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class="vector-toc-numb">2</span> <span>Formal definition and discussion</span> </div> </a> <button aria-controls="toc-Formal_definition_and_discussion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Formal definition and discussion subsection</span> </button> <ul id="toc-Formal_definition_and_discussion-sublist" class="vector-toc-list"> <li id="toc-Isometries_of_the_Euclidean_plane" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isometries_of_the_Euclidean_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Isometries of the Euclidean plane</span> </div> </a> <ul id="toc-Isometries_of_the_Euclidean_plane-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_independent_translations_condition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_independent_translations_condition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>The independent translations condition</span> </div> </a> <ul id="toc-The_independent_translations_condition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_discreteness_condition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_discreteness_condition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>The discreteness condition</span> </div> </a> <ul id="toc-The_discreteness_condition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notations_for_wallpaper_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notations_for_wallpaper_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Notations for wallpaper groups</span> </div> </a> <ul id="toc-Notations_for_wallpaper_groups-sublist" class="vector-toc-list"> <li id="toc-Crystallographic_notation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Crystallographic_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.1</span> <span>Crystallographic notation</span> </div> </a> <ul id="toc-Crystallographic_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbifold_notation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orbifold_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.2</span> <span>Orbifold notation</span> </div> </a> <ul id="toc-Orbifold_notation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Why_there_are_exactly_seventeen_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Why_there_are_exactly_seventeen_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Why there are exactly seventeen groups</span> </div> </a> <ul id="toc-Why_there_are_exactly_seventeen_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Guide_to_recognizing_wallpaper_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Guide_to_recognizing_wallpaper_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Guide to recognizing wallpaper groups</span> </div> </a> <ul id="toc-Guide_to_recognizing_wallpaper_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_seventeen_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_seventeen_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>The seventeen groups</span> </div> </a> <button aria-controls="toc-The_seventeen_groups-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle The seventeen groups subsection</span> </button> <ul id="toc-The_seventeen_groups-sublist" class="vector-toc-list"> <li id="toc-Group_p1_(o)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p1_(o)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Group <i>p</i>1 (o)</span> </div> </a> <ul id="toc-Group_p1_(o)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p2_(2222)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p2_(2222)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Group <i>p</i>2 (2222)</span> </div> </a> <ul id="toc-Group_p2_(2222)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_pm_(**)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_pm_(**)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Group <i>pm</i> (**)</span> </div> </a> <ul id="toc-Group_pm_(**)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_pg_(××)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_pg_(××)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Group <i>pg</i> (××)</span> </div> </a> <ul id="toc-Group_pg_(××)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_cm_(*×)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_cm_(*×)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Group <i>cm</i> (*×)</span> </div> </a> <ul id="toc-Group_cm_(*×)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_pmm_(*2222)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_pmm_(*2222)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Group <i>pmm</i> (*2222)</span> </div> </a> <ul id="toc-Group_pmm_(*2222)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_pmg_(22*)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_pmg_(22*)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Group <i>pmg</i> (22*)</span> </div> </a> <ul id="toc-Group_pmg_(22*)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_pgg_(22×)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_pgg_(22×)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Group <i>pgg</i> (22×)</span> </div> </a> <ul id="toc-Group_pgg_(22×)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_cmm_(2*22)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_cmm_(2*22)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Group <i>cmm</i> (2*22)</span> </div> </a> <ul id="toc-Group_cmm_(2*22)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p4_(442)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p4_(442)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.10</span> <span>Group <i>p</i>4 (442)</span> </div> </a> <ul id="toc-Group_p4_(442)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p4m_(*442)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p4m_(*442)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.11</span> <span>Group <i>p</i>4<i>m</i> (*442)</span> </div> </a> <ul id="toc-Group_p4m_(*442)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p4g_(4*2)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p4g_(4*2)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.12</span> <span>Group <i>p</i>4<i>g</i> (4*2)</span> </div> </a> <ul id="toc-Group_p4g_(4*2)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p3_(333)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p3_(333)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.13</span> <span>Group <i>p</i>3 (333)</span> </div> </a> <ul id="toc-Group_p3_(333)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p3m1_(*333)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p3m1_(*333)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.14</span> <span>Group <i>p</i>3<i>m</i>1 (*333)</span> </div> </a> <ul id="toc-Group_p3m1_(*333)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p31m_(3*3)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p31m_(3*3)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.15</span> <span>Group <i>p</i>31<i>m</i> (3*3)</span> </div> </a> <ul id="toc-Group_p31m_(3*3)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p6_(632)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p6_(632)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.16</span> <span>Group <i>p</i>6 (632)</span> </div> </a> <ul id="toc-Group_p6_(632)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_p6m_(*632)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_p6m_(*632)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.17</span> <span>Group <i>p</i>6<i>m</i> (*632)</span> </div> </a> <ul id="toc-Group_p6m_(*632)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lattice_types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lattice_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Lattice types</span> </div> </a> <ul id="toc-Lattice_types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Symmetry_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Symmetry groups</span> </div> </a> <ul id="toc-Symmetry_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dependence_of_wallpaper_groups_on_transformations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dependence_of_wallpaper_groups_on_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Dependence of wallpaper groups on transformations</span> </div> </a> <ul id="toc-Dependence_of_wallpaper_groups_on_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Web_demo_and_software" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Web_demo_and_software"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Web demo and software</span> </div> </a> <ul id="toc-Web_demo_and_software-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <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-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <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-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Wallpaper group</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 13 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-13" 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">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gr%C5%B5p_papur_wal" title="Grŵp papur wal – Welsh" lang="cy" hreflang="cy" data-title="Grŵp papur wal" 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/Tapetgruppe" title="Tapetgruppe – Danish" lang="da" hreflang="da" data-title="Tapetgruppe" 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/Ebene_kristallographische_Gruppe" title="Ebene kristallographische Gruppe – German" lang="de" hreflang="de" data-title="Ebene kristallographische Gruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_del_papel_pintado" title="Grupo del papel pintado – Spanish" lang="es" hreflang="es" data-title="Grupo del papel pintado" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_de_papier_peint" title="Groupe de papier peint – French" lang="fr" hreflang="fr" data-title="Groupe de papier peint" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%EB%A9%B4%EC%9D%98_%EA%B2%B0%EC%A0%95%EA%B5%B0" title="평면의 결정군 – Korean" lang="ko" hreflang="ko" data-title="평면의 결정군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_kertas_dinding" title="Grup kertas dinding – Indonesian" lang="id" hreflang="id" data-title="Grup kertas dinding" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Behangpatroongroep" title="Behangpatroongroep – Dutch" lang="nl" hreflang="nl" data-title="Behangpatroongroep" 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/%E6%96%87%E6%A7%98%E7%BE%A4" title="文様群 – Japanese" lang="ja" hreflang="ja" data-title="文様群" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%BE%D1%80%D0%BD%D0%B0%D0%BC%D0%B5%D0%BD%D1%82%D0%B0" title="Группа орнамента – Russian" lang="ru" hreflang="ru" data-title="Группа орнамента" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Symmetrier_i_planet" title="Symmetrier i planet – Swedish" lang="sv" hreflang="sv" data-title="Symmetrier i planet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%B3%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AF%80%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="தளச் சமச்சீர்க் குலம் – Tamil" lang="ta" hreflang="ta" data-title="தளச் சமச்சீர்க் குலம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_%D0%BE%D1%80%D0%BD%D0%B0%D0%BC%D0%B5%D0%BD%D1%82%D1%83" title="Група орнаменту – Ukrainian" lang="uk" hreflang="uk" data-title="Група орнаменту" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q906867#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Wallpaper_group-p4m-5.jpg/375px-Wallpaper_group-p4m-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Wallpaper_group-p4m-5.jpg/500px-Wallpaper_group-p4m-5.jpg 2x" data-file-width="1074" data-file-height="1063" /></a><figcaption>Example of an <a href="/wiki/Egypt" title="Egypt">Egyptian</a> design with wallpaper group <a href="#Group_p4mm"><b><i>p</i>4<i>m</i></b></a> </figcaption></figure> <p>A <b>wallpaper group</b> (or <b>plane symmetry group</b> or <b>plane crystallographic group</b>) is a mathematical classification of a two-dimensional repetitive pattern, based on the <a href="/wiki/Symmetry" title="Symmetry">symmetries</a> in the pattern. Such patterns occur frequently in <a href="/wiki/Architecture" title="Architecture">architecture</a> and <a href="/wiki/Decorative_art" class="mw-redirect" title="Decorative art">decorative art</a>, especially in <a href="/wiki/Textile" title="Textile">textiles</a>, <a href="/wiki/Tile" title="Tile">tiles</a>, and <a href="/wiki/Wallpaper" title="Wallpaper">wallpaper</a>. </p><p>The simplest wallpaper group, Group <i>p</i>1, applies when there is no symmetry beyond simple translation of a pattern in two dimensions. The following patterns have more forms of symmetry, including some rotational and reflectional symmetries: </p> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-2.jpg" class="mw-file-description" title="Example A: Cloth, Tahiti"><img alt="Example A: Cloth, Tahiti" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wallpaper_group-p4m-2.jpg/120px-Wallpaper_group-p4m-2.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wallpaper_group-p4m-2.jpg/180px-Wallpaper_group-p4m-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wallpaper_group-p4m-2.jpg/240px-Wallpaper_group-p4m-2.jpg 2x" data-file-width="997" data-file-height="986" /></a></span></div> <div class="gallerytext">Example <b>A</b>: Cloth, <a href="/wiki/Tahiti" title="Tahiti">Tahiti</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-1.jpg" class="mw-file-description" title="Example B: Ornamental painting, Nineveh, Assyria"><img alt="Example B: Ornamental painting, Nineveh, Assyria" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/120px-Wallpaper_group-p4m-1.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/180px-Wallpaper_group-p4m-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/240px-Wallpaper_group-p4m-1.jpg 2x" data-file-width="1065" data-file-height="1057" /></a></span></div> <div class="gallerytext">Example <b>B</b>: Ornamental painting, <a href="/wiki/Nineveh" title="Nineveh">Nineveh</a>, <a href="/wiki/Assyria" title="Assyria">Assyria</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4g-2.jpg" class="mw-file-description" title="Example C: Painted porcelain, China"><img alt="Example C: Painted porcelain, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/118px-Wallpaper_group-p4g-2.jpg" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/177px-Wallpaper_group-p4g-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/237px-Wallpaper_group-p4g-2.jpg 2x" data-file-width="1054" data-file-height="1069" /></a></span></div> <div class="gallerytext">Example <b>C</b>: Painted <a href="/wiki/Porcelain" title="Porcelain">porcelain</a>, <a href="/wiki/China" title="China">China</a></div> </li> </ul> <p>Examples <b>A</b> and <b>B</b> have the same wallpaper group; it is called <a href="#Group_p4mm"><b><i>p</i>4<i>m</i></b></a> in the <a href="/wiki/International_Union_of_Crystallography#IUCr_Symmetry_notation" title="International Union of Crystallography">IUCr notation</a> and <a href="#Group_p4mm">*442</a> in the <a href="/wiki/Orbifold_notation" title="Orbifold notation">orbifold notation</a>. Example <b>C</b> has a different wallpaper group, called <a href="#Group_p4gm"><b><i>p</i>4<i>g</i></b></a> or <a href="#Group_p4gm">4*2</a> . The fact that <b>A</b> and <b>B</b> have the same wallpaper group means that they have the same symmetries, regardless of the designs' superficial details; whereas <b>C</b> has a different set of symmetries. </p><p>The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a> and the three-dimensional <a href="/wiki/Space_group" title="Space group">space groups</a>. </p><p>A <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> that there are only 17 distinct <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> of such planar symmetries was first carried out by <a href="/wiki/Yevgraf_Fyodorov" class="mw-redirect" title="Yevgraf Fyodorov">Evgraf Fedorov</a> in 1891<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and then derived independently by <a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a> in 1924.<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> The proof that the list of wallpaper groups is complete came only after the much harder case of <a href="/wiki/Space_group" title="Space group">space groups</a> had been done. The seventeen wallpaper groups are listed below; see <a href="#The_seventeen_groups">§ The seventeen groups</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Symmetries_of_patterns">Symmetries of patterns</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=1" title="Edit section: Symmetries of patterns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> of a pattern is, loosely speaking, a way of transforming the pattern so that it looks exactly the same after the transformation. For example, <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a> is present when the pattern can be <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translated</a> (in other words, shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored. </p><p>The types of transformations that are relevant here are called <a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean plane isometries</a>. For example: </p> <ul><li>If one <i>shifts</i> example <b>B</b> one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is <i>exactly the same</i> as the starting pattern. This type of symmetry is called a <b><a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a></b>. Examples <b>A</b> and <b>C</b> are similar, except that the smallest possible shifts are in diagonal directions.</li> <li>If one <i>turns</i> example <b>B</b> clockwise by 90°, around the centre of one of the squares, again one obtains exactly the same pattern. This is called a <b><a href="/wiki/Rotation" title="Rotation">rotation</a></b>. Examples <b>A</b> and <b>C</b> also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for <b>C</b>.</li> <li>One can also <i>flip</i> example <b>B</b> across a horizontal axis that runs across the middle of the image. This is called a <b><a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a></b>. Example <b>B</b> also has reflections across a vertical axis, and across two diagonal axes. The same can be said for <b>A</b>.</li></ul> <p>However, example <b>C</b> is <i>different</i>. It only has reflections in horizontal and vertical directions, <i>not</i> across diagonal axes. If one flips across a diagonal line, one does <i>not</i> get the same pattern back, but the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of <b>A</b> and <b>B</b> is different from the wallpaper group of <b>C</b>. </p><p>Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Krok_6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Krok_6.svg/300px-Krok_6.svg.png" decoding="async" width="300" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Krok_6.svg/450px-Krok_6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Krok_6.svg/600px-Krok_6.svg.png 2x" data-file-width="2279" data-file-height="412" /></a><figcaption>A glide reflection will map a set of left and right footprints into each other</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Formal_definition_and_discussion">Formal definition and discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=2" title="Edit section: Formal definition and discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematically, a wallpaper group or plane crystallographic group is a type of <a href="/wiki/Discrete_space" title="Discrete space">topologically discrete</a> <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">isometries of the Euclidean plane</a> that contains two <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>. </p><p>Two such <a href="/wiki/Isometry_group" title="Isometry group">isometry groups</a> are of the same type (of the same wallpaper group) if they are <a href="/wiki/Conjugation_of_isometries_in_Euclidean_space#Isometry_groups" title="Conjugation of isometries in Euclidean space">the same up to an affine transformation of the plane</a>. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no <a href="/wiki/Glide_reflection" title="Glide reflection">glide reflections</a>, and <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> is at most of order 2). </p><p>Unlike in <a href="/wiki/Space_group#Group_theory" title="Space group">the three-dimensional case</a>, one can equivalently restrict the affine transformations to those that preserve <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>. </p><p>It follows from the <a href="/wiki/Bieberbach_conjecture" class="mw-redirect" title="Bieberbach conjecture">Bieberbach conjecture</a> that all wallpaper groups are different even as abstract groups (as opposed to e.g. <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a>, of which two are isomorphic with <b>Z</b>). </p><p>2D patterns with double translational symmetry can be categorized according to their <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> type. </p> <div class="mw-heading mw-heading3"><h3 id="Isometries_of_the_Euclidean_plane">Isometries of the Euclidean plane</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=3" title="Edit section: Isometries of the Euclidean plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Isometries of the Euclidean plane fall into four categories (see the article <a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean plane isometry</a> for more information). </p> <ul><li><b><a href="/wiki/Translation_(geometry)" title="Translation (geometry)">Translations</a></b>, denoted by <i>T</i><sub><i>v</i></sub>, where <i>v</i> is a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> in <b>R</b><sup>2</sup>. This has the effect of shifting the plane applying <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> vector <i>v</i>.</li> <li><b><a href="/wiki/Coordinate_rotation" class="mw-redirect" title="Coordinate rotation">Rotations</a></b>, denoted by <i>R</i><sub><i>c</i>,<i>θ</i></sub>, where <i>c</i> is a point in the plane (the centre of rotation), and <i>θ</i> is the angle of rotation.</li> <li><b><a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">Reflections</a></b>, or <b>mirror isometries</b>, denoted by <i>F</i><sub><i>L</i></sub>, where <i>L</i> is a line in <b>R</b><sup>2</sup>. (<i>F</i> is for "flip"). This has the effect of reflecting the plane in the line <i>L</i>, called the <b>reflection axis</b> or the associated <b>mirror</b>.</li> <li><b><a href="/wiki/Glide_reflection" title="Glide reflection">Glide reflections</a></b>, denoted by <i>G</i><sub><i>L</i>,<i>d</i></sub>, where <i>L</i> is a line in <b>R</b><sup>2</sup> and <i>d</i> is a distance. This is a combination of a reflection in the line <i>L</i> and a translation along <i>L</i> by a distance <i>d</i>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="The_independent_translations_condition">The independent translations condition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=4" title="Edit section: The independent translations condition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The condition on linearly independent translations means that there exist linearly independent vectors <i>v</i> and <i>w</i> (in <b>R</b><sup>2</sup>) such that the group contains both <i>T</i><sub><i>v</i></sub> and <i>T</i><sub><i>w</i></sub>. </p><p>The purpose of this condition is to distinguish wallpaper groups from <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a>, which possess a translation but not two linearly independent ones, and from <a href="/wiki/Symmetry_group#Two_dimensions" title="Symmetry group">two-dimensional discrete point groups</a>, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in <i>two</i> distinct directions, in contrast to frieze groups, which only repeat along a single axis. </p><p>(It is possible to generalise this situation. One could for example study discrete groups of isometries of <b>R</b><sup><i>n</i></sup> with <i>m</i> linearly independent translations, where <i>m</i> is any integer in the range 0 ≤ <i>m</i> ≤ <i>n</i>.) </p> <div class="mw-heading mw-heading3"><h3 id="The_discreteness_condition">The discreteness condition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=5" title="Edit section: The discreteness condition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The discreteness condition means that there is some positive real number ε, such that for every translation <i>T</i><sub><i>v</i></sub> in the group, the vector <i>v</i> has length <i>at least</i> ε (except of course in the case that <i>v</i> is the zero vector, but the independent translations condition prevents this, since any set that contains the zero vector is linearly dependent by definition and thus disallowed). </p><p>The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, one might have for example a group containing the translation <i>T</i><sub><i>x</i></sub> for every <a href="/wiki/Rational_number" title="Rational number">rational number</a> <i>x</i>, which would not correspond to any reasonable wallpaper pattern. </p><p>One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180°, 120°, 90°, or 60°. This fact is known as the <a href="/wiki/Crystallographic_restriction_theorem" title="Crystallographic restriction theorem">crystallographic restriction theorem</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and can be generalised to higher-dimensional cases. </p> <div class="mw-heading mw-heading3"><h3 id="Notations_for_wallpaper_groups">Notations for wallpaper groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=6" title="Edit section: Notations for wallpaper groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Crystallographic_notation">Crystallographic notation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=7" title="Edit section: Crystallographic notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Crystallography has 230 <a href="/wiki/Space_group" title="Space group">space groups</a> to distinguish, far more than the 17 wallpaper groups, but many of the symmetries in the groups are the same. Thus one can use a similar notation for both kinds of groups, that of <a href="/wiki/Carl_Hermann" class="mw-redirect" title="Carl Hermann">Carl Hermann</a> and <a href="/wiki/Charles-Victor_Mauguin" title="Charles-Victor Mauguin">Charles-Victor Mauguin</a>. An example of a full wallpaper name in Hermann-Mauguin style (also called <a href="/wiki/International_Union_of_Crystallography#IUCr_Symmetry_notation" title="International Union of Crystallography">IUCr notation</a>) is <a href="#Group_p31m"><b><i>p</i>31<i>m</i></b></a>, with four letters or digits; more usual is a shortened name like <a href="#Group_c2mm"><i><b>cmm</b></i></a> or <a href="#Group_pg"><i><b>pg</b></i></a>. </p><p>For wallpaper groups the full notation begins with either <i><b>p</b></i> or <i><b>c</b></i>, for a <i><a href="/wiki/Primitive_cell" class="mw-redirect" title="Primitive cell">primitive cell</a></i> or a <i>face-centred cell</i>; these are explained below. This is followed by a digit, <i><b>n</b></i>, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis that is the main one (or if there are two, one of them). The symbols are either <i><b>m</b></i>, <i><b>g</b></i>, or <b>1</b>, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180°/<i>n</i> (when <i>n</i> > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an <i><b>m</b></i> that can be deduced, so long as that leaves no confusion with another group. </p><p>A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal <a href="/wiki/Space_group" title="Space group">space groups</a> uses additional cell types. </p> <dl><dt>Examples</dt></dl> <ul><li><b><a href="#Group_p2"><i>p</i>2</a></b> (<b><i>p</i>2</b>): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.</li> <li><b><a href="#Group_p4gm"><i>p</i>4<i>gm</i></a></b> (<b><i>p</i>4<i>gm</i></b>): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.</li> <li><b><a href="#Group_c2mm"><i>c</i>2<i>mm</i></a></b> (<b><i>c</i>2<i>mm</i></b>): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.</li> <li><b><a href="#Group_p31m"><i>p</i>31<i>m</i></a></b> (<b><i>p</i>31<i>m</i></b>): Primitive cell, 3-fold rotation, mirror axis at 60°.</li></ul> <p>Here are all the names that differ in short and full notation. </p> <dl><dd><table class="wikitable"> <caption><b>Crystallographic short and full names</b> </caption> <tbody><tr> <th><i>Short</i> </th> <th><a href="#Group_pm"><i>pm</i></a> </th> <th><a href="#Group_pg"><i>pg</i></a> </th> <th><a href="#Group_cm"><i>cm</i></a> </th> <th><a href="#Group_p2mm"><i>pmm</i></a> </th> <th><a href="#Group_p2mg"><i>pmg</i></a> </th> <th><a href="#Group_p2gg"><i>pgg</i></a> </th> <th><a href="#Group_c2mm"><i>cmm</i></a> </th> <th><a href="#Group_p4mm"><i>p</i>4<i>m</i></a> </th> <th><a href="#Group_p4gm"><i>p</i>4<i>g</i></a> </th> <th><a href="#Group_p6mm"><i>p</i>6<i>m</i></a> </th></tr> <tr style="background:white"> <th><i>Full</i></th> <th><i>p</i>1<i>m</i>1</th> <th><i>p</i>1<i>g</i>1</th> <th><i>c</i>1<i>m</i>1</th> <th><i>p</i>2<i>mm</i></th> <th><i>p</i>2<i>mg</i></th> <th><i>p</i>2<i>gg</i></th> <th><i>c</i>2<i>mm</i></th> <th><i>p</i>4<i>mm</i></th> <th><i>p</i>4<i>gm</i></th> <th><i>p</i>6<i>mm</i> </th></tr></tbody></table></dd></dl> <p>The remaining names are <b><a href="#Group_p1"><i>p</i>1</a></b>, <b><a href="#Group_p2"><i>p</i>2</a></b>, <b><a href="#Group_p3"><i>p</i>3</a></b>, <b><a href="#Group_p3m1"><i>p</i>3<i>m</i>1</a></b>, <b><a href="#Group_p31m"><i>p</i>31<i>m</i></a></b>, <b><a href="#Group_p4"><i>p</i>4</a></b>, and <b><a href="#Group_p6"><i>p</i>6</a></b>. </p> <div class="mw-heading mw-heading4"><h4 id="Orbifold_notation">Orbifold notation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=8" title="Edit section: Orbifold notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orbifold notation</a> for wallpaper groups, advocated by <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a> (Conway, 1992) (Conway 2008), is based not on crystallography, but on topology. One can fold the infinite periodic tiling of the plane into its essence, an <a href="/wiki/Orbifold" title="Orbifold">orbifold</a>, then describe that with a few symbols. </p> <ul><li>A digit, <i><b>n</b></i>, indicates a centre of <i>n</i>-fold rotation corresponding to a cone point on the orbifold. By the crystallographic restriction theorem, <i>n</i> must be 2, 3, 4, or 6.</li> <li>An asterisk, <b>*</b>, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the digits as follows: <ol><li>Digits before <b>*</b> denote centres of pure rotation (<a href="/wiki/Point_group" title="Point group">cyclic</a>).</li> <li>Digits after <b>*</b> denote centres of rotation with mirrors through them, corresponding to "corners" on the boundary of the orbifold (<a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a>).</li></ol></li> <li>A cross, <b>×</b>, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so need no notation.</li> <li>The "no symmetry" symbol, <b>o</b>, stands alone, and indicates there are only lattice translations with no other symmetry. The orbifold with this symbol is a torus; in general the symbol <b>o</b> denotes a handle on the orbifold.</li></ul> <p>The group denoted in crystallographic notation by <a href="#Group_c2mm"><i><b>cmm</b></i></a> will, in Conway's notation, be <b>2*22</b>. The <b>2</b> before the <b>*</b> says there is a 2-fold rotation centre with no mirror through it. The <b>*</b> itself says there is a mirror. The first <b>2</b> after the <b>*</b> says there is a 2-fold rotation centre on a mirror. The final <b>2</b> says there is an independent second 2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries. </p><p>The group denoted by <a href="#Group_p2gg"><i><b>pgg</b></i></a> will be <b>22×</b>. There are two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with <a href="#Group_p2mg"><i><b>pmg</b></i></a>, Conway <b>22*</b>, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold. </p><p><a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a>'s <a href="/wiki/Coxeter_notation" title="Coxeter notation">bracket notation</a> is also included, based on reflectional <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a>, and modified with plus superscripts accounting for rotations, <a href="/wiki/Improper_rotation" title="Improper rotation">improper rotations</a> and translations. </p> <table class="wikitable"> <caption><b>Conway, Coxeter and crystallographic correspondence</b> </caption> <tbody><tr align="center" style="background:lightyellow"> <th><i>Conway</i> </th> <td><span style="color:blue">o</span></td> <td><span style="color: red;">××</span></td> <td><span style="color: red;">*×</span></td> <td><span style="color: red;">**</span></td> <td><span style="color:blue">632</span></td> <td><span style="color: red;">*632</span> </td></tr> <tr align="center" style="background:lightgray"> <th><i>Coxeter</i> </th> <td>[∞<sup>+</sup>,2,∞<sup>+</sup>]</td> <td>[(∞,2)<sup>+</sup>,∞<sup>+</sup>]</td> <td>[∞,2<sup>+</sup>,∞<sup>+</sup>]</td> <td>[∞,2,∞<sup>+</sup>]</td> <td>[6,3]<sup>+</sup></td> <td>[6,3] </td></tr> <tr align="center" style="background:white"> <th><i>Crystallographic</i> </th> <td><a href="#Group_p1"><i>p</i>1</a></td> <td><a href="#Group_pg"><i>pg</i></a></td> <td><a href="#Group_cm"><i>cm</i></a></td> <td><a href="#Group_pm"><i>pm</i></a></td> <td><a href="#Group_p6"><i>p</i>6</a></td> <td><a href="#Group_p6mm"><i>p</i>6<i>m</i></a> </td></tr></tbody></table> <table class="wikitable"> <tbody><tr align="center" style="background:lightyellow"> <th><i>Conway</i> </th> <td><span style="color:blue;">333</span></td> <td><span style="color:red;">*333</span></td> <td><span style="color:blue;">3</span><span style="color:red;">*3</span></td> <td><span style="color:blue;">442</span></td> <td><span style="color:red;">*442</span></td> <td><span style="color:blue;">4</span><span style="color:red;">*2</span> </td></tr> <tr align="center" style="background:lightgray"> <th><i>Coxeter</i> </th> <td>[3<sup>[3]</sup>]<sup>+</sup></td> <td>[3<sup>[3]</sup>]</td> <td>[3<sup>+</sup>,6]</td> <td>[4,4]<sup>+</sup> </td> <td>[4,4]</td> <td>[4<sup>+</sup>,4] </td></tr> <tr align="center" style="background:white"> <th><i>Crystallographic</i> </th> <td><a href="#Group_p3"><i>p</i>3</a></td> <td><a href="#Group_p3m1"><i>p</i>3<i>m</i>1</a></td> <td><a href="#Group_p31m"><i>p</i>31<i>m</i></a></td> <td><a href="#Group_p4"><i>p</i>4</a> </td> <td><a href="#Group_p4mm"><i>p</i>4<i>m</i></a></td> <td><a href="#Group_p4gm"><i>p</i>4<i>g</i></a> </td></tr></tbody></table> <table class="wikitable"> <tbody><tr align="center" style="background:lightyellow"> <th><i>Conway</i> </th> <td><span style="color:blue;">2222</span></td> <td><span style="color:blue;">22</span><span style="color:red;">×</span></td> <td><span style="color:blue;">22</span><span style="color:red;">*</span></td> <td><span style="color:red;">*2222</span></td> <td><span style="color:blue;">2</span><span style="color:red;">*22</span> </td></tr> <tr align="center" style="background:lightgray"> <th><i>Coxeter</i> </th> <td>[∞,2,∞]<sup>+</sup></td> <td>[((∞,2)<sup>+</sup>,(∞,2)<sup>+</sup>)]</td> <td>[(∞,2)<sup>+</sup>,∞]</td> <td>[∞,2,∞]</td> <td>[∞,2<sup>+</sup>,∞] </td></tr> <tr align="center" style="background:white"> <th><i>Crystallographic</i> </th> <td><a href="#Group_p2"><i>p</i>2</a></td> <td><a href="#Group_p2gg"><i>pgg</i></a></td> <td><a href="#Group_p2mg"><i>pmg</i></a></td> <td><a href="#Group_p2mm"><i>pmm</i></a></td> <td><a href="#Group_c2mm"><i>cmm</i></a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Why_there_are_exactly_seventeen_groups">Why there are exactly seventeen groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=9" title="Edit section: Why there are exactly seventeen groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An orbifold can be viewed as a <a href="/wiki/Polygon" title="Polygon">polygon</a> with face, edges, and vertices which can be unfolded to form a possibly infinite set of polygons which tile either the <a href="/wiki/Sphere" title="Sphere">sphere</a>, the plane or the <a href="/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane" title="Hyperbolic geometry">hyperbolic plane</a>. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a <a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">spherical symmetry group</a> or <a href="/wiki/Orbifold_notation#Hyperbolic_plane" title="Orbifold notation">Hyperbolic symmetry group</a>. The type of space the polygons tile can be found by calculating the <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a>, <i>χ</i> = <i>V</i> − <i>E</i> + <i>F</i>, where <i>V</i> is the number of corners (vertices), <i>E</i> is the number of edges and <i>F</i> is the number of faces. If the Euler characteristic is positive then the orbifold has an elliptic (spherical) structure; if it is zero then it has a parabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full set of possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0. </p><p>When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces, which must be consistent with the Euler characteristic. Reversing the process, one can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of the symmetry group. </p><p>The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows: </p> <ul><li>A digit <i><b>n</b></i> without or before a * counts as <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"><i>n</i> − 1</span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span>.</li> <li>A digit <i><b>n</b></i> after a * counts as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i> − 1</span><span class="sr-only">/</span><span class="den">2<i>n</i></span></span>⁠</span>.</li> <li>Both * and × count as 1.</li> <li>The "no symmetry" o counts as 2.</li></ul> <p>For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2. </p> <dl><dt>Examples</dt></dl> <ul><li>632: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> = 2</li> <li>3*3: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> + 1 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> = 2</li> <li>4*2: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> + 1 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> = 2</li> <li>22×: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> + 1 = 2</li></ul> <p>Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2. </p><p>Feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is <a href="/wiki/Orbifold_notation#Hyperbolic_plane" title="Orbifold notation">hyperbolic</a>; when positive, <a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">spherical</a> or <i><a href="/w/index.php?title=Bad_orbifold&action=edit&redlink=1" class="new" title="Bad orbifold (page does not exist)">bad</a></i>). </p> <div class="mw-heading mw-heading2"><h2 id="Guide_to_recognizing_wallpaper_groups">Guide to recognizing wallpaper groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=10" title="Edit section: Guide to recognizing wallpaper groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To work out which wallpaper group corresponds to a given design, one may use the following table.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="text-align: center;margin: 1em auto;"> <tbody><tr> <th rowspan="2">Size of smallest<br />rotation</th> <th colspan="6">Has reflection? </th></tr> <tr> <td colspan="4">Yes</td> <td colspan="2">No </td></tr> <tr> <td>360° / 6</td> <td colspan="4"><a href="#Group_p6mm"><i>p</i>6<i>m</i> (*632)</a></td> <td colspan="2"><a href="#Group_p6"><i>p</i>6 (632)</a> </td></tr> <tr> <td rowspan="2">360° / 4</td> <td colspan="4"><b>Has mirrors at 45°?</b></td> <td rowspan="2" colspan="2"><a href="#Group_p4"><i>p</i>4 (442)</a> </td></tr> <tr> <td colspan="2">Yes: <a href="#Group_p4mm"><i>p</i>4<i>m</i> (*442)</a></td> <td colspan="2">No: <a href="#Group_p4gm"><i>p</i>4<i>g</i> (4*2)</a> </td></tr> <tr> <td rowspan="2">360° / 3</td> <td colspan="4"><b>Has rot. centre off mirrors?</b></td> <td rowspan="2" colspan="2"><a href="#Group_p3"><i>p</i>3 (333)</a> </td></tr> <tr> <td colspan="2">Yes: <a href="#Group_p31m"><i>p</i>31<i>m</i> (3*3)</a></td> <td colspan="2">No: <a href="#Group_p3m1"><i>p</i>3<i>m</i>1 (*333)</a> </td></tr> <tr> <td rowspan="4">360° / 2</td> <td colspan="4"><b>Has perpendicular reflections?</b></td> <td rowspan="2" colspan="2"><b>Has glide reflection?</b> </td></tr> <tr> <td colspan="3">Yes</td> <td>No </td></tr> <tr> <td colspan="3"><b>Has rot. centre off mirrors?</b></td> <td rowspan="2"><a href="#Group_p2mg"><i>pmg</i> (22*)</a></td> <td rowspan="2">Yes: <a href="#Group_p2gg"><i>pgg</i> (22×)</a></td> <td rowspan="2">No: <a href="#Group_p2"><i>p</i>2 (2222)</a> </td></tr> <tr> <td>Yes: <a href="#Group_c2mm"><i>cmm</i> (2*22)</a></td> <td colspan="2">No: <a href="#Group_p2mm"><i>pmm</i> (*2222)</a> </td></tr> <tr> <td rowspan="2">none</td> <td colspan="4"><b>Has glide axis off mirrors?</b></td> <td colspan="2"><b>Has glide reflection?</b> </td></tr> <tr> <td colspan="2">Yes: <a href="#Group_cm"><i>cm</i> (*×)</a></td> <td colspan="2">No: <a href="#Group_pm"><i>pm</i> (**)</a></td> <td>Yes: <a href="#Group_pg"><i>pg</i> (××)</a></td> <td>No: <a href="#Group_p1"><i>p</i>1 (o)</a> </td></tr></tbody></table> <p>See also <a href="https://commons.wikimedia.org/wiki/Wallpaper_group_diagrams" class="extiw" title="commons:Wallpaper group diagrams">this overview with diagrams</a>. </p> <div class="mw-heading mw-heading2"><h2 id="The_seventeen_groups">The seventeen groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=11" title="Edit section: The seventeen groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows (it is the shape that is significant, not the colour): </p> <table class="wikitable"> <tbody><tr> <td align="right"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_legend_rotation2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Wallpaper_group_diagram_legend_rotation2.svg/20px-Wallpaper_group_diagram_legend_rotation2.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Wallpaper_group_diagram_legend_rotation2.svg/30px-Wallpaper_group_diagram_legend_rotation2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Wallpaper_group_diagram_legend_rotation2.svg/40px-Wallpaper_group_diagram_legend_rotation2.svg.png 2x" data-file-width="59" data-file-height="59" /></a></span> </td> <td>a centre of rotation of order two (180°). </td></tr> <tr> <td align="right"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_legend_rotation3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Wallpaper_group_diagram_legend_rotation3.svg/20px-Wallpaper_group_diagram_legend_rotation3.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Wallpaper_group_diagram_legend_rotation3.svg/30px-Wallpaper_group_diagram_legend_rotation3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Wallpaper_group_diagram_legend_rotation3.svg/40px-Wallpaper_group_diagram_legend_rotation3.svg.png 2x" data-file-width="59" data-file-height="59" /></a></span> </td> <td>a centre of rotation of order three (120°). </td></tr> <tr> <td align="right"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_legend_rotation4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Wallpaper_group_diagram_legend_rotation4.svg/20px-Wallpaper_group_diagram_legend_rotation4.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Wallpaper_group_diagram_legend_rotation4.svg/30px-Wallpaper_group_diagram_legend_rotation4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Wallpaper_group_diagram_legend_rotation4.svg/40px-Wallpaper_group_diagram_legend_rotation4.svg.png 2x" data-file-width="59" data-file-height="59" /></a></span> </td> <td>a centre of rotation of order four (90°). </td></tr> <tr> <td align="right"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_legend_rotation6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Wallpaper_group_diagram_legend_rotation6.svg/20px-Wallpaper_group_diagram_legend_rotation6.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Wallpaper_group_diagram_legend_rotation6.svg/30px-Wallpaper_group_diagram_legend_rotation6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Wallpaper_group_diagram_legend_rotation6.svg/40px-Wallpaper_group_diagram_legend_rotation6.svg.png 2x" data-file-width="59" data-file-height="59" /></a></span> </td> <td>a centre of rotation of order six (60°). </td></tr> <tr> <td align="right"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_legend_reflection.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Wallpaper_group_diagram_legend_reflection.svg/120px-Wallpaper_group_diagram_legend_reflection.svg.png" decoding="async" width="120" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Wallpaper_group_diagram_legend_reflection.svg/180px-Wallpaper_group_diagram_legend_reflection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Wallpaper_group_diagram_legend_reflection.svg/240px-Wallpaper_group_diagram_legend_reflection.svg.png 2x" data-file-width="352" data-file-height="59" /></a></span> </td> <td>an axis of reflection. </td></tr> <tr> <td align="right"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_legend_glide_reflection.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group_diagram_legend_glide_reflection.svg/120px-Wallpaper_group_diagram_legend_glide_reflection.svg.png" decoding="async" width="120" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group_diagram_legend_glide_reflection.svg/180px-Wallpaper_group_diagram_legend_glide_reflection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group_diagram_legend_glide_reflection.svg/240px-Wallpaper_group_diagram_legend_glide_reflection.svg.png 2x" data-file-width="352" data-file-height="59" /></a></span> </td> <td>an axis of glide reflection. </td></tr></tbody></table> <p>On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently. </p><p>The <b>brown or yellow area</b> indicates a <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domain</a>, i.e. the smallest part of the pattern that is repeated. </p><p>The diagrams on the right show the cell of the <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> corresponding to the smallest translations; those on the left sometimes show a larger area. </p> <div class="mw-heading mw-heading3"><h3 id="Group_p1_(o)"><span id="Group_p1_.28o.29"></span><span class="anchor" id="Group_p1"></span> Group <i>p</i>1 (o)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=12" title="Edit section: Group p1 (o)"><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:SymBlend_p1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/SymBlend_p1.svg/220px-SymBlend_p1.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/SymBlend_p1.svg/330px-SymBlend_p1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/SymBlend_p1.svg/440px-SymBlend_p1.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>1</b></figcaption></figure> <table class="wikitable"> <caption>Cell structures for <i>p</i>1 by lattice type </caption> <tbody><tr> <th colspan="3"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Wallpaper_group_diagram_p1.svg/160px-Wallpaper_group_diagram_p1.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Wallpaper_group_diagram_p1.svg/240px-Wallpaper_group_diagram_p1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Wallpaper_group_diagram_p1.svg/320px-Wallpaper_group_diagram_p1.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Oblique </th> <th colspan="3"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p1_half.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Wallpaper_group_diagram_p1_half.svg/160px-Wallpaper_group_diagram_p1_half.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Wallpaper_group_diagram_p1_half.svg/240px-Wallpaper_group_diagram_p1_half.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Wallpaper_group_diagram_p1_half.svg/320px-Wallpaper_group_diagram_p1_half.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Hexagonal </th></tr> <tr> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p1_rect.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p1_rect.svg/120px-Wallpaper_group_diagram_p1_rect.svg.png" decoding="async" width="120" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p1_rect.svg/180px-Wallpaper_group_diagram_p1_rect.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p1_rect.svg/240px-Wallpaper_group_diagram_p1_rect.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Rectangular </th> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p1_rhombic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Wallpaper_group_diagram_p1_rhombic.svg/120px-Wallpaper_group_diagram_p1_rhombic.svg.png" decoding="async" width="120" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Wallpaper_group_diagram_p1_rhombic.svg/180px-Wallpaper_group_diagram_p1_rhombic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Wallpaper_group_diagram_p1_rhombic.svg/240px-Wallpaper_group_diagram_p1_rhombic.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Rhombic </th> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p1_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Wallpaper_group_diagram_p1_square.svg/80px-Wallpaper_group_diagram_p1_square.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Wallpaper_group_diagram_p1_square.svg/120px-Wallpaper_group_diagram_p1_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Wallpaper_group_diagram_p1_square.svg/160px-Wallpaper_group_diagram_p1_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span><br />Square </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:blue;">o</span></li> <li>Coxeter notation (rectangular): [∞<sup>+</sup>,2,∞<sup>+</sup>] or [∞]<sup>+</sup>×[∞]<sup>+</sup></li> <li>Lattice: oblique</li> <li>Point group: C<sub>1</sub></li> <li>The group <b><i>p</i>1</b> contains only translations; there are no rotations, reflections, or glide reflections.</li></ul> <dl><dt>Examples of group <i>p</i>1</dt></dl> <div style="clear:both;" class=""></div> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP1.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/WallpaperP1.GIF/120px-WallpaperP1.GIF" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/WallpaperP1.GIF/180px-WallpaperP1.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/WallpaperP1.GIF/240px-WallpaperP1.GIF 2x" data-file-width="250" data-file-height="250" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p1-3.jpg" class="mw-file-description" title="Medieval wall diapering"><img alt="Medieval wall diapering" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Wallpaper_group-p1-3.jpg/91px-Wallpaper_group-p1-3.jpg" decoding="async" width="91" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Wallpaper_group-p1-3.jpg/136px-Wallpaper_group-p1-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Wallpaper_group-p1-3.jpg/181px-Wallpaper_group-p1-3.jpg 2x" data-file-width="698" data-file-height="923" /></a></span></div> <div class="gallerytext"><a href="/wiki/Medieval" class="mw-redirect" title="Medieval">Medieval</a> wall <a href="/wiki/Diapering" title="Diapering">diapering</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p1.svg" class="mw-file-description" title="21 co-uniform tiling"><img alt="21 co-uniform tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Co-Uniform_Wallpaper_Example_p1.svg/120px-Co-Uniform_Wallpaper_Example_p1.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Co-Uniform_Wallpaper_Example_p1.svg/180px-Co-Uniform_Wallpaper_Example_p1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Co-Uniform_Wallpaper_Example_p1.svg/240px-Co-Uniform_Wallpaper_Example_p1.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">21 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling</div> </li> </ul> <p>The two translations (cell sides) can each have different lengths, and can form any angle. </p> <div class="mw-heading mw-heading3"><h3 id="Group_p2_(2222)"><span id="Group_p2_.282222.29"></span><span class="anchor" id="Group_p2"></span> Group <i>p</i>2 (2222)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=13" title="Edit section: Group p2 (2222)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/SymBlend_p2.svg/220px-SymBlend_p2.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/SymBlend_p2.svg/330px-SymBlend_p2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/SymBlend_p2.svg/440px-SymBlend_p2.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>2</b></figcaption></figure> <table class="wikitable"> <caption>Cell structures for <i>p</i>2 by lattice type </caption> <tbody><tr> <th colspan="3"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Wallpaper_group_diagram_p2.svg/160px-Wallpaper_group_diagram_p2.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Wallpaper_group_diagram_p2.svg/240px-Wallpaper_group_diagram_p2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Wallpaper_group_diagram_p2.svg/320px-Wallpaper_group_diagram_p2.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Oblique </th> <th colspan="3"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p2_half.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Wallpaper_group_diagram_p2_half.svg/160px-Wallpaper_group_diagram_p2_half.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Wallpaper_group_diagram_p2_half.svg/240px-Wallpaper_group_diagram_p2_half.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Wallpaper_group_diagram_p2_half.svg/320px-Wallpaper_group_diagram_p2_half.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Hexagonal </th></tr> <tr> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p2_rect.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Wallpaper_group_diagram_p2_rect.svg/120px-Wallpaper_group_diagram_p2_rect.svg.png" decoding="async" width="120" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Wallpaper_group_diagram_p2_rect.svg/180px-Wallpaper_group_diagram_p2_rect.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Wallpaper_group_diagram_p2_rect.svg/240px-Wallpaper_group_diagram_p2_rect.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Rectangular </th> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p2_rhombic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Wallpaper_group_diagram_p2_rhombic.svg/120px-Wallpaper_group_diagram_p2_rhombic.svg.png" decoding="async" width="120" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Wallpaper_group_diagram_p2_rhombic.svg/180px-Wallpaper_group_diagram_p2_rhombic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Wallpaper_group_diagram_p2_rhombic.svg/240px-Wallpaper_group_diagram_p2_rhombic.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Rhombic </th> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p2_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Wallpaper_group_diagram_p2_square.svg/80px-Wallpaper_group_diagram_p2_square.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Wallpaper_group_diagram_p2_square.svg/120px-Wallpaper_group_diagram_p2_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Wallpaper_group_diagram_p2_square.svg/160px-Wallpaper_group_diagram_p2_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span><br />Square </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:blue;">2222</span></li> <li>Coxeter notation (rectangular): [∞,2,∞]<sup>+</sup></li> <li>Lattice: oblique</li> <li>Point group: C<sub>2</sub></li> <li>The group <b><i>p</i>2</b> contains four rotation centres of order two (180°), but no reflections or glide reflections.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>2</dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP2.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/WallpaperP2.GIF/115px-WallpaperP2.GIF" decoding="async" width="115" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/WallpaperP2.GIF/173px-WallpaperP2.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/WallpaperP2.GIF/231px-WallpaperP2.GIF 2x" data-file-width="253" data-file-height="263" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p2-1.jpg" class="mw-file-description" title="Cloth, Sandwich Islands (Hawaii)"><img alt="Cloth, Sandwich Islands (Hawaii)" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Wallpaper_group-p2-1.jpg/120px-Wallpaper_group-p2-1.jpg" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Wallpaper_group-p2-1.jpg/180px-Wallpaper_group-p2-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Wallpaper_group-p2-1.jpg/240px-Wallpaper_group-p2-1.jpg 2x" data-file-width="1018" data-file-height="980" /></a></span></div> <div class="gallerytext">Cloth, <a href="/wiki/Hawaiian_Islands" title="Hawaiian Islands">Sandwich Islands</a> (<a href="/wiki/Hawaii" title="Hawaii">Hawaii</a>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p2-3.jpg" class="mw-file-description" title="Ceiling of an Egyptian tomb"><img alt="Ceiling of an Egyptian tomb" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Wallpaper_group-p2-3.jpg/120px-Wallpaper_group-p2-3.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Wallpaper_group-p2-3.jpg/180px-Wallpaper_group-p2-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Wallpaper_group-p2-3.jpg/240px-Wallpaper_group-p2-3.jpg 2x" data-file-width="1085" data-file-height="1075" /></a></span></div> <div class="gallerytext">Ceiling of an <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p2-4.jpg" class="mw-file-description" title="Wire fence, U.S."><img alt="Wire fence, U.S." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Wallpaper_group-p2-4.jpg/120px-Wallpaper_group-p2-4.jpg" decoding="async" width="120" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Wallpaper_group-p2-4.jpg/180px-Wallpaper_group-p2-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Wallpaper_group-p2-4.jpg/240px-Wallpaper_group-p2-4.jpg 2x" data-file-width="966" data-file-height="882" /></a></span></div> <div class="gallerytext"><a href="/wiki/Chain-link_fencing" title="Chain-link fencing">Wire fence</a>, U.S.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p2.svg" class="mw-file-description" title="15 co-uniform tiling"><img alt="15 co-uniform tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Co-Uniform_Wallpaper_Example_p2.svg/120px-Co-Uniform_Wallpaper_Example_p2.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Co-Uniform_Wallpaper_Example_p2.svg/180px-Co-Uniform_Wallpaper_Example_p2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Co-Uniform_Wallpaper_Example_p2.svg/240px-Co-Uniform_Wallpaper_Example_p2.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">15 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p2-2.jpg" class="mw-file-description" title="Mat on which an Egyptian king stood"><img alt="Mat on which an Egyptian king stood" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Wallpaper_group-p2-2.jpg/120px-Wallpaper_group-p2-2.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Wallpaper_group-p2-2.jpg/180px-Wallpaper_group-p2-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Wallpaper_group-p2-2.jpg/240px-Wallpaper_group-p2-2.jpg 2x" data-file-width="1070" data-file-height="1066" /></a></span></div> <div class="gallerytext">Mat on which an <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> king stood</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p2-2_detail_2.jpg" class="mw-file-description" title="Egyptian mat (detail)"><img alt="Egyptian mat (detail)" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-p2-2_detail_2.jpg/114px-Wallpaper_group-p2-2_detail_2.jpg" decoding="async" width="114" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c6/Wallpaper_group-p2-2_detail_2.jpg 1.5x" data-file-width="120" data-file-height="126" /></a></span></div> <div class="gallerytext">Egyptian mat (detail)</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_pm_(**)"><span id="Group_pm_.28.2A.2A.29"></span><span class="anchor" id="Group_pm"></span> Group <i>pm</i> (**)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=14" title="Edit section: Group pm (**)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_pm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/SymBlend_pm.svg/220px-SymBlend_pm.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/SymBlend_pm.svg/330px-SymBlend_pm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/SymBlend_pm.svg/440px-SymBlend_pm.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>pm</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structure for <i><b>pm</b></i> </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Wallpaper_group_diagram_pm.svg/160px-Wallpaper_group_diagram_pm.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Wallpaper_group_diagram_pm.svg/240px-Wallpaper_group_diagram_pm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Wallpaper_group_diagram_pm.svg/320px-Wallpaper_group_diagram_pm.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Horizontal mirrors </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pm_rotated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wallpaper_group_diagram_pm_rotated.svg/160px-Wallpaper_group_diagram_pm_rotated.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wallpaper_group_diagram_pm_rotated.svg/240px-Wallpaper_group_diagram_pm_rotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wallpaper_group_diagram_pm_rotated.svg/320px-Wallpaper_group_diagram_pm_rotated.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Vertical mirrors </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:red;">**</span></li> <li>Coxeter notation: [∞,2,∞<sup>+</sup>] or [∞<sup>+</sup>,2,∞]</li> <li>Lattice: rectangular</li> <li>Point group: D<sub>1</sub></li> <li>The group <i><b>pm</b></i> has no rotations. It has reflection axes, they are all parallel.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>pm</i></dt></dl> <p>(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.) </p> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperPM.gif" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/WallpaperPM.gif/120px-WallpaperPM.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/WallpaperPM.gif/180px-WallpaperPM.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/WallpaperPM.gif/240px-WallpaperPM.gif 2x" data-file-width="246" data-file-height="246" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pm-3.jpg" class="mw-file-description" title="Dress of a figure in a tomb at Biban el Moluk, Egypt"><img alt="Dress of a figure in a tomb at Biban el Moluk, Egypt" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Wallpaper_group-pm-3.jpg/120px-Wallpaper_group-pm-3.jpg" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Wallpaper_group-pm-3.jpg/180px-Wallpaper_group-pm-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Wallpaper_group-pm-3.jpg/240px-Wallpaper_group-pm-3.jpg 2x" data-file-width="1033" data-file-height="994" /></a></span></div> <div class="gallerytext">Dress of a figure in a <a href="/wiki/Tomb" title="Tomb">tomb</a> at <a href="/wiki/Biban_el_Moluk" class="mw-redirect" title="Biban el Moluk">Biban el Moluk</a>, <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pm-4.jpg" class="mw-file-description" title="Egyptian tomb, Thebes"><img alt="Egyptian tomb, Thebes" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Wallpaper_group-pm-4.jpg/120px-Wallpaper_group-pm-4.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Wallpaper_group-pm-4.jpg/180px-Wallpaper_group-pm-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Wallpaper_group-pm-4.jpg/240px-Wallpaper_group-pm-4.jpg 2x" data-file-width="1078" data-file-height="1075" /></a></span></div> <div class="gallerytext"><a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a>, <a href="/wiki/Thebes_(Egypt)" class="mw-redirect" title="Thebes (Egypt)">Thebes</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pm-1.jpg" class="mw-file-description" title="Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal"><img alt="Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Wallpaper_group-pm-1.jpg/120px-Wallpaper_group-pm-1.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Wallpaper_group-pm-1.jpg/180px-Wallpaper_group-pm-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Wallpaper_group-pm-1.jpg/240px-Wallpaper_group-pm-1.jpg 2x" data-file-width="1082" data-file-height="1076" /></a></span></div> <div class="gallerytext">Ceiling of a <a href="/wiki/Tomb" title="Tomb">tomb</a> at Gourna, <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a>. Reflection axis is diagonal</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pm-5.jpg" class="mw-file-description" title="Indian metalwork at the Great Exhibition in 1851. This is almost pm (ignoring short diagonal lines between ovals motifs, which make it p1)"><img alt="Indian metalwork at the Great Exhibition in 1851. This is almost pm (ignoring short diagonal lines between ovals motifs, which make it p1)" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Wallpaper_group-pm-5.jpg/119px-Wallpaper_group-pm-5.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Wallpaper_group-pm-5.jpg/178px-Wallpaper_group-pm-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Wallpaper_group-pm-5.jpg/238px-Wallpaper_group-pm-5.jpg 2x" data-file-width="1064" data-file-height="1074" /></a></span></div> <div class="gallerytext"><a href="/wiki/India" title="India">Indian</a> metalwork at the <a href="/wiki/Great_Exhibition" title="Great Exhibition">Great Exhibition</a> in 1851. This is almost <i><b>pm</b></i> (ignoring short diagonal lines between ovals motifs, which make it <a href="#Group_p1"><b><i>p</i>1</b></a>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_pm.svg" class="mw-file-description" title="6 co-uniform tiling (slab)"><img alt="6 co-uniform tiling (slab)" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Co-Uniform_Wallpaper_Example_pm.svg/120px-Co-Uniform_Wallpaper_Example_pm.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Co-Uniform_Wallpaper_Example_pm.svg/180px-Co-Uniform_Wallpaper_Example_pm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Co-Uniform_Wallpaper_Example_pm.svg/240px-Co-Uniform_Wallpaper_Example_pm.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">6 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling (slab)</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_pg_(××)"><span id="Group_pg_.28.C3.97.C3.97.29"></span><span class="anchor" id="Group_pg"></span> Group <i>pg</i> (××)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=15" title="Edit section: Group pg (××)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_pg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/SymBlend_pg.svg/220px-SymBlend_pg.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/SymBlend_pg.svg/330px-SymBlend_pg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/SymBlend_pg.svg/440px-SymBlend_pg.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>pg</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structures for <i>pg</i> </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Wallpaper_group_diagram_pg.svg/160px-Wallpaper_group_diagram_pg.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Wallpaper_group_diagram_pg.svg/240px-Wallpaper_group_diagram_pg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Wallpaper_group_diagram_pg.svg/320px-Wallpaper_group_diagram_pg.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Horizontal glides </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pg_rotated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Wallpaper_group_diagram_pg_rotated.svg/160px-Wallpaper_group_diagram_pg_rotated.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Wallpaper_group_diagram_pg_rotated.svg/240px-Wallpaper_group_diagram_pg_rotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Wallpaper_group_diagram_pg_rotated.svg/320px-Wallpaper_group_diagram_pg_rotated.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Vertical glides </th></tr> <tr> <th colspan="2">Rectangular </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:red;">××</span></li> <li>Coxeter notation: [(∞,2)<sup>+</sup>,∞<sup>+</sup>] or [∞<sup>+</sup>,(2,∞)<sup>+</sup>]</li> <li>Lattice: rectangular</li> <li>Point group: D<sub>1</sub></li> <li>The group <i><b>pg</b></i> contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>pg</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperPG.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/WallpaperPG.GIF/120px-WallpaperPG.GIF" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/WallpaperPG.GIF/180px-WallpaperPG.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/WallpaperPG.GIF/240px-WallpaperPG.GIF 2x" data-file-width="253" data-file-height="247" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pg-1.jpg" class="mw-file-description" title="Mat with herringbone pattern on which Egyptian king stood"><img alt="Mat with herringbone pattern on which Egyptian king stood" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-pg-1.jpg/120px-Wallpaper_group-pg-1.jpg" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-pg-1.jpg/180px-Wallpaper_group-pg-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-pg-1.jpg/240px-Wallpaper_group-pg-1.jpg 2x" data-file-width="1091" data-file-height="1064" /></a></span></div> <div class="gallerytext">Mat with <a href="/wiki/Herringbone_pattern" title="Herringbone pattern">herringbone pattern</a> on which <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> king stood</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pg-1_detail.jpg" class="mw-file-description" title="Egyptian mat (detail)"><img alt="Egyptian mat (detail)" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Wallpaper_group-pg-1_detail.jpg/119px-Wallpaper_group-pg-1_detail.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Wallpaper_group-pg-1_detail.jpg/178px-Wallpaper_group-pg-1_detail.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3f/Wallpaper_group-pg-1_detail.jpg 2x" data-file-width="238" data-file-height="240" /></a></span></div> <div class="gallerytext">Egyptian mat (detail)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pg-2.jpg" class="mw-file-description" title="Pavement with herringbone pattern in Salzburg. Glide reflection axis runs northeast–southwest"><img alt="Pavement with herringbone pattern in Salzburg. Glide reflection axis runs northeast–southwest" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Wallpaper_group-pg-2.jpg/120px-Wallpaper_group-pg-2.jpg" decoding="async" width="120" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Wallpaper_group-pg-2.jpg/180px-Wallpaper_group-pg-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Wallpaper_group-pg-2.jpg/240px-Wallpaper_group-pg-2.jpg 2x" data-file-width="2568" data-file-height="1944" /></a></span></div> <div class="gallerytext">Pavement with <a href="/wiki/Herringbone_pattern" title="Herringbone pattern">herringbone pattern</a> in <a href="/wiki/Salzburg" title="Salzburg">Salzburg</a>. Glide reflection axis runs northeast–southwest</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_33434.svg" class="mw-file-description" title="One of the colorings of the snub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just pg, then it is p4g (see there for this image with equally colored triangles)[5]"><img alt="One of the colorings of the snub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just pg, then it is p4g (see there for this image with equally colored triangles)[5]" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Tile_33434.svg/120px-Tile_33434.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Tile_33434.svg/180px-Tile_33434.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Tile_33434.svg/240px-Tile_33434.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext">One of the colorings of the <a href="/wiki/Snub_square_tiling" title="Snub square tiling">snub square tiling</a>; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just <i><b>pg</b></i>, then it is <b><i>p</i>4<i>g</i></b> (see there for this image with equally colored triangles)<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></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_pg.svg" class="mw-file-description" title="6 co-uniform tiling made only of pentagons"><img alt="6 co-uniform tiling made only of pentagons" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Co-Uniform_Wallpaper_Example_pg.svg/120px-Co-Uniform_Wallpaper_Example_pg.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Co-Uniform_Wallpaper_Example_pg.svg/180px-Co-Uniform_Wallpaper_Example_pg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Co-Uniform_Wallpaper_Example_pg.svg/240px-Co-Uniform_Wallpaper_Example_pg.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">6 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling made only of <a href="/wiki/Elongated_triangular_tiling#Related_tilings" title="Elongated triangular tiling">pentagons</a></div> </li> </ul> <p>Without the details inside the zigzag bands the mat is <a href="#Group_p2mg"><i><b>pmg</b></i></a>; with the details but without the distinction between brown and black it is <a href="#Group_p2gg"><i><b>pgg</b></i></a>. </p><p>Ignoring the wavy borders of the tiles, the pavement is <a href="#Group_p2gg"><i><b>pgg</b></i></a>. </p> <div class="mw-heading mw-heading3"><h3 id="Group_cm_(*×)"><span id="Group_cm_.28.2A.C3.97.29"></span><span class="anchor" id="Group_cm"></span> Group <i>cm</i> (*×)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=16" title="Edit section: Group cm (*×)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_cm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/SymBlend_cm.svg/220px-SymBlend_cm.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/SymBlend_cm.svg/330px-SymBlend_cm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/SymBlend_cm.svg/440px-SymBlend_cm.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>cm</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structure for <i>cm</i> </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_cm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Wallpaper_group_diagram_cm.svg/160px-Wallpaper_group_diagram_cm.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Wallpaper_group_diagram_cm.svg/240px-Wallpaper_group_diagram_cm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Wallpaper_group_diagram_cm.svg/320px-Wallpaper_group_diagram_cm.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Horizontal mirrors </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_cm_rotated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Wallpaper_group_diagram_cm_rotated.svg/160px-Wallpaper_group_diagram_cm_rotated.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Wallpaper_group_diagram_cm_rotated.svg/240px-Wallpaper_group_diagram_cm_rotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Wallpaper_group_diagram_cm_rotated.svg/320px-Wallpaper_group_diagram_cm_rotated.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Vertical mirrors </th></tr> <tr> <th colspan="2">Rhombic </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:red;">*×</span></li> <li>Coxeter notation: [∞<sup>+</sup>,2<sup>+</sup>,∞] or [∞,2<sup>+</sup>,∞<sup>+</sup>]</li> <li>Lattice: rhombic</li> <li>Point group: D<sub>1</sub></li> <li>The group <i><b>cm</b></i> contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is <i>not</i> a reflection axis; it is halfway between two adjacent parallel reflection axes.</li> <li>This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>cm</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperCM.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/WallpaperCM.GIF/120px-WallpaperCM.GIF" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/WallpaperCM.GIF/180px-WallpaperCM.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/WallpaperCM.GIF/240px-WallpaperCM.GIF 2x" data-file-width="249" data-file-height="245" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-1.jpg" class="mw-file-description" title="Dress of Amun, from Abu Simbel, Egypt"><img alt="Dress of Amun, from Abu Simbel, Egypt" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Wallpaper_group-cm-1.jpg/120px-Wallpaper_group-cm-1.jpg" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Wallpaper_group-cm-1.jpg/180px-Wallpaper_group-cm-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Wallpaper_group-cm-1.jpg/240px-Wallpaper_group-cm-1.jpg 2x" data-file-width="1088" data-file-height="1054" /></a></span></div> <div class="gallerytext">Dress of <a href="/wiki/Amun" title="Amun">Amun</a>, from <a href="/wiki/Abu_Simbel" title="Abu Simbel">Abu Simbel</a>, <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-2.jpg" class="mw-file-description" title="Dado from Biban el Moluk, Egypt"><img alt="Dado from Biban el Moluk, Egypt" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-cm-2.jpg/120px-Wallpaper_group-cm-2.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-cm-2.jpg/180px-Wallpaper_group-cm-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-cm-2.jpg/240px-Wallpaper_group-cm-2.jpg 2x" data-file-width="1078" data-file-height="1066" /></a></span></div> <div class="gallerytext"><a href="/wiki/Dado_(architecture)" title="Dado (architecture)">Dado</a> from <a href="/wiki/Biban_el_Moluk" class="mw-redirect" title="Biban el Moluk">Biban el Moluk</a>, <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-3.jpg" class="mw-file-description" title="Bronze vessel in Nimroud, Assyria"><img alt="Bronze vessel in Nimroud, Assyria" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Wallpaper_group-cm-3.jpg/120px-Wallpaper_group-cm-3.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Wallpaper_group-cm-3.jpg/180px-Wallpaper_group-cm-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Wallpaper_group-cm-3.jpg/240px-Wallpaper_group-cm-3.jpg 2x" data-file-width="1050" data-file-height="1052" /></a></span></div> <div class="gallerytext"><a href="/wiki/Bronze" title="Bronze">Bronze</a> vessel in <a href="/wiki/Nimroud" class="mw-redirect" title="Nimroud">Nimroud</a>, <a href="/wiki/Assyria" title="Assyria">Assyria</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-4.jpg" class="mw-file-description" title="Spandrels of arches, the Alhambra, Spain"><img alt="Spandrels of arches, the Alhambra, Spain" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wallpaper_group-cm-4.jpg/78px-Wallpaper_group-cm-4.jpg" decoding="async" width="78" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wallpaper_group-cm-4.jpg/117px-Wallpaper_group-cm-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wallpaper_group-cm-4.jpg/156px-Wallpaper_group-cm-4.jpg 2x" data-file-width="1123" data-file-height="1729" /></a></span></div> <div class="gallerytext"><a href="/wiki/Spandrel" title="Spandrel">Spandrels</a> of <a href="/wiki/Arch" title="Arch">arches</a>, the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a>, <a href="/wiki/Spain" title="Spain">Spain</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-5.jpg" class="mw-file-description" title="Soffitt of arch, the Alhambra, Spain"><img alt="Soffitt of arch, the Alhambra, Spain" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Wallpaper_group-cm-5.jpg/120px-Wallpaper_group-cm-5.jpg" decoding="async" width="120" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Wallpaper_group-cm-5.jpg/180px-Wallpaper_group-cm-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Wallpaper_group-cm-5.jpg/240px-Wallpaper_group-cm-5.jpg 2x" data-file-width="1626" data-file-height="1523" /></a></span></div> <div class="gallerytext"><a href="/wiki/Soffitt" class="mw-redirect" title="Soffitt">Soffitt</a> of arch, the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a>, <a href="/wiki/Spain" title="Spain">Spain</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-6.jpg" class="mw-file-description" title="Persian tapestry"><img alt="Persian tapestry" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group-cm-6.jpg/106px-Wallpaper_group-cm-6.jpg" decoding="async" width="106" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group-cm-6.jpg/159px-Wallpaper_group-cm-6.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group-cm-6.jpg/212px-Wallpaper_group-cm-6.jpg 2x" data-file-width="1132" data-file-height="1282" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_empire" class="mw-redirect" title="Persian empire">Persian</a> <a href="/wiki/Tapestry" title="Tapestry">tapestry</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cm-7.jpg" class="mw-file-description" title="Indian metalwork at the Great Exhibition in 1851"><img alt="Indian metalwork at the Great Exhibition in 1851" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Wallpaper_group-cm-7.jpg/120px-Wallpaper_group-cm-7.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Wallpaper_group-cm-7.jpg/180px-Wallpaper_group-cm-7.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Wallpaper_group-cm-7.jpg/240px-Wallpaper_group-cm-7.jpg 2x" data-file-width="1067" data-file-height="1064" /></a></span></div> <div class="gallerytext"><a href="/wiki/India" title="India">Indian</a> metalwork at the <a href="/wiki/Great_Exhibition" title="Great Exhibition">Great Exhibition</a> in 1851</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pm-2.jpg" class="mw-file-description" title="Dress of a figure in a tomb at Biban el Moluk, Egypt"><img alt="Dress of a figure in a tomb at Biban el Moluk, Egypt" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Wallpaper_group-pm-2.jpg/120px-Wallpaper_group-pm-2.jpg" decoding="async" width="120" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Wallpaper_group-pm-2.jpg/180px-Wallpaper_group-pm-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Wallpaper_group-pm-2.jpg/240px-Wallpaper_group-pm-2.jpg 2x" data-file-width="1080" data-file-height="946" /></a></span></div> <div class="gallerytext">Dress of a figure in a <a href="/wiki/Tomb" title="Tomb">tomb</a> at <a href="/wiki/Biban_el_Moluk" class="mw-redirect" title="Biban el Moluk">Biban el Moluk</a>, <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_cm.svg" class="mw-file-description" title="6 co-uniform tiling with hexagonal cells"><img alt="6 co-uniform tiling with hexagonal cells" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Co-Uniform_Wallpaper_Example_cm.svg/120px-Co-Uniform_Wallpaper_Example_cm.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Co-Uniform_Wallpaper_Example_cm.svg/180px-Co-Uniform_Wallpaper_Example_cm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Co-Uniform_Wallpaper_Example_cm.svg/240px-Co-Uniform_Wallpaper_Example_cm.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">6 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling with hexagonal cells</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Houndstooth.jpg" class="mw-file-description" title="Textile pattern: houndstooth"><img alt="Textile pattern: houndstooth" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Houndstooth.jpg/117px-Houndstooth.jpg" decoding="async" width="117" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Houndstooth.jpg/175px-Houndstooth.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Houndstooth.jpg/234px-Houndstooth.jpg 2x" data-file-width="390" data-file-height="400" /></a></span></div> <div class="gallerytext">Textile pattern: <a href="/wiki/Houndstooth" title="Houndstooth">houndstooth</a></div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_pmm_(*2222)"><span id="Group_pmm_.28.2A2222.29"></span><span class="anchor" id="Group_p2mm"></span> Group <i>pmm</i> (*2222)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=17" title="Edit section: Group pmm (*2222)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_pmm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/SymBlend_pmm.svg/220px-SymBlend_pmm.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/SymBlend_pmm.svg/330px-SymBlend_pmm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/SymBlend_pmm.svg/440px-SymBlend_pmm.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>pmm</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structure for <i>pmm</i> </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pmm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Wallpaper_group_diagram_pmm.svg/240px-Wallpaper_group_diagram_pmm.svg.png" decoding="async" width="240" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Wallpaper_group_diagram_pmm.svg/360px-Wallpaper_group_diagram_pmm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Wallpaper_group_diagram_pmm.svg/480px-Wallpaper_group_diagram_pmm.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />rectangular </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pmm_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wallpaper_group_diagram_pmm_square.svg/140px-Wallpaper_group_diagram_pmm_square.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wallpaper_group_diagram_pmm_square.svg/210px-Wallpaper_group_diagram_pmm_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wallpaper_group_diagram_pmm_square.svg/280px-Wallpaper_group_diagram_pmm_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span><br />square </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:red;">*2222</span></li> <li>Coxeter notation (rectangular): [∞,2,∞] or [∞]×[∞]</li> <li>Coxeter notation (square): [4,1<sup>+</sup>,4] or [1<sup>+</sup>,4,4,1<sup>+</sup>]</li> <li>Lattice: rectangular</li> <li>Point group: D<sub>2</sub></li> <li>The group <i><b>pmm</b></i> has reflections in two perpendicular directions, and four rotation centres of order two (180°) located at the intersections of the reflection axes.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>pmm</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmm-1.jpg" class="mw-file-description" title="2D image of lattice fence, U.S. (in 3D there is additional symmetry)"><img alt="2D image of lattice fence, U.S. (in 3D there is additional symmetry)" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Wallpaper_group-pmm-1.jpg/120px-Wallpaper_group-pmm-1.jpg" decoding="async" width="120" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Wallpaper_group-pmm-1.jpg/180px-Wallpaper_group-pmm-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Wallpaper_group-pmm-1.jpg/240px-Wallpaper_group-pmm-1.jpg 2x" data-file-width="1029" data-file-height="951" /></a></span></div> <div class="gallerytext">2D image of lattice <a href="/wiki/Fence" title="Fence">fence</a>, U.S. (in 3D there is additional symmetry)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmm-2.jpg" class="mw-file-description" title="Mummy case stored in The Louvre"><img alt="Mummy case stored in The Louvre" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Wallpaper_group-pmm-2.jpg/120px-Wallpaper_group-pmm-2.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Wallpaper_group-pmm-2.jpg/180px-Wallpaper_group-pmm-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Wallpaper_group-pmm-2.jpg/240px-Wallpaper_group-pmm-2.jpg 2x" data-file-width="1076" data-file-height="1069" /></a></span></div> <div class="gallerytext"><a href="/wiki/Mummy" title="Mummy">Mummy</a> case stored in <a href="/wiki/The_Louvre" class="mw-redirect" title="The Louvre">The Louvre</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmm-4.jpg" class="mw-file-description" title="Mummy case stored in The Louvre. Would be type p4m except for the mismatched coloring"><img alt="Mummy case stored in The Louvre. Would be type p4m except for the mismatched coloring" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Wallpaper_group-pmm-4.jpg/120px-Wallpaper_group-pmm-4.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Wallpaper_group-pmm-4.jpg/180px-Wallpaper_group-pmm-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Wallpaper_group-pmm-4.jpg/240px-Wallpaper_group-pmm-4.jpg 2x" data-file-width="1070" data-file-height="1068" /></a></span></div> <div class="gallerytext"><a href="/wiki/Mummy" title="Mummy">Mummy</a> case stored in <a href="/wiki/The_Louvre" class="mw-redirect" title="The Louvre">The Louvre</a>. Would be type <b><i>p</i>4<i>m</i></b> except for the mismatched coloring</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_pmm.svg" class="mw-file-description" title="8 co-uniform tiling with all non-slab planigons"><img alt="8 co-uniform tiling with all non-slab planigons" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Co-Uniform_Wallpaper_Example_pmm.svg/120px-Co-Uniform_Wallpaper_Example_pmm.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Co-Uniform_Wallpaper_Example_pmm.svg/180px-Co-Uniform_Wallpaper_Example_pmm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Co-Uniform_Wallpaper_Example_pmm.svg/240px-Co-Uniform_Wallpaper_Example_pmm.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">8 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling with all non-slab planigons</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_pmg_(22*)"><span id="Group_pmg_.2822.2A.29"></span><span class="anchor" id="Group_p2mg"></span> Group <i>pmg</i> (22*)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=18" title="Edit section: Group pmg (22*)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_pmg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/SymBlend_pmg.svg/220px-SymBlend_pmg.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/SymBlend_pmg.svg/330px-SymBlend_pmg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/SymBlend_pmg.svg/440px-SymBlend_pmg.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>pmg</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structures for <i>pmg</i> </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pmg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Wallpaper_group_diagram_pmg.svg/160px-Wallpaper_group_diagram_pmg.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Wallpaper_group_diagram_pmg.svg/240px-Wallpaper_group_diagram_pmg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Wallpaper_group_diagram_pmg.svg/320px-Wallpaper_group_diagram_pmg.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Horizontal mirrors </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pmg_rotated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Wallpaper_group_diagram_pmg_rotated.svg/160px-Wallpaper_group_diagram_pmg_rotated.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Wallpaper_group_diagram_pmg_rotated.svg/240px-Wallpaper_group_diagram_pmg_rotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Wallpaper_group_diagram_pmg_rotated.svg/320px-Wallpaper_group_diagram_pmg_rotated.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Vertical mirrors </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:blue;">22</span><span style="color:red;">*</span></li> <li>Coxeter notation: [(∞,2)<sup>+</sup>,∞] or [∞,(2,∞)<sup>+</sup>]</li> <li>Lattice: rectangular</li> <li>Point group: D<sub>2</sub></li> <li>The group <i><b>pmg</b></i> has two rotation centres of order two (180°), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>pmg</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperPMG.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/WallpaperPMG.GIF/120px-WallpaperPMG.GIF" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/WallpaperPMG.GIF/180px-WallpaperPMG.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/6/64/WallpaperPMG.GIF 2x" data-file-width="240" data-file-height="231" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmg-1.jpg" class="mw-file-description" title="Cloth, Sandwich Islands (Hawaii)"><img alt="Cloth, Sandwich Islands (Hawaii)" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Wallpaper_group-pmg-1.jpg/118px-Wallpaper_group-pmg-1.jpg" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Wallpaper_group-pmg-1.jpg/177px-Wallpaper_group-pmg-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Wallpaper_group-pmg-1.jpg/236px-Wallpaper_group-pmg-1.jpg 2x" data-file-width="993" data-file-height="1011" /></a></span></div> <div class="gallerytext">Cloth, <a href="/wiki/Hawaiian_Islands" title="Hawaiian Islands">Sandwich Islands</a> (<a href="/wiki/Hawaii" title="Hawaii">Hawaii</a>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmg-2.jpg" class="mw-file-description" title="Ceiling of Egyptian tomb"><img alt="Ceiling of Egyptian tomb" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Wallpaper_group-pmg-2.jpg/120px-Wallpaper_group-pmg-2.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Wallpaper_group-pmg-2.jpg/180px-Wallpaper_group-pmg-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Wallpaper_group-pmg-2.jpg/240px-Wallpaper_group-pmg-2.jpg 2x" data-file-width="1066" data-file-height="1050" /></a></span></div> <div class="gallerytext">Ceiling of <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmg-3.jpg" class="mw-file-description" title="Floor tiling in Prague, the Czech Republic"><img alt="Floor tiling in Prague, the Czech Republic" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-pmg-3.jpg/120px-Wallpaper_group-pmg-3.jpg" decoding="async" width="120" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-pmg-3.jpg/180px-Wallpaper_group-pmg-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-pmg-3.jpg/240px-Wallpaper_group-pmg-3.jpg 2x" data-file-width="1050" data-file-height="968" /></a></span></div> <div class="gallerytext">Floor tiling in <a href="/wiki/Prague" title="Prague">Prague</a>, the <a href="/wiki/Czech_Republic" title="Czech Republic">Czech Republic</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pmg-4.jpg" class="mw-file-description" title="Bowl from Kerma"><img alt="Bowl from Kerma" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Wallpaper_group-pmg-4.jpg/120px-Wallpaper_group-pmg-4.jpg" decoding="async" width="120" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Wallpaper_group-pmg-4.jpg/180px-Wallpaper_group-pmg-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Wallpaper_group-pmg-4.jpg/240px-Wallpaper_group-pmg-4.jpg 2x" data-file-width="1894" data-file-height="1650" /></a></span></div> <div class="gallerytext">Bowl from <a href="/wiki/Kingdom_of_Kerma" class="mw-redirect" title="Kingdom of Kerma">Kerma</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_pmg.svg" class="mw-file-description" title="4 co-uniform tiling"><img alt="4 co-uniform tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Co-Uniform_Wallpaper_Example_pmg.svg/120px-Co-Uniform_Wallpaper_Example_pmg.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Co-Uniform_Wallpaper_Example_pmg.svg/180px-Co-Uniform_Wallpaper_Example_pmg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Co-Uniform_Wallpaper_Example_pmg.svg/240px-Co-Uniform_Wallpaper_Example_pmg.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">4 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_pentagon_packing.svg" class="mw-file-description" title="Pentagon packing"><img alt="Pentagon packing" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/2-d_pentagon_packing.svg/120px-2-d_pentagon_packing.svg.png" decoding="async" width="120" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/2-d_pentagon_packing.svg/180px-2-d_pentagon_packing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/2-d_pentagon_packing.svg/240px-2-d_pentagon_packing.svg.png 2x" data-file-width="947" data-file-height="800" /></a></span></div> <div class="gallerytext">Pentagon packing</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_pgg_(22×)"><span id="Group_pgg_.2822.C3.97.29"></span><span class="anchor" id="Group_p2gg"></span> Group <i>pgg</i> (22×)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=19" title="Edit section: Group pgg (22×)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_pgg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/SymBlend_pgg.svg/220px-SymBlend_pgg.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/SymBlend_pgg.svg/330px-SymBlend_pgg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/SymBlend_pgg.svg/440px-SymBlend_pgg.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>pgg</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structures for <i>pgg</i> by lattice type </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pgg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Wallpaper_group_diagram_pgg.svg/160px-Wallpaper_group_diagram_pgg.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Wallpaper_group_diagram_pgg.svg/240px-Wallpaper_group_diagram_pgg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Wallpaper_group_diagram_pgg.svg/320px-Wallpaper_group_diagram_pgg.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Rectangular </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_pgg_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Wallpaper_group_diagram_pgg_square.svg/100px-Wallpaper_group_diagram_pgg_square.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Wallpaper_group_diagram_pgg_square.svg/150px-Wallpaper_group_diagram_pgg_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Wallpaper_group_diagram_pgg_square.svg/200px-Wallpaper_group_diagram_pgg_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span><br />Square </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:blue;">22</span><span style="color:red;">×</span></li> <li>Coxeter notation (rectangular): [((∞,2)<sup>+</sup>,(∞,2)<sup>+</sup>)]</li> <li>Coxeter notation (square): [4<sup>+</sup>,4<sup>+</sup>]</li> <li>Lattice: rectangular</li> <li>Point group: D<sub>2</sub></li> <li>The group <i><b>pgg</b></i> contains two rotation centres of order two (180°), and glide reflections in two perpendicular directions. The centres of rotation are not located on the glide reflection axes. There are no reflections.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>pgg</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperPGG.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/WallpaperPGG.GIF/112px-WallpaperPGG.GIF" decoding="async" width="112" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/WallpaperPGG.GIF/168px-WallpaperPGG.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/WallpaperPGG.GIF/224px-WallpaperPGG.GIF 2x" data-file-width="232" data-file-height="248" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pgg-1.jpg" class="mw-file-description" title="Bronze vessel in Nimroud, Assyria"><img alt="Bronze vessel in Nimroud, Assyria" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Wallpaper_group-pgg-1.jpg/117px-Wallpaper_group-pgg-1.jpg" decoding="async" width="117" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Wallpaper_group-pgg-1.jpg/175px-Wallpaper_group-pgg-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Wallpaper_group-pgg-1.jpg/234px-Wallpaper_group-pgg-1.jpg 2x" data-file-width="1041" data-file-height="1069" /></a></span></div> <div class="gallerytext"><a href="/wiki/Bronze" title="Bronze">Bronze</a> vessel in <a href="/wiki/Nimroud" class="mw-redirect" title="Nimroud">Nimroud</a>, <a href="/wiki/Assyria" title="Assyria">Assyria</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-pgg-2.jpg" class="mw-file-description" title="Pavement in Budapest, Hungary"><img alt="Pavement in Budapest, Hungary" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-pgg-2.jpg/120px-Wallpaper_group-pgg-2.jpg" decoding="async" width="120" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-pgg-2.jpg/180px-Wallpaper_group-pgg-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Wallpaper_group-pgg-2.jpg/240px-Wallpaper_group-pgg-2.jpg 2x" data-file-width="1150" data-file-height="910" /></a></span></div> <div class="gallerytext"><a href="/wiki/Pavement_(roads)" class="mw-redirect" title="Pavement (roads)">Pavement</a> in <a href="/wiki/Budapest" title="Budapest">Budapest</a>, <a href="/wiki/Hungary" title="Hungary">Hungary</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_pgg.svg" class="mw-file-description" title="4 co-uniform tiling (strictly trihexagonal)"><img alt="4 co-uniform tiling (strictly trihexagonal)" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Co-Uniform_Wallpaper_Example_pgg.svg/120px-Co-Uniform_Wallpaper_Example_pgg.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Co-Uniform_Wallpaper_Example_pgg.svg/180px-Co-Uniform_Wallpaper_Example_pgg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Co-Uniform_Wallpaper_Example_pgg.svg/240px-Co-Uniform_Wallpaper_Example_pgg.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">4 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling (strictly trihexagonal)</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_cmm_(2*22)"><span id="Group_cmm_.282.2A22.29"></span><span class="anchor" id="Group_c2mm"></span> Group <i>cmm</i> (2*22)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=20" title="Edit section: Group cmm (2*22)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_cmm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/SymBlend_cmm.svg/220px-SymBlend_cmm.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/SymBlend_cmm.svg/330px-SymBlend_cmm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/SymBlend_cmm.svg/440px-SymBlend_cmm.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <i><b>cmm</b></i></figcaption></figure> <table class="wikitable"> <caption>Cell structures for <i>cmm</i> by lattice type </caption> <tbody><tr> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_cmm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Wallpaper_group_diagram_cmm.svg/200px-Wallpaper_group_diagram_cmm.svg.png" decoding="async" width="200" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Wallpaper_group_diagram_cmm.svg/300px-Wallpaper_group_diagram_cmm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Wallpaper_group_diagram_cmm.svg/400px-Wallpaper_group_diagram_cmm.svg.png 2x" data-file-width="744" data-file-height="425" /></a></span><br />Rhombic </th> <th><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_cmm_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Wallpaper_group_diagram_cmm_square.svg/120px-Wallpaper_group_diagram_cmm_square.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Wallpaper_group_diagram_cmm_square.svg/180px-Wallpaper_group_diagram_cmm_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Wallpaper_group_diagram_cmm_square.svg/240px-Wallpaper_group_diagram_cmm_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span><br />Square </th></tr></tbody></table> <ul><li>Orbifold signature: <span style="color:blue;">2</span><span style="color:red;">*22</span></li> <li>Coxeter notation (rhombic): [∞,2<sup>+</sup>,∞]</li> <li>Coxeter notation (square): [(4,4,2<sup>+</sup>)]</li> <li>Lattice: rhombic</li> <li>Point group: D<sub>2</sub></li> <li>The group <i><b>cmm</b></i> has reflections in two perpendicular directions, and a rotation of order two (180°) whose centre is <i>not</i> on a reflection axis. It also has two rotations whose centres <i>are</i> on a reflection axis.</li> <li>This group is frequently seen in everyday life, since the most common arrangement of <a href="/wiki/Brick" title="Brick">bricks</a> in a brick building (<a href="/wiki/Running_bond" class="mw-redirect" title="Running bond">running bond</a>) utilises this group (see example below).</li></ul> <p>The rotational symmetry of order 2 with centres of rotation at the centres of the sides of the rhombus is a consequence of the other properties. </p><p>The pattern corresponds to each of the following: </p> <ul><li>symmetrically staggered rows of identical doubly symmetric objects</li> <li>a checkerboard pattern of two alternating rectangular tiles, of which each, by itself, is doubly symmetric</li> <li>a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>cmm</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperCMM.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/WallpaperCMM.GIF/120px-WallpaperCMM.GIF" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/WallpaperCMM.GIF/180px-WallpaperCMM.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/WallpaperCMM.GIF/240px-WallpaperCMM.GIF 2x" data-file-width="249" data-file-height="238" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n8.svg" class="mw-file-description" title="Elongated triangular tiling"><img alt="Elongated triangular tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/1-uniform_n8.svg/120px-1-uniform_n8.svg.png" decoding="async" width="120" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/1-uniform_n8.svg/180px-1-uniform_n8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/1-uniform_n8.svg/240px-1-uniform_n8.svg.png 2x" data-file-width="1000" data-file-height="930" /></a></span></div> <div class="gallerytext"><a href="/wiki/Elongated_triangular_tiling" title="Elongated triangular tiling">Elongated triangular tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cmm-1.jpg" class="mw-file-description" title="Suburban brick wall using running bond arrangement, U.S."><img alt="Suburban brick wall using running bond arrangement, U.S." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Wallpaper_group-cmm-1.jpg/120px-Wallpaper_group-cmm-1.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Wallpaper_group-cmm-1.jpg/180px-Wallpaper_group-cmm-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Wallpaper_group-cmm-1.jpg/240px-Wallpaper_group-cmm-1.jpg 2x" data-file-width="1528" data-file-height="1528" /></a></span></div> <div class="gallerytext">Suburban <a href="/wiki/Brick" title="Brick">brick</a> wall using <a href="/wiki/Running_bond" class="mw-redirect" title="Running bond">running bond</a> arrangement, U.S.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cmm-2.jpg" class="mw-file-description" title="Ceiling of Egyptian tomb. Ignoring colors, this would be p4g"><img alt="Ceiling of Egyptian tomb. Ignoring colors, this would be p4g" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Wallpaper_group-cmm-2.jpg/120px-Wallpaper_group-cmm-2.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Wallpaper_group-cmm-2.jpg/180px-Wallpaper_group-cmm-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Wallpaper_group-cmm-2.jpg/240px-Wallpaper_group-cmm-2.jpg 2x" data-file-width="1058" data-file-height="1050" /></a></span></div> <div class="gallerytext">Ceiling of <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a>. Ignoring colors, this would be <a href="#Group_p4gm"><b><i>p</i>4<i>g</i></b></a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cmm-3.jpg" class="mw-file-description" title="Egyptian"><img alt="Egyptian" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Wallpaper_group-cmm-3.jpg/120px-Wallpaper_group-cmm-3.jpg" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Wallpaper_group-cmm-3.jpg/180px-Wallpaper_group-cmm-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Wallpaper_group-cmm-3.jpg/240px-Wallpaper_group-cmm-3.jpg 2x" data-file-width="1095" data-file-height="1056" /></a></span></div> <div class="gallerytext"><a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cmm-4.jpg" class="mw-file-description" title="Persian tapestry"><img alt="Persian tapestry" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Wallpaper_group-cmm-4.jpg/74px-Wallpaper_group-cmm-4.jpg" decoding="async" width="74" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Wallpaper_group-cmm-4.jpg/111px-Wallpaper_group-cmm-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Wallpaper_group-cmm-4.jpg/149px-Wallpaper_group-cmm-4.jpg 2x" data-file-width="1086" data-file-height="1752" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_empire" class="mw-redirect" title="Persian empire">Persian</a> <a href="/wiki/Tapestry" title="Tapestry">tapestry</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cmm-5.jpg" class="mw-file-description" title="Egyptian tomb"><img alt="Egyptian tomb" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Wallpaper_group-cmm-5.jpg/120px-Wallpaper_group-cmm-5.jpg" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Wallpaper_group-cmm-5.jpg/180px-Wallpaper_group-cmm-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Wallpaper_group-cmm-5.jpg/240px-Wallpaper_group-cmm-5.jpg 2x" data-file-width="1084" data-file-height="1054" /></a></span></div> <div class="gallerytext"><a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-cmm-6.jpg" class="mw-file-description" title="Turkish dish"><img alt="Turkish dish" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Wallpaper_group-cmm-6.jpg/117px-Wallpaper_group-cmm-6.jpg" decoding="async" width="117" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Wallpaper_group-cmm-6.jpg/175px-Wallpaper_group-cmm-6.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Wallpaper_group-cmm-6.jpg/234px-Wallpaper_group-cmm-6.jpg 2x" data-file-width="1656" data-file-height="1701" /></a></span></div> <div class="gallerytext"><a href="/wiki/Turkic_peoples" title="Turkic peoples">Turkish</a> dish</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r1.svg" class="mw-file-description" title="A compact packing of two sizes of circle"><img alt="A compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/2-d_dense_packing_r1.svg/120px-2-d_dense_packing_r1.svg.png" decoding="async" width="120" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/2-d_dense_packing_r1.svg/180px-2-d_dense_packing_r1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/2-d_dense_packing_r1.svg/240px-2-d_dense_packing_r1.svg.png 2x" data-file-width="1160" data-file-height="1003" /></a></span></div> <div class="gallerytext">A compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r3.svg" class="mw-file-description" title="Another compact packing of two sizes of circle"><img alt="Another compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/2-d_dense_packing_r3.svg/111px-2-d_dense_packing_r3.svg.png" decoding="async" width="111" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/2-d_dense_packing_r3.svg/167px-2-d_dense_packing_r3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/2-d_dense_packing_r3.svg/223px-2-d_dense_packing_r3.svg.png 2x" data-file-width="1000" data-file-height="1078" /></a></span></div> <div class="gallerytext">Another compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r7.svg" class="mw-file-description" title="Another compact packing of two sizes of circle"><img alt="Another compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/2-d_dense_packing_r7.svg/112px-2-d_dense_packing_r7.svg.png" decoding="async" width="112" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/2-d_dense_packing_r7.svg/169px-2-d_dense_packing_r7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/2-d_dense_packing_r7.svg/225px-2-d_dense_packing_r7.svg.png 2x" data-file-width="937" data-file-height="1000" /></a></span></div> <div class="gallerytext">Another compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_cmm.svg" class="mw-file-description" title="3 co-uniform tiling (Krötenheerdt)"><img alt="3 co-uniform tiling (Krötenheerdt)" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Co-Uniform_Wallpaper_Example_cmm.svg/120px-Co-Uniform_Wallpaper_Example_cmm.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Co-Uniform_Wallpaper_Example_cmm.svg/180px-Co-Uniform_Wallpaper_Example_cmm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Co-Uniform_Wallpaper_Example_cmm.svg/240px-Co-Uniform_Wallpaper_Example_cmm.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">3 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling (Krötenheerdt)</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p4_(442)"><span id="Group_p4_.28442.29"></span><span class="anchor" id="Group_p4"></span> Group <i>p</i>4 (442)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=21" title="Edit section: Group p4 (442)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/SymBlend_p4.svg/220px-SymBlend_p4.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/SymBlend_p4.svg/330px-SymBlend_p4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/SymBlend_p4.svg/440px-SymBlend_p4.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>4</b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p4_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p4_square.svg/220px-Wallpaper_group_diagram_p4_square.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p4_square.svg/330px-Wallpaper_group_diagram_p4_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p4_square.svg/440px-Wallpaper_group_diagram_p4_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>4</b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:blue;">442</span></li> <li>Coxeter notation: [4,4]<sup>+</sup></li> <li>Lattice: square</li> <li>Point group: C<sub>4</sub></li> <li>The group <b><i>p</i>4</b> has two rotation centres of order four (90°), and one rotation centre of order two (180°). It has no reflections or glide reflections.</li></ul> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>4</dt></dl> <p>A <b><i>p</i>4</b> pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a <a href="/wiki/Checkerboard" title="Checkerboard">checkerboard</a> pattern of two such tiles, a factor <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> smaller and rotated 45°. </p> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP4.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/WallpaperP4.GIF/119px-WallpaperP4.GIF" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/WallpaperP4.GIF/178px-WallpaperP4.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/WallpaperP4.GIF/237px-WallpaperP4.GIF 2x" data-file-width="265" data-file-height="268" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4-1.jpg" class="mw-file-description" title="Ceiling of Egyptian tomb; ignoring colors this is p4, otherwise p2"><img alt="Ceiling of Egyptian tomb; ignoring colors this is p4, otherwise p2" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Wallpaper_group-p4-1.jpg/119px-Wallpaper_group-p4-1.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Wallpaper_group-p4-1.jpg/179px-Wallpaper_group-p4-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Wallpaper_group-p4-1.jpg/239px-Wallpaper_group-p4-1.jpg 2x" data-file-width="1064" data-file-height="1070" /></a></span></div> <div class="gallerytext">Ceiling of <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a>; ignoring colors this is <b><i>p</i>4</b>, otherwise <a href="#Group_p2"><b><i>p</i>2</b></a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4-2.jpg" class="mw-file-description" title="Ceiling of Egyptian tomb"><img alt="Ceiling of Egyptian tomb" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Wallpaper_group-p4-2.jpg/119px-Wallpaper_group-p4-2.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Wallpaper_group-p4-2.jpg/178px-Wallpaper_group-p4-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Wallpaper_group-p4-2.jpg/238px-Wallpaper_group-p4-2.jpg 2x" data-file-width="1063" data-file-height="1073" /></a></span></div> <div class="gallerytext">Ceiling of <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:A_wallpaper_pattern_Overlaid_patterns.svg" class="mw-file-description" title="Overlaid patterns"><img alt="Overlaid patterns" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/A_wallpaper_pattern_Overlaid_patterns.svg/120px-A_wallpaper_pattern_Overlaid_patterns.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/A_wallpaper_pattern_Overlaid_patterns.svg/180px-A_wallpaper_pattern_Overlaid_patterns.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/A_wallpaper_pattern_Overlaid_patterns.svg/240px-A_wallpaper_pattern_Overlaid_patterns.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div> <div class="gallerytext">Overlaid patterns</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4-3.jpg" class="mw-file-description" title="Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections"><img alt="Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Wallpaper_group-p4-3.jpg/120px-Wallpaper_group-p4-3.jpg" decoding="async" width="120" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Wallpaper_group-p4-3.jpg/180px-Wallpaper_group-p4-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Wallpaper_group-p4-3.jpg/240px-Wallpaper_group-p4-3.jpg 2x" data-file-width="1498" data-file-height="1205" /></a></span></div> <div class="gallerytext">Frieze, the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a>, <a href="/wiki/Spain" title="Spain">Spain</a>. Requires close inspection to see why there are no reflections</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4-4.jpg" class="mw-file-description" title="Viennese cane"><img alt="Viennese cane" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Wallpaper_group-p4-4.jpg/120px-Wallpaper_group-p4-4.jpg" decoding="async" width="120" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Wallpaper_group-p4-4.jpg/180px-Wallpaper_group-p4-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Wallpaper_group-p4-4.jpg/240px-Wallpaper_group-p4-4.jpg 2x" data-file-width="1050" data-file-height="958" /></a></span></div> <div class="gallerytext">Viennese cane</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4-5.jpg" class="mw-file-description" title="Renaissance earthenware"><img alt="Renaissance earthenware" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wallpaper_group-p4-5.jpg/120px-Wallpaper_group-p4-5.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wallpaper_group-p4-5.jpg/180px-Wallpaper_group-p4-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wallpaper_group-p4-5.jpg/240px-Wallpaper_group-p4-5.jpg 2x" data-file-width="1098" data-file-height="1076" /></a></span></div> <div class="gallerytext">Renaissance earthenware</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:A_tri-colored_Pythagorean_tiling_View_4.svg" class="mw-file-description" title="Pythagorean tiling"><img alt="Pythagorean tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/A_tri-colored_Pythagorean_tiling_View_4.svg/120px-A_tri-colored_Pythagorean_tiling_View_4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/A_tri-colored_Pythagorean_tiling_View_4.svg/180px-A_tri-colored_Pythagorean_tiling_View_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/A_tri-colored_Pythagorean_tiling_View_4.svg/240px-A_tri-colored_Pythagorean_tiling_View_4.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div> <div class="gallerytext"><a href="/wiki/Pythagorean_tiling" title="Pythagorean tiling">Pythagorean tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Lizard_p4_p4.png" class="mw-file-description" title="Generated from a photograph"><img alt="Generated from a photograph" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Lizard_p4_p4.png/120px-Lizard_p4_p4.png" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Lizard_p4_p4.png/180px-Lizard_p4_p4.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Lizard_p4_p4.png/240px-Lizard_p4_p4.png 2x" data-file-width="300" data-file-height="298" /></a></span></div> <div class="gallerytext">Generated from a photograph</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p4.svg" class="mw-file-description" title="4 co-uniform tiling"><img alt="4 co-uniform tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Co-Uniform_Wallpaper_Example_p4.svg/120px-Co-Uniform_Wallpaper_Example_p4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Co-Uniform_Wallpaper_Example_p4.svg/180px-Co-Uniform_Wallpaper_Example_p4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Co-Uniform_Wallpaper_Example_p4.svg/240px-Co-Uniform_Wallpaper_Example_p4.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">4 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p4m_(*442)"><span id="Group_p4m_.28.2A442.29"></span><span class="anchor" id="Group_p4mm"></span> Group <i>p</i>4<i>m</i> (*442)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=22" title="Edit section: Group p4m (*442)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p4m.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/SymBlend_p4m.svg/220px-SymBlend_p4m.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/SymBlend_p4m.svg/330px-SymBlend_p4m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/SymBlend_p4m.svg/440px-SymBlend_p4m.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>4<i>m</i></b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p4m_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/220px-Wallpaper_group_diagram_p4m_square.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/330px-Wallpaper_group_diagram_p4m_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/440px-Wallpaper_group_diagram_p4m_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>4<i>m</i></b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:red;">*442</span></li> <li>Coxeter notation: [4,4]</li> <li>Lattice: square</li> <li>Point group: D<sub>4</sub></li> <li>The group <b><i>p</i>4<i>m</i></b> has two rotation centres of order four (90°), and reflections in four distinct directions (horizontal, vertical, and diagonals). It has additional glide reflections whose axes are not reflection axes; rotations of order two (180°) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes.</li></ul> <p>This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to a <a href="/wiki/Checkerboard" title="Checkerboard">checkerboard</a> pattern of two of such squares. </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>4<i>m</i></dt></dl> <p>Examples displayed with the smallest translations horizontal and vertical (like in the diagram): </p> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP4M.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/WallpaperP4M.GIF/120px-WallpaperP4M.GIF" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/WallpaperP4M.GIF/180px-WallpaperP4M.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/WallpaperP4M.GIF/240px-WallpaperP4M.GIF 2x" data-file-width="281" data-file-height="280" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n5.svg" class="mw-file-description" title="Square tiling"><img alt="Square tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/1-uniform_n5.svg/120px-1-uniform_n5.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/1-uniform_n5.svg/180px-1-uniform_n5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/1-uniform_n5.svg/240px-1-uniform_n5.svg.png 2x" data-file-width="384" data-file-height="384" /></a></span></div> <div class="gallerytext"><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_V488.svg" class="mw-file-description" title="Tetrakis square tiling; ignoring colors, this is p4m, otherwise cmm"><img alt="Tetrakis square tiling; ignoring colors, this is p4m, otherwise cmm" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tile_V488.svg/120px-Tile_V488.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tile_V488.svg/180px-Tile_V488.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tile_V488.svg/240px-Tile_V488.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Tetrakis_square_tiling" title="Tetrakis square tiling">Tetrakis square tiling</a>; ignoring colors, this is <b><i>p</i>4<i>m</i></b>, otherwise <a href="#Group_c2mm"><i><b>cmm</b></i></a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_488.svg" class="mw-file-description" title="Truncated square tiling (ignoring color also, with smaller translations)"><img alt="Truncated square tiling (ignoring color also, with smaller translations)" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Tile_488.svg/120px-Tile_488.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Tile_488.svg/180px-Tile_488.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Tile_488.svg/240px-Tile_488.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Truncated_square_tiling" title="Truncated square tiling">Truncated square tiling</a> (ignoring color also, with smaller translations)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-1.jpg" class="mw-file-description" title="Ornamental painting, Nineveh, Assyria"><img alt="Ornamental painting, Nineveh, Assyria" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/120px-Wallpaper_group-p4m-1.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/180px-Wallpaper_group-p4m-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wallpaper_group-p4m-1.jpg/240px-Wallpaper_group-p4m-1.jpg 2x" data-file-width="1065" data-file-height="1057" /></a></span></div> <div class="gallerytext">Ornamental painting, <a href="/wiki/Nineveh" title="Nineveh">Nineveh</a>, <a href="/wiki/Assyria" title="Assyria">Assyria</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-3.jpg" class="mw-file-description" title="Storm drain, U.S."><img alt="Storm drain, U.S." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Wallpaper_group-p4m-3.jpg/120px-Wallpaper_group-p4m-3.jpg" decoding="async" width="120" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Wallpaper_group-p4m-3.jpg/180px-Wallpaper_group-p4m-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Wallpaper_group-p4m-3.jpg/240px-Wallpaper_group-p4m-3.jpg 2x" data-file-width="1389" data-file-height="1242" /></a></span></div> <div class="gallerytext"><a href="/wiki/Storm_drain" title="Storm drain">Storm drain</a>, U.S.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-5.jpg" class="mw-file-description" title="Egyptian mummy case"><img alt="Egyptian mummy case" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Wallpaper_group-p4m-5.jpg/120px-Wallpaper_group-p4m-5.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Wallpaper_group-p4m-5.jpg/180px-Wallpaper_group-p4m-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Wallpaper_group-p4m-5.jpg/240px-Wallpaper_group-p4m-5.jpg 2x" data-file-width="1074" data-file-height="1063" /></a></span></div> <div class="gallerytext"><a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Mummy" title="Mummy">mummy</a> case</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-6.jpg" class="mw-file-description" title="Persian glazed tile"><img alt="Persian glazed tile" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Wallpaper_group-p4m-6.jpg/120px-Wallpaper_group-p4m-6.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Wallpaper_group-p4m-6.jpg/179px-Wallpaper_group-p4m-6.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Wallpaper_group-p4m-6.jpg/239px-Wallpaper_group-p4m-6.jpg 2x" data-file-width="1070" data-file-height="1074" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_Empire" class="mw-redirect" title="Persian Empire">Persian</a> <a href="/wiki/Glaze_(painting_technique)" title="Glaze (painting technique)">glazed</a> tile</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r4.svg" class="mw-file-description" title="Compact packing of two sizes of circle"><img alt="Compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/2-d_dense_packing_r4.svg/120px-2-d_dense_packing_r4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/2-d_dense_packing_r4.svg/180px-2-d_dense_packing_r4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/2-d_dense_packing_r4.svg/240px-2-d_dense_packing_r4.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext">Compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p4m.svg" class="mw-file-description" title="4 co-uniform tiling (Krötenheerdt)"><img alt="4 co-uniform tiling (Krötenheerdt)" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Co-Uniform_Wallpaper_Example_p4m.svg/120px-Co-Uniform_Wallpaper_Example_p4m.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Co-Uniform_Wallpaper_Example_p4m.svg/180px-Co-Uniform_Wallpaper_Example_p4m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Co-Uniform_Wallpaper_Example_p4m.svg/240px-Co-Uniform_Wallpaper_Example_p4m.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">4 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling (Krötenheerdt)</div> </li> </ul> <p>Examples displayed with the smallest translations diagonal: </p> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_4,4.svg" class="mw-file-description" title="checkerboard"><img alt="checkerboard" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tile_4%2C4.svg/120px-Tile_4%2C4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tile_4%2C4.svg/180px-Tile_4%2C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tile_4%2C4.svg/240px-Tile_4%2C4.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext">checkerboard</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-2.jpg" class="mw-file-description" title="Cloth, Otaheite (Tahiti)"><img alt="Cloth, Otaheite (Tahiti)" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wallpaper_group-p4m-2.jpg/120px-Wallpaper_group-p4m-2.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wallpaper_group-p4m-2.jpg/180px-Wallpaper_group-p4m-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wallpaper_group-p4m-2.jpg/240px-Wallpaper_group-p4m-2.jpg 2x" data-file-width="997" data-file-height="986" /></a></span></div> <div class="gallerytext">Cloth, <a href="/wiki/Otaheite" class="mw-redirect" title="Otaheite">Otaheite</a> (<a href="/wiki/Tahiti" title="Tahiti">Tahiti</a>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-4.jpg" class="mw-file-description" title="Egyptian tomb"><img alt="Egyptian tomb" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Wallpaper_group-p4m-4.jpg/120px-Wallpaper_group-p4m-4.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Wallpaper_group-p4m-4.jpg/180px-Wallpaper_group-p4m-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Wallpaper_group-p4m-4.jpg/240px-Wallpaper_group-p4m-4.jpg 2x" data-file-width="1090" data-file-height="1078" /></a></span></div> <div class="gallerytext"><a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptian</a> <a href="/wiki/Tomb" title="Tomb">tomb</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-7.jpg" class="mw-file-description" title="Cathedral of Bourges"><img alt="Cathedral of Bourges" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Wallpaper_group-p4m-7.jpg/120px-Wallpaper_group-p4m-7.jpg" decoding="async" width="120" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Wallpaper_group-p4m-7.jpg/180px-Wallpaper_group-p4m-7.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Wallpaper_group-p4m-7.jpg/240px-Wallpaper_group-p4m-7.jpg 2x" data-file-width="1390" data-file-height="935" /></a></span></div> <div class="gallerytext">Cathedral of <a href="/wiki/Bourges" title="Bourges">Bourges</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4m-8.jpg" class="mw-file-description" title="Dish from Turkey, Ottoman period"><img alt="Dish from Turkey, Ottoman period" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group-p4m-8.jpg/120px-Wallpaper_group-p4m-8.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group-p4m-8.jpg/179px-Wallpaper_group-p4m-8.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group-p4m-8.jpg/239px-Wallpaper_group-p4m-8.jpg 2x" data-file-width="1479" data-file-height="1485" /></a></span></div> <div class="gallerytext">Dish from <a href="/wiki/Turkey" title="Turkey">Turkey</a>, <a href="/wiki/Ottoman_Empire" title="Ottoman Empire">Ottoman</a> period</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p4g_(4*2)"><span id="Group_p4g_.284.2A2.29"></span><span class="anchor" id="Group_p4gm"></span> Group <i>p</i>4<i>g</i> (4*2)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=23" title="Edit section: Group p4g (4*2)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p4g.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SymBlend_p4g.svg/220px-SymBlend_p4g.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SymBlend_p4g.svg/330px-SymBlend_p4g.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SymBlend_p4g.svg/440px-SymBlend_p4g.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>4<i>g</i></b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p4g_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wallpaper_group_diagram_p4g_square.svg/220px-Wallpaper_group_diagram_p4g_square.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wallpaper_group_diagram_p4g_square.svg/330px-Wallpaper_group_diagram_p4g_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wallpaper_group_diagram_p4g_square.svg/440px-Wallpaper_group_diagram_p4g_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>4<i>g</i></b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:blue;">4</span><span style="color:red;">*2</span></li> <li>Coxeter notation: [4<sup>+</sup>,4]</li> <li>Lattice: square</li> <li>Point group: D<sub>4</sub></li> <li>The group <b><i>p</i>4<i>g</i></b> has two centres of rotation of order four (90°), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180°) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45° with these.</li></ul> <p>A <b><i>p</i>4<i>g</i></b> pattern can be looked upon as a <a href="/wiki/Checkerboard" title="Checkerboard">checkerboard</a> pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group <b><a href="#Group_p4mm"><i>p</i>4<i>m</i></a></b> (with diagonal translation cells). </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>4<i>g</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4g-1.jpg" class="mw-file-description" title="Bathroom linoleum, U.S."><img alt="Bathroom linoleum, U.S." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Wallpaper_group-p4g-1.jpg/120px-Wallpaper_group-p4g-1.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Wallpaper_group-p4g-1.jpg/180px-Wallpaper_group-p4g-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Wallpaper_group-p4g-1.jpg/240px-Wallpaper_group-p4g-1.jpg 2x" data-file-width="1587" data-file-height="1587" /></a></span></div> <div class="gallerytext">Bathroom <a href="/wiki/Linoleum" title="Linoleum">linoleum</a>, U.S.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4g-2.jpg" class="mw-file-description" title="Painted porcelain, China"><img alt="Painted porcelain, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/118px-Wallpaper_group-p4g-2.jpg" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/177px-Wallpaper_group-p4g-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wallpaper_group-p4g-2.jpg/237px-Wallpaper_group-p4g-2.jpg 2x" data-file-width="1054" data-file-height="1069" /></a></span></div> <div class="gallerytext">Painted <a href="/wiki/Porcelain" title="Porcelain">porcelain</a>, China</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4g-3.jpg" class="mw-file-description" title="Fly screen, U.S."><img alt="Fly screen, U.S." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Wallpaper_group-p4g-3.jpg/116px-Wallpaper_group-p4g-3.jpg" decoding="async" width="116" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Wallpaper_group-p4g-3.jpg/175px-Wallpaper_group-p4g-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Wallpaper_group-p4g-3.jpg/233px-Wallpaper_group-p4g-3.jpg 2x" data-file-width="434" data-file-height="447" /></a></span></div> <div class="gallerytext">Fly screen, U.S.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p4g-4.jpg" class="mw-file-description" title="Painting, China"><img alt="Painting, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Wallpaper_group-p4g-4.jpg/119px-Wallpaper_group-p4g-4.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Wallpaper_group-p4g-4.jpg/178px-Wallpaper_group-p4g-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Wallpaper_group-p4g-4.jpg/237px-Wallpaper_group-p4g-4.jpg 2x" data-file-width="1068" data-file-height="1080" /></a></span></div> <div class="gallerytext">Painting, China</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_44-h01.png" class="mw-file-description" title="one of the colorings of the snub square tiling (see also at pg)"><img alt="one of the colorings of the snub square tiling (see also at pg)" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Uniform_tiling_44-h01.png/120px-Uniform_tiling_44-h01.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Uniform_tiling_44-h01.png/180px-Uniform_tiling_44-h01.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Uniform_tiling_44-h01.png/240px-Uniform_tiling_44-h01.png 2x" data-file-width="618" data-file-height="617" /></a></span></div> <div class="gallerytext">one of the colorings of the <a href="/wiki/Snub_square_tiling" title="Snub square tiling">snub square tiling</a> (see also at <i><b>pg</b></i>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p4g.svg" class="mw-file-description" title="4 co-uniform tiling (fractalization of snub square tiling)"><img alt="4 co-uniform tiling (fractalization of snub square tiling)" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Co-Uniform_Wallpaper_Example_p4g.svg/120px-Co-Uniform_Wallpaper_Example_p4g.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Co-Uniform_Wallpaper_Example_p4g.svg/180px-Co-Uniform_Wallpaper_Example_p4g.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Co-Uniform_Wallpaper_Example_p4g.svg/240px-Co-Uniform_Wallpaper_Example_p4g.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">4 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling (<a href="/wiki/Euclidean_tilings_by_convex_regular_polygons#Fractalizing_k-uniform_tilings" title="Euclidean tilings by convex regular polygons">fractalization</a> of snub square tiling)</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p3_(333)"><span id="Group_p3_.28333.29"></span><span class="anchor" id="Group_p3"></span> Group <i>p</i>3 (333)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=24" title="Edit section: Group p3 (333)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/SymBlend_p3.svg/220px-SymBlend_p3.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/SymBlend_p3.svg/330px-SymBlend_p3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/SymBlend_p3.svg/440px-SymBlend_p3.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>3</b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Wallpaper_group_diagram_p3.svg/220px-Wallpaper_group_diagram_p3.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Wallpaper_group_diagram_p3.svg/330px-Wallpaper_group_diagram_p3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Wallpaper_group_diagram_p3.svg/440px-Wallpaper_group_diagram_p3.svg.png 2x" data-file-width="744" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>3</b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:blue;">333</span></li> <li>Coxeter notation: [(3,3,3)]<sup>+</sup> or [3<sup>[3]</sup>]<sup>+</sup></li> <li>Lattice: hexagonal</li> <li>Point group: C<sub>3</sub></li> <li>The group <b><i>p</i>3</b> has three different rotation centres of order three (120°), but no reflections or glide reflections.</li></ul> <p>Imagine a <a href="/wiki/Tessellation" title="Tessellation">tessellation</a> of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal it is <b><i>p</i>6</b>, if they are each other's mirror image it is <b><i>p</i>31<i>m</i></b>, if they are both symmetric it is <b><i>p</i>3<i>m</i>1</b>; if two of the three apply then the third also, and it is <b><i>p</i>6<i>m</i></b>). For a given image, three of these tessellations are possible, each with rotation centres as vertices, i.e. for any tessellation two shifts are possible. In terms of the image: the vertices can be the red, the blue or the green triangles. </p><p>Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six it is <b><i>p</i>6</b>, if they are symmetric with respect to the main diagonals it is <b><i>p</i>31<i>m</i></b>, if they are symmetric with respect to lines perpendicular to the sides it is <b><i>p</i>3<i>m</i>1</b>; if two of the three apply then the third also, it is <b><i>p</i>6<i>m</i></b>). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles. </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>3</dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP3.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/WallpaperP3.GIF/120px-WallpaperP3.GIF" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/WallpaperP3.GIF/180px-WallpaperP3.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/WallpaperP3.GIF/240px-WallpaperP3.GIF 2x" data-file-width="250" data-file-height="250" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_33336.svg" class="mw-file-description" title="Snub trihexagonal tiling (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image"><img alt="Snub trihexagonal tiling (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Tile_33336.svg/120px-Tile_33336.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Tile_33336.svg/180px-Tile_33336.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Tile_33336.svg/240px-Tile_33336.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Snub_trihexagonal_tiling" title="Snub trihexagonal tiling">Snub trihexagonal tiling</a> (ignoring the colors: <b><i>p</i>6</b>); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p3-1.jpg" class="mw-file-description" title="Street pavement in Zakopane, Poland"><img alt="Street pavement in Zakopane, Poland" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Wallpaper_group-p3-1.jpg/120px-Wallpaper_group-p3-1.jpg" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Wallpaper_group-p3-1.jpg/180px-Wallpaper_group-p3-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Wallpaper_group-p3-1.jpg/240px-Wallpaper_group-p3-1.jpg 2x" data-file-width="1000" data-file-height="960" /></a></span></div> <div class="gallerytext">Street pavement in <a href="/wiki/Zakopane" title="Zakopane">Zakopane</a>, <a href="/wiki/Poland" title="Poland">Poland</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Alhambra-p3-closeup.jpg" class="mw-file-description" title="Wall tiling in the Alhambra, Spain (and the whole wall); ignoring all colors this is p3 (ignoring only star colors it is p1)"><img alt="Wall tiling in the Alhambra, Spain (and the whole wall); ignoring all colors this is p3 (ignoring only star colors it is p1)" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Alhambra-p3-closeup.jpg/120px-Alhambra-p3-closeup.jpg" decoding="async" width="120" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Alhambra-p3-closeup.jpg/180px-Alhambra-p3-closeup.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Alhambra-p3-closeup.jpg/240px-Alhambra-p3-closeup.jpg 2x" data-file-width="809" data-file-height="680" /></a></span></div> <div class="gallerytext">Wall tiling in the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a>, <a href="/wiki/Spain" title="Spain">Spain</a> (and the <a href="/wiki/File:Alhambra-p3-wall.jpg" title="File:Alhambra-p3-wall.jpg">whole wall</a>); ignoring all colors this is <b><i>p</i>3</b> (ignoring only star colors it is <a href="#Group_p1"><b><i>p</i>1</b></a>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p3.svg" class="mw-file-description" title="6 co-uniform tiling, each rotation point surrounded by a 3-fold cluster"><img alt="6 co-uniform tiling, each rotation point surrounded by a 3-fold cluster" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Co-Uniform_Wallpaper_Example_p3.svg/120px-Co-Uniform_Wallpaper_Example_p3.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Co-Uniform_Wallpaper_Example_p3.svg/180px-Co-Uniform_Wallpaper_Example_p3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Co-Uniform_Wallpaper_Example_p3.svg/240px-Co-Uniform_Wallpaper_Example_p3.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">6 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling, each rotation point surrounded by a 3-fold cluster</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p3m1_(*333)"><span id="Group_p3m1_.28.2A333.29"></span><span class="anchor" id="Group_p3m1"></span> Group <i>p</i>3<i>m</i>1 (*333)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=25" title="Edit section: Group p3m1 (*333)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p3m1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/SymBlend_p3m1.svg/220px-SymBlend_p3m1.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/SymBlend_p3m1.svg/330px-SymBlend_p3m1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/SymBlend_p3m1.svg/440px-SymBlend_p3m1.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>3<i>m</i>1</b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p3m1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Wallpaper_group_diagram_p3m1.svg/220px-Wallpaper_group_diagram_p3m1.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Wallpaper_group_diagram_p3m1.svg/330px-Wallpaper_group_diagram_p3m1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Wallpaper_group_diagram_p3m1.svg/440px-Wallpaper_group_diagram_p3m1.svg.png 2x" data-file-width="744" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>3<i>m</i>1</b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:red;">*333</span></li> <li>Coxeter notation: [(3,3,3)] or [3<sup>[3]</sup>]</li> <li>Lattice: hexagonal</li> <li>Point group: D<sub>3</sub></li> <li>The group <b><i>p</i>3<i>m</i>1</b> has three different rotation centres of order three (120°). It has reflections in the three sides of an equilateral triangle. The centre of every rotation lies on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.</li></ul> <p>Like for <b><a href="#Group_p3"><i>p</i>3</a></b>, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the blue or the green triangles. </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>3<i>m</i>1</dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_3,6.svg" class="mw-file-description" title="Triangular tiling (ignoring colors: p6m)"><img alt="Triangular tiling (ignoring colors: p6m)" 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> (ignoring colors: <b><i>p</i>6<i>m</i></b>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_6,3.svg" class="mw-file-description" title="Hexagonal tiling (ignoring colors: p6m)"><img alt="Hexagonal tiling (ignoring colors: p6m)" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Tile_6%2C3.svg/120px-Tile_6%2C3.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Tile_6%2C3.svg/180px-Tile_6%2C3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Tile_6%2C3.svg/240px-Tile_6%2C3.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">Hexagonal tiling</a> (ignoring colors: <b><i>p</i>6<i>m</i></b>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_3bb.svg" class="mw-file-description" title="Truncated hexagonal tiling (ignoring colors: p6m)"><img alt="Truncated hexagonal tiling (ignoring colors: p6m)" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Tile_3bb.svg/120px-Tile_3bb.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Tile_3bb.svg/180px-Tile_3bb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Tile_3bb.svg/240px-Tile_3bb.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Truncated_hexagonal_tiling" title="Truncated hexagonal tiling">Truncated hexagonal tiling</a> (ignoring colors: <b><i>p</i>6<i>m</i></b>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p3m1-1.jpg" class="mw-file-description" title="Persian glazed tile (ignoring colors: p6m)"><img alt="Persian glazed tile (ignoring colors: p6m)" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Wallpaper_group-p3m1-1.jpg/98px-Wallpaper_group-p3m1-1.jpg" decoding="async" width="98" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Wallpaper_group-p3m1-1.jpg/147px-Wallpaper_group-p3m1-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Wallpaper_group-p3m1-1.jpg/196px-Wallpaper_group-p3m1-1.jpg 2x" data-file-width="1077" data-file-height="1317" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_Empire" class="mw-redirect" title="Persian Empire">Persian</a> <a href="/wiki/Glaze_(painting_technique)" title="Glaze (painting technique)">glazed</a> tile (ignoring colors: <b><i>p</i>6<i>m</i></b>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p3m1-3.jpg" class="mw-file-description" title="Persian ornament"><img alt="Persian ornament" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Wallpaper_group-p3m1-3.jpg/120px-Wallpaper_group-p3m1-3.jpg" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Wallpaper_group-p3m1-3.jpg/180px-Wallpaper_group-p3m1-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Wallpaper_group-p3m1-3.jpg/240px-Wallpaper_group-p3m1-3.jpg 2x" data-file-width="1058" data-file-height="1028" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_Empire" class="mw-redirect" title="Persian Empire">Persian</a> ornament</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p3m1-2.jpg" class="mw-file-description" title="Painting, China (see detailed image)"><img alt="Painting, China (see detailed image)" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Wallpaper_group-p3m1-2.jpg/120px-Wallpaper_group-p3m1-2.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Wallpaper_group-p3m1-2.jpg/180px-Wallpaper_group-p3m1-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Wallpaper_group-p3m1-2.jpg/240px-Wallpaper_group-p3m1-2.jpg 2x" data-file-width="1075" data-file-height="1075" /></a></span></div> <div class="gallerytext">Painting, <a href="/wiki/China" title="China">China</a> (see detailed image)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p3m1.svg" class="mw-file-description" title="6 co-uniform tiling (smallest one containing 3 different 3-clusters)"><img alt="6 co-uniform tiling (smallest one containing 3 different 3-clusters)" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Co-Uniform_Wallpaper_Example_p3m1.svg/120px-Co-Uniform_Wallpaper_Example_p3m1.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Co-Uniform_Wallpaper_Example_p3m1.svg/180px-Co-Uniform_Wallpaper_Example_p3m1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Co-Uniform_Wallpaper_Example_p3m1.svg/240px-Co-Uniform_Wallpaper_Example_p3m1.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">6 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling (smallest one containing 3 different 3-clusters)</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p31m_(3*3)"><span id="Group_p31m_.283.2A3.29"></span><span class="anchor" id="Group_p31m"></span> Group <i>p</i>31<i>m</i> (3*3)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=26" title="Edit section: Group p31m (3*3)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p31m.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/SymBlend_p31m.svg/220px-SymBlend_p31m.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/SymBlend_p31m.svg/330px-SymBlend_p31m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/SymBlend_p31m.svg/440px-SymBlend_p31m.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>31<i>m</i></b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p31m.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Wallpaper_group_diagram_p31m.svg/220px-Wallpaper_group_diagram_p31m.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Wallpaper_group_diagram_p31m.svg/330px-Wallpaper_group_diagram_p31m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Wallpaper_group_diagram_p31m.svg/440px-Wallpaper_group_diagram_p31m.svg.png 2x" data-file-width="744" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>31<i>m</i></b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:blue;">3</span><span style="color:red;">*3</span></li> <li>Coxeter notation: [6,3<sup>+</sup>]</li> <li>Lattice: hexagonal</li> <li>Point group: D<sub>3</sub></li> <li>The group <b><i>p</i>31<i>m</i></b> has three different rotation centres of order three (120°), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does <i>not</i> lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.</li></ul> <p>Like for <b><i>p</i>3</b> and <b><i>p</i>3<i>m</i>1</b>, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices must be the red triangles, <i>not</i> the blue triangles. </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>31<i>m</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p31m-1.jpg" class="mw-file-description" title="Persian glazed tile"><img alt="Persian glazed tile" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Wallpaper_group-p31m-1.jpg/105px-Wallpaper_group-p31m-1.jpg" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Wallpaper_group-p31m-1.jpg/157px-Wallpaper_group-p31m-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Wallpaper_group-p31m-1.jpg/210px-Wallpaper_group-p31m-1.jpg 2x" data-file-width="1090" data-file-height="1246" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_empire" class="mw-redirect" title="Persian empire">Persian</a> <a href="/wiki/Glaze_(painting_technique)" title="Glaze (painting technique)">glazed</a> tile</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p31m-2.jpg" class="mw-file-description" title="Painted porcelain, China"><img alt="Painted porcelain, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Wallpaper_group-p31m-2.jpg/119px-Wallpaper_group-p31m-2.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Wallpaper_group-p31m-2.jpg/178px-Wallpaper_group-p31m-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Wallpaper_group-p31m-2.jpg/238px-Wallpaper_group-p31m-2.jpg 2x" data-file-width="1075" data-file-height="1086" /></a></span></div> <div class="gallerytext">Painted <a href="/wiki/Porcelain" title="Porcelain">porcelain</a>, <a href="/wiki/China" title="China">China</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p31m-3.jpg" class="mw-file-description" title="Painting, China"><img alt="Painting, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Wallpaper_group-p31m-3.jpg/120px-Wallpaper_group-p31m-3.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Wallpaper_group-p31m-3.jpg/180px-Wallpaper_group-p31m-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Wallpaper_group-p31m-3.jpg/240px-Wallpaper_group-p31m-3.jpg 2x" data-file-width="1074" data-file-height="1059" /></a></span></div> <div class="gallerytext">Painting, <a href="/wiki/China" title="China">China</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r2.svg" class="mw-file-description" title="Compact packing of two sizes of circle"><img alt="Compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2-d_dense_packing_r2.svg/104px-2-d_dense_packing_r2.svg.png" decoding="async" width="104" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2-d_dense_packing_r2.svg/156px-2-d_dense_packing_r2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2-d_dense_packing_r2.svg/208px-2-d_dense_packing_r2.svg.png 2x" data-file-width="1180" data-file-height="1362" /></a></span></div> <div class="gallerytext">Compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p31m.svg" class="mw-file-description" title="4 co-uniform tiling"><img alt="4 co-uniform tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Co-Uniform_Wallpaper_Example_p31m.svg/120px-Co-Uniform_Wallpaper_Example_p31m.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Co-Uniform_Wallpaper_Example_p31m.svg/180px-Co-Uniform_Wallpaper_Example_p31m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Co-Uniform_Wallpaper_Example_p31m.svg/240px-Co-Uniform_Wallpaper_Example_p31m.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">4 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p6_(632)"><span id="Group_p6_.28632.29"></span><span class="anchor" id="Group_p6"></span> Group <i>p</i>6 (632)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=27" title="Edit section: Group p6 (632)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/SymBlend_p6.svg/220px-SymBlend_p6.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/SymBlend_p6.svg/330px-SymBlend_p6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/SymBlend_p6.svg/440px-SymBlend_p6.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>6</b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Wallpaper_group_diagram_p6.svg/220px-Wallpaper_group_diagram_p6.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Wallpaper_group_diagram_p6.svg/330px-Wallpaper_group_diagram_p6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Wallpaper_group_diagram_p6.svg/440px-Wallpaper_group_diagram_p6.svg.png 2x" data-file-width="744" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>6</b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:blue;">632</span></li> <li>Coxeter notation: [6,3]<sup>+</sup></li> <li>Lattice: hexagonal</li> <li>Point group: C<sub>6</sub></li> <li>The group <b><i>p</i>6</b> has one rotation centre of order six (60°); two rotation centres of order three (120°), which are each other's images under a rotation of 60°; and three rotation centres of order two (180°) which are also each other's images under a rotation of 60°. It has no reflections or glide reflections.</li></ul> <p>A pattern with this symmetry can be looked upon as a <a href="/wiki/Tessellation" title="Tessellation">tessellation</a> of the plane with equal triangular tiles with <a href="/wiki/Symmetry_group#Two_dimensions" title="Symmetry group">C<sub>3</sub></a> symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C<sub>6</sub> symmetry (with the edges of the tiles not necessarily part of the pattern). </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>6</dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP6.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/WallpaperP6.GIF/120px-WallpaperP6.GIF" decoding="async" width="120" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/WallpaperP6.GIF/180px-WallpaperP6.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/2/26/WallpaperP6.GIF 2x" data-file-width="238" data-file-height="219" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg" class="mw-file-description" title="Regular polygons"><img alt="Regular polygons" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg/120px-A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg/180px-A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg/240px-A_periodic_tiling_by_regular_hexagons_and_equilateral_triangles.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div> <div class="gallerytext"><a href="/wiki/Tiling_by_regular_polygons" class="mw-redirect" title="Tiling by regular polygons">Regular polygons</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6-1.jpg" class="mw-file-description" title="Wall panelling, the Alhambra, Spain"><img alt="Wall panelling, the Alhambra, Spain" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Wallpaper_group-p6-1.jpg/120px-Wallpaper_group-p6-1.jpg" decoding="async" width="120" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Wallpaper_group-p6-1.jpg/180px-Wallpaper_group-p6-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Wallpaper_group-p6-1.jpg/240px-Wallpaper_group-p6-1.jpg 2x" data-file-width="1612" data-file-height="1152" /></a></span></div> <div class="gallerytext">Wall panelling, the <a href="/wiki/Alhambra" title="Alhambra">Alhambra</a>, <a href="/wiki/Spain" title="Spain">Spain</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6-2.jpg" class="mw-file-description" title="Persian ornament"><img alt="Persian ornament" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Wallpaper_group-p6-2.jpg/118px-Wallpaper_group-p6-2.jpg" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Wallpaper_group-p6-2.jpg/178px-Wallpaper_group-p6-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Wallpaper_group-p6-2.jpg/237px-Wallpaper_group-p6-2.jpg 2x" data-file-width="1022" data-file-height="1036" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_Empire" class="mw-redirect" title="Persian Empire">Persian</a> ornament</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Floret_Pentagonal_with_RPs.svg" class="mw-file-description" title="Floret pentagonal tiling"><img alt="Floret pentagonal tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Floret_Pentagonal_with_RPs.svg/120px-Floret_Pentagonal_with_RPs.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Floret_Pentagonal_with_RPs.svg/180px-Floret_Pentagonal_with_RPs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Floret_Pentagonal_with_RPs.svg/240px-Floret_Pentagonal_with_RPs.svg.png 2x" data-file-width="304" data-file-height="304" /></a></span></div> <div class="gallerytext"><a href="/wiki/Floret_pentagonal_tiling" class="mw-redirect" title="Floret pentagonal tiling">Floret pentagonal tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p6.svg" class="mw-file-description" title="7 co-uniform tiling with horizontal and 60° translations"><img alt="7 co-uniform tiling with horizontal and 60° translations" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Co-Uniform_Wallpaper_Example_p6.svg/120px-Co-Uniform_Wallpaper_Example_p6.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Co-Uniform_Wallpaper_Example_p6.svg/180px-Co-Uniform_Wallpaper_Example_p6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Co-Uniform_Wallpaper_Example_p6.svg/240px-Co-Uniform_Wallpaper_Example_p6.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">7 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling with horizontal and 60° translations</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Group_p6m_(*632)"><span id="Group_p6m_.28.2A632.29"></span><span class="anchor" id="Group_p6mm"></span> Group <i>p</i>6<i>m</i> (*632)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=28" title="Edit section: Group p6m (*632)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SymBlend_p6m.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SymBlend_p6m.svg/220px-SymBlend_p6m.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SymBlend_p6m.svg/330px-SymBlend_p6m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SymBlend_p6m.svg/440px-SymBlend_p6m.svg.png 2x" data-file-width="990" data-file-height="354" /></a><figcaption>Example and diagram for <b><i>p</i>6<i>m</i></b></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p6m.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wallpaper_group_diagram_p6m.svg/220px-Wallpaper_group_diagram_p6m.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wallpaper_group_diagram_p6m.svg/330px-Wallpaper_group_diagram_p6m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wallpaper_group_diagram_p6m.svg/440px-Wallpaper_group_diagram_p6m.svg.png 2x" data-file-width="744" data-file-height="425" /></a><figcaption>Cell structure for <b><i>p</i>6<i>m</i></b></figcaption></figure> <ul><li>Orbifold signature: <span style="color:red;">*632</span></li> <li>Coxeter notation: [6,3]</li> <li>Lattice: hexagonal</li> <li>Point group: D<sub>6</sub></li> <li>The group <b><i>p</i>6<i>m</i></b> has one rotation centre of order six (60°); it has two rotation centres of order three, which only differ by a rotation of 60° (or, equivalently, 180°), and three of order two, which only differ by a rotation of 60°. It has also reflections in six distinct directions. There are additional glide reflections in six distinct directions, whose axes are located halfway between adjacent parallel reflection axes.</li></ul> <p>A pattern with this symmetry can be looked upon as a <a href="/wiki/Tessellation" title="Tessellation">tessellation</a> of the plane with equal triangular tiles with <a href="/wiki/Dihedral_group#The_dihedral_group_as_symmetry_group_in_2D_and_rotation_group_in_3D" title="Dihedral group">D<sub>3</sub></a> symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D<sub>6</sub> symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a <a href="/wiki/Triangular_lattice" class="mw-redirect" title="Triangular lattice">triangular lattice</a> with or without connecting lines, and a <a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">hexagonal tiling</a> with one color for outlining the hexagons and one for the background. </p> <div style="clear:both;" class=""></div> <dl><dt>Examples of group <i>p</i>6<i>m</i></dt></dl> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:WallpaperP6M.GIF" class="mw-file-description" title="Computer generated"><img alt="Computer generated" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/WallpaperP6M.GIF/120px-WallpaperP6M.GIF" decoding="async" width="120" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/WallpaperP6M.GIF/180px-WallpaperP6M.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/WallpaperP6M.GIF/240px-WallpaperP6M.GIF 2x" data-file-width="284" data-file-height="254" /></a></span></div> <div class="gallerytext">Computer generated</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_3636.svg" class="mw-file-description" title="Trihexagonal tiling"><img alt="Trihexagonal tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Tile_3636.svg/120px-Tile_3636.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Tile_3636.svg/180px-Tile_3636.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Tile_3636.svg/240px-Tile_3636.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Trihexagonal_tiling" title="Trihexagonal tiling">Trihexagonal tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_3464.svg" class="mw-file-description" title="Small rhombitrihexagonal tiling"><img alt="Small rhombitrihexagonal tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Tile_3464.svg/120px-Tile_3464.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Tile_3464.svg/180px-Tile_3464.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Tile_3464.svg/240px-Tile_3464.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Small_rhombitrihexagonal_tiling" class="mw-redirect" title="Small rhombitrihexagonal tiling">Small rhombitrihexagonal tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tile_46b.svg" class="mw-file-description" title="Great rhombitrihexagonal tiling"><img alt="Great rhombitrihexagonal tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Tile_46b.svg/120px-Tile_46b.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Tile_46b.svg/180px-Tile_46b.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Tile_46b.svg/240px-Tile_46b.svg.png 2x" data-file-width="320" data-file-height="320" /></a></span></div> <div class="gallerytext"><a href="/wiki/Great_rhombitrihexagonal_tiling" class="mw-redirect" title="Great rhombitrihexagonal tiling">Great rhombitrihexagonal tiling</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6m-1.jpg" class="mw-file-description" title="Persian glazed tile"><img alt="Persian glazed tile" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Wallpaper_group-p6m-1.jpg/120px-Wallpaper_group-p6m-1.jpg" decoding="async" width="120" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Wallpaper_group-p6m-1.jpg/180px-Wallpaper_group-p6m-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Wallpaper_group-p6m-1.jpg/240px-Wallpaper_group-p6m-1.jpg 2x" data-file-width="1590" data-file-height="1073" /></a></span></div> <div class="gallerytext"><a href="/wiki/Persian_empire" class="mw-redirect" title="Persian empire">Persian</a> <a href="/wiki/Glaze_(painting_technique)" title="Glaze (painting technique)">glazed</a> tile</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6m-3.jpg" class="mw-file-description" title="Bronze vessel in Nimroud, Assyria"><img alt="Bronze vessel in Nimroud, Assyria" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Wallpaper_group-p6m-3.jpg/119px-Wallpaper_group-p6m-3.jpg" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Wallpaper_group-p6m-3.jpg/179px-Wallpaper_group-p6m-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Wallpaper_group-p6m-3.jpg/238px-Wallpaper_group-p6m-3.jpg 2x" data-file-width="1050" data-file-height="1057" /></a></span></div> <div class="gallerytext"><a href="/wiki/Bronze" title="Bronze">Bronze</a> vessel in <a href="/wiki/Nimroud" class="mw-redirect" title="Nimroud">Nimroud</a>, <a href="/wiki/Assyria" title="Assyria">Assyria</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6m-4.jpg" class="mw-file-description" title="Byzantine marble pavement, Rome"><img alt="Byzantine marble pavement, Rome" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Wallpaper_group-p6m-4.jpg/120px-Wallpaper_group-p6m-4.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Wallpaper_group-p6m-4.jpg/180px-Wallpaper_group-p6m-4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Wallpaper_group-p6m-4.jpg/240px-Wallpaper_group-p6m-4.jpg 2x" data-file-width="1134" data-file-height="1134" /></a></span></div> <div class="gallerytext"><a href="/wiki/Byzantine_art" title="Byzantine art">Byzantine</a> <a href="/wiki/Marble" title="Marble">marble</a> pavement, <a href="/wiki/Rome" title="Rome">Rome</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6m-5.jpg" class="mw-file-description" title="Painted porcelain, China"><img alt="Painted porcelain, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wallpaper_group-p6m-5.jpg/117px-Wallpaper_group-p6m-5.jpg" decoding="async" width="117" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wallpaper_group-p6m-5.jpg/176px-Wallpaper_group-p6m-5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wallpaper_group-p6m-5.jpg/235px-Wallpaper_group-p6m-5.jpg 2x" data-file-width="1055" data-file-height="1078" /></a></span></div> <div class="gallerytext">Painted <a href="/wiki/Porcelain" title="Porcelain">porcelain</a>, <a href="/wiki/China" title="China">China</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6m-6.jpg" class="mw-file-description" title="Painted porcelain, China"><img alt="Painted porcelain, China" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Wallpaper_group-p6m-6.jpg/120px-Wallpaper_group-p6m-6.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Wallpaper_group-p6m-6.jpg/180px-Wallpaper_group-p6m-6.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Wallpaper_group-p6m-6.jpg/240px-Wallpaper_group-p6m-6.jpg 2x" data-file-width="1064" data-file-height="1044" /></a></span></div> <div class="gallerytext">Painted <a href="/wiki/Porcelain" title="Porcelain">porcelain</a>, <a href="/wiki/China" title="China">China</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r5.svg" class="mw-file-description" title="Compact packing of two sizes of circle"><img alt="Compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/2-d_dense_packing_r5.svg/120px-2-d_dense_packing_r5.svg.png" decoding="async" width="120" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/2-d_dense_packing_r5.svg/180px-2-d_dense_packing_r5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/2-d_dense_packing_r5.svg/240px-2-d_dense_packing_r5.svg.png 2x" data-file-width="1199" data-file-height="1039" /></a></span></div> <div class="gallerytext">Compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2-d_dense_packing_r6.svg" class="mw-file-description" title="Another compact packing of two sizes of circle"><img alt="Another compact packing of two sizes of circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2-d_dense_packing_r6.svg/120px-2-d_dense_packing_r6.svg.png" decoding="async" width="120" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2-d_dense_packing_r6.svg/180px-2-d_dense_packing_r6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2-d_dense_packing_r6.svg/240px-2-d_dense_packing_r6.svg.png 2x" data-file-width="1042" data-file-height="902" /></a></span></div> <div class="gallerytext">Another compact packing of two sizes of circle</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Co-Uniform_Wallpaper_Example_p6m.svg" class="mw-file-description" title="7 co-uniform tiling"><img alt="7 co-uniform tiling" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Co-Uniform_Wallpaper_Example_p6m.svg/120px-Co-Uniform_Wallpaper_Example_p6m.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Co-Uniform_Wallpaper_Example_p6m.svg/180px-Co-Uniform_Wallpaper_Example_p6m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Co-Uniform_Wallpaper_Example_p6m.svg/240px-Co-Uniform_Wallpaper_Example_p6m.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="gallerytext">7 <a href="/wiki/Planigon" title="Planigon">co-uniform</a> tiling</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group-p6m-2.jpg" class="mw-file-description" title="King's dress, Khorsabad, Assyria; this is almost p6m (ignoring inner parts of flowers, which make it cmm)"><img alt="King's dress, Khorsabad, Assyria; this is almost p6m (ignoring inner parts of flowers, which make it cmm)" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-p6m-2.jpg/118px-Wallpaper_group-p6m-2.jpg" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-p6m-2.jpg/178px-Wallpaper_group-p6m-2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group-p6m-2.jpg/237px-Wallpaper_group-p6m-2.jpg 2x" data-file-width="1042" data-file-height="1056" /></a></span></div> <div class="gallerytext">King's dress, <a href="/wiki/Khorsabad" class="mw-redirect" title="Khorsabad">Khorsabad</a>, <a href="/wiki/Assyria" title="Assyria">Assyria</a>; this is almost <b><i>p</i>6<i>m</i></b> (ignoring inner parts of flowers, which make it <a href="#Group_c2mm"><i><b>cmm</b></i></a>)</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Lattice_types">Lattice types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=29" title="Edit section: Lattice types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are five <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> types or <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattices</a>, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. </p> <ul><li>In the 5 cases of rotational symmetry of order 3 or 6, the unit cell consists of two equilateral triangles (hexagonal lattice, itself <b><i>p</i>6<i>m</i></b>). They form a rhombus with angles 60° and 120°.</li> <li>In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself <b><i>p</i>4<i>m</i></b>).</li> <li>In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself <i><b>pmm</b></i>). It may also be interpreted as a centered rhombic lattice. Special cases: square.</li> <li>In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself <i><b>cmm</b></i>). It may also be interpreted as a centered rectangular lattice. Special cases: square, hexagonal unit cell.</li> <li>In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic or oblique lattice, itself <b><i>p</i>2</b>). Special cases: rectangle, square, rhombus, hexagonal unit cell.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Symmetry_groups">Symmetry groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=30" title="Edit section: Symmetry groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The actual <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers. </p><p>The numbers of <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> are: </p> <ul><li>6 for <b><i>p</i>2</b></li> <li>5 for <i><b>pmm</b></i>, <i><b>pmg</b></i>, <i><b>pgg</b></i>, and <i><b>cmm</b></i></li> <li>4 for the rest.</li></ul> <p>However, within each wallpaper group, all symmetry groups are algebraically isomorphic. </p><p>Some symmetry group isomorphisms: </p> <ul><li><b><i>p</i>1</b>: <b>Z</b><sup>2</sup></li> <li><i><b>pm</b></i>: <b>Z</b> × <a href="/wiki/Dihedral_group#Generalizations" title="Dihedral group">D<sub>∞</sub></a></li> <li><i><b>pmm</b></i>: D<sub>∞</sub> × D<sub>∞</sub>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Dependence_of_wallpaper_groups_on_transformations">Dependence of wallpaper groups on transformations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=31" title="Edit section: Dependence of wallpaper groups on transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The wallpaper group of a pattern is invariant under isometries and uniform <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaling</a> (<a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity transformations</a>).</li> <li>Translational symmetry is preserved under arbitrary bijective <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>.</li> <li>Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.</li> <li>Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changes <b><i>p</i>6<i>m</i></b>, <b><i>p</i>4<i>g</i></b>, and <b><i>p</i>3<i>m</i>1</b> into <i><b>cmm</b></i>, <b><i>p</i>3<i>m</i>1</b> into <i><b>cm</b></i>, and <b><i>p</i>4<i>m</i></b>, depending on direction of expansion/contraction, into <i><b>pmm</b></i> or <i><b>cmm</b></i>. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from <b><i>p</i>6<i>m</i></b> to <b><i>p</i>4<i>m</i></b>.</li></ul> <p>Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry. </p><p>Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change. </p><p>If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change. </p> <div class="mw-heading mw-heading2"><h2 id="Web_demo_and_software">Web demo and software</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=32" title="Edit section: Web demo and software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several software graphic tools will let you create 2D patterns using wallpaper symmetry groups. Usually you can edit the original tile and its copies in the entire pattern are updated automatically. </p> <ul><li><a rel="nofollow" class="external text" href="http://www.madpattern.com/">MadPattern</a>, a free set of Adobe Illustrator templates that support the 17 wallpaper groups</li> <li><a rel="nofollow" class="external text" href="http://www.peda.com/tess/">Tess</a>, a <a href="/wiki/Shareware" title="Shareware">shareware</a> tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.</li> <li><a rel="nofollow" class="external text" href="http://math.hws.edu/eck/js/symmetry/wallpaper.html">Wallpaper Symmetry</a> is a free online JavaScript drawing tool supporting the 17 groups. The <a rel="nofollow" class="external text" href="http://math.hws.edu/eck/js/symmetry/symmetry-info.html">main page</a> has an explanation of the wallpaper groups, as well as drawing tools and explanations for the other <a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">planar symmetry groups</a> as well.</li> <li><a rel="nofollow" class="external text" href="https://en.oiler.education/tales">TALES GAME</a>, a free software designed for educational purposes which includes the tessellation function.</li> <li><a rel="nofollow" class="external text" href="http://www.scienceu.com/geometry/handson/kali/">Kali</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181216031531/http://www.scienceu.com/geometry/handson/kali/">Archived</a> 2018-12-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, online graphical symmetry editor <a href="/wiki/Java_applet" title="Java applet">Java applet</a> (not supported by default in browsers).</li> <li><a rel="nofollow" class="external text" href="http://www.geometrygames.org/Kali/index.html">Kali</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201121143626/http://www.geometrygames.org/Kali/index.html">Archived</a> 2020-11-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, free downloadable Kali for Windows and Mac Classic.</li> <li><a href="/wiki/Inkscape" title="Inkscape">Inkscape</a>, a <a href="/wiki/Free_software" title="Free software">free</a> <a href="/wiki/Vector_graphics_editor" title="Vector graphics editor">vector graphics editor</a>, supports all 17 groups plus arbitrary scales, shifts, rotates, and color changes per row or per column, optionally randomized to a given degree. (See <a rel="nofollow" class="external autonumber" href="http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Tiles-Symmetries.html">[1]</a>)</li> <li><a rel="nofollow" class="external text" href="http://www.artlandia.com/products/SymmetryWorks/">SymmetryWorks</a> is a commercial plugin for <a href="/wiki/Adobe_Illustrator" title="Adobe Illustrator">Adobe Illustrator</a>, supports all 17 groups.</li> <li><a rel="nofollow" class="external text" href="https://eschersket.ch/">EscherSketch</a> is a free online JavaScript drawing tool supporting the 17 groups.</li> <li><a rel="nofollow" class="external text" href="https://repper.app/">Repper</a> is a commercial online drawing tool supporting the 17 groups plus a number of non-periodic tilings</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=Wallpaper_group&action=edit&section=33" title="Edit section: See also"><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"><span><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" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Wallpaper_group_diagrams" class="extiw" title="commons:Category:Wallpaper group diagrams">Wallpaper group diagrams</a></span>.</div></div> </div> <ul><li><a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">List of planar symmetry groups</a> (summary of this page)</li> <li><a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">Aperiodic tiling</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li> <li><a href="/wiki/Layer_group" title="Layer group">Layer group</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a></li> <li><a href="/wiki/Point_group" title="Point group">Point group</a></li> <li><a href="/wiki/Symmetry_groups_in_one_dimension" class="mw-redirect" title="Symmetry groups in one dimension">Symmetry groups in one dimension</a></li> <li><a href="/wiki/Tessellation" title="Tessellation">Tessellation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wallpaper_group&action=edit&section=34" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">E. Fedorov (1891) <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=umn.31951t00080576a;view=1up;seq=357">"Симметрія на плоскости"</a> (<i>Simmetrija na ploskosti</i>, Symmetry in the plane), <i>Записки Императорского С.-Петербургского минералогического общества</i> (<i>Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva</i>, Proceedings of the Imperial St. Petersburg Mineralogical Society), series 2, <b>28</b> : 345–390 (in Russian).</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"><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="CITEREFPólya1924" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/George_P%C3%B3lya" title="George Pólya">Pólya, George</a> (November 1924). "Über die Analogie der Kristallsymmetrie in der Ebene" [On the analog of crystal symmetry in the plane]. <i>Zeitschrift für Kristallographie</i> (in German). <b>60</b> (1–6): 278–282. <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.1924.60.1.278">10.1524/zkri.1924.60.1.278</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:102174323">102174323</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&rft.atitle=%C3%9Cber+die+Analogie+der+Kristallsymmetrie+in+der+Ebene&rft.volume=60&rft.issue=1%E2%80%936&rft.pages=278-282&rft.date=1924-11&rft_id=info%3Adoi%2F10.1524%2Fzkri.1924.60.1.278&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A102174323%23id-name%3DS2CID&rft.aulast=P%C3%B3lya&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWallpaper+group" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlarreich2013" class="citation web cs1">Klarreich, Erica (5 March 2013). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/how-to-make-impossible-wallpaper-20130305/">"How to Make Impossible Wallpaper"</a>. <i>Quanta Magazine</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-04-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=How+to+Make+Impossible+Wallpaper&rft.date=2013-03-05&rft.aulast=Klarreich&rft.aufirst=Erica&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fhow-to-make-impossible-wallpaper-20130305%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWallpaper+group" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRadaelli" class="citation book cs1">Radaelli, Paulo G. <i>Symmetry in Crystallography</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+in+Crystallography&rft.pub=Oxford+University+Press&rft.aulast=Radaelli&rft.aufirst=Paulo+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWallpaper+group" 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">If one thinks of the squares as the background, then one can see a simple patterns of rows of rhombuses.</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=Wallpaper_group&action=edit&section=35" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://search.library.wisc.edu/digital/ALXEMQRWNML2C48G"><i>The Grammar of Ornament</i></a> (1856), by <a href="/wiki/Owen_Jones_(architect)" title="Owen Jones (architect)">Owen Jones</a>. Many of the images in this article are from this book; it contains many more.</li> <li><a href="/wiki/John_H._Conway" class="mw-redirect" title="John H. Conway">John H. Conway</a> (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), <i>Groups, Combinatorics and Geometry</i>, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc. Lecture Notes Series <b>165</b>. Cambridge University Press, Cambridge. pp. 438–447</li> <li><a href="/wiki/John_H._Conway" class="mw-redirect" title="John H. Conway">John H. Conway</a>, Heidi Burgiel and <a href="/wiki/Chaim_Goodman-Strauss" title="Chaim Goodman-Strauss">Chaim Goodman-Strauss</a> (2008): <i><a href="/wiki/The_Symmetries_of_Things" title="The Symmetries of Things">The Symmetries of Things</a></i>. Worcester MA: A.K. Peters. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-220-5" title="Special:BookSources/1-56881-220-5">1-56881-220-5</a>.</li> <li><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a> and G. C. Shephard (1987): <i><a href="/wiki/Tilings_and_patterns" title="Tilings and patterns">Tilings and patterns</a></i>. New York: Freeman. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-1193-1" title="Special:BookSources/0-7167-1193-1">0-7167-1193-1</a>.</li> <li>Pattern Design, Lewis F. Day</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=Wallpaper_group&action=edit&section=36" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://it.iucr.org/A/">International Tables for Crystallography Volume A: Space-group symmetry</a> by the International Union of Crystallography</li> <li><a rel="nofollow" class="external text" href="http://www.clarku.edu/~djoyce/wallpaper/seventeen.html">The 17 plane symmetry groups</a> by David E. Joyce</li> <li><a rel="nofollow" class="external text" href="http://www.geom.uiuc.edu/education/math5337/Wallpaper/introduction.html">Introduction to wallpaper patterns</a> by Chaim Goodman-Strauss and Heidi Burgiel</li> <li><a rel="nofollow" class="external text" href="http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node12.html">Description</a> by Silvio Levy</li> <li><a rel="nofollow" class="external text" href="http://clowder.net/hop/17walppr/17walppr.html">Example tiling for each group, with dynamic demos of properties</a></li> <li><a rel="nofollow" class="external text" href="http://www.math.toronto.edu/~drorbn/Gallery/Symmetry/Tilings/Sanderson/index.html">Overview with example tiling for each group, by Brian Sanderson</a></li> <li><a rel="nofollow" class="external text" href="http://escher.epfl.ch/escher/">Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups</a></li> <li><a rel="nofollow" class="external text" href="http://www-viz.tamu.edu/faculty/ergun/research/artisticdepiction/symmetric/program/index.html">Burak, a Java applet for drawing symmetry groups.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090218071905/http://www-viz.tamu.edu/faculty/ergun/research/artisticdepiction/symmetric/program/index.html">Archived</a> 2009-02-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://math.hws.edu/eck/js/symmetry/wallpaper.html">A JavaScript app for drawing wallpaper patterns</a></li> <li><a rel="nofollow" class="external text" href="http://circle-pattern.kankeleit.de/">Circle-Pattern on Roman Mosaics in Greece</a></li> <li><a rel="nofollow" class="external text" href="http://faculty.ms.u-tokyo.ac.jp/users/urabe/pattrn/PatternE.html">Seventeen Kinds of Wallpaper Patterns</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171012000713/http://faculty.ms.u-tokyo.ac.jp/users/urabe/pattrn/PatternE.html">Archived</a> 2017-10-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> the 17 symmetries found in traditional Japanese patterns.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaloglou2002" class="citation web cs1">Baloglou, George (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180807113612/http://www.oswego.edu/~baloglou/103/seventeen.html">"An elementary, purely geometrical classification of the 17 planar crystallographic groups (wallpaper patterns)"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.oswego.edu/~baloglou/103/seventeen.html">the original</a> on 2018-08-07<span class="reference-accessdate">. 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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="Tessellation" style="font-size:114%;margin:0 4em"><a href="/wiki/Tessellation" title="Tessellation">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="Periodic" 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 class="mw-selflink selflink">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:2C_3_1979.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/2C_3_1979.JPG/90px-2C_3_1979.JPG" decoding="async" width="90" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/2C_3_1979.JPG/135px-2C_3_1979.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/2C_3_1979.JPG/180px-2C_3_1979.JPG 2x" data-file-width="1562" data-file-height="1688" /></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="Aperiodic" 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="Other" 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_type" 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="Mathematics_and_art" 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: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 class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematics_and_art" title="Special:EditPage/Template:Mathematics and art"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematics_and_art" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</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/Algorithm" title="Algorithm">Algorithm</a></li> <li><a href="/wiki/Catenary" title="Catenary">Catenary</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a></li> <li><a href="/wiki/Hyperboloid_structure" title="Hyperboloid structure">Hyperboloid structure</a></li> <li><a href="/wiki/Minimal_surface" title="Minimal surface">Minimal 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 href="/wiki/Tessellation" title="Tessellation">Tessellation</a></li> <li><a class="mw-selflink selflink">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/Tessellation" title="Tessellation">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/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" 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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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