, giving the order-4 hexagonal tiling. For every n ≥ 5 there is a hyperbolic tiling by right-angled regular n-gons, dual to the tiling with n squares about each vertex.[132]" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/H2_tiling_246-1.png/180px-H2_tiling_246-1.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/H2_tiling_246-1.png/270px-H2_tiling_246-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/H2_tiling_246-1.png/360px-H2_tiling_246-1.png 2x" data-file-width="2520" data-file-height="2520" /></a></span></div> <div class="gallerytext">Regular hexagons with right angles can tile the hyperbolic plane with four hexagons meeting at each vertex, giving the <a href="/wiki/Order-4_hexagonal_tiling" title="Order-4 hexagonal tiling">order-4 hexagonal tiling</a>. For every <span class="texhtml"><i>n</i> ≥ 5</span> there is a hyperbolic tiling by right-angled regular <span class="texhtml mvar" style="font-style:italic;">n</span>-gons, <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> to the tiling with <span class="texhtml mvar" style="font-style:italic;">n</span> squares about each vertex.<sup id="cite_ref-singer_132-3" class="reference"><a href="#cite_note-singer-132"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup></div> </li> </ul> <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=Square&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Finsler%E2%80%93Hadwiger_theorem" title="Finsler–Hadwiger theorem">Finsler–Hadwiger theorem</a> on a square derived from two squares sharing a vertex</li> <li><a href="/wiki/Midsquare_quadrilateral" title="Midsquare quadrilateral">Midsquare quadrilateral</a>, a polygon whose edge midpoints form a square</li> <li><a href="/wiki/Monsky%27s_theorem" title="Monsky's theorem">Monsky's theorem</a>, on subdividing a square into an odd number of equal-area triangles</li> <li><a href="/wiki/Square_planar_molecular_geometry" title="Square planar molecular geometry">Square planar molecular geometry</a>, chemical structure with atoms at the corners of a square</li> <li><a href="/wiki/Square_trisection" title="Square trisection">Square trisection</a>, a problem of cutting and reassembling one square into three squares</li> <li><a href="/wiki/Squircle" title="Squircle">Squircle</a>, a shape intermediate between a square and a circle</li> <li><a href="/wiki/Tarski%27s_circle-squaring_problem" title="Tarski's circle-squaring problem">Tarski's circle-squaring problem</a>, dividing a disk into sets that can be rearranged into a square</li> <li><a href="/wiki/Van_Aubel%27s_theorem" title="Van Aubel's theorem">Van Aubel's theorem</a> and <a href="/wiki/Th%C3%A9bault%27s_theorem" title="Thébault's theorem">Thébault's theorem</a>, on squares placed on the sides of a quadrilateral</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=Square&action=edit&section=17" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-zalgri-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-zalgri_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-zalgri_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-zalgri_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-zalgri_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-zalgri_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-zalgri_1-5"><sup><i><b>f</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="CITEREFUsiskinGriffin2008" class="citation book cs1"><a href="/wiki/Zalman_Usiskin" title="Zalman Usiskin">Usiskin, Zalman</a>; Griffin, Jennifer (2008). <i>The Classification of Quadrilaterals: A Study of Definition</i>. Information Age Publishing. p. 59. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-59311-695-8" title="Special:BookSources/978-1-59311-695-8"><bdi>978-1-59311-695-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Classification+of+Quadrilaterals%3A+A+Study+of+Definition&rft.pages=59&rft.pub=Information+Age+Publishing&rft.date=2008&rft.isbn=978-1-59311-695-8&rft.aulast=Usiskin&rft.aufirst=Zalman&rft.au=Griffin%2C+Jennifer&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" 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="CITEREFWilson2010" class="citation web cs1">Wilson, Jim (Summer 2010). <a rel="nofollow" class="external text" href="http://jwilson.coe.uga.edu/MATH7200/ProblemSet1.3.html">"Problem Set 1.3, problem 10"</a>. <i>Math 5200/7200 Foundations of Geometry I</i>. University of Georgia<span class="reference-accessdate">. Retrieved <span class="nowrap">2025-02-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+5200%2F7200+Foundations+of+Geometry+I&rft.atitle=Problem+Set+1.3%2C+problem+10&rft.ssn=summer&rft.date=2010&rft.aulast=Wilson&rft.aufirst=Jim&rft_id=http%3A%2F%2Fjwilson.coe.uga.edu%2FMATH7200%2FProblemSet1.3.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAlsinaNelsen2020" class="citation book cs1">Alsina, Claudi; Nelsen, Roger B. (2020). "Theorem 9.2.1". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA22"><i>A Cornucopia of Quadrilaterals</i></a>. Dolciani Mathematical Expositions. Vol. 55. American Mathematical Society. p. 186. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781470453121" title="Special:BookSources/9781470453121"><bdi>9781470453121</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theorem+9.2.1&rft.btitle=A+Cornucopia+of+Quadrilaterals&rft.series=Dolciani+Mathematical+Expositions&rft.pages=186&rft.pub=American+Mathematical+Society&rft.date=2020&rft.isbn=9781470453121&rft.aulast=Alsina&rft.aufirst=Claudi&rft.au=Nelsen%2C+Roger+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCGDSDwAAQBAJ%26pg%3DPA22&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-rich-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-rich_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rich_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-rich_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-rich_4-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRich1963" class="citation book cs1">Rich, Barnett (1963). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.131938/page/n147"><i>Principles And Problems Of Plane Geometry</i></a>. Schaum. p. 132.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+And+Problems+Of+Plane+Geometry&rft.pages=132&rft.pub=Schaum&rft.date=1963&rft.aulast=Rich&rft.aufirst=Barnett&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.131938%2Fpage%2Fn147&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-godsid-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-godsid_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-godsid_5-1"><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="CITEREFGodfreySiddons1919" class="citation book cs1">Godfrey, Charles; Siddons, A. W. (1919). <a rel="nofollow" class="external text" href="https://archive.org/details/elementarygeomet00godfuoft/page/40"><i>Elementary Geometry: Practical and Theoretical</i></a> (3rd ed.). Cambridge University Press. p. 40.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Geometry%3A+Practical+and+Theoretical&rft.pages=40&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=1919&rft.aulast=Godfrey&rft.aufirst=Charles&rft.au=Siddons%2C+A.+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarygeomet00godfuoft%2Fpage%2F40&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchorlingClarkCarter1935" class="citation book cs1">Schorling, R.; Clark, John P.; Carter, H. W. (1935). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.84435/page/n127"><i>Modern Mathematics: An Elementary Course</i></a>. George G. Harrap & Co. pp. <span class="nowrap">124–</span>125.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Mathematics%3A+An+Elementary+Course&rft.pages=%3Cspan+class%3D%22nowrap%22%3E124-%3C%2Fspan%3E125&rft.pub=George+G.+Harrap+%26+Co.&rft.date=1935&rft.aulast=Schorling&rft.aufirst=R.&rft.au=Clark%2C+John+P.&rft.au=Carter%2C+H.+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.84435%2Fpage%2Fn127&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGodfreySiddons1919135-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGodfreySiddons1919135_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGodfreySiddons1919">Godfrey & Siddons (1919)</a>, p. 135.</span> </li> <li id="cite_note-FOOTNOTESchorlingClarkCarter1935101-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESchorlingClarkCarter1935101_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchorlingClarkCarter1935">Schorling, Clark & Carter (1935)</a>, p. 101.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFApostol1990" class="citation book cs1"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/02-the-story-of-pi/01%20Similarity/page/9/mode/1up"><i>Project Mathematics! Program Guide and Workbook: Similarity</i></a>. California Institute of Technology. p. 8–9.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Project+Mathematics%21+Program+Guide+and+Workbook%3A+Similarity&rft.pages=8-9&rft.pub=California+Institute+of+Technology&rft.date=1990&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2F02-the-story-of-pi%2F01%2520Similarity%2Fpage%2F9%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> Workbook accompanying <i><a href="/wiki/Project_Mathematics!" title="Project Mathematics!">Project Mathematics!</a></i> <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=vpxWyJg4_1A">Ep. 1: "Similarity"</a> (Video).</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="CITEREFGellertGottwaldHellwichKästner1989" class="citation book cs1">Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; Küstner, H. (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/vnrconciseencycl00gell/page/161/mode/1up?q=%22square+is+given%22">"Quadrilaterals"</a>. <i>The VNR Concise Encyclopedia of Mathematics</i> (2nd ed.). New York: Van Nostrand Reinhold. § 7.5, p. 161. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-442-20590-2" title="Special:BookSources/0-442-20590-2"><bdi>0-442-20590-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quadrilaterals&rft.btitle=The+VNR+Concise+Encyclopedia+of+Mathematics&rft.place=New+York&rft.pages=%C2%A7+7.5%2C+p.+161&rft.edition=2nd&rft.pub=Van+Nostrand+Reinhold&rft.date=1989&rft.isbn=0-442-20590-2&rft.aulast=Gellert&rft.aufirst=W.&rft.au=Gottwald%2C+S.&rft.au=Hellwich%2C+M.&rft.au=K%C3%A4stner%2C+H.&rft.au=K%C3%BCstner%2C+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fvnrconciseencycl00gell%2Fpage%2F161%2Fmode%2F1up%3Fq%3D%2522square%2Bis%2Bgiven%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" 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="CITEREFHenrici1879" class="citation book cs1"><a href="/wiki/Olaus_Henrici" title="Olaus Henrici">Henrici, Olaus</a> (1879). <a rel="nofollow" class="external text" href="https://archive.org/details/elementarygeome00henrgoog/page/n164"><i>Elementary Geometry: Congruent Figures</i></a>. Longmans, Green. p. 134.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Geometry%3A+Congruent+Figures&rft.pages=134&rft.pub=Longmans%2C+Green&rft.date=1879&rft.aulast=Henrici&rft.aufirst=Olaus&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarygeome00henrgoog%2Fpage%2Fn164&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen147_131]-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen147_131]_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRich1963">Rich (1963)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.131938/page/n147">131</a>.</span> </li> <li id="cite_note-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERich1963[httpsarchiveorgdetailsinernetdli2015131938pagen135_120]_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRich1963">Rich (1963)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.131938/page/n135">120</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConwayGuy1996" class="citation book cs1"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, J. 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London: Longman, Brown, Green, and Longmans. p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Elementary+Treatise+on+Algebra%3A+Theoretical+and+Practical&rft.place=London&rft.pages=4&rft.pub=Longman%2C+Brown%2C+Green%2C+and+Longmans&rft.date=1845&rft.aulast=Thomson&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fanelementarytre01thomgoog%2Fpage%2Fn15&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEConwayGuy199630–33,_38–40-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEConwayGuy199630–33,_38–40_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFConwayGuy1996">Conway & Guy (1996)</a>, pp. 30–33, 38–40.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKonhauserVellemanWagon1997" class="citation book cs1"><a href="/wiki/Joseph_Konhauser" title="Joseph Konhauser">Konhauser, Joseph D. E.</a>; Velleman, Dan; <a href="/wiki/Stan_Wagon" title="Stan Wagon">Wagon, Stan</a> (1997). "95. When does the perimeter equal the area?". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29"><i>Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries</i></a>. Dolciani Mathematical Expositions. Vol. 18. Cambridge University Press. p. 29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780883853252" title="Special:BookSources/9780883853252"><bdi>9780883853252</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=95.+When+does+the+perimeter+equal+the+area%3F&rft.btitle=Which+Way+Did+the+Bicycle+Go%3F%3A+And+Other+Intriguing+Mathematical+Mysteries&rft.series=Dolciani+Mathematical+Expositions&rft.pages=29&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=9780883853252&rft.aulast=Konhauser&rft.aufirst=Joseph+D.+E.&rft.au=Velleman%2C+Dan&rft.au=Wagon%2C+Stan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DElSi5V5uS2MC%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Page 147 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChakerian1979" class="citation book cs1">Chakerian, G. 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Informa UK Limited: <span class="nowrap">209–</span>228. <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%2F00681288.1850.11886925">10.1080/00681288.1850.11886925</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+the+British+Archaeological+Association&rft.atitle=On+the+structure+of+the+Norman+Fortress+in+England&rft.volume=6&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E209-%3C%2Fspan%3E228&rft.date=1850-10&rft_id=info%3Adoi%2F10.1080%2F00681288.1850.11886925&rft.aulast=Bruce&rft.aufirst=J.+Collingwood&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjournalbritishar06brituoft%2Fpage%2F208&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> See <a rel="nofollow" class="external text" href="https://archive.org/details/journalbritishar06brituoft/page/212">p. 213</a>.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChoi2000" class="citation thesis cs1">Choi, Yongsun (2000). <i>A Study on Planning and Development of Tall Building: The Exploration of Planning Considerations</i> (Ph.D. thesis). Illinois Institute of Technology. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ProQuest" title="ProQuest">ProQuest</a> <a rel="nofollow" class="external text" href="https://www.proquest.com/docview/304600838">304600838</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=A+Study+on+Planning+and+Development+of+Tall+Building%3A+The+Exploration+of+Planning+Considerations&rft.inst=Illinois+Institute+of+Technology&rft.date=2000&rft.aulast=Choi&rft.aufirst=Yongsun&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> See in particular pp. 88–90</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFXu2010" class="citation journal cs1">Xu, Ping (Fall 2010). 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"The outmoded instant: From Instagram to Polaroid". <i>Afterimage</i>. <b>45</b> (5). University of California Press: <span class="nowrap">10–</span>15. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1525%2Faft.2018.45.5.10">10.1525/aft.2018.45.5.10</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Afterimage&rft.atitle=The+outmoded+instant%3A+From+Instagram+to+Polaroid&rft.volume=45&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E10-%3C%2Fspan%3E15&rft.date=2018-09&rft_id=info%3Adoi%2F10.1525%2Faft.2018.45.5.10&rft.aulast=Chester&rft.aufirst=Alicia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdams1980" class="citation book cs1"><a href="/wiki/Ansel_Adams" title="Ansel Adams">Adams, Ansel</a> (1980). "Medium-Format Cameras". <i>The Camera</i>. Boston: New York Graphic Society. Ch. 3, pp. 21–28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Medium-Format+Cameras&rft.btitle=The+Camera&rft.place=Boston&rft.pages=Ch.+3%2C+pp.-21-28&rft.pub=New+York+Graphic+Society&rft.date=1980&rft.aulast=Adams&rft.aufirst=Ansel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMai2016" class="citation conference cs1">Mai, James (2016). <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2016/bridges2016-233.html">"Planes and frames: spatial layering in Josef Albers' <i>Homage to the Square</i> paintings"</a>. 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"Mathematics in Baseball". <i>The Mathematics Teacher</i>. <b>86</b> (4): <span class="nowrap">336–</span>342. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fmt.86.4.0336">10.5951/mt.86.4.0336</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27968332">27968332</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=Mathematics+in+Baseball&rft.volume=86&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E336-%3C%2Fspan%3E342&rft.date=1993-04&rft_id=info%3Adoi%2F10.5951%2Fmt.86.4.0336&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27968332%23id-name%3DJSTOR&rft.aulast=Battista&rft.aufirst=Michael+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> See p. 339.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChetwynd2016" class="citation book cs1">Chetwynd, Josh (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xmPhCwAAQBAJ&pg=PA122"><i>The Field Guide to Sports Metaphors: A Compendium of Competitive Words and Idioms</i></a>. Ten Speed Press. p. 122. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781607748113" title="Special:BookSources/9781607748113"><bdi>9781607748113</bdi></a>. <q>The decision to go oxymoron with a squared "ring" had taken place by the late 1830s ... Despite the geometric shift, the language was set.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Field+Guide+to+Sports+Metaphors%3A+A+Compendium+of+Competitive+Words+and+Idioms&rft.pages=122&rft.pub=Ten+Speed+Press&rft.date=2016&rft.isbn=9781607748113&rft.aulast=Chetwynd&rft.aufirst=Josh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxmPhCwAAQBAJ%26pg%3DPA122&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSciarappaHenle2022" class="citation journal cs1">Sciarappa, Luke; Henle, Jim (2022). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8889875">"Square Dance from a Mathematical Perspective"</a>. <i>The Mathematical Intelligencer</i>. <b>44</b> (1): <span class="nowrap">58–</span>64. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-021-10151-0">10.1007/s00283-021-10151-0</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/PMC8889875">8889875</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/35250151">35250151</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=Square+Dance+from+a+Mathematical+Perspective&rft.volume=44&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E58-%3C%2Fspan%3E64&rft.date=2022&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8889875%23id-name%3DPMC&rft_id=info%3Apmid%2F35250151&rft_id=info%3Adoi%2F10.1007%2Fs00283-021-10151-0&rft.aulast=Sciarappa&rft.aufirst=Luke&rft.au=Henle%2C+Jim&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8889875&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWorthen2010" class="citation book cs1">Worthen, William B. (2010). <a rel="nofollow" class="external text" href="https://www.textures-archiv.geisteswissenschaften.fu-berlin.de/wp-content/uploads/2010/08/worthen_bill_2010_08.pdf">"Quad: Euclidean Dramaturgies"</a> <span class="cs1-format">(PDF)</span>. <i>Drama: Between Poetry and Performance</i>. Wiley. Ch. 4.i, pp. 196–204. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-405-15342-3" title="Special:BookSources/978-1-405-15342-3"><bdi>978-1-405-15342-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quad%3A+Euclidean+Dramaturgies&rft.btitle=Drama%3A+Between+Poetry+and+Performance&rft.pages=Ch.+4.i%2C+pp.-196-204&rft.pub=Wiley&rft.date=2010&rft.isbn=978-1-405-15342-3&rft.aulast=Worthen&rft.aufirst=William+B.&rft_id=https%3A%2F%2Fwww.textures-archiv.geisteswissenschaften.fu-berlin.de%2Fwp-content%2Fuploads%2F2010%2F08%2Fworthen_bill_2010_08.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLangLiangzhi2024" class="citation book cs1">Lang, Ye; Liangzhi, Zhu (2024). 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Springer Nature Singapore. pp. <span class="nowrap">469–</span>476. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-97-4511-1_38">10.1007/978-981-97-4511-1_38</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789819745111" title="Special:BookSources/9789819745111"><bdi>9789819745111</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Weiqi%3A+A+Game+of+Wits&rft.btitle=Insights+into+Chinese+Culture&rft.pages=%3Cspan+class%3D%22nowrap%22%3E469-%3C%2Fspan%3E476&rft.pub=Springer+Nature+Singapore&rft.date=2024&rft_id=info%3Adoi%2F10.1007%2F978-981-97-4511-1_38&rft.isbn=9789819745111&rft.aulast=Lang&rft.aufirst=Ye&rft.au=Liangzhi%2C+Zhu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> See page 472.</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNewman1961" class="citation journal cs1">Newman, James R. (August 1961). "About the rich lore of games played on boards and tables (review of <i>Board and Table Games From Many Civilizations</i> by R. C. Bell)". <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. <b>205</b> (2): <span class="nowrap">155–</span>161. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0861-155">10.1038/scientificamerican0861-155</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24937045">24937045</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=About+the+rich+lore+of+games+played+on+boards+and+tables+%28review+of+Board+and+Table+Games+From+Many+Civilizations+by+R.+C.+Bell%29&rft.volume=205&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E155-%3C%2Fspan%3E161&rft.date=1961-08&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0861-155&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24937045%23id-name%3DJSTOR&rft.aulast=Newman&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDonovan2017" class="citation book cs1">Donovan, Tristan (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PAyrDgAAQBAJ&pg=PA10"><i>It's All a Game: The History of Board Games from Monopoly to Settlers of Catan</i></a>. St. Martin's. pp. <span class="nowrap">10–</span>14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781250082725" title="Special:BookSources/9781250082725"><bdi>9781250082725</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=It%27s+All+a+Game%3A+The+History+of+Board+Games+from+Monopoly+to+Settlers+of+Catan&rft.pages=%3Cspan+class%3D%22nowrap%22%3E10-%3C%2Fspan%3E14&rft.pub=St.+Martin%27s&rft.date=2017&rft.isbn=9781250082725&rft.aulast=Donovan&rft.aufirst=Tristan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPAyrDgAAQBAJ%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKlarreich2004" class="citation journal cs1"><a href="/wiki/Erica_Klarreich" title="Erica Klarreich">Klarreich, Erica</a> (May 15, 2004). <a rel="nofollow" class="external text" href="https://www.sciencenews.org/article/glimpses-genius">"Glimpses of Genius: mathematicians and historians piece together a puzzle that Archimedes pondered"</a>. <i>Science News</i>: <span class="nowrap">314–</span>315. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F4015223">10.2307/4015223</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4015223">4015223</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+News&rft.atitle=Glimpses+of+Genius%3A+mathematicians+and+historians+piece+together+a+puzzle+that+Archimedes+pondered&rft.pages=%3Cspan+class%3D%22nowrap%22%3E314-%3C%2Fspan%3E315&rft.date=2004-05-15&rft_id=info%3Adoi%2F10.2307%2F4015223&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4015223%23id-name%3DJSTOR&rft.aulast=Klarreich&rft.aufirst=Erica&rft_id=https%3A%2F%2Fwww.sciencenews.org%2Farticle%2Fglimpses-genius&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGolomb1994" class="citation book cs1">Golomb, Solomon W. 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Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08573-0" title="Special:BookSources/0-691-08573-0"><bdi>0-691-08573-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1291821">1291821</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyominoes%3A+Puzzles%2C+Patterns%2C+Problems%2C+and+Packings&rft.edition=2nd&rft.pub=Princeton+University+Press&rft.date=1994&rft.isbn=0-691-08573-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1291821%23id-name%3DMR&rft.aulast=Golomb&rft.aufirst=Solomon+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThomann2008" class="citation book cs1">Thomann, Johannes (2008). "Chapter Five: Square Horoscope Diagrams In Middle Eastern Astrology And Chinese Cosmological Diagrams: Were These Designs Transmitted Through The Silk Road?". In Forêt, Philippe; Kaplony, Andreas (eds.). <i>The Journey of Maps and Images on the Silk Road</i>. Brill's Inner Asian Library. Vol. 21. BRILL. pp. <span class="nowrap">97–</span>118. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1163%2Fej.9789004171657.i-248.45">10.1163/ej.9789004171657.i-248.45</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789004171657" title="Special:BookSources/9789004171657"><bdi>9789004171657</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+Five%3A+Square+Horoscope+Diagrams+In+Middle+Eastern+Astrology+And+Chinese+Cosmological+Diagrams%3A+Were+These+Designs+Transmitted+Through+The+Silk+Road%3F&rft.btitle=The+Journey+of+Maps+and+Images+on+the+Silk+Road&rft.series=Brill%27s+Inner+Asian+Library&rft.pages=%3Cspan+class%3D%22nowrap%22%3E97-%3C%2Fspan%3E118&rft.pub=BRILL&rft.date=2008&rft_id=info%3Adoi%2F10.1163%2Fej.9789004171657.i-248.45&rft.isbn=9789004171657&rft.aulast=Thomann&rft.aufirst=Johannes&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCipra" class="citation journal cs1">Cipra, Barry A. <a rel="nofollow" class="external text" href="https://mpadocuments.s3.amazonaws.com/origami/InTheFold.pdf">"In the Fold: Origami Meets Mathematics"</a> <span class="cs1-format">(PDF)</span>. <i>SIAM News</i>. <b>34</b> (8).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+News&rft.atitle=In+the+Fold%3A+Origami+Meets+Mathematics&rft.volume=34&rft.issue=8&rft.aulast=Cipra&rft.aufirst=Barry+A.&rft_id=https%3A%2F%2Fmpadocuments.s3.amazonaws.com%2Forigami%2FInTheFold.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWickstrom2014" class="citation journal cs1">Wickstrom, Megan H. (November 2014). "Piecing it together". <i>Teaching Children Mathematics</i>. <b>21</b> (4): <span class="nowrap">220–</span>227. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fteacchilmath.21.4.0220">10.5951/teacchilmath.21.4.0220</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/10.5951/teacchilmath.21.4.0220">10.5951/teacchilmath.21.4.0220</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Teaching+Children+Mathematics&rft.atitle=Piecing+it+together&rft.volume=21&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E220-%3C%2Fspan%3E227&rft.date=2014-11&rft_id=info%3Adoi%2F10.5951%2Fteacchilmath.21.4.0220&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F10.5951%2Fteacchilmath.21.4.0220%23id-name%3DJSTOR&rft.aulast=Wickstrom&rft.aufirst=Megan+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNyamweya2024" class="citation book cs1">Nyamweya, Jeff (2024). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3404EQAAQBAJ&pg=PA76"><i>Everything Graphic Design: A Comprehensive Understanding of Visual Communications for Beginners & Creatives</i></a>. Bogano. p. 78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789914371413" title="Special:BookSources/9789914371413"><bdi>9789914371413</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Everything+Graphic+Design%3A+A+Comprehensive+Understanding+of+Visual+Communications+for+Beginners+%26+Creatives&rft.pages=78&rft.pub=Bogano&rft.date=2024&rft.isbn=9789914371413&rft.aulast=Nyamweya&rft.aufirst=Jeff&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3404EQAAQBAJ%26pg%3DPA76&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoutell1864" class="citation book cs1">Boutell, Charles (1864). <i>Heraldry, Historical and Popular</i> (2nd ed.). London: Bentley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/heraldryhistori00boutgoog/page/n58">31</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/heraldryhistori00boutgoog/page/88">89</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Heraldry%2C+Historical+and+Popular&rft.place=London&rft.pages=31%2C+89&rft.edition=2nd&rft.pub=Bentley&rft.date=1864&rft.aulast=Boutell&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=aIEvEQAAQBAJ&pg=PA200"><i>Complete Flags of the World: The Ultimate Pocket Guide</i></a> (7th ed.). DK Penguin Random House. 2021. pp. <span class="nowrap">200–</span>206. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7440-6001-0" title="Special:BookSources/978-0-7440-6001-0"><bdi>978-0-7440-6001-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complete+Flags+of+the+World%3A+The+Ultimate+Pocket+Guide&rft.pages=%3Cspan+class%3D%22nowrap%22%3E200-%3C%2Fspan%3E206&rft.edition=7th&rft.pub=DK+Penguin+Random+House&rft.date=2021&rft.isbn=978-0-7440-6001-0&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaIEvEQAAQBAJ%26pg%3DPA200&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKanTengChen2011" class="citation book cs1">Kan, Tai-Wei; Teng, Chin-Hung; Chen, Mike Y. (2011). "QR code based augmented reality applications". In Furht, Borko (ed.). <i>Handbook of Augmented Reality</i>. Springer. pp. <span class="nowrap">339–</span>354. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-0064-6_16">10.1007/978-1-4614-0064-6_16</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781461400646" title="Special:BookSources/9781461400646"><bdi>9781461400646</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=QR+code+based+augmented+reality+applications&rft.btitle=Handbook+of+Augmented+Reality&rft.pages=%3Cspan+class%3D%22nowrap%22%3E339-%3C%2Fspan%3E354&rft.pub=Springer&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2F978-1-4614-0064-6_16&rft.isbn=9781461400646&rft.aulast=Kan&rft.aufirst=Tai-Wei&rft.au=Teng%2C+Chin-Hung&rft.au=Chen%2C+Mike+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> See especially <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fG8JUdrScsYC&pg=PA341">Section 2.1, Appearance, pp. 341–342</a>.</span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRybczynski2000" class="citation book cs1">Rybczynski, Witold (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nv9L_FxyhuIC&pg=PA80"><i>One Good Turn: A Natural History of the Screwdriver and the Screw</i></a>. Scribner. pp. <span class="nowrap">80–</span>83. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-684-86730-4" title="Special:BookSources/978-0-684-86730-4"><bdi>978-0-684-86730-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=One+Good+Turn%3A+A+Natural+History+of+the+Screwdriver+and+the+Screw&rft.pages=%3Cspan+class%3D%22nowrap%22%3E80-%3C%2Fspan%3E83&rft.pub=Scribner&rft.date=2000&rft.isbn=978-0-684-86730-4&rft.aulast=Rybczynski&rft.aufirst=Witold&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dnv9L_FxyhuIC%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCharlesworthLind1990" class="citation book cs1">Charlesworth, Rosalind; Lind, Karen (1990). <i>Math and Science for Young Children</i>. Delmar Publishers. p. 195. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780827334021" title="Special:BookSources/9780827334021"><bdi>9780827334021</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Math+and+Science+for+Young+Children&rft.pages=195&rft.pub=Delmar+Publishers&rft.date=1990&rft.isbn=9780827334021&rft.aulast=Charlesworth&rft.aufirst=Rosalind&rft.au=Lind%2C+Karen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYanagihara2014" class="citation book cs1">Yanagihara, Dawn (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vkr0AgAAQBAJ&pg=PA11"><i>Waffles: Sweet, Savory, Simple</i></a>. Chronicle Books. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781452138411" title="Special:BookSources/9781452138411"><bdi>9781452138411</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Waffles%3A+Sweet%2C+Savory%2C+Simple&rft.pages=11&rft.pub=Chronicle+Books&rft.date=2014&rft.isbn=9781452138411&rft.aulast=Yanagihara&rft.aufirst=Dawn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dvkr0AgAAQBAJ%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKraigSen2013" class="citation book cs1">Kraig, Bruce; Sen, Colleen Taylor (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VhXHEAAAQBAJ&pg=PA50"><i>Street Food around the World: An Encyclopedia of Food and Culture</i></a>. Bloomsbury Publishing USA. p. 50. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781598849554" title="Special:BookSources/9781598849554"><bdi>9781598849554</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Street+Food+around+the+World%3A+An+Encyclopedia+of+Food+and+Culture&rft.pages=50&rft.pub=Bloomsbury+Publishing+USA&rft.date=2013&rft.isbn=9781598849554&rft.aulast=Kraig&rft.aufirst=Bruce&rft.au=Sen%2C+Colleen+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVhXHEAAAQBAJ%26pg%3DPA50&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJesperson1989" class="citation book cs1">Jesperson, Ivan F. (1989). <i>Fat-Back and Molasses</i>. Breakwater Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780920502044" title="Special:BookSources/9780920502044"><bdi>9780920502044</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fat-Back+and+Molasses&rft.pub=Breakwater+Books&rft.date=1989&rft.isbn=9780920502044&rft.aulast=Jesperson&rft.aufirst=Ivan+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> Caramel squares and date squares, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xWEjlBCsSaMC&pg=PA134">p. 134</a>; lemon squares, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xWEjlBCsSaMC&pg=PA104">p. 104</a>.</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAllen2015" class="citation book cs1">Allen, Gary (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nz0pCgAAQBAJ&pg=PA57"><i>Sausage: A Global History</i></a>. Reaktion Books. p. 57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781780235554" title="Special:BookSources/9781780235554"><bdi>9781780235554</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sausage%3A+A+Global+History&rft.pages=57&rft.pub=Reaktion+Books&rft.date=2015&rft.isbn=9781780235554&rft.aulast=Allen&rft.aufirst=Gary&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dnz0pCgAAQBAJ%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHarbutt2015" class="citation book cs1">Harbutt, Juliet (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kMvlBwAAQBAJ&pg=PA45"><i>World Cheese Book</i></a>. Penguin. p. 45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781465443724" title="Special:BookSources/9781465443724"><bdi>9781465443724</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=World+Cheese+Book&rft.pages=45&rft.pub=Penguin&rft.date=2015&rft.isbn=9781465443724&rft.aulast=Harbutt&rft.aufirst=Juliet&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkMvlBwAAQBAJ%26pg%3DPA45&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRosenthalRosenthalRosenthal2018" class="citation book cs1">Rosenthal, Daniel; Rosenthal, David; Rosenthal, Peter (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JQGQDwAAQBAJ&pg=PA108"><i>A Readable Introduction to Real Mathematics</i></a>. Undergraduate Texts in Mathematics (2nd ed.). Springer International Publishing. p. 108. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-00632-7">10.1007/978-3-030-00632-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783030006327" title="Special:BookSources/9783030006327"><bdi>9783030006327</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Readable+Introduction+to+Real+Mathematics&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=108&rft.edition=2nd&rft.pub=Springer+International+Publishing&rft.date=2018&rft_id=info%3Adoi%2F10.1007%2F978-3-030-00632-7&rft.isbn=9783030006327&rft.aulast=Rosenthal&rft.aufirst=Daniel&rft.au=Rosenthal%2C+David&rft.au=Rosenthal%2C+Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJQGQDwAAQBAJ%26pg%3DPA108&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-iobst-76"><span class="mw-cite-backlink">^ <a href="#cite_ref-iobst_76-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-iobst_76-1"><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="CITEREFIobst2018" class="citation journal cs1">Iobst, Christopher Simon (14 June 2018). <a rel="nofollow" class="external text" href="https://ojs.library.osu.edu/index.php/OJSM/article/view/6367">"Shapes and Their Equations: Experimentation with Desmos"</a>. <i>Ohio Journal of School Mathematics</i>. <b>79</b> (1): <span class="nowrap">27–</span>31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ohio+Journal+of+School+Mathematics&rft.atitle=Shapes+and+Their+Equations%3A+Experimentation+with+Desmos&rft.volume=79&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E27-%3C%2Fspan%3E31&rft.date=2018-06-14&rft.aulast=Iobst&rft.aufirst=Christopher+Simon&rft_id=https%3A%2F%2Fojs.library.osu.edu%2Findex.php%2FOJSM%2Farticle%2Fview%2F6367&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVince2011" class="citation cs1">Vince, John (2011). <i>Rotation Transforms for Computer Graphics</i>. London: Springer. p. 11. <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/2011rtfc.book.....V">2011rtfc.book.....V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-85729-154-7">10.1007/978-0-85729-154-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780857291547" title="Special:BookSources/9780857291547"><bdi>9780857291547</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rotation+Transforms+for+Computer+Graphics&rft.place=London&rft.pages=11&rft.pub=Springer&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2F978-0-85729-154-7&rft_id=info%3Abibcode%2F2011rtfc.book.....V&rft.isbn=9780857291547&rft.aulast=Vince&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-78">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNahin2010" class="citation book cs1">Nahin, Paul (2010). <i>An Imaginary Tale: The Story 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 {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" /></span></i>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OPyPwaElDvUC&pg=PA54">54</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781400833894" title="Special:BookSources/9781400833894"><bdi>9781400833894</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Imaginary+Tale%3A+The+Story+of+MATH+RENDER+ERROR&rft.pages=54&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=9781400833894&rft.aulast=Nahin&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-numberverse-79"><span class="mw-cite-backlink">^ <a href="#cite_ref-numberverse_79-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-numberverse_79-1"><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="CITEREFMcLemanMcNicholasStarr2022" class="citation book cs1">McLeman, Cam; McNicholas, Erin; Starr, Colin (2022). <i>Explorations in Number Theory: Commuting through the Numberverse</i>. Undergraduate Texts in Mathematics. Springer International Publishing. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G7OiEAAAQBAJ&pg=PA7">7</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-98931-6">10.1007/978-3-030-98931-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783030989316" title="Special:BookSources/9783030989316"><bdi>9783030989316</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Explorations+in+Number+Theory%3A+Commuting+through+the+Numberverse&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=7&rft.pub=Springer+International+Publishing&rft.date=2022&rft_id=info%3Adoi%2F10.1007%2F978-3-030-98931-6&rft.isbn=9783030989316&rft.aulast=McLeman&rft.aufirst=Cam&rft.au=McNicholas%2C+Erin&rft.au=Starr%2C+Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-80">^</a></b></span> <span class="reference-text"><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, Book I, Proposition 46. <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/elements/bookI/propI46.html">Online English version</a> by <a href="/wiki/David_E._Joyce" title="David E. Joyce">David E. Joyce</a>.</span> </li> <li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMartin1998" class="citation book cs1">Martin, George E. (1998). <a href="/wiki/Geometric_Constructions" title="Geometric Constructions"><i>Geometric Constructions</i></a>. Undergraduate Texts in Mathematics. Springer-Verlag, New York. p. 46. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98276-0" title="Special:BookSources/0-387-98276-0"><bdi>0-387-98276-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Constructions&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=46&rft.pub=Springer-Verlag%2C+New+York&rft.date=1998&rft.isbn=0-387-98276-0&rft.aulast=Martin&rft.aufirst=George+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text"><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, Book IV, <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/elements/bookIV/propIV6.html">Proposition 6</a>, <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/elements/bookIV/propIV7.html">Proposition 7</a>. Online English version by <a href="/wiki/David_E._Joyce" title="David E. Joyce">David E. Joyce</a>.</span> </li> <li id="cite_note-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-83">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoxeter1948" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1948). <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)"><i>Regular Polytopes</i></a>. Methuen and Co. p. 2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Polytopes&rft.pages=2&rft.pub=Methuen+and+Co.&rft.date=1948&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTECoxeter1948148-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1948148_84-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1948">Coxeter (1948)</a>, p. 148.</span> </li> <li id="cite_note-FOOTNOTECoxeter1948122–123-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1948122–123_85-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1948">Coxeter (1948)</a>, pp. 122–123.</span> </li> <li id="cite_note-FOOTNOTECoxeter1948121–122-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1948121–122_86-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1948">Coxeter (1948)</a>, pp. 121–122.</span> </li> <li id="cite_note-FOOTNOTECoxeter1948122,_126-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1948122,_126_87-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1948">Coxeter (1948)</a>, pp. 122, 126.</span> </li> <li id="cite_note-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-88">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarkerHowe2007" class="citation book cs1">Barker, William; <a href="/wiki/Roger_Evans_Howe" title="Roger Evans Howe">Howe, Roger</a> (2007). <i>Continuous Symmetry: From Euclid to Klein</i>. 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New York: Springer-Verlag. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0871-6">10.1007/978-1-4612-0871-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94265-3" title="Special:BookSources/0-387-94265-3"><bdi>0-387-94265-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1299533">1299533</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space-Filling+Curves&rft.place=New+York&rft.series=Universitext&rft.pub=Springer-Verlag&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1299533%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0871-6&rft.isbn=0-387-94265-3&rft.aulast=Sagan&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span> For the Hilbert curve, see p. 10; for the Peano curve, see p. 35; for the Sierpiński curve, see p. 51.</span> </li> <li id="cite_note-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-90">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBursteddeHolkeIsaac2019" class="citation journal cs1">Burstedde, Carsten; Holke, Johannes; Isaac, Tobin (2019). 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Springer. pp. <span class="nowrap">3–</span>4. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-10-1804-6">10.1007/978-981-10-1804-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-10-1804-6" title="Special:BookSources/978-981-10-1804-6"><bdi>978-981-10-1804-6</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3728290">3728290</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+II&rft.series=Texts+and+Readings+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E4&rft.pub=Springer&rft.date=2016&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3728290%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-981-10-1804-6&rft.isbn=978-981-10-1804-6&rft.aulast=Tao&rft.aufirst=Terence&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare" class="Z3988"></span></span> </li> <li id="cite_note-maraner-131"><span class="mw-cite-backlink">^ <a href="#cite_ref-maraner_131-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-maraner_131-1"><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="CITEREFMaraner2010" class="citation journal cs1">Maraner, Paolo (2010). 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class="nv-view"><a href="/wiki/Template:Polytopes" title="Template:Polytopes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polytopes" title="Template talk:Polytopes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polytopes" title="Special:EditPage/Template:Polytopes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div class="navbar-ct-mini">Fundamental convex <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a> and <a href="/wiki/Uniform_polytope" title="Uniform polytope">uniform polytopes</a> in dimensions 2–10</div> </th></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Coxeter_group#Finite_Coxeter_groups" title="Coxeter group">Family</a> </th> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group"><span class="texhtml"><i>A</i><sub><i>n</i></sub></span></a> </td> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group"><span class="texhtml"><i>B</i><sub><i>n</i></sub></span></a> </td> <td style="background:gainsboro;"><span style="background-color: #f0f0e0; color:;"><span class="texhtml"><i>I</i><sub>2</sub>(p) / <a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group"><i>D</i><sub><i>n</i></sub></a></span></span> </td> <td style="background:gainsboro;"><span class="texhtml"><span style="background-color: #f0e0e0; color:;"><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)"><i>E</i><sub>6</sub></a> / <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)"><i>E</i><sub>7</sub></a> / <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)"><i>E</i><sub>8</sub></a></span> / <span style="background-color: #e0f0e0; color:;"><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)"><i>F</i><sub>4</sub></a></span> / <span style="background-color: #e0e0f0; color:;"><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)"><i>G</i><sub>2</sub></a></span></span> </td> <td style="background:gainsboro;"><a href="/wiki/H4_(mathematics)" class="mw-redirect" title="H4 (mathematics)"><span class="texhtml mvar" style="font-style:italic;">H<sub>n</sub></span></a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a> </th> <td><a href="/wiki/Equilateral_triangle" title="Equilateral triangle">Triangle</a> </td> <td><a class="mw-selflink selflink">Square</a> </td> <td style="background:#f0f0e0;"><a href="/wiki/Regular_polygon" title="Regular polygon">p-gon</a> </td> <td style="background:#e0e0f0;"><a href="/wiki/Hexagon" title="Hexagon">Hexagon</a> </td> <td><a href="/wiki/Pentagon" title="Pentagon">Pentagon</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">Uniform polyhedron</a> </th> <td style=""><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a> </td> <td style=""><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a> • <a href="/wiki/Cube" title="Cube">Cube</a> </td> <td style=""><a href="/wiki/Tetrahedron" title="Tetrahedron">Demicube</a> </td> <td style=""> </td> <td style=""><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Dodecahedron</a> • <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">Icosahedron</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polychoron" class="mw-redirect" title="Uniform polychoron">Uniform polychoron</a> </th> <td><a href="/wiki/5-cell" title="5-cell">Pentachoron</a> </td> <td><a href="/wiki/16-cell" title="16-cell">16-cell</a> • <a href="/wiki/Tesseract" title="Tesseract">Tesseract</a> </td> <td><a href="/wiki/16-cell" title="16-cell">Demitesseract</a> </td> <td style="background:#e0f0e0;"><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td> <td><a href="/wiki/120-cell" title="120-cell">120-cell</a> • <a href="/wiki/600-cell" title="600-cell">600-cell</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_5-polytope" title="Uniform 5-polytope">Uniform 5-polytope</a> </th> <td style=""><a href="/wiki/5-simplex" title="5-simplex">5-simplex</a> </td> <td style=""><a href="/wiki/5-orthoplex" title="5-orthoplex">5-orthoplex</a> • <a href="/wiki/5-cube" title="5-cube">5-cube</a> </td> <td style=""><a href="/wiki/5-demicube" title="5-demicube">5-demicube</a> </td> <td style=""> </td> <td style=""> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_6-polytope" title="Uniform 6-polytope">Uniform 6-polytope</a> </th> <td><a href="/wiki/6-simplex" title="6-simplex">6-simplex</a> </td> <td><a href="/wiki/6-orthoplex" title="6-orthoplex">6-orthoplex</a> • <a href="/wiki/6-cube" title="6-cube">6-cube</a> </td> <td><a href="/wiki/6-demicube" title="6-demicube">6-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_22_polytope" title="1 22 polytope">1<sub>22</sub></a> • <a href="/wiki/2_21_polytope" title="2 21 polytope">2<sub>21</sub></a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_7-polytope" title="Uniform 7-polytope">Uniform 7-polytope</a> </th> <td style=""><a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> </td> <td style=""><a href="/wiki/7-orthoplex" title="7-orthoplex">7-orthoplex</a> • <a href="/wiki/7-cube" title="7-cube">7-cube</a> </td> <td style=""><a href="/wiki/7-demicube" title="7-demicube">7-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_32_polytope" title="1 32 polytope">1<sub>32</sub></a> • <a href="/wiki/2_31_polytope" title="2 31 polytope">2<sub>31</sub></a> • <a href="/wiki/3_21_polytope" title="3 21 polytope">3<sub>21</sub></a> </td> <td style=""> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_8-polytope" title="Uniform 8-polytope">Uniform 8-polytope</a> </th> <td><a href="/wiki/8-simplex" title="8-simplex">8-simplex</a> </td> <td><a href="/wiki/8-orthoplex" title="8-orthoplex">8-orthoplex</a> • <a href="/wiki/8-cube" title="8-cube">8-cube</a> </td> <td><a href="/wiki/8-demicube" title="8-demicube">8-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_42_polytope" title="1 42 polytope">1<sub>42</sub></a> • <a href="/wiki/2_41_polytope" title="2 41 polytope">2<sub>41</sub></a> • <a href="/wiki/4_21_polytope" title="4 21 polytope">4<sub>21</sub></a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_9-polytope" title="Uniform 9-polytope">Uniform 9-polytope</a> </th> <td style=""><a href="/wiki/9-simplex" title="9-simplex">9-simplex</a> </td> <td style=""><a href="/wiki/9-orthoplex" title="9-orthoplex">9-orthoplex</a> • <a href="/wiki/9-cube" title="9-cube">9-cube</a> </td> <td style=""><a href="/wiki/9-demicube" title="9-demicube">9-demicube</a> </td> <td style=""> </td> <td style=""> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_10-polytope" title="Uniform 10-polytope">Uniform 10-polytope</a> </th> <td><a href="/wiki/10-simplex" title="10-simplex">10-simplex</a> </td> <td><a href="/wiki/10-orthoplex" title="10-orthoplex">10-orthoplex</a> • <a href="/wiki/10-cube" title="10-cube">10-cube</a> </td> <td><a href="/wiki/10-demicube" title="10-demicube">10-demicube</a> </td> <td> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;">Uniform <i>n</i>-<a href="/wiki/Polytope" title="Polytope">polytope</a> </th> <td style=""><i>n</i>-<a href="/wiki/Simplex" title="Simplex">simplex</a> </td> <td style=""><i>n</i>-<a href="/wiki/Cross-polytope" title="Cross-polytope">orthoplex</a> • <i>n</i>-<a href="/wiki/Hypercube" title="Hypercube">cube</a> </td> <td style=""><i>n</i>-<a href="/wiki/Demihypercube" title="Demihypercube">demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/Uniform_1_k2_polytope" title="Uniform 1 k2 polytope">1<sub>k2</sub></a> • <a href="/wiki/Uniform_2_k1_polytope" title="Uniform 2 k1 polytope">2<sub>k1</sub></a> • <a href="/wiki/Uniform_k_21_polytope" title="Uniform k 21 polytope">k<sub>21</sub></a> </td> <td style=""><i>n</i>-<a href="/wiki/Pentagonal_polytope" title="Pentagonal polytope">pentagonal polytope</a> </td></tr> <tr style="text-align:center;"> <th colspan="13" style="background:gainsboro;" class="skin-invert">Topics: <a href="/wiki/Polytope_families" class="mw-redirect" title="Polytope families">Polytope families</a> • <a href="/wiki/Regular_polytope" title="Regular polytope">Regular polytope</a> • <a href="/wiki/List_of_regular_polytopes_and_compounds" class="mw-redirect" title="List of regular polytopes and compounds">List of regular polytopes and compounds</a> </th></tr></tbody></table> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output 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title="Template:Polygons"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polygons" title="Template talk:Polygons"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polygons" title="Special:EditPage/Template:Polygons"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Polygons_(List)123" style="font-size:114%;margin:0 4em"><a href="/wiki/Polygon" title="Polygon">Polygons</a> (<a href="/wiki/List_of_polygons" title="List of polygons">List</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Triangle" title="Triangle">Triangles</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Acute</a></li> <li><a href="/wiki/Equilateral_triangle" title="Equilateral triangle">Equilateral</a></li> <li><a href="/wiki/Ideal_triangle" title="Ideal triangle">Ideal</a></li> <li><a href="/wiki/Isosceles_triangle" title="Isosceles triangle">Isosceles</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler</a></li> <li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li> <li><a href="/wiki/Right_triangle" title="Right triangle">Right</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilaterals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Apollonius_quadrilateral" title="Apollonius quadrilateral">Apollonius</a></li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li> <li><a href="/wiki/Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li> <li><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li> <li><a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal</a></li> <li><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a></li> <li><a href="/wiki/Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a class="mw-selflink selflink">Square</a></li> <li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By number <br />of sides</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">1–10 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monogon" title="Monogon">Monogon (1)</a></li> <li><a href="/wiki/Digon" title="Digon">Digon (2)</a></li> <li><a href="/wiki/Triangle" title="Triangle">Triangle (3)</a></li> <li><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral (4)</a></li> <li><a href="/wiki/Pentagon" title="Pentagon">Pentagon (5)</a></li> <li><a href="/wiki/Hexagon" title="Hexagon">Hexagon (6)</a></li> <li><a href="/wiki/Heptagon" title="Heptagon">Heptagon (7)</a></li> <li><a href="/wiki/Octagon" title="Octagon">Octagon (8)</a></li> <li><a href="/wiki/Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li> <li><a href="/wiki/Decagon" title="Decagon">Decagon (10)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 sides</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecagon" title="Hendecagon">Hendecagon (11)</a></li> <li><a href="/wiki/Dodecagon" title="Dodecagon">Dodecagon (12)</a></li> <li><a href="/wiki/Tridecagon" title="Tridecagon">Tridecagon (13)</a></li> <li><a href="/wiki/Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li> <li><a href="/wiki/Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li> <li><a href="/wiki/Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li> <li><a href="/wiki/Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li> <li><a href="/wiki/Octadecagon" title="Octadecagon">Octadecagon (18)</a></li> <li><a href="/wiki/Icosagon" title="Icosagon">Icosagon (20)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositrigon" title="Icositrigon">Icositrigon (23)</a></li> <li><a href="/wiki/Icositetragon" title="Icositetragon">Icositetragon (24)</a></li> <li><a href="/wiki/Triacontagon" title="Triacontagon">Triacontagon (30)</a></li> <li><a href="/wiki/257-gon" title="257-gon">257-gon</a></li> <li><a href="/wiki/Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li> <li><a href="/wiki/Myriagon" title="Myriagon">Myriagon (10,000)</a></li> <li><a href="/wiki/65537-gon" title="65537-gon">65537-gon</a></li> <li><a href="/wiki/Megagon" title="Megagon">Megagon (1,000,000)</a></li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon (∞)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_polygon" title="Star polygon">Star polygons</a><br /></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentagram" title="Pentagram">Pentagram</a></li> <li><a href="/wiki/Hexagram" title="Hexagram">Hexagram</a></li> <li><a href="/wiki/Heptagram" title="Heptagram">Heptagram</a></li> <li><a href="/wiki/Octagram" title="Octagram">Octagram</a></li> <li><a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li> <li><a href="/wiki/Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li> <li><a href="/wiki/Hendecagram" title="Hendecagram">Hendecagram</a></li> <li><a href="/wiki/Dodecagram" title="Dodecagram">Dodecagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concave_polygon" title="Concave polygon">Concave</a></li> <li><a href="/wiki/Convex_polygon" title="Convex polygon">Convex</a></li> <li><a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li> <li><a href="/wiki/Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li> <li><a href="/wiki/Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li> <li><a href="/wiki/Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a></li> <li><a href="/wiki/Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li> <li><a href="/wiki/Magic_polygon" title="Magic polygon">Magic</a></li> <li><a href="/wiki/Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li> <li><a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular</a></li> <li><a href="/wiki/Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li> <li><a href="/wiki/Simple_polygon" title="Simple polygon">Simple</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew</a></li> <li><a href="/wiki/Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li> <li><a href="/wiki/Tangential_polygon" title="Tangential polygon">Tangential</a></li> <li><a href="/wiki/Weakly_simple_polygon" title="Weakly simple polygon">Weakly simple</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox1201" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" 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Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"0.989","limit":"10.000"},"limitreport-memusage":{"value":8291707,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAbrahamsenStade2024\"] = 1,\n [\"CITEREFAdams1980\"] = 1,\n [\"CITEREFAllen2015\"] = 1,\n [\"CITEREFAlsinaNelsen2020\"] = 1,\n [\"CITEREFApostol1990\"] = 1,\n [\"CITEREFBarkerHowe2007\"] = 1,\n [\"CITEREFBattista1993\"] = 1,\n [\"CITEREFBeardon2012\"] = 1,\n [\"CITEREFBerendonk2017\"] = 1,\n [\"CITEREFBerger2010\"] = 1,\n [\"CITEREFBoutell1864\"] = 1,\n [\"CITEREFBright1978\"] = 1,\n [\"CITEREFBruce1850\"] = 1,\n [\"CITEREFBursteddeHolkeIsaac2019\"] = 1,\n [\"CITEREFChakerian1979\"] = 1,\n [\"CITEREFCharlesworthLind1990\"] = 1,\n [\"CITEREFChester2018\"] = 1,\n [\"CITEREFChetwynd2016\"] = 1,\n [\"CITEREFChoi2000\"] = 1,\n [\"CITEREFChungGraham2020\"] = 1,\n [\"CITEREFCipra\"] = 1,\n [\"CITEREFConwayBurgielGoodman-Strauss2008\"] = 1,\n [\"CITEREFConwayGuy1996\"] = 1,\n [\"CITEREFCox1978\"] = 1,\n [\"CITEREFCoxeter1937\"] = 1,\n [\"CITEREFCoxeter1948\"] = 1,\n [\"CITEREFCroftFalconer,_Kenneth_J.Guy,_Richard_K.1991\"] = 1,\n [\"CITEREFDonovan2017\"] = 1,\n [\"CITEREFDuffinPatchettAdamsonSimmons1984\"] = 1,\n [\"CITEREFDuijvestijn1978\"] = 1,\n [\"CITEREFEggleston1958\"] = 1,\n [\"CITEREFEstévezRoldánSegerman2023\"] = 1,\n [\"CITEREFFink2014\"] = 1,\n [\"CITEREFFisher2003\"] = 1,\n [\"CITEREFFriedman2009\"] = 1,\n [\"CITEREFGardner1980\"] = 1,\n [\"CITEREFGardner1997\"] = 1,\n [\"CITEREFGarg\"] = 1,\n [\"CITEREFGellertGottwaldHellwichKästner1989\"] = 1,\n [\"CITEREFGhirshman1961\"] = 1,\n [\"CITEREFGodfreySiddons1919\"] = 1,\n [\"CITEREFGolomb1994\"] = 1,\n [\"CITEREFGroveBenson1985\"] = 1,\n [\"CITEREFGrünbaumShephard1987\"] = 1,\n [\"CITEREFGuo2004\"] = 1,\n [\"CITEREFGutierrez\"] = 1,\n [\"CITEREFHarbutt2015\"] = 1,\n [\"CITEREFHenleHenle2008\"] = 1,\n [\"CITEREFHenrici1879\"] = 1,\n [\"CITEREFIobst2018\"] = 1,\n [\"CITEREFJesperson1989\"] = 1,\n [\"CITEREFKanTengChen2011\"] = 1,\n [\"CITEREFKasner1933\"] = 1,\n [\"CITEREFKlarreich2004\"] = 1,\n [\"CITEREFKonhauserVellemanWagon1997\"] = 1,\n [\"CITEREFKraigSen2013\"] = 1,\n [\"CITEREFLambers2016\"] = 1,\n [\"CITEREFLangLiangzhi2024\"] = 1,\n [\"CITEREFLuecking2010\"] = 1,\n [\"CITEREFMai2016\"] = 1,\n [\"CITEREFMaor2019\"] = 1,\n [\"CITEREFMaraner2010\"] = 1,\n [\"CITEREFMartin1998\"] = 1,\n [\"CITEREFMatschke2014\"] = 1,\n [\"CITEREFMcLemanMcNicholasStarr2022\"] = 1,\n [\"CITEREFMeskhishvili2021\"] = 1,\n [\"CITEREFMillard1972\"] = 1,\n [\"CITEREFMiller1903\"] = 1,\n [\"CITEREFMontanherNeumaierMarkótDomes2019\"] = 1,\n [\"CITEREFNahin2010\"] = 1,\n [\"CITEREFNakamuraOkazaki2016\"] = 1,\n [\"CITEREFNelsen2003\"] = 1,\n [\"CITEREFNewman1961\"] = 1,\n [\"CITEREFNyamweya2024\"] = 1,\n [\"CITEREFOtt2002\"] = 1,\n [\"CITEREFPakSchlenker2010\"] = 1,\n [\"CITEREFPark2016\"] = 1,\n [\"CITEREFPopko2012\"] = 1,\n [\"CITEREFPostnikov2000\"] = 1,\n [\"CITEREFRich1963\"] = 1,\n [\"CITEREFRichardson2002\"] = 1,\n [\"CITEREFRoberts2023\"] = 1,\n [\"CITEREFRosenthalRosenthalRosenthal2018\"] = 1,\n [\"CITEREFRybczynski2000\"] = 1,\n [\"CITEREFSagan1994\"] = 1,\n [\"CITEREFSalomon2011\"] = 1,\n [\"CITEREFSamet2006\"] = 1,\n [\"CITEREFSchattschneider1978\"] = 1,\n [\"CITEREFScheid1961\"] = 1,\n [\"CITEREFSchorlingClarkCarter1935\"] = 1,\n [\"CITEREFSciarappaHenle2022\"] = 1,\n [\"CITEREFSeaton2021\"] = 1,\n [\"CITEREFSinger1998\"] = 1,\n [\"CITEREFSmith1995\"] = 1,\n [\"CITEREFStillwell1992\"] = 1,\n [\"CITEREFStinyMitchell1980\"] = 1,\n [\"CITEREFSugiyama1993\"] = 1,\n [\"CITEREFTao2016\"] = 1,\n [\"CITEREFThomann2008\"] = 1,\n [\"CITEREFThomson1845\"] = 1,\n [\"CITEREFThorpe1979\"] = 1,\n [\"CITEREFTreese2018\"] = 1,\n [\"CITEREFTrustrum1965\"] = 1,\n [\"CITEREFUsiskinGriffin2008\"] = 1,\n [\"CITEREFVafea2002\"] = 1,\n [\"CITEREFVince2011\"] = 1,\n [\"CITEREFVlăduț1991\"] = 1,\n [\"CITEREFWickstrom2014\"] = 1,\n [\"CITEREFWilson2010\"] = 1,\n [\"CITEREFWorthen2010\"] = 1,\n [\"CITEREFXu2010\"] = 1,\n [\"CITEREFYanagihara2014\"] = 1,\n}\ntemplate_list = table#1 {\n [\"-\"] = 5,\n [\"About\"] = 1,\n [\"Also\"] = 1,\n [\"Authority control\"] = 1,\n [\"Bi\"] = 2,\n [\"Block indent\"] = 3,\n [\"CS1 config\"] = 1,\n [\"Citation\"] = 1,\n [\"Cite OEIS\"] = 3,\n [\"Cite book\"] = 55,\n [\"Cite conference\"] = 3,\n [\"Cite journal\"] = 48,\n [\"Cite news\"] = 1,\n [\"Cite tech report\"] = 1,\n [\"Cite thesis\"] = 1,\n [\"Cite web\"] = 3,\n [\"Commons category\"] = 1,\n [\"Infobox polygon\"] = 1,\n [\"Main\"] = 4,\n [\"Math\"] = 4,\n [\"Multiple image\"] = 9,\n [\"Mvar\"] = 6,\n [\"Nowrap\"] = 2,\n [\"Pgs\"] = 2,\n [\"Pi\"] = 4,\n [\"Plainlist\"] = 1,\n [\"Polygons\"] = 1,\n [\"Polytopes\"] = 1,\n [\"Portal\"] = 1,\n [\"Pp-pc\"] = 1,\n [\"ProQuest\"] = 1,\n [\"R\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfnp\"] = 15,\n [\"Short description\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-5c6f46dcf-98mb7","timestamp":"20250331072423","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Square","url":"https:\/\/en.wikipedia.org\/wiki\/Square","sameAs":"http:\/\/www.wikidata.org\/entity\/Q164","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q164","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-05-17T01:15:38Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/a5\/Regular_polygon_4_annotated.svg","headline":"regular quadrilateral"}</script> </body> </html>