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cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In mathematics subsection</span> </button> <ul id="toc-In_mathematics-sublist" class="vector-toc-list"> <li id="toc-Introduction_to_tessellations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introduction_to_tessellations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Introduction to tessellations</span> </div> </a> <ul id="toc-Introduction_to_tessellations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wallpaper_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wallpaper_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Wallpaper groups</span> </div> </a> <ul id="toc-Wallpaper_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aperiodic_tilings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aperiodic_tilings"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Aperiodic tilings</span> </div> </a> <ul id="toc-Aperiodic_tilings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tessellations_and_colour" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tessellations_and_colour"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Tessellations and colour</span> </div> </a> <ul id="toc-Tessellations_and_colour-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tessellations_with_polygons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tessellations_with_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Tessellations with polygons</span> </div> </a> <ul id="toc-Tessellations_with_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voronoi_tilings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Voronoi_tilings"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Voronoi tilings</span> </div> </a> <ul id="toc-Voronoi_tilings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tessellations_in_higher_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tessellations_in_higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Tessellations in higher dimensions</span> </div> </a> <ul id="toc-Tessellations_in_higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tessellations_in_non-Euclidean_geometries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tessellations_in_non-Euclidean_geometries"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Tessellations in non-Euclidean geometries</span> </div> </a> <ul id="toc-Tessellations_in_non-Euclidean_geometries-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_art" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_art"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In art</span> </div> </a> <ul id="toc-In_art-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_manufacturing" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_manufacturing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In manufacturing</span> </div> </a> <ul id="toc-In_manufacturing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_nature" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_nature"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In nature</span> </div> </a> <ul id="toc-In_nature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_puzzles_and_recreational_mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_puzzles_and_recreational_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In puzzles and recreational mathematics</span> </div> </a> <ul id="toc-In_puzzles_and_recreational_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Explanatory_footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Explanatory_footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Explanatory footnotes</span> </div> </a> <ul id="toc-Explanatory_footnotes-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">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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Available in 42 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-42" 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">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B3%D9%8A%D9%81%D8%B3%D8%A7%D8%A1_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="فسيفساء (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="فسيفساء (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9F%E0%A6%BE%E0%A6%B2%E0%A6%BF%E0%A6%95%E0%A6%B0%E0%A6%A3" title="টালিকরণ – Bangla" lang="bn" hreflang="bn" data-title="টালিকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tessel%C2%B7laci%C3%B3" title="Tessel·lació – Catalan" lang="ca" hreflang="ca" data-title="Tessel·lació" 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%9F%D0%BB%D0%B0%D0%BD%D0%B8%D0%B3%D0%BE%D0%BD" title="Планигон – Chuvash" lang="cv" hreflang="cv" data-title="Планигон" 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/Teselace" title="Teselace – Czech" lang="cs" hreflang="cs" data-title="Teselace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Brithwaith" title="Brithwaith – Welsh" lang="cy" hreflang="cy" data-title="Brithwaith" 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/Tessellation" title="Tessellation – Danish" lang="da" hreflang="da" data-title="Tessellation" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Parkettierung" title="Parkettierung – German" lang="de" hreflang="de" data-title="Parkettierung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tesselatsioon" title="Tesselatsioon – Estonian" lang="et" hreflang="et" data-title="Tesselatsioon" 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%A8%CE%B7%CF%86%CE%B9%CE%B4%CE%BF%CE%B8%CE%AD%CF%84%CE%B7%CF%83%CE%B7" title="Ψηφιδοθέτηση – Greek" lang="el" hreflang="el" data-title="Ψηφιδοθέτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teselado" title="Teselado – Spanish" lang="es" hreflang="es" data-title="Teselado" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Teselazio" title="Teselazio – Basque" lang="eu" hreflang="eu" data-title="Teselazio" 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/%DA%A9%D8%A7%D8%B4%DB%8C%E2%80%8C%DA%A9%D8%A7%D8%B1%DB%8C" title="کاشی‌کاری – Persian" lang="fa" hreflang="fa" data-title="کاشی‌کاری" 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/Pavage_du_plan" title="Pavage du plan – French" lang="fr" hreflang="fr" data-title="Pavage du plan" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teselaci%C3%B3n" title="Teselación – Galician" lang="gl" hreflang="gl" data-title="Teselación" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%85%8C%EC%85%80%EB%A0%88%EC%9D%B4%EC%85%98" title="테셀레이션 – Korean" lang="ko" hreflang="ko" data-title="테셀레이션" 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%84%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%BF%D5%A1%D5%AF_(%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Մանրահատակ (երկրաչափություն) – Armenian" lang="hy" hreflang="hy" data-title="Մանրահատակ (երկրաչափություն)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teselasi" title="Teselasi – Indonesian" lang="id" hreflang="id" data-title="Teselasi" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tassellatura" title="Tassellatura – Italian" lang="it" hreflang="it" data-title="Tassellatura" 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%A6%D7%95%D7%A3_%D7%A9%D7%9C_%D7%94%D7%9E%D7%99%D7%A9%D7%95%D7%A8" title="ריצוף של המישור – Hebrew" lang="he" hreflang="he" data-title="ריצוף של המישור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Tesszal%C3%A1ci%C3%B3" title="Tesszaláció – Hungarian" lang="hu" hreflang="hu" data-title="Tesszaláció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Penteselan" title="Penteselan – Malay" lang="ms" hreflang="ms" data-title="Penteselan" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Betegeling" title="Betegeling – Dutch" lang="nl" hreflang="nl" data-title="Betegeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%BF%E3%82%A4%E3%83%AB%E5%BC%B5%E3%82%8A" title="タイル張り – Japanese" lang="ja" hreflang="ja" data-title="タイル張り" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tesselering" title="Tesselering – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tesselering" 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/Tessellering" title="Tessellering – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tessellering" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Pavimentassion" title="Pavimentassion – Piedmontese" lang="pms" hreflang="pms" data-title="Pavimentassion" 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/Parkieta%C5%BC" title="Parkietaż – Polish" lang="pl" hreflang="pl" data-title="Parkietaż" 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/Tessela%C3%A7%C3%A3o" title="Tesselação – Portuguese" lang="pt" hreflang="pt" data-title="Tesselação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Pavare" title="Pavare – Romanian" lang="ro" hreflang="ro" data-title="Pavare" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BC%D0%BE%D1%89%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Замощение (геометрия) – Russian" lang="ru" hreflang="ru" data-title="Замощение (геометрия)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tessellation" title="Tessellation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tessellation" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teselacija" title="Teselacija – Slovenian" lang="sl" hreflang="sl" data-title="Teselacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D1%81%D0%B5%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Теселација – Serbian" lang="sr" hreflang="sr" data-title="Теселација" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tessellaatio" title="Tessellaatio – Finnish" lang="fi" hreflang="fi" data-title="Tessellaatio" 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/Tessellation" title="Tessellation – Swedish" lang="sv" hreflang="sv" data-title="Tessellation" 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class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Good_articles*" title="This is a good article. Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Tiling of a plane in mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Tessellate" redirects here. For the song, see <a href="/wiki/Tessellate_(song)" title="Tessellate (song)">Tessellate (song)</a>. For the computer graphics technique, see <a href="/wiki/Tessellation_(computer_graphics)" title="Tessellation (computer graphics)">Tessellation (computer graphics)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">"Mathematical tiling" redirects here. For the building material, see <a href="/wiki/Mathematical_tile" title="Mathematical tile">Mathematical tile</a>.</div> <p class="mw-empty-elt"> </p> <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 tright"><div class="thumbinner multiimageinner" style="width:212px;max-width:212px"><div class="trow"><div class="tsingle" style="width:210px;max-width:210px"><div class="thumbimage" style="height:140px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Ceramic_Tile_Tessellations_in_Marrakech.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Ceramic_Tile_Tessellations_in_Marrakech.jpg/208px-Ceramic_Tile_Tessellations_in_Marrakech.jpg" decoding="async" width="208" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Ceramic_Tile_Tessellations_in_Marrakech.jpg/312px-Ceramic_Tile_Tessellations_in_Marrakech.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Ceramic_Tile_Tessellations_in_Marrakech.jpg/416px-Ceramic_Tile_Tessellations_in_Marrakech.jpg 2x" data-file-width="2263" data-file-height="1525" /></a></span></div><div class="thumbcaption"><a href="/wiki/Zellige" class="mw-redirect" title="Zellige">Zellige</a> <a href="/wiki/Terracotta" title="Terracotta">terracotta</a> tiles in <a href="/wiki/Marrakech" class="mw-redirect" title="Marrakech">Marrakech</a>, forming edge‑to‑edge, regular and other tessellations</div></div></div><div class="trow"><div class="tsingle" style="width:210px;max-width:210px"><div class="thumbimage" style="height:156px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Leeuwarden_-_Tegeltableau_Escher.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Leeuwarden_-_Tegeltableau_Escher.jpg/208px-Leeuwarden_-_Tegeltableau_Escher.jpg" decoding="async" width="208" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Leeuwarden_-_Tegeltableau_Escher.jpg/312px-Leeuwarden_-_Tegeltableau_Escher.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Leeuwarden_-_Tegeltableau_Escher.jpg/416px-Leeuwarden_-_Tegeltableau_Escher.jpg 2x" data-file-width="4000" data-file-height="3000" /></a></span></div><div class="thumbcaption">A wall sculpture in <a href="/wiki/Leeuwarden" title="Leeuwarden">Leeuwarden</a> celebrating the artistic tessellations of <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a></div></div></div><div class="trow"><div class="tsingle" style="width:210px;max-width:210px"><div class="thumbimage" style="height:208px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg/208px-Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg.png" decoding="async" width="208" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg/312px-Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg/416px-Birds_on_identical_tiles_edge-to-edge_like_puzzle_pieces_Non-periodic_tiling.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div><div class="thumbcaption">An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles</div></div></div></div></div> <p>A <b>tessellation</b> or <b>tiling</b> is the covering of a <a href="/wiki/Surface" title="Surface">surface</a>, often a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>, using one or more <a href="/wiki/Geometric_shape" class="mw-redirect" title="Geometric shape">geometric shapes</a>, called <i>tiles</i>, with no overlaps and no gaps. In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, tessellation can be generalized to <a href="/wiki/High-dimensional_spaces" class="mw-redirect" title="High-dimensional spaces">higher dimensions</a> and a variety of geometries. </p><p>A <b>periodic tiling</b> has a repeating pattern. Some special kinds include <i><a href="/wiki/Regular_tilings" class="mw-redirect" title="Regular tilings">regular tilings</a></i> with <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygonal</a> tiles all of the same shape, and <i><a href="/wiki/Semiregular_tiling" class="mw-redirect" title="Semiregular tiling">semiregular tilings</a></i> with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 <a href="/wiki/Wallpaper_groups" class="mw-redirect" title="Wallpaper groups">wallpaper groups</a>. A tiling that lacks a repeating pattern is called "non-periodic". An <i><a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">aperiodic tiling</a></i> uses a small set of tile shapes that cannot form a repeating pattern (an <a href="/wiki/Aperiodic_set_of_prototiles" title="Aperiodic set of prototiles">aperiodic set of prototiles</a>). A <i><a href="/wiki/Tessellation_of_space" class="mw-redirect" title="Tessellation of space">tessellation of space</a></i>, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. </p><p>A real physical tessellation is a tiling made of materials such as <a href="/wiki/Cement" title="Cement">cemented</a> <a href="/wiki/Ceramic" title="Ceramic">ceramic</a> squares or hexagons. Such tilings may be decorative <a href="/wiki/Patterns" class="mw-redirect" title="Patterns">patterns</a>, or may have functions such as providing durable and water-resistant <a href="/wiki/Pavers_(flooring)" title="Pavers (flooring)">pavement</a>, floor, or wall coverings. Historically, tessellations were used in <a href="/wiki/Ancient_Rome" title="Ancient Rome">Ancient Rome</a> and in <a href="/wiki/Islamic_art" title="Islamic art">Islamic art</a> such as in the <a href="/wiki/Moroccan_architecture" title="Moroccan architecture">Moroccan architecture</a> and <a href="/wiki/Islamic_geometric_patterns" title="Islamic geometric patterns">decorative geometric tiling</a> of the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a> palace. In the twentieth century, the work of <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a> often made use of tessellations, both in ordinary <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> and in <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, for artistic effect. Tessellations are sometimes employed for decorative effect in <a href="/wiki/Quilting" title="Quilting">quilting</a>. Tessellations form a class of <a href="/wiki/Patterns_in_nature" title="Patterns in nature">patterns in nature</a>, for example in the arrays of <a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">hexagonal cells</a> found in <a href="/wiki/Honeycomb" title="Honeycomb">honeycombs</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Stone-Cone_Temple_mosaics,_Pergamon_Museum.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Stone-Cone_Temple_mosaics%2C_Pergamon_Museum.JPG/220px-Stone-Cone_Temple_mosaics%2C_Pergamon_Museum.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Stone-Cone_Temple_mosaics%2C_Pergamon_Museum.JPG/330px-Stone-Cone_Temple_mosaics%2C_Pergamon_Museum.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Stone-Cone_Temple_mosaics%2C_Pergamon_Museum.JPG/440px-Stone-Cone_Temple_mosaics%2C_Pergamon_Museum.JPG 2x" data-file-width="2272" data-file-height="1704" /></a><figcaption>A temple mosaic from the ancient Sumerian city of <a href="/wiki/Uruk" title="Uruk">Uruk</a> IV (3400–3100 BC), showing a tessellation pattern in coloured tiles</figcaption></figure> <p>Tessellations were used by the <a href="/wiki/Architecture_of_Mesopotamia" title="Architecture of Mesopotamia">Sumerians</a> (about 4000 BC) in building wall decorations formed by patterns of clay tiles.<sup id="cite_ref-Pickover2009_1-0" class="reference"><a href="#cite_note-Pickover2009-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Decorative <a href="/wiki/Mosaic" title="Mosaic">mosaic</a> tilings made of small squared blocks called <a href="/wiki/Tessera" title="Tessera">tesserae</a> were widely employed in <a href="/wiki/Classical_antiquity" title="Classical antiquity">classical antiquity</a>,<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> sometimes displaying geometric patterns.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Field1988_4-0" class="reference"><a href="#cite_note-Field1988-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In 1619, <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his <span title="Latin-language text"><i lang="la"><a href="/wiki/Harmonices_Mundi" class="mw-redirect" title="Harmonices Mundi">Harmonices Mundi</a></i></span>; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and <a href="/wiki/Snowflakes" class="mw-redirect" title="Snowflakes">snowflakes</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEGullberg1997395_6-0" class="reference"><a href="#cite_note-FOOTNOTEGullberg1997395-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEStewart200113_7-0" class="reference"><a href="#cite_note-FOOTNOTEStewart200113-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Roman_geometric_mosaic.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Roman_geometric_mosaic.jpg/280px-Roman_geometric_mosaic.jpg" decoding="async" width="280" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Roman_geometric_mosaic.jpg/420px-Roman_geometric_mosaic.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Roman_geometric_mosaic.jpg/560px-Roman_geometric_mosaic.jpg 2x" data-file-width="1548" data-file-height="681" /></a><figcaption><a href="/wiki/Ancient_Rome" title="Ancient Rome">Roman</a> geometric mosaic</figcaption></figure> <p>Some two hundred years later in 1891, the Russian crystallographer <a href="/wiki/Yevgraf_Fyodorov" class="mw-redirect" title="Yevgraf Fyodorov">Yevgraf Fyodorov</a> proved that every periodic tiling of the plane features one of seventeen different groups of isometries.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><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> Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include <a href="/wiki/Alexei_Vasilievich_Shubnikov" title="Alexei Vasilievich Shubnikov">Alexei Vasilievich Shubnikov</a> and <a href="/wiki/Nikolay_Belov_(geochemist)" title="Nikolay Belov (geochemist)">Nikolai Belov</a> in their book <i><a href="/wiki/Colored_Symmetry_(book)" title="Colored Symmetry (book)">Colored Symmetry</a></i> (1964),<sup id="cite_ref-ShubnikovBelov1964_10-0" class="reference"><a href="#cite_note-ShubnikovBelov1964-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Heinrich_Heesch" title="Heinrich Heesch">Heinrich Heesch</a> and Otto Kienzle (1963).<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="Etymology">Etymology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=2" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Latin, <i>tessella</i> is a small cubical piece of <a href="/wiki/Clay" title="Clay">clay</a>, <a href="/wiki/Rock_(geology)" title="Rock (geology)">stone</a>, or <a href="/wiki/Glass" title="Glass">glass</a> used to make mosaics.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The word "tessella" means "small square" (from <i>tessera</i>, square, which in turn is from the Greek word τέσσερα for <i>four</i>). It corresponds to the everyday term <i>tiling</i>, which refers to applications of tessellations, often made of <a href="/wiki/Ceramic_glaze" title="Ceramic glaze">glazed</a> clay. </p> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=3" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Semi-regular-floor-3464.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Semi-regular-floor-3464.JPG/170px-Semi-regular-floor-3464.JPG" decoding="async" width="170" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Semi-regular-floor-3464.JPG/255px-Semi-regular-floor-3464.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Semi-regular-floor-3464.JPG/340px-Semi-regular-floor-3464.JPG 2x" data-file-width="960" data-file-height="1280" /></a><figcaption>A <a href="/wiki/Rhombitrihexagonal_tiling" title="Rhombitrihexagonal tiling">rhombitrihexagonal tiling</a>: tiled floor in the <a href="/wiki/Archeological_Museum_of_Seville" title="Archeological Museum of Seville">Archeological Museum of Seville</a>, Spain, using square, triangle, and hexagon prototiles</figcaption></figure> <p>Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as <i>tiles</i>, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The tessellations created by <a href="/wiki/Bond_(brick)" class="mw-redirect" title="Bond (brick)">bonded brickwork</a> do not obey this rule. Among those that do, a <a href="/wiki/Regular_tessellation" class="mw-redirect" title="Regular tessellation">regular tessellation</a> has both identical<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Regular_polygon" title="Regular polygon">regular tiles</a> and identical regular corners or vertices, having the same angle between adjacent edges for every tile.<sup id="cite_ref-FOOTNOTECoxeter1973_15-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> There are only three shapes that can form such regular tessellations: the equilateral <a href="/wiki/Triangle" title="Triangle">triangle</a>, <a href="/wiki/Square" title="Square">square</a> and the regular <a href="/wiki/Hexagon" title="Hexagon">hexagon</a>. Any one of these three shapes can be duplicated infinitely to fill a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> with no gaps.<sup id="cite_ref-FOOTNOTEGullberg1997395_6-1" class="reference"><a href="#cite_note-FOOTNOTEGullberg1997395-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Irregular tessellations can also be made from other shapes such as <a href="/wiki/Pentagons" class="mw-redirect" title="Pentagons">pentagons</a>, <a href="/wiki/Polyominoes" class="mw-redirect" title="Polyominoes">polyominoes</a> and in fact almost any kind of geometric shape. The artist <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a> is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects.<sup id="cite_ref-FOOTNOTEEscher197411–12,_15–16_17-0" class="reference"><a href="#cite_note-FOOTNOTEEscher197411–12,_15–16-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tassellatura_alhambra.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Tassellatura_alhambra.jpg/220px-Tassellatura_alhambra.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Tassellatura_alhambra.jpg/330px-Tassellatura_alhambra.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Tassellatura_alhambra.jpg/440px-Tassellatura_alhambra.jpg 2x" data-file-width="1280" data-file-height="853" /></a><figcaption>The elaborate and colourful <a href="/wiki/Zellige" class="mw-redirect" title="Zellige">zellige</a> tessellations of glazed tiles at the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a> in Spain that attracted the attention of <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a></figcaption></figure> <p>More formally, a tessellation or tiling is a <a href="/wiki/Cover_(topology)" title="Cover (topology)">cover</a> of the Euclidean plane by a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> number of closed sets, called <i>tiles</i>, such that the tiles intersect only on their <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundaries</a>. These tiles may be polygons or any other shapes.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> Many tessellations are formed from a finite number of <a href="/wiki/Prototile" title="Prototile">prototiles</a> in which all tiles in the tessellation are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to <i>tessellate</i> or to <i>tile the plane</i>. The <a href="/wiki/Conway_criterion" title="Conway criterion">Conway criterion</a> is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> No general rule has been found for determining whether a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198759_19-1" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198759-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Mathematically, tessellations can be extended to spaces other than the Euclidean plane.<sup id="cite_ref-FOOTNOTEGullberg1997395_6-2" class="reference"><a href="#cite_note-FOOTNOTEGullberg1997395-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Switzerland" title="Switzerland">Swiss</a> <a href="/wiki/Geometry" title="Geometry">geometer</a> <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> pioneered this by defining <i>polyschemes</i>, which mathematicians nowadays call <a href="/wiki/Polytopes" class="mw-redirect" title="Polytopes">polytopes</a>. These are the analogues to polygons and <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> in spaces with more dimensions. He further defined the <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is {3}, while that for a square is {4}.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is {6,3}.<sup id="cite_ref-MathWorldTessellation_23-0" class="reference"><a href="#cite_note-MathWorldTessellation-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the <a href="/wiki/Vertex_configuration" title="Vertex configuration">vertex configuration</a>, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 4<sup>4</sup>. The tiling of regular hexagons is noted 6.6.6, or 6<sup>3</sup>.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198759_19-2" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198759-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_mathematics">In mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=4" title="Edit section: In mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Introduction_to_tessellations">Introduction to tessellations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=5" title="Edit section: Introduction to tessellations"><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">Further information: <a href="/wiki/Euclidean_tilings_by_convex_regular_polygons" title="Euclidean tilings by convex regular polygons">Euclidean tilings by convex regular polygons</a>, <a href="/wiki/Uniform_tiling" title="Uniform tiling">Uniform tiling</a>, and <a href="/wiki/List_of_Euclidean_uniform_tilings" title="List of Euclidean uniform tilings">List of Euclidean uniform tilings</a></div> <p>Mathematicians use some technical terms when discussing tilings. An <i><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edge</a></i> is the intersection between two bordering tiles; it is often a straight line. A <i><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a></i> is the point of intersection of three or more bordering tiles. Using these terms, an <i>isogonal</i> or <a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a> tiling is a tiling where every vertex point is identical; that is, the arrangement of <a href="/wiki/Polygon" title="Polygon">polygons</a> about each vertex is the same.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198759_19-3" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198759-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental region</a> is a shape such as a rectangle that is repeated to form the tessellation.<sup id="cite_ref-EmmerSchattschneider2007_24-0" class="reference"><a href="#cite_note-EmmerSchattschneider2007-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> For example, a regular tessellation of the plane with squares has a meeting of <a href="/wiki/Square_tiling" title="Square tiling">four squares at every vertex</a>.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198759_19-4" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198759-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The sides of the polygons are not necessarily identical to the edges of the tiles. An <b>edge-to-edge tiling</b> is any polygonal tessellation where adjacent tiles only share one full side, i.e., no tile shares a partial side or more than one side with any other tile. In an edge-to-edge tiling, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198759_19-5" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198759-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>A <i>normal tiling</i> is a tessellation for which every tile is <a href="/wiki/Topologically" class="mw-redirect" title="Topologically">topologically</a> equivalent to a <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a>, the intersection of any two tiles is a <a href="/wiki/Connected_set" class="mw-redirect" title="Connected set">connected set</a> or the <a href="/wiki/Empty_set" title="Empty set">empty set</a>, and all tiles are <a href="/wiki/Uniformly_bounded" class="mw-redirect" title="Uniformly bounded">uniformly bounded</a>. This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling; the condition disallows tiles that are pathologically long or thin.<sup id="cite_ref-Horne2000_25-0" class="reference"><a href="#cite_note-Horne2000-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:P5-type15-chiral_coloring.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/P5-type15-chiral_coloring.png/170px-P5-type15-chiral_coloring.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/P5-type15-chiral_coloring.png/255px-P5-type15-chiral_coloring.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/P5-type15-chiral_coloring.png/340px-P5-type15-chiral_coloring.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>An example of a non-edge‑to‑edge tiling: the 15th convex <a href="https://en.wiktionary.org/wiki/monohedral" class="extiw" title="wikt:monohedral"><i>monohedral</i></a> <a href="/wiki/Pentagonal_tiling" title="Pentagonal tiling">pentagonal tiling</a>, discovered in 2015</figcaption></figure> <p><span class="anchor" id="Monohedral"></span>A <span class="nowrap"><b>monohedral tiling</b></span> is a tessellation in which all tiles are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a>; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936; the <a href="/wiki/Voderberg_tiling" title="Voderberg tiling">Voderberg tiling</a> has a unit tile that is a nonconvex <a href="/wiki/Enneagon" class="mw-redirect" title="Enneagon">enneagon</a>.<sup id="cite_ref-Pickover2009_1-1" class="reference"><a href="#cite_note-Pickover2009-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The <b>Hirschhorn tiling</b>, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a <a href="/wiki/Pentagon_tiling" class="mw-redirect" title="Pentagon tiling">pentagon tiling</a> using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the <a href="/wiki/Internal_angle" class="mw-redirect" title="Internal angle">internal angle</a> of a regular pentagon, <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">3<span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, is not a divisor of 2<span class="texhtml mvar" style="font-style:italic;">π</span>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hirschhorn1985_27-0" class="reference"><a href="#cite_note-Hirschhorn1985-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> group of the tiling.<sup id="cite_ref-Horne2000_25-1" class="reference"><a href="#cite_note-Horne2000-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms <a href="/wiki/Anisohedral_tiling" title="Anisohedral tiling">anisohedral tilings</a>. </p><p>A <a href="/wiki/Tiling_by_regular_polygons" class="mw-redirect" title="Tiling by regular polygons">regular tessellation</a> is a highly <a href="/wiki/Symmetry" title="Symmetry">symmetric</a>, edge-to-edge tiling made up of <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>, all of the same shape. There are only three regular tessellations: those made up of <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>, <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">squares</a>, or regular <a href="/wiki/Hexagon" title="Hexagon">hexagons</a>. All three of these tilings are isogonal and monohedral.<sup id="cite_ref-threeregular_28-0" class="reference"><a href="#cite_note-threeregular-28"><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:2005-06-25_Tiles_together.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/2005-06-25_Tiles_together.jpg/170px-2005-06-25_Tiles_together.jpg" decoding="async" width="170" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/2005-06-25_Tiles_together.jpg/255px-2005-06-25_Tiles_together.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/2005-06-25_Tiles_together.jpg/340px-2005-06-25_Tiles_together.jpg 2x" data-file-width="1520" data-file-height="1080" /></a><figcaption>A <a href="/wiki/Pythagorean_tiling" title="Pythagorean tiling">Pythagorean tiling</a> is not an edge‑to‑edge tiling.</figcaption></figure> <p>A <a href="/wiki/Tiling_by_regular_polygons#Archimedean,_uniform_or_semiregular_tilings" class="mw-redirect" title="Tiling by regular polygons">semi-regular (or Archimedean) tessellation</a> uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two).<sup id="cite_ref-FOOTNOTEStewart200175_29-0" class="reference"><a href="#cite_note-FOOTNOTEStewart200175-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> These can be described by their <a href="/wiki/Vertex_configuration" title="Vertex configuration">vertex configuration</a>; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.8<sup>2</sup> (each vertex has one square and two octagons).<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of <a href="/wiki/Pythagorean_tiling" title="Pythagorean tiling">Pythagorean tilings</a>, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> An <a href="/wiki/Edge_tessellation" title="Edge tessellation">edge tessellation</a> is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Wallpaper_groups">Wallpaper groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=6" title="Edit section: Wallpaper groups"><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/Wallpaper_group" title="Wallpaper group">Wallpaper group</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group-p3-1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Wallpaper_group-p3-1.jpg/170px-Wallpaper_group-p3-1.jpg" decoding="async" width="170" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Wallpaper_group-p3-1.jpg/255px-Wallpaper_group-p3-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Wallpaper_group-p3-1.jpg/340px-Wallpaper_group-p3-1.jpg 2x" data-file-width="1000" data-file-height="960" /></a><figcaption>This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.</figcaption></figure> <p>Tilings with <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a> in two independent directions can be categorized by <a href="/wiki/Wallpaper_group" title="Wallpaper group">wallpaper groups</a>, of which 17 exist.<sup id="cite_ref-armstrong_33-0" class="reference"><a href="#cite_note-armstrong-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> It has been claimed that all seventeen of these groups are represented in the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a> palace in <a href="/wiki/Granada" title="Granada">Granada</a>, <a href="/wiki/Spain" title="Spain">Spain</a>. Although this is disputed,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> the variety and sophistication of the Alhambra tilings have interested modern researchers.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Of the three regular tilings two are in the <i>p6m</i> wallpaper group and one is in <i>p4m</i>. Tilings in 2-D with translational symmetry in just one direction may be categorized by the seven frieze groups describing the possible <a href="/wiki/Frieze_pattern" class="mw-redirect" title="Frieze pattern">frieze patterns</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Orbifold_notation" title="Orbifold notation">Orbifold notation</a> can be used to describe wallpaper groups of the Euclidean plane.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Aperiodic_tilings">Aperiodic tilings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=7" title="Edit section: Aperiodic tilings"><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/Aperiodic_tiling" title="Aperiodic tiling">Aperiodic tiling</a> and <a href="/wiki/List_of_aperiodic_sets_of_tiles" title="List of aperiodic sets of tiles">List of aperiodic sets of tiles</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Penrose_Tiling_(Rhombi).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Penrose_Tiling_%28Rhombi%29.svg/170px-Penrose_Tiling_%28Rhombi%29.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Penrose_Tiling_%28Rhombi%29.svg/255px-Penrose_Tiling_%28Rhombi%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Penrose_Tiling_%28Rhombi%29.svg/340px-Penrose_Tiling_%28Rhombi%29.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>A <a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose tiling</a>, with several symmetries, but no periodic repetitions</figcaption></figure> <p><a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose tilings</a>, which use two different quadrilateral prototiles, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of <a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">aperiodic tilings</a>, which use tiles that cannot tessellate periodically. The <a href="/wiki/Recursion" title="Recursion">recursive process</a> of <a href="/wiki/Substitution_tiling" title="Substitution tiling">substitution tiling</a> is a method of generating aperiodic tilings. One class that can be generated in this way is the <a href="/wiki/Rep-tile" title="Rep-tile">rep-tiles</a>; these tilings have unexpected <a href="/wiki/Self-replication" title="Self-replication">self-replicating</a> properties.<sup id="cite_ref-FOOTNOTEGardner19891–18_38-0" class="reference"><a href="#cite_note-FOOTNOTEGardner19891–18-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pinwheel_tiling" title="Pinwheel tiling">Pinwheel tilings</a> are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a>, do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches.<sup id="cite_ref-Austin_40-0" class="reference"><a href="#cite_note-Austin-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> A substitution rule, such as can be used to generate Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Fibonacci_word" title="Fibonacci word">Fibonacci word</a> can be used to build an aperiodic tiling, and to study <a href="/wiki/Quasicrystal" title="Quasicrystal">quasicrystals</a>, which are structures with aperiodic order.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wang_tiles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Wang_tiles.svg/220px-Wang_tiles.svg.png" decoding="async" width="220" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Wang_tiles.svg/330px-Wang_tiles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Wang_tiles.svg/440px-Wang_tiles.svg.png 2x" data-file-width="500" data-file-height="130" /></a><figcaption>A set of 13 <a href="/wiki/Wang_tile" title="Wang tile">Wang tiles</a> that tile the plane only <a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">aperiodically</a> </figcaption></figure> <p><a href="/wiki/Wang_tile" title="Wang tile">Wang tiles</a> are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wang <a href="/wiki/Dominoes" title="Dominoes">dominoes</a>. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> can be represented as a set of Wang dominoes that tile the plane if, and only if, the Turing machine does not halt. Since the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a> is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Truchet_base_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Truchet_base_tiling.svg/170px-Truchet_base_tiling.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Truchet_base_tiling.svg/255px-Truchet_base_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Truchet_base_tiling.svg/340px-Truchet_base_tiling.svg.png 2x" data-file-width="640" data-file-height="640" /></a><figcaption>Random <a href="/wiki/Truchet_tile" class="mw-redirect" title="Truchet tile">Truchet tiling</a> </figcaption></figure> <p><a href="/wiki/Truchet_tiles" title="Truchet tiles">Truchet tiles</a> are square tiles decorated with patterns so they do not have <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a>; in 1704, <a href="/wiki/S%C3%A9bastien_Truchet" title="Sébastien Truchet">Sébastien Truchet</a> used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p><p>An <i><a href="/wiki/Einstein_tile" class="mw-redirect" title="Einstein tile">einstein tile</a></i> is a single shape that forces aperiodic tiling. The first such tile, dubbed a "hat", was discovered in 2023 by David Smith, a hobbyist mathematician.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> The discovery is under professional review and, upon confirmation, will be credited as solving a longstanding <a href="/wiki/Einstein_problem" title="Einstein problem">mathematical problem</a>.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tessellations_and_colour">Tessellations and colour</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=8" title="Edit section: Tessellations and colour"><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">Further information: <a href="/wiki/Four_color_theorem" title="Four color theorem">Four colour theorem</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Torus_with_seven_colours.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Torus_with_seven_colours.svg/220px-Torus_with_seven_colours.svg.png" decoding="async" width="220" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Torus_with_seven_colours.svg/330px-Torus_with_seven_colours.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Torus_with_seven_colours.svg/440px-Torus_with_seven_colours.svg.png 2x" data-file-width="991" data-file-height="450" /></a><figcaption>At least seven colors are required if the colours of this tiling are to form a pattern by repeating this rectangle as the <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domain</a>; more generally, at least <a href="/wiki/Four_color_theorem" title="Four color theorem">four colours</a> are needed.</figcaption></figure> <p>Sometimes the colour of a tile is understood as part of the tiling; at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity, one needs to specify whether the colours are part of the tiling or just part of its illustration. This affects whether tiles with the same shape, but different colours, are considered identical, which in turn affects questions of symmetry. The <a href="/wiki/Four_color_theorem" title="Four color theorem">four colour theorem</a> states that for every tessellation of a normal <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The colouring guaranteed by the four colour theorem does not generally respect the symmetries of the tessellation. To produce a colouring that does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as demonstrated in the image at left.<sup id="cite_ref-Hazewinkel2001_53-0" class="reference"><a href="#cite_note-Hazewinkel2001-53"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tessellations_with_polygons">Tessellations with polygons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=9" title="Edit section: Tessellations with polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Next to the various <a href="/wiki/Tilings_by_regular_polygons" class="mw-redirect" title="Tilings by regular polygons">tilings by regular polygons</a>, tilings by other polygons have also been studied. </p><p>Any triangle or <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> (even <a href="/wiki/Concave_polygon" title="Concave polygon">non-convex</a>) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to <a href="/wiki/Wallpaper_group#Group_p2" title="Wallpaper group">wallpaper group p2</a>. As <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domain</a> we have the quadrilateral. Equivalently, we can construct a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Texas_tessellation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Texas_tessellation.svg/170px-Texas_tessellation.svg.png" decoding="async" width="170" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Texas_tessellation.svg/255px-Texas_tessellation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Texas_tessellation.svg/340px-Texas_tessellation.svg.png 2x" data-file-width="1733" data-file-height="1485" /></a><figcaption>Tessellation using <a href="/wiki/Texas" title="Texas">Texas</a>-shaped non-convex 12-sided polygons</figcaption></figure> <p>If only one shape of tile is allowed, tilings exist with convex <i>N</i>-gons for <i>N</i> equal to 3, 4, 5, and 6. For <span class="nowrap"><i>N</i> = 5</span>, see <a href="/wiki/Pentagonal_tiling" title="Pentagonal tiling">Pentagonal tiling</a>, for <span class="nowrap"><i>N</i> = 6</span>, see <a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">Hexagonal tiling</a>, for <span class="nowrap"><i>N</i> = 7</span>, see <a href="/wiki/Heptagonal_tiling" title="Heptagonal tiling">Heptagonal tiling</a> and for <span class="nowrap"><i>N</i> = 8</span>, see <a href="/wiki/Octagonal_tiling" title="Octagonal tiling">octagonal tiling</a>. </p><p>With non-convex polygons, there are far fewer limitations in the number of sides, even if only one shape is allowed. </p><p><a href="/wiki/Polyomino" title="Polyomino">Polyominoes</a> are examples of tiles that are either convex of non-convex, for which various combinations, rotations, and reflections can be used to tile a plane. For results on tiling the plane with <a href="/wiki/Polyomino" title="Polyomino">polyominoes</a>, see <a href="/wiki/Polyomino_tiling" class="mw-redirect" title="Polyomino tiling">Polyomino § Uses of polyominoes</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Voronoi_tilings">Voronoi tilings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=10" title="Edit section: Voronoi tilings"><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:Coloured_Voronoi_2D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Coloured_Voronoi_2D.svg/170px-Coloured_Voronoi_2D.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Coloured_Voronoi_2D.svg/255px-Coloured_Voronoi_2D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Coloured_Voronoi_2D.svg/340px-Coloured_Voronoi_2D.svg.png 2x" data-file-width="699" data-file-height="699" /></a><figcaption>A <a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi tiling</a>, in which the cells are always convex polygons</figcaption></figure> <p><a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi or Dirichlet</a> tilings are tessellations where each tile is defined as the set of points closest to one of the points in a discrete set of defining points. (Think of geographical regions where each region is defined as all the points closest to a given city or post office.)<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> The <i>Voronoi cell</i> for each defining point is a convex polygon. The <a href="/wiki/Delaunay_triangulation" title="Delaunay triangulation">Delaunay triangulation</a> is a tessellation that is the <a href="/wiki/Dual_graph" title="Dual graph">dual graph</a> of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges.<sup id="cite_ref-George1998_57-0" class="reference"><a href="#cite_note-George1998-57"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tessellations_in_higher_dimensions">Tessellations in higher dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=11" title="Edit section: Tessellations in higher dimensions"><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/Honeycomb_(geometry)" title="Honeycomb (geometry)">Honeycomb (geometry)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HC_R1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/HC_R1.png/170px-HC_R1.png" decoding="async" width="170" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/HC_R1.png/255px-HC_R1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/HC_R1.png/340px-HC_R1.png 2x" data-file-width="640" data-file-height="840" /></a><figcaption>Tessellating three-dimensional (3-D) space: the <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> is one of the solids that can be stacked to <a href="/wiki/Rhombic_dodecahedral_honeycomb" title="Rhombic dodecahedral honeycomb">fill space exactly</a>.</figcaption></figure> <p>Tessellation can be extended to three dimensions. Certain <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> can be stacked in a regular <a href="/wiki/Bravais_lattice" title="Bravais lattice">crystal pattern</a> to fill (or tile) three-dimensional space, including the <a href="/wiki/Cube" title="Cube">cube</a> (the only <a href="/wiki/Platonic_polyhedron" class="mw-redirect" title="Platonic polyhedron">Platonic polyhedron</a> to do so), the <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a>, the <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a>, and triangular, quadrilateral, and hexagonal <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a>, among others.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Any polyhedron that fits this criterion is known as a <a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedron</a>, and may possess between 4 and 38 faces.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> Naturally occurring rhombic dodecahedra are found as <a href="/wiki/Crystal" title="Crystal">crystals</a> of <a href="/wiki/Andradite" title="Andradite">andradite</a> (a kind of <a href="/wiki/Garnet" title="Garnet">garnet</a>) and <a href="/wiki/Fluorite" title="Fluorite">fluorite</a>.<sup id="cite_ref-Oldershaw2003_61-0" class="reference"><a href="#cite_note-Oldershaw2003-61"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:SCD_tile.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/SCD_tile.svg/170px-SCD_tile.svg.png" decoding="async" width="170" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/SCD_tile.svg/255px-SCD_tile.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/SCD_tile.svg/340px-SCD_tile.svg.png 2x" data-file-width="124" data-file-height="123" /></a><figcaption>Illustration of a Schmitt–Conway biprism, also called a Schmitt–Conway–Danzer tile</figcaption></figure> <p>Tessellations in three or more dimensions are called <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">honeycombs</a>. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> honeycomb, which has eight <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedra</a> and six <a href="/wiki/Octahedron" title="Octahedron">octahedra</a> at each polyhedron vertex. However, there are many possible <a href="/wiki/Convex_uniform_honeycomb" title="Convex uniform honeycomb">semiregular honeycombs</a> in three dimensions.<sup id="cite_ref-CoxeterSherk1995_64-0" class="reference"><a href="#cite_note-CoxeterSherk1995-64"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Uniform honeycombs can be constructed using the <a href="/wiki/Wythoff_construction" title="Wythoff construction">Wythoff construction</a>.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Gyrobifastigium#Schmitt–Conway–Danzer_biprism" title="Gyrobifastigium">Schmitt-Conway biprism</a> is a convex polyhedron with the property of tiling space only aperiodically.<sup id="cite_ref-Senechal1996_66-0" class="reference"><a href="#cite_note-Senechal1996-66"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Schwarz_triangle" title="Schwarz triangle">Schwarz triangle</a> is a <a href="/wiki/Spherical_triangle" class="mw-redirect" title="Spherical triangle">spherical triangle</a> that can be used to tile a <a href="/wiki/Sphere" title="Sphere">sphere</a>.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tessellations_in_non-Euclidean_geometries">Tessellations in non-Euclidean geometries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=12" title="Edit section: Tessellations in non-Euclidean geometries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Rhombitriheptagonal_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Rhombitriheptagonal_tiling.svg/170px-Rhombitriheptagonal_tiling.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Rhombitriheptagonal_tiling.svg/255px-Rhombitriheptagonal_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Rhombitriheptagonal_tiling.svg/340px-Rhombitriheptagonal_tiling.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a><figcaption><a href="/wiki/Rhombitriheptagonal_tiling" title="Rhombitriheptagonal tiling">Rhombitriheptagonal tiling</a> in hyperbolic plane, seen in <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a> projection</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:H3_353_CC_center.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/H3_353_CC_center.png/220px-H3_353_CC_center.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/H3_353_CC_center.png/330px-H3_353_CC_center.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/H3_353_CC_center.png/440px-H3_353_CC_center.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>The regular <a href="/wiki/Icosahedral_honeycomb" title="Icosahedral honeycomb">{3,5,3} icosahedral honeycomb</a>, one of four regular compact honeycombs in <a href="/wiki/Hyperbolic_3-space" class="mw-redirect" title="Hyperbolic 3-space">hyperbolic 3-space</a> </figcaption></figure> <p>It is possible to tessellate in <a href="/wiki/Non-Euclidean" class="mw-redirect" title="Non-Euclidean">non-Euclidean</a> geometries such as <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>. A <a href="/wiki/Uniform_tilings_in_hyperbolic_plane" title="Uniform tilings in hyperbolic plane">uniform tiling in the hyperbolic plane</a> (that may be regular, quasiregular, or semiregular) is an edge-to-edge filling of the hyperbolic plane, with <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> as <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>; these are <a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a> (<a href="/wiki/Transitive_group_action" class="mw-redirect" title="Transitive group action">transitive</a> on its <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>), and isogonal (there is an <a href="/wiki/Isometry" title="Isometry">isometry</a> mapping any vertex onto any other).<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Uniform_honeycombs_in_hyperbolic_space" title="Uniform honeycombs in hyperbolic space">uniform honeycomb in hyperbolic space</a> is a uniform tessellation of <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform polyhedral</a> <a href="/wiki/Cell_(geometry)" class="mw-redirect" title="Cell (geometry)">cells</a>. In three-dimensional (3-D) hyperbolic space there are nine <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a> families of compact <a href="/wiki/Convex_uniform_honeycomb" title="Convex uniform honeycomb">convex uniform honeycombs</a>, generated as <a href="/wiki/Wythoff_construction" title="Wythoff construction">Wythoff constructions</a>, and represented by <a href="/wiki/Permutation" title="Permutation">permutations</a> of <a href="/wiki/Coxeter-Dynkin_diagram#Application_with_uniform_polytopes" class="mw-redirect" title="Coxeter-Dynkin diagram">rings</a> of the <a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagrams</a> for each family.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_art">In art</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=13" title="Edit section: In art"><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">Further information: <a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Mosaic_floor_panel_-_Google_Art_Project.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Mosaic_floor_panel_-_Google_Art_Project.jpg/170px-Mosaic_floor_panel_-_Google_Art_Project.jpg" decoding="async" width="170" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Mosaic_floor_panel_-_Google_Art_Project.jpg/255px-Mosaic_floor_panel_-_Google_Art_Project.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Mosaic_floor_panel_-_Google_Art_Project.jpg/340px-Mosaic_floor_panel_-_Google_Art_Project.jpg 2x" data-file-width="3470" data-file-height="3804" /></a><figcaption>Roman <a href="/wiki/Mosaic" title="Mosaic">mosaic</a> floor panel of stone, tile, and glass, from a villa near <a href="/wiki/Antakya" title="Antakya">Antioch</a> in Roman Syria. second century AD</figcaption></figure> <p>In architecture, tessellations have been used to create decorative motifs since ancient times. <a href="/wiki/Mosaic" title="Mosaic">Mosaic</a> tilings often had geometric patterns.<sup id="cite_ref-Field1988_4-1" class="reference"><a href="#cite_note-Field1988-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the <a href="/wiki/Moorish" class="mw-redirect" title="Moorish">Moorish</a> wall tilings of <a href="/wiki/Islamic_architecture" title="Islamic architecture">Islamic architecture</a>, using <a href="/wiki/Girih_tiles" title="Girih tiles">Girih</a> and <a href="/wiki/Zellige" class="mw-redirect" title="Zellige">Zellige</a> tiles in buildings such as the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a><sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Mosque%E2%80%93Cathedral_of_C%C3%B3rdoba" title="Mosque–Cathedral of Córdoba">La Mezquita</a>.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> </p><p>Tessellations frequently appeared in the graphic art of <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited <a href="/wiki/Spain" title="Spain">Spain</a> in 1936.<sup id="cite_ref-FOOTNOTEEscher19745,_17_73-0" class="reference"><a href="#cite_note-FOOTNOTEEscher19745,_17-73"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> Escher made four "<a href="/wiki/Circle_Limit_III" title="Circle Limit III">Circle Limit</a>" drawings of tilings that use hyperbolic geometry.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> For his <a href="/wiki/Woodcut" title="Woodcut">woodcut</a> "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.<sup id="cite_ref-FOOTNOTEEscher1974142–143_76-0" class="reference"><a href="#cite_note-FOOTNOTEEscher1974142–143-76"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line."<sup id="cite_ref-FOOTNOTEEscher197416_77-0" class="reference"><a href="#cite_note-FOOTNOTEEscher197416-77"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ralli_Quilt.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/9d/Ralli_Quilt.jpg/220px-Ralli_Quilt.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/9d/Ralli_Quilt.jpg/330px-Ralli_Quilt.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/9d/Ralli_Quilt.jpg/440px-Ralli_Quilt.jpg 2x" data-file-width="480" data-file-height="360" /></a><figcaption>A quilt showing a regular tessellation pattern</figcaption></figure> <p>Tessellated designs often appear on textiles, whether woven, stitched in, or printed. Tessellation patterns have been used to design interlocking <a href="/wiki/Motif_(textile_arts)" title="Motif (textile arts)">motifs</a> of patch shapes in <a href="/wiki/Quilt" title="Quilt">quilts</a>.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> </p><p>Tessellations are also a main genre in <a href="/wiki/Origami#tessellation" title="Origami">origami</a> (paper folding), where pleats are used to connect molecules, such as twist folds, together in a repeating fashion.<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_manufacturing">In manufacturing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=14" title="Edit section: In manufacturing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tessellation is used in <a href="/wiki/Manufacturing_industry" class="mw-redirect" title="Manufacturing industry">manufacturing industry</a> to reduce the wastage of material (yield losses) such as <a href="/wiki/Sheet_metal" title="Sheet metal">sheet metal</a> when cutting out shapes for objects such as <a href="/wiki/Car_door" title="Car door">car doors</a> or <a href="/wiki/Drinks_can" class="mw-redirect" title="Drinks can">drink cans</a>.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> </p><p>Tessellation is apparent in the <a href="/wiki/Mudcrack" title="Mudcrack">mudcrack</a>-like <a href="/wiki/Fracture" title="Fracture">cracking</a> of <a href="/wiki/Thin_film" title="Thin film">thin films</a><sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> – with a degree of <a href="/wiki/Self-organisation" class="mw-redirect" title="Self-organisation">self-organisation</a> being observed using <a href="/wiki/Microtechnology" title="Microtechnology">micro</a> and <a href="/wiki/Nanotechnology" title="Nanotechnology">nanotechnologies</a>.<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_nature">In nature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=15" title="Edit section: In nature"><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/Patterns_in_nature#Tessellations" title="Patterns in nature">Patterns in nature § Tessellations</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Buckfast_bee.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Buckfast_bee.jpg/220px-Buckfast_bee.jpg" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Buckfast_bee.jpg/330px-Buckfast_bee.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Buckfast_bee.jpg/440px-Buckfast_bee.jpg 2x" data-file-width="700" data-file-height="507" /></a><figcaption>A <a href="/wiki/Honeycomb" title="Honeycomb">honeycomb</a> is a natural tessellated structure.</figcaption></figure> <p>The <a href="/wiki/Honeycomb" title="Honeycomb">honeycomb</a> is a well-known example of tessellation in nature with its hexagonal cells.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> </p><p>In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the <a href="/wiki/Fritillaria" title="Fritillaria">fritillary</a>,<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> and some species of <i>Colchicum</i>, are characteristically tessellate.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> </p><p>Many <a href="/wiki/Patterns_in_nature" title="Patterns in nature">patterns in nature</a> are formed by cracks in sheets of materials. These patterns can be described by <a href="/wiki/Gilbert_tessellation" title="Gilbert tessellation">Gilbert tessellations</a>,<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> also known as random crack networks.<sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> The Gilbert tessellation is a mathematical model for the formation of <a href="/wiki/Mudcrack" title="Mudcrack">mudcracks</a>, needle-like <a href="/wiki/Crystal" title="Crystal">crystals</a>, and similar structures. The model, named after <a href="/wiki/Edgar_Gilbert" title="Edgar Gilbert">Edgar Gilbert</a>, allows cracks to form starting from being randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons.<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Basalt" title="Basalt">Basaltic</a> <a href="/wiki/Lava_flow" class="mw-redirect" title="Lava flow">lava flows</a> often display <a href="/wiki/Columnar_basalt" class="mw-redirect" title="Columnar basalt">columnar jointing</a> as a result of <a href="/wiki/Thermal_expansion#Contraction_effects_(negative_thermal_expansion)" title="Thermal expansion">contraction</a> forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the <a href="/wiki/Giant%27s_Causeway" title="Giant's Causeway">Giant's Causeway</a> in Northern Ireland.<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Tessellated_pavement" title="Tessellated pavement">Tessellated pavement</a>, a characteristic example of which is found at <a href="/wiki/Eaglehawk_Neck,_Tasmania" class="mw-redirect" title="Eaglehawk Neck, Tasmania">Eaglehawk Neck</a> on the <a href="/wiki/Tasman_Peninsula" title="Tasman Peninsula">Tasman Peninsula</a> of <a href="/wiki/Tasmania" title="Tasmania">Tasmania</a>, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Colchicum_-_unknown_species.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Colchicum_-_unknown_species.jpg/220px-Colchicum_-_unknown_species.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Colchicum_-_unknown_species.jpg/330px-Colchicum_-_unknown_species.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Colchicum_-_unknown_species.jpg/440px-Colchicum_-_unknown_species.jpg 2x" data-file-width="3648" data-file-height="2736" /></a><figcaption>Tessellate pattern in a <i><a href="/wiki/Colchicum" title="Colchicum">Colchicum</a></i> flower</figcaption></figure> <p>Other natural patterns occur in <a href="/wiki/Foam" title="Foam">foams</a>; these are packed according to <a href="/wiki/Plateau%27s_laws" title="Plateau's laws">Plateau's laws</a>, which require <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surfaces</a>. Such foams present a problem in how to pack cells as tightly as possible: in 1887, <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a> proposed a packing using only one solid, the <a href="/wiki/Bitruncated_cubic_honeycomb" title="Bitruncated cubic honeycomb">bitruncated cubic honeycomb</a> with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the <a href="/wiki/Weaire%E2%80%93Phelan_structure" title="Weaire–Phelan structure">Weaire–Phelan structure</a>, which uses less surface area to separate cells of equal volume than Kelvin's foam.<sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_puzzles_and_recreational_mathematics">In puzzles and recreational mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=16" title="Edit section: In puzzles and recreational mathematics"><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:Tangram_set_00.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Tangram_set_00.jpg/170px-Tangram_set_00.jpg" decoding="async" width="170" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Tangram_set_00.jpg/255px-Tangram_set_00.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Tangram_set_00.jpg/340px-Tangram_set_00.jpg 2x" data-file-width="1440" data-file-height="1432" /></a><figcaption>Traditional <a href="/wiki/Tangram" title="Tangram">tangram</a> <a href="/wiki/Dissection_puzzle" title="Dissection puzzle">dissection puzzle</a> </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/Tiling_puzzle" title="Tiling puzzle">Tiling puzzle</a> and <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a></div> <p>Tessellations have given rise to many types of <a href="/wiki/Tiling_puzzle" title="Tiling puzzle">tiling puzzle</a>, from traditional <a href="/wiki/Jigsaw_puzzle" title="Jigsaw puzzle">jigsaw puzzles</a> (with irregular pieces of wood or cardboard)<sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Tangram" title="Tangram">tangram</a>,<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> to more modern puzzles that often have a mathematical basis. For example, <a href="/wiki/Polyiamond" title="Polyiamond">polyiamonds</a> and <a href="/wiki/Polyomino" title="Polyomino">polyominoes</a> are figures of regular triangles and squares, often used in tiling puzzles.<sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> Authors such as <a href="/wiki/Henry_Dudeney" title="Henry Dudeney">Henry Dudeney</a> and <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> have made many uses of tessellation in <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>. For example, Dudeney invented the <a href="/wiki/Hinged_dissection" title="Hinged dissection">hinged dissection</a>,<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> while Gardner wrote about the "<a href="/wiki/Rep-tile" title="Rep-tile">rep-tile</a>", a shape that can be <a href="/wiki/Dissection_(geometry)" class="mw-redirect" title="Dissection (geometry)">dissected</a> into smaller copies of the same shape.<sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gardner2006_100-0" class="reference"><a href="#cite_note-Gardner2006-100"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> Inspired by Gardner's articles in <a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a>, the amateur mathematician <a href="/wiki/Marjorie_Rice" title="Marjorie Rice">Marjorie Rice</a> found four new tessellations with pentagons.<sup id="cite_ref-101" class="reference"><a href="#cite_note-101"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Squaring_the_square" title="Squaring the square">Squaring the square</a> is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.<sup id="cite_ref-103" class="reference"><a href="#cite_note-103"><span class="cite-bracket">[</span>100<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-104" class="reference"><a href="#cite_note-104"><span class="cite-bracket">[</span>101<span class="cite-bracket">]</span></a></sup> An extension is squaring the plane, tiling it by squares whose sizes are all natural numbers without repetitions; James and Frederick Henle proved that this was possible.<sup id="cite_ref-105" class="reference"><a href="#cite_note-105"><span class="cite-bracket">[</span>102<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=17" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Tile_3,6.svg" class="mw-file-description" title="Triangular tiling, one of the three regular tilings of the plane"><img alt="Triangular tiling, one of the three regular tilings of the plane" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Tile_3%2C6.svg/120px-Tile_3%2C6.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Tile_3%2C6.svg/180px-Tile_3%2C6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Tile_3%2C6.svg/240px-Tile_3%2C6.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Triangular_tiling" title="Triangular tiling">Triangular tiling</a>, one of the three <a href="/wiki/Regular_tiling" class="mw-redirect" title="Regular tiling">regular tilings</a> of the plane</div> </li> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg" class="mw-file-description" title="Snub hexagonal tiling, a semiregular tiling of the plane"><img alt="Snub hexagonal tiling, a semiregular tiling of the plane" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg/120px-Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg/180px-Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg/240px-Academ_Periodic_tiling_where_eighteen_triangles_encircle_each_hexagon.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div> <div class="gallerytext"><a href="/wiki/Snub_hexagonal_tiling" class="mw-redirect" title="Snub hexagonal tiling">Snub hexagonal tiling</a>, a <a href="/wiki/Semiregular_tiling" class="mw-redirect" title="Semiregular tiling">semiregular tiling</a> of the plane </div> </li> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg" class="mw-file-description" title="Floret pentagonal tiling, dual to a semiregular tiling and one of 15 monohedral pentagon tilings"><img alt="Floret pentagonal tiling, dual to a semiregular tiling and one of 15 monohedral pentagon tilings" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg/120px-Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg/180px-Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg/240px-Tiling_Dual_Semiregular_V3-3-3-3-6_Floret_Pentagonal.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext"><a href="/wiki/Floret_pentagonal_tiling" class="mw-redirect" title="Floret pentagonal tiling">Floret pentagonal tiling</a>, dual to a semiregular tiling and one of 15 monohedral <a href="/wiki/Pentagon_tiling" class="mw-redirect" title="Pentagon tiling">pentagon tilings</a> </div> </li> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg" class="mw-file-description" title="All tiling elements are identical pseudo‑triangles by disregarding their colors and ornaments"><img alt="All tiling elements are identical pseudo‑triangles by disregarding their colors and ornaments" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg/120px-A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg.png" decoding="async" width="120" height="74" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg/180px-A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg/240px-A_variation_on_a_tiling_in_the_Alhambra_of_Spain.svg.png 2x" data-file-width="987" data-file-height="610" /></a></span></div> <div class="gallerytext">All tiling elements are <a href="#Monohedral">identical pseudo‑triangles</a> by disregarding their colors and ornaments</div> </li> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Voderberg.png" class="mw-file-description" title="The Voderberg tiling, a spiral, monohedral tiling made of enneagons"><img alt="The Voderberg tiling, a spiral, monohedral tiling made of enneagons" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Voderberg.png/120px-Voderberg.png" decoding="async" width="120" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Voderberg.png/180px-Voderberg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Voderberg.png/240px-Voderberg.png 2x" data-file-width="1792" data-file-height="1600" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Voderberg_tiling" title="Voderberg tiling">Voderberg tiling</a>, a spiral, <a href="#Monohedral">monohedral</a> tiling made of <a href="/wiki/Enneagon" class="mw-redirect" title="Enneagon">enneagons</a> </div> </li> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_433-t0.png" class="mw-file-description" title="Alternated octagonal or tritetragonal tiling is a uniform tiling of the hyperbolic plane"><img alt="Alternated octagonal or tritetragonal tiling is a uniform tiling of the hyperbolic plane" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Uniform_tiling_433-t0.png/120px-Uniform_tiling_433-t0.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Uniform_tiling_433-t0.png/180px-Uniform_tiling_433-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Uniform_tiling_433-t0.png/240px-Uniform_tiling_433-t0.png 2x" data-file-width="698" data-file-height="698" /></a></span></div> <div class="gallerytext"><a href="/wiki/Alternated_octagonal_tiling" title="Alternated octagonal tiling">Alternated octagonal or tritetragonal tiling</a> is a uniform tiling of the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic plane</a> </div> </li> <li class="gallerybox" style="width: 125px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Capital_I_tiling-4color.svg" class="mw-file-description" title="Topological square tiling, isohedrally distorted into I shapes"><img alt="Topological square tiling, isohedrally distorted into I shapes" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Capital_I_tiling-4color.svg/120px-Capital_I_tiling-4color.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Capital_I_tiling-4color.svg/180px-Capital_I_tiling-4color.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Capital_I_tiling-4color.svg/240px-Capital_I_tiling-4color.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Topology" title="Topology">Topological</a> square tiling, isohedrally distorted into I shapes </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=Tessellation&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Discrete_global_grid" title="Discrete global grid">Discrete global grid</a></li> <li><a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">Honeycomb (geometry)</a></li> <li><a href="/wiki/Space_partitioning" title="Space partitioning">Space partitioning</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Explanatory_footnotes">Explanatory footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=19" title="Edit section: Explanatory footnotes"><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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">The mathematical term for identical shapes is "congruent" – in mathematics, "identical" means they are the same tile.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">The tiles are usually required to be <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> (topologically equivalent) to a <a href="/wiki/Closed_disk" class="mw-redirect" title="Closed disk">closed disk</a>, which means bizarre shapes with holes, dangling line segments, or infinite areas are excluded.<sup id="cite_ref-FOOTNOTEGrünbaumShephard198759_19-0" class="reference"><a href="#cite_note-FOOTNOTEGrünbaumShephard198759-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></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">In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.</span> </li> </ol></div></div> <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=Tessellation&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Pickover2009-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Pickover2009_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Pickover2009_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPickover2009" class="citation book cs1">Pickover, Clifford A. (2009). <i>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics</i>. <a href="/wiki/Sterling_Publishing" title="Sterling Publishing">Sterling</a>. p. 372. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4027-5796-9" title="Special:BookSources/978-1-4027-5796-9"><bdi>978-1-4027-5796-9</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+Math+Book%3A+From+Pythagoras+to+the+57th+Dimension%2C+250+Milestones+in+the+History+of+Mathematics&rft.pages=372&rft.pub=Sterling&rft.date=2009&rft.isbn=978-1-4027-5796-9&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFDunbabin2006" class="citation book cs1">Dunbabin, Katherine M. D. (2006). <i>Mosaics of the Greek and Roman world</i>. Cambridge University Press. p. 280.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mosaics+of+the+Greek+and+Roman+world&rft.pages=280&rft.pub=Cambridge+University+Press&rft.date=2006&rft.aulast=Dunbabin&rft.aufirst=Katherine+M.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.hullcc.gov.uk/museumcollections/collections/storydetail.php?irn=410">"The Brantingham Geometric Mosaics"</a>. Hull City Council. 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">26 May</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Brantingham+Geometric+Mosaics&rft.pub=Hull+City+Council&rft.date=2008&rft_id=http%3A%2F%2Fwww.hullcc.gov.uk%2Fmuseumcollections%2Fcollections%2Fstorydetail.php%3Firn%3D410&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-Field1988-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Field1988_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Field1988_4-1"><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="CITEREFField,_Robert1988" class="citation book cs1">Field, Robert (1988). <i>Geometric Patterns from Roman Mosaics</i>. Tarquin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-906-21263-9" title="Special:BookSources/978-0-906-21263-9"><bdi>978-0-906-21263-9</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+Patterns+from+Roman+Mosaics&rft.pub=Tarquin&rft.date=1988&rft.isbn=978-0-906-21263-9&rft.au=Field%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKepler,_Johannes1619" class="citation book cs1"><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler, Johannes</a> (1619). <a href="/wiki/Harmonices_Mundi" class="mw-redirect" title="Harmonices Mundi"><i>Harmonices Mundi</i></a> [<i>Harmony of the Worlds</i>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmonices+Mundi&rft.date=1619&rft.au=Kepler%2C+Johannes&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGullberg1997395-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGullberg1997395_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGullberg1997395_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGullberg1997395_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGullberg1997">Gullberg 1997</a>, p. 395.</span> </li> <li id="cite_note-FOOTNOTEStewart200113-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStewart200113_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStewart2001">Stewart 2001</a>, p. 13.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDjidjev,_HristoPotkonjak,_Miodrag2012" class="citation web cs1">Djidjev, Hristo; Potkonjak, Miodrag (2012). <a rel="nofollow" class="external text" href="http://public.lanl.gov/djidjev/papers/coverage_chapter.pdf">"Dynamic Coverage Problems in Sensor Networks"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/Los_Alamos_National_Laboratory" title="Los Alamos National Laboratory">Los Alamos National Laboratory</a>. p. 2<span class="reference-accessdate">. Retrieved <span class="nowrap">6 April</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dynamic+Coverage+Problems+in+Sensor+Networks&rft.pages=2&rft.pub=Los+Alamos+National+Laboratory&rft.date=2012&rft.au=Djidjev%2C+Hristo&rft.au=Potkonjak%2C+Miodrag&rft_id=http%3A%2F%2Fpublic.lanl.gov%2Fdjidjev%2Fpapers%2Fcoverage_chapter.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFyodorov,_Y.1891" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Yevgraf_Fyodorov" class="mw-redirect" title="Yevgraf Fyodorov">Fyodorov, Y.</a> (1891). "Simmetrija na ploskosti [Symmetry in the plane]". <i>Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society]</i>. 2 (in Russian). <b>28</b>: <span class="nowrap">245–</span>291.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zapiski+Imperatorskogo+Sant-Petersburgskogo+Mineralogicheskogo+Obshchestva+%5BProceedings+of+the+Imperial+St.+Petersburg+Mineralogical+Society%5D&rft.atitle=Simmetrija+na+ploskosti+%5BSymmetry+in+the+plane%5D&rft.volume=28&rft.pages=%3Cspan+class%3D%22nowrap%22%3E245-%3C%2Fspan%3E291&rft.date=1891&rft.au=Fyodorov%2C+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-ShubnikovBelov1964-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-ShubnikovBelov1964_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShubnikovBelov1964" class="citation book cs1">Shubnikov, Alekseĭ Vasilʹevich; Belov, Nikolaĭ Vasilʹevich (1964). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QKk9AAAAIAAJ"><i>Colored Symmetry</i></a>. <a href="/wiki/Macmillan_Publishers" title="Macmillan Publishers">Macmillan</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Colored+Symmetry&rft.pub=Macmillan&rft.date=1964&rft.aulast=Shubnikov&rft.aufirst=Alekse%C4%AD+Vasil%CA%B9evich&rft.au=Belov%2C+Nikola%C4%AD+Vasil%CA%B9evich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQKk9AAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFHeeschKienzle1963" class="citation book cs1 cs1-prop-foreign-lang-source">Heesch, H.; Kienzle, O. 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S. M. Coxeter">Coxeter, H. S. M.</a> (1948). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iWvXsVInpgMC&pg=PP1"><i>Regular Polytopes</i></a>. <a href="/wiki/Methuen_Publishing" title="Methuen Publishing">Methuen</a>. pp. 14, 69, 149. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61480-9" title="Special:BookSources/978-0-486-61480-9"><bdi>978-0-486-61480-9</bdi></a>.</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=14%2C+69%2C+149&rft.pub=Methuen&rft.date=1948&rft.isbn=978-0-486-61480-9&rft.au=Coxeter%2C+H.+S.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiWvXsVInpgMC%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-MathWorldTessellation-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-MathWorldTessellation_23-0">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Tessellation"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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Berlin Heidelberg: Springer. p. 325. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-28849-7" title="Special:BookSources/978-3-540-28849-7"><bdi>978-3-540-28849-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=M.C.+Escher%27s+Legacy%3A+A+Centennial+Celebration&rft.place=Berlin+Heidelberg&rft.pages=325&rft.pub=Springer&rft.date=2007-05-08&rft.isbn=978-3-540-28849-7&rft.aulast=Emmer&rft.aufirst=Michele&rft.au=Schattschneider%2C+Doris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5DDyBwAAQBAJ%26pg%3DPA325&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-Horne2000-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-Horne2000_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Horne2000_25-1"><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="CITEREFHorne2000" class="citation book cs1">Horne, Clare E. 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"The tiling patterns of Sebastian Truchet and the topology of structural hierarchy". <i><a href="/wiki/Leonardo_(journal)" title="Leonardo (journal)">Leonardo</a></i>. <b>20</b> (4): <span class="nowrap">373–</span>385. <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%2F1578535">10.2307/1578535</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/1578535">1578535</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:192944820">192944820</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Leonardo&rft.atitle=The+tiling+patterns+of+Sebastian+Truchet+and+the+topology+of+structural+hierarchy&rft.volume=20&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E373-%3C%2Fspan%3E385&rft.date=1987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A192944820%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1578535%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1578535&rft.aulast=Smith&rft.aufirst=Cyril+Stanley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConover2023" class="citation web cs1">Conover, Emily (24 March 2023). <a rel="nofollow" class="external text" href="https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile">"Mathematicians have finally discovered an elusive 'einstein' tile"</a>. <i><a href="/wiki/Science_News" title="Science News">Science News</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">25 March</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Science+News&rft.atitle=Mathematicians+have+finally+discovered+an+elusive+%27einstein%27+tile&rft.date=2023-03-24&rft.aulast=Conover&rft.aufirst=Emily&rft_id=https%3A%2F%2Fwww.sciencenews.org%2Farticle%2Fmathematicians-discovered-einstein-tile&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span> with image of the pattern</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text">Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (March 2023). "An aperiodic monotile". arXiv:2303.10798</span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text">Roberts, Soibhan, <i><a rel="nofollow" class="external text" href="https://www.nytimes.com/2023/03/28/science/mathematics-tiling-einstein.html">Elusive 'Einstein' Solves a Longstanding Mathematical Problem</a></i>, the New York Times, March 28, 2023, with image of the pattern</span> </li> <li id="cite_note-Hazewinkel2001-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hazewinkel2001_53-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Four-colour_problem">"Four-colour problem"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Four-colour+problem&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFour-colour_problem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFJones1910" class="citation book cs1"><a href="/wiki/Owen_Jones_(architect)" title="Owen Jones (architect)">Jones, Owen</a> (1910) [1856]. <a rel="nofollow" class="external text" href="http://digital.library.wisc.edu/1711.dl/DLDecArts.GramOrnJones"><i>The Grammar of Ornament</i></a> (folio ed.). <a href="/wiki/Bernard_Quaritch" title="Bernard Quaritch">Bernard Quaritch</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Grammar+of+Ornament&rft.edition=folio&rft.pub=Bernard+Quaritch&rft.date=1910&rft.aulast=Jones&rft.aufirst=Owen&rft_id=http%3A%2F%2Fdigital.library.wisc.edu%2F1711.dl%2FDLDecArts.GramOrnJones&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFAurenhammer,_Franz1991" class="citation journal cs1"><a href="/wiki/Franz_Aurenhammer" title="Franz Aurenhammer">Aurenhammer, Franz</a> (1991). 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Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-2652-9" title="Special:BookSources/978-1-4612-2652-9"><bdi>978-1-4612-2652-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Random+Voronoi+Tessellations&rft.pub=Springer&rft.date=1994&rft.isbn=978-1-4612-2652-9&rft.aulast=Moller&rft.aufirst=Jesper&rft_id=https%3A%2F%2Fwww.springer.com%2Fgp%2Fbook%2F9780387942643&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFGrünbaum,_Branko1994" class="citation journal cs1">Grünbaum, Branko (1994). "Uniform tilings of 3-space". <i>Geombinatorics</i>. <b>4</b> (2): <span class="nowrap">49–</span>56.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geombinatorics&rft.atitle=Uniform+tilings+of+3-space&rft.volume=4&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E49-%3C%2Fspan%3E56&rft.date=1994&rft.au=Gr%C3%BCnbaum%2C+Branko&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFEngel1981" class="citation journal cs1">Engel, Peter (1981). "Über Wirkungsbereichsteilungen von kubischer Symmetrie". <i>Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie</i>. <b>154</b> (<span class="nowrap">3–</span>4): <span class="nowrap">199–</span>215. <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/1981ZK....154..199E">1981ZK....154..199E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1524%2Fzkri.1981.154.3-4.199">10.1524/zkri.1981.154.3-4.199</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=0598811">0598811</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Kristallographie%2C+Kristallgeometrie%2C+Kristallphysik%2C+Kristallchemie&rft.atitle=%C3%9Cber+Wirkungsbereichsteilungen+von+kubischer+Symmetrie&rft.volume=154&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E199-%3C%2Fspan%3E215&rft.date=1981&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D598811%23id-name%3DMR&rft_id=info%3Adoi%2F10.1524%2Fzkri.1981.154.3-4.199&rft_id=info%3Abibcode%2F1981ZK....154..199E&rft.aulast=Engel&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span>.</span> </li> <li id="cite_note-Oldershaw2003-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-Oldershaw2003_61-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOldershaw2003" class="citation book cs1">Oldershaw, Cally (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fireflyguidetoge0000olde"><i>Firefly Guide to Gems</i></a></span>. Firefly Books. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fireflyguidetoge0000olde/page/107">107</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55297-814-6" title="Special:BookSources/978-1-55297-814-6"><bdi>978-1-55297-814-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Firefly+Guide+to+Gems&rft.pages=107&rft.pub=Firefly+Books&rft.date=2003&rft.isbn=978-1-55297-814-6&rft.aulast=Oldershaw&rft.aufirst=Cally&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffireflyguidetoge0000olde&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFKirkaldy1968" class="citation book cs1">Kirkaldy, J. F. (1968). <i>Minerals and Rocks in Colour</i> (2nd ed.). Blandford. pp. <span class="nowrap">138–</span>139.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Minerals+and+Rocks+in+Colour&rft.pages=%3Cspan+class%3D%22nowrap%22%3E138-%3C%2Fspan%3E139&rft.edition=2nd&rft.pub=Blandford&rft.date=1968&rft.aulast=Kirkaldy&rft.aufirst=J.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-CoxeterSherk1995-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-CoxeterSherk1995_64-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoxeterSherkCanadian_Mathematical_Society1995" class="citation book cs1">Coxeter, Harold Scott Macdonald; Sherk, F. Arthur; Canadian Mathematical Society (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/kaleidoscopessel0000coxe"><i>Kaleidoscopes: Selected Writings of H.S.M. Coxeter</i></a></span>. John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/kaleidoscopessel0000coxe/page/3">3</a> and passim. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-01003-6" title="Special:BookSources/978-0-471-01003-6"><bdi>978-0-471-01003-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kaleidoscopes%3A+Selected+Writings+of+H.S.M.+Coxeter&rft.pages=3+and+passim&rft.pub=John+Wiley+%26+Sons&rft.date=1995&rft.isbn=978-0-471-01003-6&rft.aulast=Coxeter&rft.aufirst=Harold+Scott+Macdonald&rft.au=Sherk%2C+F.+Arthur&rft.au=Canadian+Mathematical+Society&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fkaleidoscopessel0000coxe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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"><span class="citation mathworld" id="Reference-Mathworld-Wythoff_construction"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WythoffConstruction.html">"Wythoff construction"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Wythoff+construction&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FWythoffConstruction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span></span> </li> <li id="cite_note-Senechal1996-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-Senechal1996_66-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSenechal1996" class="citation book cs1">Senechal, Marjorie (26 September 1996). <a href="/wiki/Quasicrystals_and_Geometry" title="Quasicrystals and Geometry"><i>Quasicrystals and Geometry</i></a>. 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(1873). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002155206">"Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt"</a>. <i><a href="/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a></i>. <b>1873</b> (75): <span class="nowrap">292–</span>335. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1873.75.292">10.1515/crll.1873.75.292</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121698536">121698536</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Ueber+diejenigen+F%C3%A4lle+in+welchen+die+Gaussichen+hypergeometrische+Reihe+eine+algebraische+Function+ihres+vierten+Elementes+darstellt&rft.volume=1873&rft.issue=75&rft.pages=%3Cspan+class%3D%22nowrap%22%3E292-%3C%2Fspan%3E335&rft.date=1873&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121698536%23id-name%3DS2CID&rft.issn=0075-4102&rft_id=info%3Adoi%2F10.1515%2Fcrll.1873.75.292&rft.aulast=Schwarz&rft.aufirst=H.+A.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3FGDZPPN002155206&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></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="CITEREFMargenstern2011" class="citation arxiv cs1">Margenstern, Maurice (4 January 2011). "Coordinates for a new triangular tiling of the hyperbolic plane". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1101.0530">1101.0530</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.FL">cs.FL</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Coordinates+for+a+new+triangular+tiling+of+the+hyperbolic+plane&rft.date=2011-01-04&rft_id=info%3Aarxiv%2F1101.0530&rft.aulast=Margenstern&rft.aufirst=Maurice&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFZadnik,_Gašper" class="citation web cs1">Zadnik, Gašper. <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/TilingTheHyperbolicPlaneWithRegularPolygons/">"Tiling the Hyperbolic Plane with Regular Polygons"</a>. Wolfram<span class="reference-accessdate">. Retrieved <span class="nowrap">27 May</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Tiling+the+Hyperbolic+Plane+with+Regular+Polygons&rft.pub=Wolfram&rft.au=Zadnik%2C+Ga%C5%A1per&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FTilingTheHyperbolicPlaneWithRegularPolygons%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFCoxeter,_H.S.M.1999" class="citation book cs1">Coxeter, H.S.M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". <i>The Beauty of Geometry: Twelve Essays</i>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. pp. <span class="nowrap">212–</span>213. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-40919-1" title="Special:BookSources/978-0-486-40919-1"><bdi>978-0-486-40919-1</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+10%3A+Regular+honeycombs+in+hyperbolic+space&rft.btitle=The+Beauty+of+Geometry%3A+Twelve+Essays&rft.pages=%3Cspan+class%3D%22nowrap%22%3E212-%3C%2Fspan%3E213&rft.pub=Dover+Publications&rft.date=1999&rft.isbn=978-0-486-40919-1&rft.au=Coxeter%2C+H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml#Symmetry">"Mathematics in Art and Architecture"</a>. National University of Singapore<span class="reference-accessdate">. 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M. <a rel="nofollow" class="external text" href="http://www.math.utah.edu/~sg/Papers/banff.pdf">"Introduction to Hyperbolic and Automatic Groups"</a> <span class="cs1-format">(PDF)</span>. University of Utah<span class="reference-accessdate">. Retrieved <span class="nowrap">27 May</span> 2015</span>. <q>Figure 1 is part of a tiling of the Euclidean plane, which we imagine as continued in all directions, and Figure 2 [Circle Limit IV] is a beautiful tesselation of the Poincaré unit disc model of the hyperbolic plane by white tiles representing angels and black tiles representing devils. An important feature of the second is that all white tiles are mutually congruent as are all black tiles; of course this is not true for the Euclidean metric, but holds for the Poincaré metric</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction+to+Hyperbolic+and+Automatic+Groups&rft.pub=University+of+Utah&rft.aulast=Gersten&rft.aufirst=S.+M.&rft_id=http%3A%2F%2Fwww.math.utah.edu%2F~sg%2FPapers%2Fbanff.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" 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="CITEREFLeys2015" class="citation web cs1">Leys, Jos (2015). <a rel="nofollow" class="external text" href="http://www.josleys.com/show_gallery.php?galid=325">"Hyperbolic Escher"</a><span class="reference-accessdate">. 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"Mathematical Games". <i>Scientific American</i>.</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=Mathematical+Games&rft.date=1958-11&rft.aulast=Gardner&rft.aufirst=Martin&rft.au=Tutte%2C+William+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> <li id="cite_note-105"><span class="mw-cite-backlink"><b><a href="#cite_ref-105">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHenleHenle2008" class="citation journal cs1">Henle, Frederick V.; Henle, James M. (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060620150515/http://maven.smith.edu/%7Ejhenle/stp/stp.pdf">"Squaring the plane"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>115</b> (1): <span class="nowrap">3–</span>12. <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%2F00029890.2008.11920491">10.1080/00029890.2008.11920491</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/27642387">27642387</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:26663945">26663945</a>. Archived from <a rel="nofollow" class="external text" href="http://maven.smith.edu/~jhenle/stp/stp.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 20 June 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Squaring+the+plane&rft.volume=115&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E12&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A26663945%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27642387%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F00029890.2008.11920491&rft.aulast=Henle&rft.aufirst=Frederick+V.&rft.au=Henle%2C+James+M.&rft_id=http%3A%2F%2Fmaven.smith.edu%2F~jhenle%2Fstp%2Fstp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=21" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoxeter1973" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1973). "Section IV : Tessellations and Honeycombs". <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)"><i>Regular Polytopes</i></a>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61480-9" title="Special:BookSources/978-0-486-61480-9"><bdi>978-0-486-61480-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+IV+%3A+Tessellations+and+Honeycombs&rft.btitle=Regular+Polytopes&rft.pub=Dover+Publications&rft.date=1973&rft.isbn=978-0-486-61480-9&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEscher1974" class="citation book cs1"><a href="/wiki/M._C._Escher" title="M. C. Escher">Escher, M. C.</a> (1974). J. L. Locher (ed.). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/worldofmcescher00esch"><i>The World of M. C. Escher</i></a></span> (New Concise NAL ed.). Abrams. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-451-79961-6" title="Special:BookSources/978-0-451-79961-6"><bdi>978-0-451-79961-6</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+World+of+M.+C.+Escher&rft.edition=New+Concise+NAL&rft.pub=Abrams&rft.date=1974&rft.isbn=978-0-451-79961-6&rft.aulast=Escher&rft.aufirst=M.+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fworldofmcescher00esch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGardner1989" class="citation book cs1"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (1989). <i>Penrose Tiles to Trapdoor Ciphers</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-521-8" title="Special:BookSources/978-0-88385-521-8"><bdi>978-0-88385-521-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Penrose+Tiles+to+Trapdoor+Ciphers&rft.pub=Cambridge+University+Press&rft.date=1989&rft.isbn=978-0-88385-521-8&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrünbaumShephard1987" class="citation book cs1"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a>; Shephard, G. C. (1987). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_0716711931"><i>Tilings and Patterns</i></a></span>. W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-1193-3" title="Special:BookSources/978-0-7167-1193-3"><bdi>978-0-7167-1193-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tilings+and+Patterns&rft.pub=W.+H.+Freeman&rft.date=1987&rft.isbn=978-0-7167-1193-3&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rft.au=Shephard%2C+G.+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_0716711931&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGullberg1997" class="citation book cs1"><a href="/wiki/Jan_Gullberg" title="Jan Gullberg">Gullberg, Jan</a> (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull"><i>Mathematics From the Birth of Numbers</i></a></span>. Norton. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04002-9" title="Special:BookSources/978-0-393-04002-9"><bdi>978-0-393-04002-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+From+the+Birth+of+Numbers&rft.pub=Norton&rft.date=1997&rft.isbn=978-0-393-04002-9&rft.aulast=Gullberg&rft.aufirst=Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsfromb1997gull&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStewart2001" class="citation book cs1"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (2001). <i>What Shape Is a Snowflake?</i>. Weidenfeld and Nicolson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-297-60723-6" title="Special:BookSources/978-0-297-60723-6"><bdi>978-0-297-60723-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=What+Shape+Is+a+Snowflake%3F&rft.pub=Weidenfeld+and+Nicolson&rft.date=2001&rft.isbn=978-0-297-60723-6&rft.aulast=Stewart&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tessellation&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><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="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Tiling" class="extiw" title="commons:Tiling"><span style="font-style:italic; font-weight:bold;">Tiling</span></a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://ab.inf.uni-tuebingen.de/software/tegula">Tegula</a> (open-source software for exploring two-dimensional tilings of the plane, sphere and hyperbolic plane; includes databases containing millions of tilings)</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Tessellation.html">Wolfram MathWorld: Tessellation</a> (good bibliography, drawings of regular, semiregular and demiregular tessellations)</li> <li>Dirk Frettlöh and <a href="/wiki/Edmund_Harriss" title="Edmund Harriss">Edmund Harriss</a>. "<a rel="nofollow" class="external text" href="http://tilings.math.uni-bielefeld.de/">Tilings Encyclopedia</a>" (extensive information on substitution tilings, including drawings, people, and references)</li> <li><a rel="nofollow" class="external text" href="http://www.tessellations.org/">Tessellations.org</a> (how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEppstein,_David" class="citation web cs1"><a href="/wiki/David_Eppstein" title="David Eppstein">Eppstein, David</a>. <a rel="nofollow" class="external text" href="http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html">"The Geometry Junkyard: Hyperbolic Tiling"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Geometry+Junkyard%3A+Hyperbolic+Tiling&rft.au=Eppstein%2C+David&rft_id=http%3A%2F%2Fwww.ics.uci.edu%2F~eppstein%2Fjunkyard%2Fhypertile.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATessellation" class="Z3988"></span> (list of web resources including articles and galleries)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist 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a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematics_and_art" title="Template:Mathematics and art"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematics_and_art" title="Template talk:Mathematics and art"><abbr title="Discuss this template">t</abbr></a></li><li 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surface</a></li> <li><a href="/wiki/Paraboloid" title="Paraboloid">Paraboloid</a></li> <li><a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">Perspective</a> <ul><li><a href="/wiki/Camera_lucida" title="Camera lucida">Camera lucida</a></li> <li><a href="/wiki/Camera_obscura" title="Camera obscura">Camera obscura</a></li></ul></li> <li><a href="/wiki/Plastic_ratio" title="Plastic ratio">Plastic ratio</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective geometry</a></li> <li>Proportion <ul><li><a href="/wiki/Proportion_(architecture)" title="Proportion (architecture)">Architecture</a></li> <li><a href="/wiki/Body_proportions" title="Body proportions">Human</a></li></ul></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li> <li><a class="mw-selflink selflink">Tessellation</a></li> <li><a href="/wiki/Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:FWF_Samuel_Monnier_(vertical_detail).jpg" class="mw-file-description" title="Fibonacci word: detail of artwork by Samuel Monnier, 2009"><img alt="Fibonacci word: detail of artwork by Samuel Monnier, 2009" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/FWF_Samuel_Monnier_%28vertical_detail%29.jpg/75px-FWF_Samuel_Monnier_%28vertical_detail%29.jpg" decoding="async" width="75" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/FWF_Samuel_Monnier_%28vertical_detail%29.jpg/113px-FWF_Samuel_Monnier_%28vertical_detail%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/FWF_Samuel_Monnier_%28vertical_detail%29.jpg/150px-FWF_Samuel_Monnier_%28vertical_detail%29.jpg 2x" data-file-width="281" data-file-height="700" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Forms</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/Algorithmic_art" title="Algorithmic art">Algorithmic art</a></li> <li><a href="/wiki/Anamorphosis" title="Anamorphosis">Anamorphic art</a></li> <li><a href="/wiki/Mathematics_and_architecture" title="Mathematics and architecture">Architecture</a> <ul><li><a href="/wiki/Geodesic_dome" title="Geodesic dome">Geodesic dome</a></li> <li><a href="/wiki/Pyramid" title="Pyramid">Pyramid</a></li> <li><a href="/wiki/Vastu_shastra" title="Vastu shastra">Vastu shastra</a></li></ul></li> <li><a href="/wiki/Computer_art" title="Computer art">Computer art</a></li> <li><a href="/wiki/Mathematics_and_fiber_arts" title="Mathematics and fiber arts">Fiber arts</a></li> <li><a href="/wiki/Fourth_dimension_in_art" title="Fourth dimension in art">4D art</a></li> <li><a href="/wiki/Fractal_art" title="Fractal art">Fractal art</a></li> <li><a href="/wiki/Islamic_geometric_patterns" title="Islamic geometric patterns">Islamic geometric patterns</a> <ul><li><a href="/wiki/Girih" title="Girih">Girih</a></li> <li><a href="/wiki/Jali" title="Jali">Jali</a></li> <li><a href="/wiki/Muqarnas" title="Muqarnas">Muqarnas</a></li> <li><a href="/wiki/Zellij" title="Zellij">Zellij</a></li></ul></li> <li><a href="/wiki/Knot" title="Knot">Knotting</a> <ul><li><a href="/wiki/Celtic_knot" title="Celtic knot">Celtic knot</a></li> <li><a href="/wiki/Croatian_interlace" title="Croatian interlace">Croatian interlace</a></li> <li><a href="/wiki/Interlace_(art)" title="Interlace (art)">Interlace</a></li></ul></li> <li><a href="/wiki/Music_and_mathematics" title="Music and mathematics">Music</a></li> <li><a href="/wiki/Origami" title="Origami">Origami</a> <ul><li><a href="/wiki/Mathematics_of_paper_folding" title="Mathematics of paper folding">Mathematics</a></li></ul></li> <li><a href="/wiki/Mathematical_sculpture" title="Mathematical sculpture">Sculpture</a></li> <li><a href="/wiki/String_art" title="String art">String art</a></li> <li><a href="/wiki/String_figure" title="String figure">String figure</a></li> <li><a class="mw-selflink selflink">Tiling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Artworks</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/List_of_works_designed_with_the_golden_ratio" title="List of works designed with the golden ratio">List of works designed with the golden ratio</a></li> <li><i><a href="/wiki/Continuum_(sculpture)" title="Continuum (sculpture)">Continuum</a></i></li> <li><i><a href="/wiki/Mathemalchemy" title="Mathemalchemy">Mathemalchemy</a></i></li> <li><i><a href="/wiki/Mathematica:_A_World_of_Numbers..._and_Beyond" title="Mathematica: A World of Numbers... and Beyond">Mathematica: A World of Numbers... and Beyond</a></i></li> <li><i><a href="/wiki/Octacube_(sculpture)" title="Octacube (sculpture)">Octacube</a></i></li> <li><i><a href="/wiki/Pi_(art_project)" title="Pi (art project)">Pi</a></i></li> <li><i><a href="/wiki/Pi_in_the_Sky" title="Pi in the Sky">Pi in the Sky</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematics_and_architecture" title="Mathematics and architecture">Buildings</a></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/Cathedral_of_Saint_Mary_of_the_Assumption_(San_Francisco)" title="Cathedral of Saint Mary of the Assumption (San Francisco)">Cathedral of Saint Mary of the Assumption</a></li> <li><a href="/wiki/Hagia_Sophia" title="Hagia Sophia">Hagia Sophia</a></li> <li><a href="/wiki/Kresge_Auditorium" title="Kresge Auditorium">Kresge Auditorium</a></li> <li><a href="/wiki/Pantheon,_Rome" title="Pantheon, Rome">Pantheon</a></li> <li><a href="/wiki/Parthenon" title="Parthenon">Parthenon</a></li> <li><a href="/wiki/Great_Pyramid_of_Giza" title="Great Pyramid of Giza">Pyramid of Khufu</a></li> <li><a href="/wiki/Sagrada_Fam%C3%ADlia" title="Sagrada Família">Sagrada Família</a></li> <li><a href="/wiki/Sydney_Opera_House" title="Sydney Opera House">Sydney Opera House</a></li> <li><a href="/wiki/Taj_Mahal" title="Taj Mahal">Taj Mahal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_mathematical_artists" title="List of mathematical artists">Artists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" 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%">Renaissance</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/Paolo_Uccello" title="Paolo Uccello">Paolo Uccello</a></li> <li><a href="/wiki/Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a></li> <li><a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> <ul><li><i><a href="/wiki/Vitruvian_Man" title="Vitruvian Man">Vitruvian Man</a></i></li></ul></li> <li><a href="/wiki/Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a></li> <li><a href="/wiki/Parmigianino" title="Parmigianino">Parmigianino</a> <ul><li><i><a href="/wiki/Self-portrait_in_a_Convex_Mirror" class="mw-redirect" title="Self-portrait in a Convex Mirror">Self-portrait in a Convex Mirror</a></i></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">19th–20th<br />Century</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/William_Blake" title="William Blake">William Blake</a> <ul><li><i><a href="/wiki/The_Ancient_of_Days" title="The Ancient of Days">The Ancient of Days</a></i></li> <li><i><a href="/wiki/Newton_(Blake)" title="Newton (Blake)">Newton</a></i></li></ul></li> <li><a href="/wiki/Jean_Metzinger" title="Jean Metzinger">Jean Metzinger</a> <ul><li><i><a href="/wiki/Dancer_in_a_Caf%C3%A9" title="Dancer in a Café">Danseuse au café</a></i></li> <li><i><a href="/wiki/L%27Oiseau_bleu_(Metzinger)" class="mw-redirect" title="L'Oiseau bleu (Metzinger)">L'Oiseau bleu</a></i></li></ul></li> <li><a href="/wiki/Giorgio_de_Chirico" title="Giorgio de Chirico">Giorgio de Chirico</a></li> <li><a href="/wiki/Man_Ray" title="Man Ray">Man Ray</a></li> <li><a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a> <ul><li><i><a href="/wiki/Circle_Limit_III" title="Circle Limit III">Circle Limit III</a></i></li> <li><i><a href="/wiki/Print_Gallery_(M._C._Escher)" title="Print Gallery (M. C. Escher)">Print Gallery</a></i></li> <li><i><a href="/wiki/Relativity_(M._C._Escher)" title="Relativity (M. C. Escher)">Relativity</a></i></li> <li><i><a href="/wiki/Reptiles_(M._C._Escher)" title="Reptiles (M. C. Escher)">Reptiles</a></i></li> <li><i><a href="/wiki/Waterfall_(M._C._Escher)" title="Waterfall (M. C. Escher)">Waterfall</a></i></li></ul></li> <li><a href="/wiki/Ren%C3%A9_Magritte" title="René Magritte">René Magritte</a> <ul><li><i><a href="/wiki/The_Human_Condition_(Magritte)" title="The Human Condition (Magritte)">La condition humaine</a></i></li></ul></li> <li><a href="/wiki/Salvador_Dal%C3%AD" title="Salvador Dalí">Salvador Dalí</a> <ul><li><i><a href="/wiki/Crucifixion_(Corpus_Hypercubus)" title="Crucifixion (Corpus Hypercubus)">Crucifixion</a></i></li> <li><i><a href="/wiki/The_Swallow%27s_Tail" title="The Swallow's Tail">The Swallow's Tail</a></i></li></ul></li> <li><a href="/wiki/Crockett_Johnson" title="Crockett Johnson">Crockett Johnson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Contemporary</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/Max_Bill" title="Max Bill">Max Bill</a></li> <li><a href="/wiki/Martin_Demaine" title="Martin Demaine">Martin</a> and <a href="/wiki/Erik_Demaine" title="Erik Demaine">Erik Demaine</a></li> <li><a href="/wiki/Scott_Draves" title="Scott Draves">Scott Draves</a></li> <li><a href="/wiki/Jan_Dibbets" title="Jan Dibbets">Jan Dibbets</a></li> <li><a href="/wiki/John_Ernest" title="John Ernest">John Ernest</a></li> <li><a href="/wiki/Helaman_Ferguson" title="Helaman Ferguson">Helaman Ferguson</a></li> <li><a href="/wiki/Peter_Forakis" title="Peter Forakis">Peter Forakis</a></li> <li><a href="/wiki/Susan_Goldstine" title="Susan Goldstine">Susan Goldstine</a></li> <li><a href="/wiki/Bathsheba_Grossman" title="Bathsheba Grossman">Bathsheba Grossman</a></li> <li><a href="/wiki/George_W._Hart" title="George W. Hart">George W. Hart</a></li> <li><a href="/wiki/Desmond_Paul_Henry" title="Desmond Paul Henry">Desmond Paul Henry</a></li> <li><a href="/wiki/Anthony_Hill_(artist)" title="Anthony Hill (artist)">Anthony Hill</a></li> <li><a href="/wiki/Charles_Jencks" title="Charles Jencks">Charles Jencks</a> <ul><li><i><a href="/wiki/Garden_of_Cosmic_Speculation" title="Garden of Cosmic Speculation">Garden of Cosmic Speculation</a></i></li></ul></li> <li><a href="/wiki/Andy_Lomas" title="Andy Lomas">Andy Lomas</a></li> <li><a href="/wiki/Robert_Longhurst" title="Robert Longhurst">Robert Longhurst</a></li> <li><a href="/wiki/Jeanette_McLeod" title="Jeanette McLeod">Jeanette McLeod</a></li> <li><a href="/wiki/Hamid_Naderi_Yeganeh" title="Hamid Naderi Yeganeh">Hamid Naderi Yeganeh</a></li> <li><a href="/wiki/Istv%C3%A1n_Orosz" title="István Orosz">István Orosz</a></li> <li><a href="/wiki/Hinke_Osinga" title="Hinke Osinga">Hinke Osinga</a></li> <li><a href="/wiki/Antoine_Pevsner" title="Antoine Pevsner">Antoine Pevsner</a></li> <li><a href="/wiki/Tony_Robbin" title="Tony Robbin">Tony Robbin</a></li> <li><a href="/wiki/Alba_Rojo_Cama" title="Alba Rojo Cama">Alba Rojo Cama</a></li> <li><a href="/wiki/Reza_Sarhangi" class="mw-redirect" title="Reza Sarhangi">Reza Sarhangi</a></li> <li><a href="/wiki/Oliver_Sin" title="Oliver Sin">Oliver Sin</a></li> <li><a href="/wiki/Hiroshi_Sugimoto" title="Hiroshi Sugimoto">Hiroshi Sugimoto</a></li> <li><a href="/wiki/Daina_Taimi%C5%86a" title="Daina Taimiņa">Daina Taimiņa</a></li> <li><a href="/wiki/Roman_Verostko" title="Roman Verostko">Roman Verostko</a></li> <li><a href="/wiki/Margaret_Wertheim" title="Margaret Wertheim">Margaret Wertheim</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorists</th><td class="navbox-list-with-group navbox-list navbox-odd" 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%">Ancient</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/Polykleitos" title="Polykleitos">Polykleitos</a> <ul><li><i>Canon</i></li></ul></li> <li><a href="/wiki/Vitruvius" title="Vitruvius">Vitruvius</a> <ul><li><i><a href="/wiki/De_architectura" title="De architectura">De architectura</a></i></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Renaissance</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/Filippo_Brunelleschi" title="Filippo Brunelleschi">Filippo Brunelleschi</a></li> <li><a href="/wiki/Leon_Battista_Alberti" title="Leon Battista Alberti">Leon Battista Alberti</a> <ul><li><i><a href="/wiki/De_pictura" title="De pictura">De pictura</a></i></li> <li><i><a href="/wiki/De_re_aedificatoria" title="De re aedificatoria">De re aedificatoria</a></i></li></ul></li> <li><a href="/wiki/Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a> <ul><li><i><a href="/wiki/De_prospectiva_pingendi" title="De prospectiva pingendi">De prospectiva pingendi</a></i></li></ul></li> <li><a href="/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a> <ul><li><i><a href="/wiki/Divina_proportione" title="Divina proportione">De divina proportione</a></i></li></ul></li> <li><a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> <ul><li><i><a href="/wiki/A_Treatise_on_Painting" title="A Treatise on Painting">A Treatise on Painting</a></i></li></ul></li> <li><a href="/wiki/Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a> <ul><li><i>Vier Bücher von Menschlicher Proportion</i></li></ul></li> <li><a href="/wiki/Sebastiano_Serlio" title="Sebastiano Serlio">Sebastiano Serlio</a> <ul><li><i>Regole generali d'architettura</i></li></ul></li> <li><a href="/wiki/Andrea_Palladio" title="Andrea Palladio">Andrea Palladio</a> <ul><li><i><a href="/wiki/I_quattro_libri_dell%27architettura" title="I quattro libri dell'architettura">I quattro libri dell'architettura</a></i></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Romantic</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/Samuel_Colman" title="Samuel Colman">Samuel Colman</a> <ul><li><i>Nature's Harmonic Unity</i></li></ul></li> <li><a href="/wiki/Frederik_Macody_Lund" title="Frederik Macody Lund">Frederik Macody Lund</a> <ul><li><i>Ad Quadratum</i></li></ul></li> <li><a href="/wiki/Jay_Hambidge" title="Jay Hambidge">Jay Hambidge</a> <ul><li><i>The Greek Vase</i></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern</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/Owen_Jones_(architect)" title="Owen Jones (architect)">Owen Jones</a> <ul><li><i><a href="/wiki/Owen_Jones_(architect)#The_Grammar_of_Ornament" title="Owen Jones (architect)">The Grammar of Ornament</a></i></li></ul></li> <li><a href="/wiki/Ernest_Hanbury_Hankin" title="Ernest Hanbury Hankin">Ernest Hanbury Hankin</a> <ul><li><i>The Drawing of Geometric Patterns in Saracenic Art</i></li></ul></li> <li><a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a> <ul><li><i><a href="/wiki/A_Mathematician%27s_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a></i></li></ul></li> <li><a href="/wiki/George_David_Birkhoff" title="George David Birkhoff">George David Birkhoff</a> <ul><li><i>Aesthetic Measure</i></li></ul></li> <li><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a> <ul><li><i><a href="/wiki/G%C3%B6del,_Escher,_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></i></li></ul></li> <li><a href="/wiki/Nikos_Salingaros" title="Nikos Salingaros">Nikos Salingaros</a> <ul><li><i>The 'Life' of a Carpet</i></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Publications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Journal_of_Mathematics_and_the_Arts" title="Journal of Mathematics and the Arts">Journal of Mathematics and the Arts</a></i></li> <li><i><a href="/wiki/Lumen_Naturae" title="Lumen Naturae">Lumen Naturae</a></i></li> <li><i><a href="/wiki/Making_Mathematics_with_Needlework" title="Making Mathematics with Needlework">Making Mathematics with Needlework</a></i></li> <li><i><a href="/wiki/Rhythm_of_Structure" title="Rhythm of Structure">Rhythm of Structure</a></i></li> <li><i><a href="/wiki/Viewpoints:_Mathematical_Perspective_and_Fractal_Geometry_in_Art" title="Viewpoints: Mathematical Perspective and Fractal Geometry in Art">Viewpoints: Mathematical Perspective and Fractal Geometry in Art</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</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/Ars_Mathematica_(organization)" title="Ars Mathematica (organization)">Ars Mathematica</a></li> <li><a href="/wiki/The_Bridges_Organization" title="The Bridges Organization">The Bridges Organization</a></li> <li><a href="/wiki/European_Society_for_Mathematics_and_the_Arts" title="European Society for Mathematics and the Arts">European Society for Mathematics and the Arts</a></li> <li><a href="/wiki/Goudreau_Museum_of_Mathematics_in_Art_and_Science" title="Goudreau Museum of Mathematics in Art and Science">Goudreau Museum of Mathematics in Art and Science</a></li> <li><a href="/wiki/Institute_For_Figuring" title="Institute For Figuring">Institute For Figuring</a></li> <li><a href="/wiki/Mathemalchemy" title="Mathemalchemy">Mathemalchemy</a></li> <li><a href="/wiki/National_Museum_of_Mathematics" title="National Museum of Mathematics">National Museum of Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Droste_effect" title="Droste effect">Droste effect</a></li> <li><a href="/wiki/Mathematical_beauty" title="Mathematical beauty">Mathematical beauty</a></li> <li><a href="/wiki/Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></li> <li><a href="/wiki/Sacred_geometry" title="Sacred geometry">Sacred geometry</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Mathematics_and_art" title="Category:Mathematics and art">Category</a></b></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" /></div><div role="navigation" class="navbox" aria-labelledby="Tessellation147" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tessellation" title="Template:Tessellation"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tessellation" title="Template talk:Tessellation"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tessellation" title="Special:EditPage/Template:Tessellation"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tessellation147" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Tessellation</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Periodic147" style="font-size:114%;margin:0 4em"><a href="/wiki/Euclidean_tilings_by_convex_regular_polygons" title="Euclidean tilings by convex regular polygons">Periodic</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pythagorean_tiling" title="Pythagorean tiling">Pythagorean</a></li> <li><a href="/wiki/Rhombille_tiling" title="Rhombille tiling">Rhombille</a></li> <li><a href="/wiki/Schwarz_triangle" title="Schwarz triangle">Schwarz triangle</a></li> <li><a href="/wiki/Tiling_with_rectangles" title="Tiling with rectangles">Rectangle</a> <ul><li><a href="/wiki/Domino_tiling" title="Domino tiling">Domino</a></li></ul></li> <li><a href="/wiki/Uniform_tiling" title="Uniform tiling">Uniform tiling</a> and <a href="/wiki/Uniform_honeycomb" title="Uniform honeycomb">honeycomb</a> <ul><li><a href="/wiki/Uniform_coloring" title="Uniform coloring">Coloring</a></li> <li><a href="/wiki/List_of_Euclidean_uniform_tilings" title="List of Euclidean uniform tilings">Convex</a></li> <li><a href="/wiki/Kisrhombille" title="Kisrhombille">Kisrhombille</a></li></ul></li> <li><a href="/wiki/Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li> <li><a href="/wiki/Wythoff_construction" title="Wythoff construction">Wythoff</a></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg/90px-Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg" decoding="async" width="90" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg/135px-Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg/180px-Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg 2x" data-file-width="543" data-file-height="566" /></a></span><br /><br /> <span typeof="mw:File"><a href="/wiki/File:Caris_Tessellation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Caris_Tessellation.svg/90px-Caris_Tessellation.svg.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Caris_Tessellation.svg/135px-Caris_Tessellation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Caris_Tessellation.svg/180px-Caris_Tessellation.svg.png 2x" data-file-width="1624" data-file-height="1624" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Aperiodic147" style="font-size:114%;margin:0 4em"><a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">Aperiodic</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ammann%E2%80%93Beenker_tiling" title="Ammann–Beenker tiling">Ammann–Beenker</a></li> <li><a href="/wiki/Aperiodic_set_of_prototiles" title="Aperiodic set of prototiles">Aperiodic set of prototiles</a> <ul><li><a href="/wiki/List_of_aperiodic_sets_of_tiles" title="List of aperiodic sets of tiles">List</a></li></ul></li> <li><a href="/wiki/Einstein_problem" title="Einstein problem">Einstein problem</a> <ul><li><a href="/wiki/Socolar%E2%80%93Taylor_tile" title="Socolar–Taylor tile">Socolar–Taylor</a></li></ul></li> <li><a href="/wiki/Gilbert_tessellation" title="Gilbert tessellation">Gilbert</a></li> <li><a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose</a></li> <li><a href="/wiki/Pinwheel_tiling" title="Pinwheel tiling">Pinwheel</a></li> <li><a href="/wiki/Quaquaversal_tiling" title="Quaquaversal tiling">Quaquaversal</a></li> <li><a href="/wiki/Rep-tile" title="Rep-tile">Rep-tile</a> and <a href="/wiki/Self-tiling_tile_set" title="Self-tiling tile set">Self-tiling</a> <ul><li><a href="/wiki/Sphinx_tiling" title="Sphinx tiling">Sphinx</a></li></ul></li> <li><a href="/wiki/Socolar_tiling" title="Socolar tiling">Socolar</a></li> <li><a href="/wiki/Truchet_tiles" title="Truchet tiles">Truchet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other147" style="font-size:114%;margin:0 4em">Other</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anisohedral_tiling" title="Anisohedral tiling">Anisohedral</a> and <a href="/wiki/Isohedral_figure" title="Isohedral figure">Isohedral</a></li> <li><a href="/wiki/Architectonic_and_catoptric_tessellation" title="Architectonic and catoptric tessellation">Architectonic and catoptric</a></li> <li><i><a href="/wiki/Circle_Limit_III" title="Circle Limit III">Circle Limit III</a></i></li> <li><a href="/wiki/Tessellation_(computer_graphics)" title="Tessellation (computer graphics)">Computer graphics</a></li> <li><a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">Honeycomb</a></li> <li><a href="/wiki/List_of_isotoxal_polyhedra_and_tilings" title="List of isotoxal polyhedra and tilings">Isotoxal</a></li> <li><a href="/wiki/List_of_tessellations" title="List of tessellations">List</a></li> <li><a href="/wiki/Packing_problems" title="Packing problems">Packing</a></li> <li><a href="/wiki/Pentagonal_tiling" title="Pentagonal tiling">Pentagonal</a></li> <li>Problems <ul><li><a href="/wiki/Domino_problem" class="mw-redirect" title="Domino problem">Domino</a> <ul><li><a href="/wiki/Wang_tile" title="Wang tile">Wang</a></li></ul></li> <li><a href="/wiki/Heesch%27s_problem" title="Heesch's problem">Heesch's</a></li> <li><a href="/wiki/Squaring_the_square" title="Squaring the square">Squaring</a></li> <li><a href="/wiki/Dividing_a_square_into_similar_rectangles" title="Dividing a square into similar rectangles">Dividing a square into similar rectangles</a></li></ul></li> <li><a href="/wiki/Prototile" title="Prototile">Prototile</a> <ul><li><a href="/wiki/Conway_criterion" title="Conway criterion">Conway criterion</a></li> <li><a href="/wiki/Girih_tiles" title="Girih tiles">Girih</a></li></ul></li> <li><i><a href="/wiki/Regular_Division_of_the_Plane" title="Regular Division of the Plane">Regular Division of the Plane</a></i></li> <li><a href="/wiki/Regular_grid" title="Regular grid">Regular grid</a></li> <li><a href="/wiki/Substitution_tiling" title="Substitution tiling">Substitution</a></li> <li><a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi</a></li> <li><a href="/wiki/Voderberg_tiling" title="Voderberg tiling">Voderberg</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="By_vertex_type147" style="font-size:114%;margin:0 4em">By <a href="/wiki/Vertex_configuration" title="Vertex configuration">vertex type</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" 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%"><a href="/wiki/Spherical_polyhedron" title="Spherical polyhedron">Spherical</a></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/Hosohedron" title="Hosohedron">2<sup>n</sup></a></li> <li><a href="/wiki/Antiprism" title="Antiprism">3<sup>3</sup>.n</a></li> <li><a href="/wiki/Trapezohedron" title="Trapezohedron">V3<sup>3</sup>.n</a></li> <li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">4<sup>2</sup>.n</a></li> <li><a href="/wiki/Bipyramid" title="Bipyramid">V4<sup>2</sup>.n</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Regular</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/Apeirogonal_hosohedron" title="Apeirogonal hosohedron">2<sup>∞</sup></a></li> <li><a href="/wiki/Triangular_tiling" title="Triangular tiling">3<sup>6</sup></a></li> <li><a href="/wiki/Square_tiling" title="Square tiling">4<sup>4</sup></a></li> <li><a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">6<sup>3</sup></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Semi-<br />regular</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/Snub_square_tiling" title="Snub square tiling">3<sup>2</sup>.4.3.4</a></li> <li><a href="/wiki/Cairo_pentagonal_tiling" title="Cairo pentagonal tiling">V3<sup>2</sup>.4.3.4</a></li> <li><a href="/wiki/Elongated_triangular_tiling" title="Elongated triangular tiling">3<sup>3</sup>.4<sup>2</sup></a></li> <li><a href="/wiki/Apeirogonal_antiprism" title="Apeirogonal antiprism">3<sup>3</sup>.∞</a></li> <li><a href="/wiki/Snub_trihexagonal_tiling" title="Snub trihexagonal tiling">3<sup>4</sup>.6</a></li> <li><a href="/wiki/Floret_pentagonal_tiling" class="mw-redirect" title="Floret pentagonal tiling">V3<sup>4</sup>.6</a></li> <li><a href="/wiki/Rhombitrihexagonal_tiling" title="Rhombitrihexagonal tiling">3.4.6.4</a></li> <li><a href="/wiki/Trihexagonal_tiling" title="Trihexagonal tiling">(3.6)<sup>2</sup></a></li> <li><a href="/wiki/Truncated_hexagonal_tiling" title="Truncated hexagonal tiling">3.12<sup>2</sup></a></li> <li><a href="/wiki/Apeirogonal_prism" title="Apeirogonal prism">4<sup>2</sup>.∞</a></li> <li><a href="/wiki/Truncated_trihexagonal_tiling" title="Truncated trihexagonal tiling">4.6.12</a></li> <li><a href="/wiki/Truncated_square_tiling" title="Truncated square tiling">4.8<sup>2</sup></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Uniform_tilings_in_hyperbolic_plane" title="Uniform tilings in hyperbolic plane">Hyper-<br />bolic</a></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/Snub_tetrapentagonal_tiling" title="Snub tetrapentagonal tiling">3<sup>2</sup>.4.3.5</a></li> <li><a href="/wiki/Snub_tetrahexagonal_tiling" title="Snub tetrahexagonal tiling">3<sup>2</sup>.4.3.6</a></li> <li><a href="/wiki/Snub_tetraheptagonal_tiling" title="Snub tetraheptagonal tiling">3<sup>2</sup>.4.3.7</a></li> <li><a href="/wiki/Snub_tetraoctagonal_tiling" title="Snub tetraoctagonal tiling">3<sup>2</sup>.4.3.8</a></li> <li><a href="/wiki/Snub_tetraapeirogonal_tiling" title="Snub tetraapeirogonal tiling">3<sup>2</sup>.4.3.∞</a></li> <li><a href="/wiki/Snub_pentapentagonal_tiling" title="Snub pentapentagonal tiling">3<sup>2</sup>.5.3.5</a></li> <li><a href="/wiki/Snub_pentahexagonal_tiling" title="Snub pentahexagonal tiling">3<sup>2</sup>.5.3.6</a></li> <li><a href="/wiki/Snub_hexahexagonal_tiling" title="Snub hexahexagonal tiling">3<sup>2</sup>.6.3.6</a></li> <li><a href="/wiki/Snub_hexaoctagonal_tiling" title="Snub hexaoctagonal tiling">3<sup>2</sup>.6.3.8</a></li> <li><a href="/wiki/Snub_heptaheptagonal_tiling" title="Snub heptaheptagonal tiling">3<sup>2</sup>.7.3.7</a></li> <li><a href="/wiki/Snub_octaoctagonal_tiling" title="Snub octaoctagonal tiling">3<sup>2</sup>.8.3.8</a></li> <li><a href="/wiki/Snub_order-6_square_tiling" title="Snub order-6 square tiling">3<sup>3</sup>.4.3.4</a></li> <li><a href="/wiki/Snub_apeiroapeirogonal_tiling" title="Snub apeiroapeirogonal tiling">3<sup>2</sup>.∞.3.∞</a></li> <li><a href="/wiki/Snub_triheptagonal_tiling" title="Snub triheptagonal tiling">3<sup>4</sup>.7</a></li> <li><a href="/wiki/Snub_trioctagonal_tiling" title="Snub trioctagonal tiling">3<sup>4</sup>.8</a></li> <li><a href="/wiki/Snub_triapeirogonal_tiling" title="Snub triapeirogonal tiling">3<sup>4</sup>.∞</a></li> <li><a href="/wiki/Snub_order-8_triangular_tiling" title="Snub order-8 triangular tiling">3<sup>5</sup>.4</a></li> <li><a href="/wiki/Order-7_triangular_tiling" title="Order-7 triangular tiling">3<sup>7</sup></a></li> <li><a href="/wiki/Order-8_triangular_tiling" title="Order-8 triangular tiling">3<sup>8</sup></a></li> <li><a href="/wiki/Infinite-order_triangular_tiling" title="Infinite-order triangular tiling">3<sup>∞</sup></a></li> <li><a href="/wiki/Alternated_octagonal_tiling" title="Alternated octagonal tiling">(3.4)<sup>3</sup></a></li> <li><a href="/wiki/Alternated_order-4_hexagonal_tiling" title="Alternated order-4 hexagonal tiling">(3.4)<sup>4</sup></a></li> <li><a href="/wiki/Quarter_order-6_square_tiling" title="Quarter order-6 square tiling">3.4.6<sup>2</sup>.4</a></li> <li><a href="/wiki/Rhombitriheptagonal_tiling" title="Rhombitriheptagonal tiling">3.4.7.4</a></li> <li><a href="/wiki/Rhombitrioctagonal_tiling" title="Rhombitrioctagonal tiling">3.4.8.4</a></li> <li><a href="/wiki/Rhombitriapeirogonal_tiling" title="Rhombitriapeirogonal tiling">3.4.∞.4</a></li> <li><a href="/wiki/Cantic_octagonal_tiling" title="Cantic octagonal tiling">3.6.4.6</a></li> <li><a href="/wiki/Triheptagonal_tiling" title="Triheptagonal tiling">(3.7)<sup>2</sup></a></li> <li><a href="/wiki/Trioctagonal_tiling" title="Trioctagonal tiling">(3.8)<sup>2</sup></a></li> <li><a href="/wiki/Truncated_heptagonal_tiling" title="Truncated heptagonal tiling">3.14<sup>2</sup></a></li> <li><a href="/wiki/Truncated_octagonal_tiling" title="Truncated octagonal tiling">3.16<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-3_apeirogonal_tiling" title="Truncated order-3 apeirogonal tiling">3.∞<sup>2</sup></a></li> <li><a href="/wiki/Rhombitetrapentagonal_tiling" title="Rhombitetrapentagonal tiling">4<sup>2</sup>.5.4</a></li> <li><a href="/wiki/Rhombitetrahexagonal_tiling" title="Rhombitetrahexagonal tiling">4<sup>2</sup>.6.4</a></li> <li><a href="/wiki/Rhombitetraheptagonal_tiling" title="Rhombitetraheptagonal tiling">4<sup>2</sup>.7.4</a></li> <li><a href="/wiki/Rhombitetraoctagonal_tiling" title="Rhombitetraoctagonal tiling">4<sup>2</sup>.8.4</a></li> <li><a href="/wiki/Rhombitetraapeirogonal_tiling" title="Rhombitetraapeirogonal tiling">4<sup>2</sup>.∞.4</a></li> <li><a href="/wiki/Order-5_square_tiling" title="Order-5 square tiling">4<sup>5</sup></a></li> <li><a href="/wiki/Order-6_square_tiling" title="Order-6 square tiling">4<sup>6</sup></a></li> <li><a href="/wiki/Order-7_square_tiling" title="Order-7 square tiling">4<sup>7</sup></a></li> <li><a href="/wiki/Order-8_square_tiling" title="Order-8 square tiling">4<sup>8</sup></a></li> <li><a href="/wiki/Infinite-order_square_tiling" title="Infinite-order square tiling">4<sup>∞</sup></a></li> <li><a href="/wiki/Tetrapentagonal_tiling" title="Tetrapentagonal tiling">(4.5)<sup>2</sup></a></li> <li><a href="/wiki/Tetrahexagonal_tiling" title="Tetrahexagonal tiling">(4.6)<sup>2</sup></a></li> <li><a href="/wiki/Truncated_trihexagonal_tiling" title="Truncated trihexagonal tiling">4.6.12</a></li> <li><a href="/wiki/Truncated_triheptagonal_tiling" title="Truncated triheptagonal tiling">4.6.14</a></li> <li><a href="/wiki/3-7_kisrhombille" title="3-7 kisrhombille">V4.6.14</a></li> <li><a href="/wiki/Truncated_trioctagonal_tiling" title="Truncated trioctagonal tiling">4.6.16</a></li> <li><a href="/wiki/Order_3-8_kisrhombille" class="mw-redirect" title="Order 3-8 kisrhombille">V4.6.16</a></li> <li><a href="/wiki/Truncated_triapeirogonal_tiling" title="Truncated triapeirogonal tiling">4.6.∞</a></li> <li><a href="/wiki/Tetraheptagonal_tiling" title="Tetraheptagonal tiling">(4.7)<sup>2</sup></a></li> <li><a href="/wiki/Tetraoctagonal_tiling" title="Tetraoctagonal tiling">(4.8)<sup>2</sup></a></li> <li><a href="/wiki/Truncated_tetrapentagonal_tiling" title="Truncated tetrapentagonal tiling">4.8.10</a></li> <li><a href="/wiki/4-5_kisrhombille" title="4-5 kisrhombille">V4.8.10</a></li> <li><a href="/wiki/Truncated_tetrahexagonal_tiling" title="Truncated tetrahexagonal tiling">4.8.12</a></li> <li><a href="/wiki/Truncated_tetraheptagonal_tiling" title="Truncated tetraheptagonal tiling">4.8.14</a></li> <li><a href="/wiki/Truncated_tetraoctagonal_tiling" title="Truncated tetraoctagonal tiling">4.8.16</a></li> <li><a href="/wiki/Truncated_tetraapeirogonal_tiling" title="Truncated tetraapeirogonal tiling">4.8.∞</a></li> <li><a href="/wiki/Truncated_order-4_pentagonal_tiling" title="Truncated order-4 pentagonal tiling">4.10<sup>2</sup></a></li> <li><a href="/wiki/Truncated_pentahexagonal_tiling" title="Truncated pentahexagonal tiling">4.10.12</a></li> <li><a href="/wiki/Truncated_order-4_hexagonal_tiling" title="Truncated order-4 hexagonal tiling">4.12<sup>2</sup></a></li> <li><a href="/wiki/Truncated_hexaoctagonal_tiling" title="Truncated hexaoctagonal tiling">4.12.16</a></li> <li><a href="/wiki/Truncated_order-4_heptagonal_tiling" title="Truncated order-4 heptagonal tiling">4.14<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-4_octagonal_tiling" title="Truncated order-4 octagonal tiling">4.16<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-4_apeirogonal_tiling" title="Truncated order-4 apeirogonal tiling">4.∞<sup>2</sup></a></li> <li><a href="/wiki/Order-4_pentagonal_tiling" title="Order-4 pentagonal tiling">5<sup>4</sup></a></li> <li><a href="/wiki/Order-5_pentagonal_tiling" title="Order-5 pentagonal tiling">5<sup>5</sup></a></li> <li><a href="/wiki/Order-6_pentagonal_tiling" title="Order-6 pentagonal tiling">5<sup>6</sup></a></li> <li><a href="/wiki/Infinite-order_pentagonal_tiling" title="Infinite-order pentagonal tiling">5<sup>∞</sup></a></li> <li><a href="/wiki/Rhombipentahexagonal_tiling" title="Rhombipentahexagonal tiling">5.4.6.4</a></li> <li><a href="/wiki/Pentahexagonal_tiling" title="Pentahexagonal tiling">(5.6)<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-5_square_tiling" title="Truncated order-5 square tiling">5.8<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-5_pentagonal_tiling" title="Truncated order-5 pentagonal tiling">5.10<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-5_hexagonal_tiling" title="Truncated order-5 hexagonal tiling">5.12<sup>2</sup></a></li> <li><a href="/wiki/Order-4_hexagonal_tiling" title="Order-4 hexagonal tiling">6<sup>4</sup></a></li> <li><a href="/wiki/Order-5_hexagonal_tiling" title="Order-5 hexagonal tiling">6<sup>5</sup></a></li> <li><a href="/wiki/Order-6_hexagonal_tiling" title="Order-6 hexagonal tiling">6<sup>6</sup></a></li> <li><a href="/wiki/Order-8_hexagonal_tiling" title="Order-8 hexagonal tiling">6<sup>8</sup></a></li> <li><a href="/wiki/Rhombihexaoctagonal_tiling" title="Rhombihexaoctagonal tiling">6.4.8.4</a></li> <li><a href="/wiki/Hexaoctagonal_tiling" title="Hexaoctagonal tiling">(6.8)<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-6_square_tiling" title="Truncated order-6 square tiling">6.8<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-6_pentagonal_tiling" title="Truncated order-6 pentagonal tiling">6.10<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-6_hexagonal_tiling" title="Truncated order-6 hexagonal tiling">6.12<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-6_octagonal_tiling" title="Truncated order-6 octagonal tiling">6.16<sup>2</sup></a></li> <li><a href="/wiki/Heptagonal_tiling" title="Heptagonal tiling">7<sup>3</sup></a></li> <li><a href="/wiki/Order-4_heptagonal_tiling" title="Order-4 heptagonal tiling">7<sup>4</sup></a></li> <li><a href="/wiki/Order-7_heptagonal_tiling" title="Order-7 heptagonal tiling">7<sup>7</sup></a></li> <li><a href="/wiki/Truncated_order-7_triangular_tiling" title="Truncated order-7 triangular tiling">7.6<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-7_square_tiling" title="Truncated order-7 square tiling">7.8<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-7_heptagonal_tiling" title="Truncated order-7 heptagonal tiling">7.14<sup>2</sup></a></li> <li><a href="/wiki/Octagonal_tiling" title="Octagonal tiling">8<sup>3</sup></a></li> <li><a href="/wiki/Order-4_octagonal_tiling" title="Order-4 octagonal tiling">8<sup>4</sup></a></li> <li><a href="/wiki/Order-6_octagonal_tiling" title="Order-6 octagonal tiling">8<sup>6</sup></a></li> <li><a href="/wiki/Order-8_octagonal_tiling" title="Order-8 octagonal tiling">8<sup>8</sup></a></li> <li><a href="/wiki/Truncated_order-8_triangular_tiling" title="Truncated order-8 triangular tiling">8.6<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-8_hexagonal_tiling" title="Truncated order-8 hexagonal tiling">8.12<sup>2</sup></a></li> <li><a href="/wiki/Truncated_order-8_octagonal_tiling" title="Truncated order-8 octagonal tiling">8.16<sup>2</sup></a></li> <li><a href="/wiki/Order-3_apeirogonal_tiling" title="Order-3 apeirogonal tiling">∞<sup>3</sup></a></li> <li><a href="/wiki/Order-4_apeirogonal_tiling" title="Order-4 apeirogonal tiling">∞<sup>4</sup></a></li> <li><a href="/wiki/Order-5_apeirogonal_tiling" title="Order-5 apeirogonal tiling">∞<sup>5</sup></a></li> <li><a href="/wiki/Infinite-order_apeirogonal_tiling" title="Infinite-order apeirogonal tiling">∞<sup>∞</sup></a></li> <li><a href="/wiki/Truncated_infinite-order_triangular_tiling" title="Truncated infinite-order triangular tiling">∞.6<sup>2</sup></a></li> <li><a href="/wiki/Truncated_infinite-order_square_tiling" title="Truncated infinite-order square tiling">∞.8<sup>2</sup></a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></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" /></div><div role="navigation" class="navbox" aria-labelledby="Patterns_in_nature49" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Patterns_in_nature" title="Template:Patterns in nature"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Patterns_in_nature" title="Template talk:Patterns in nature"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Patterns_in_nature" title="Special:EditPage/Template:Patterns in nature"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Patterns_in_nature49" style="font-size:114%;margin:0 4em"><a href="/wiki/Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</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/Fracture" title="Fracture">Crack</a></li> <li><a href="/wiki/Dune" title="Dune">Dune</a></li> <li><a href="/wiki/Foam" title="Foam">Foam</a></li> <li><a href="/wiki/Meander" title="Meander">Meander</a></li> <li><a href="/wiki/Parastichy" title="Parastichy">Parastichy</a></li> <li><a href="/wiki/Phyllotaxis" title="Phyllotaxis">Phyllotaxis</a></li> <li><a href="/wiki/Soap_bubble" title="Soap bubble">Soap bubble</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a> <ul><li><a href="/wiki/Crystal_structure" title="Crystal structure">in crystals</a></li> <li><a href="/wiki/Quasicrystal" title="Quasicrystal">Quasicrystals</a></li> <li><a href="/wiki/Floral_symmetry" title="Floral symmetry">in flowers</a></li> <li><a href="/wiki/Symmetry_in_biology" title="Symmetry in biology">in biology</a></li></ul></li> <li><a class="mw-selflink selflink">Tessellation</a></li> <li><a href="/wiki/K%C3%A1rm%C3%A1n_vortex_street" title="Kármán vortex street">Vortex street</a></li> <li><a href="/wiki/Wave" title="Wave">Wave</a></li> <li><a href="/wiki/Widmanst%C3%A4tten_pattern" title="Widmanstätten pattern">Widmanstätten pattern</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Yemen_Chameleon_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Yemen_Chameleon_%28cropped%29.jpg/80px-Yemen_Chameleon_%28cropped%29.jpg" decoding="async" width="80" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Yemen_Chameleon_%28cropped%29.jpg/120px-Yemen_Chameleon_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Yemen_Chameleon_%28cropped%29.jpg/160px-Yemen_Chameleon_%28cropped%29.jpg 2x" data-file-width="2086" data-file-height="1411" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Causes</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/Pattern_formation" title="Pattern formation">Pattern formation</a></li> <li><a href="/wiki/Biology" title="Biology">Biology</a> <ul><li><a href="/wiki/Natural_selection" title="Natural selection">Natural selection</a></li> <li><a href="/wiki/Camouflage" title="Camouflage">Camouflage</a></li> <li><a href="/wiki/Mimicry" title="Mimicry">Mimicry</a></li> <li><a href="/wiki/Sexual_selection" title="Sexual selection">Sexual selection</a></li></ul></li> <li><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">Logarithmic spiral</a></li></ul></li> <li><a href="/wiki/Physics" title="Physics">Physics</a> <ul><li><a href="/wiki/Crystal" title="Crystal">Crystal</a></li> <li><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></li> <li><a href="/wiki/Plateau%27s_laws" title="Plateau's laws">Plateau's laws</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</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/Plato" title="Plato">Plato</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Empedocles" title="Empedocles">Empedocles</a></li> <li><a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> <ul><li><i><a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i></li></ul></li> <li><a href="/wiki/Adolf_Zeising" title="Adolf Zeising">Adolf Zeising</a></li> <li><a href="/wiki/Ernst_Haeckel" title="Ernst Haeckel">Ernst Haeckel</a></li> <li><a href="/wiki/Joseph_Plateau" title="Joseph Plateau">Joseph Plateau</a></li> <li><a href="/wiki/Wilson_Bentley" title="Wilson Bentley">Wilson Bentley</a></li> <li><a href="/wiki/D%27Arcy_Wentworth_Thompson" title="D'Arcy Wentworth Thompson">D'Arcy Wentworth Thompson</a> <ul><li><i><a href="/wiki/On_Growth_and_Form" title="On Growth and Form">On Growth and Form</a></i></li></ul></li> <li><a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> <ul><li><i><a href="/wiki/The_Chemical_Basis_of_Morphogenesis" title="The Chemical Basis of Morphogenesis">The Chemical Basis of Morphogenesis</a></i></li></ul></li> <li><a href="/wiki/Aristid_Lindenmayer" title="Aristid Lindenmayer">Aristid Lindenmayer</a></li> <li><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoît Mandelbrot</a> <ul><li><i><a href="/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension" class="mw-redirect" title="How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension">How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension</a></i></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Pattern_recognition_(psychology)" title="Pattern recognition (psychology)">Pattern recognition</a></li> <li><a href="/wiki/Emergence" title="Emergence">Emergence</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</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" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox1014" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q214856#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4126296-7">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Tessellations (Mathematics)"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85134138">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="teselace"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph1247622&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007531732305171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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