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<span>Geometry</span> </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Topology</span> </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-sheeted_cover" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Two-sheeted_cover"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Two-sheeted cover</span> </div> </a> <ul id="toc-Two-sheeted_cover-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n-dimensional_torus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#n-dimensional_torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span><i>n</i>-dimensional torus</span> </div> </a> <button aria-controls="toc-n-dimensional_torus-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 <i>n</i>-dimensional torus subsection</span> </button> <ul id="toc-n-dimensional_torus-sublist" class="vector-toc-list"> <li id="toc-Configuration_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Configuration_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Configuration space</span> </div> </a> <ul id="toc-Configuration_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Flat_torus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flat_torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Flat torus</span> </div> </a> <button aria-controls="toc-Flat_torus-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 Flat torus subsection</span> </button> <ul id="toc-Flat_torus-sublist" class="vector-toc-list"> <li id="toc-Conformal_classification_of_flat_tori" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conformal_classification_of_flat_tori"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Conformal classification of flat tori</span> </div> </a> <ul id="toc-Conformal_classification_of_flat_tori-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Genus_g_surface" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Genus_g_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Genus <i>g</i> surface</span> </div> </a> <ul id="toc-Genus_g_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Toroidal_polyhedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Toroidal_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Toroidal polyhedra</span> </div> </a> <ul id="toc-Toroidal_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Automorphisms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Automorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Automorphisms</span> </div> </a> <ul id="toc-Automorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coloring_a_torus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coloring_a_torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Coloring a torus</span> </div> </a> <ul id="toc-Coloring_a_torus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-de_Bruijn_torus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#de_Bruijn_torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>de Bruijn torus</span> </div> </a> <ul id="toc-de_Bruijn_torus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cutting_a_torus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cutting_a_torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Cutting a torus</span> </div> </a> <ul id="toc-Cutting_a_torus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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">Torus</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 69 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-69" 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">69 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Torus" title="Torus – Afrikaans" lang="af" hreflang="af" data-title="Torus" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D8%B1%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="طارة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="طارة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B9%D5%B8%D6%80" title="Թոր – Western Armenian" lang="hyw" hreflang="hyw" data-title="Թոր" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Tor_(h%C9%99nd%C9%99si_fiqur)" title="Tor (həndəsi fiqur) – Azerbaijani" lang="az" hreflang="az" data-title="Tor (həndəsi fiqur)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F)" title="Тор (геаметрыя) – Belarusian" lang="be" hreflang="be" data-title="Тор (геаметрыя)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Тор (геометрия) – Bulgarian" lang="bg" hreflang="bg" data-title="Тор (геометрия)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Torus" title="Torus – Bosnian" lang="bs" hreflang="bs" data-title="Torus" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tor_(geometria)" title="Tor (geometria) – Catalan" lang="ca" hreflang="ca" data-title="Tor (geometria)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%C3%A7%D0%B8%D0%B9)" title="Тор (çий) – Chuvash" lang="cv" hreflang="cv" data-title="Тор (çий)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Torus" title="Torus – Czech" lang="cs" hreflang="cs" data-title="Torus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Toru_(giumitria)" title="Toru (giumitria) – Corsican" lang="co" hreflang="co" data-title="Toru (giumitria)" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Torus" title="Torus – Danish" lang="da" hreflang="da" data-title="Torus" 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/Torus" title="Torus – German" lang="de" hreflang="de" data-title="Torus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Toor" title="Toor – Estonian" lang="et" hreflang="et" data-title="Toor" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%8C%CF%81%CE%BF%CF%82" title="Τόρος – Greek" lang="el" hreflang="el" data-title="Τόρος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Toro_(geometr%C3%ADa)" title="Toro (geometría) – Spanish" lang="es" hreflang="es" data-title="Toro (geometría)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Toro_(geometrio)" title="Toro (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Toro (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Toru" title="Toru – Basque" lang="eu" hreflang="eu" data-title="Toru" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%A8%D8%B1%D9%87" title="چنبره – Persian" lang="fa" hreflang="fa" data-title="چنبره" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tore" title="Tore – French" lang="fr" hreflang="fr" data-title="Tore" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/T%C3%B3ras" title="Tóras – Irish" lang="ga" hreflang="ga" data-title="Tóras" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Toro_(xeometr%C3%ADa_e_topolox%C3%ADa)" title="Toro (xeometría e topoloxía) – Galician" lang="gl" hreflang="gl" data-title="Toro (xeometría e topoloxía)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Mbiriga_(mathabu)" title="Mbiriga (mathabu) – Kikuyu" lang="ki" hreflang="ki" data-title="Mbiriga (mathabu)" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80" title="Тор – Kalmyk" lang="xal" hreflang="xal" data-title="Тор" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%ED%99%98%EB%A9%B4" title="원환면 – Korean" lang="ko" hreflang="ko" data-title="원환면" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%B8%D6%80_(%D5%B4%D5%A1%D5%AF%D5%A5%D6%80%D6%87%D5%B8%D6%82%D5%B5%D5%A9)" title="Տոր (մակերևույթ) – Armenian" lang="hy" hreflang="hy" data-title="Տոր (մակերևույթ)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9F%E0%A5%89%E0%A4%B0%E0%A4%B8" title="टॉरस – Hindi" lang="hi" hreflang="hi" data-title="टॉरस" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Torus" title="Torus – Croatian" lang="hr" hreflang="hr" data-title="Torus" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Toro" title="Toro – Ido" lang="io" hreflang="io" data-title="Toro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Torus" title="Torus – Indonesian" lang="id" hreflang="id" data-title="Torus" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Toro" title="Toro – Interlingua" lang="ia" hreflang="ia" data-title="Toro" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Toro_(geometria)" title="Toro (geometria) – Italian" lang="it" hreflang="it" data-title="Toro (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%95%D7%A8%D7%95%D7%A1" title="טורוס – Hebrew" lang="he" hreflang="he" data-title="טורוס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98_(%E1%83%92%E1%83%94%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90)" title="ტორი (გეომეტრია) – Georgian" lang="ka" hreflang="ka" data-title="ტორი (გეომეტრია)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Torus_(mathematica)" title="Torus (mathematica) – Latin" lang="la" hreflang="la" data-title="Torus (mathematica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Tors_(%C4%A3eometrija)" title="Tors (ģeometrija) – Latvian" lang="lv" hreflang="lv" data-title="Tors (ģeometrija)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Torus" title="Torus – Luxembourgish" lang="lb" hreflang="lb" data-title="Torus" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Toras_(geometrija)" title="Toras (geometrija) – Lithuanian" lang="lt" hreflang="lt" data-title="Toras (geometrija)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B3rusz" title="Tórusz – Hungarian" lang="hu" hreflang="hu" data-title="Tórusz" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Тор (геометрија) – Macedonian" lang="mk" hreflang="mk" data-title="Тор (геометрија)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9F%E0%B5%8B%E0%B4%B1%E0%B4%B8%E0%B5%8D" title="ടോറസ് – Malayalam" lang="ml" hreflang="ml" data-title="ടോറസ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Torus" title="Torus – Malay" lang="ms" hreflang="ms" data-title="Torus" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Torus" title="Torus – Dutch" lang="nl" hreflang="nl" data-title="Torus" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%88%E3%83%BC%E3%83%A9%E3%82%B9" title="トーラス – Japanese" lang="ja" hreflang="ja" data-title="トーラス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Ring_(Geometrii)" title="Ring (Geometrii) – Northern Frisian" lang="frr" hreflang="frr" data-title="Ring (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Torus" title="Torus – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Torus" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Torus" title="Torus – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Torus" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/T%C3%B2r_(geometria)" title="Tòr (geometria) – Occitan" lang="oc" hreflang="oc" data-title="Tòr (geometria)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tor_(jism)" title="Tor (jism) – Uzbek" lang="uz" hreflang="uz" data-title="Tor (jism)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Torus_(matematyka)" title="Torus (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Torus (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Toro_(topologia)" title="Toro (topologia) – Portuguese" lang="pt" hreflang="pt" data-title="Toro (topologia)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Tor" title="Tor – Romanian" lang="ro" hreflang="ro" data-title="Tor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%BF%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D1%8C)" title="Тор (поверхность) – Russian" lang="ru" hreflang="ru" data-title="Тор (поверхность)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Torusi" title="Torusi – Albanian" lang="sq" hreflang="sq" data-title="Torusi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Toru_(giometr%C3%ACa)" title="Toru (giometrìa) – Sicilian" lang="scn" hreflang="scn" data-title="Toru (giometrìa)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Torus" title="Torus – Simple English" lang="en-simple" hreflang="en-simple" data-title="Torus" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Torus_(geometria)" title="Torus (geometria) – Slovak" lang="sk" hreflang="sk" data-title="Torus (geometria)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Torus" title="Torus – Slovenian" lang="sl" hreflang="sl" data-title="Torus" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80%D1%83%D1%81" title="Торус – Serbian" lang="sr" hreflang="sr" data-title="Торус" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Torus" title="Torus – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Torus" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Torus" title="Torus – Finnish" lang="fi" hreflang="fi" data-title="Torus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Torus" title="Torus – Swedish" lang="sv" hreflang="sv" data-title="Torus" 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%89%E0%AE%B0%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="உருள்வளையம் – Tamil" lang="ta" hreflang="ta" data-title="உருள்வளையம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%AD%E0%B8%A3%E0%B8%B1%E0%B8%AA_(%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95)" title="ทอรัส (เรขาคณิต) – Thai" lang="th" hreflang="th" data-title="ทอรัส (เรขาคณิต)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Simit_(geometri)" title="Simit (geometri) – Turkish" lang="tr" hreflang="tr" data-title="Simit (geometri)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F)" title="Тор (геометрія) – Ukrainian" lang="uk" hreflang="uk" data-title="Тор (геометрія)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_xuy%E1%BA%BFn" title="Hình xuyến – Vietnamese" lang="vi" hreflang="vi" data-title="Hình xuyến" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8E%AF%E9%9D%A2" title="环面 – Wu" lang="wuu" hreflang="wuu" data-title="环面" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Doughnut-shaped surface of revolution</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Taurus_(disambiguation)" class="mw-redirect mw-disambig" title="Taurus (disambiguation)">Taurus</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical surface. For the volume, see <a href="/wiki/Solid_torus" title="Solid torus">Solid torus</a>. For other uses, see <a href="/wiki/Torus_(disambiguation)" class="mw-disambig" title="Torus (disambiguation)">Torus (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">For the meteorological research project, see <a href="/wiki/TORUS_Project" title="TORUS Project">TORUS Project</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tesseract_torus.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tesseract_torus.png/220px-Tesseract_torus.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tesseract_torus.png/330px-Tesseract_torus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tesseract_torus.png/440px-Tesseract_torus.png 2x" data-file-width="4000" data-file-height="2680" /></a><figcaption>A ring torus with a selection of circles on its surface</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ring_Torus_to_Degenerate_Torus_(Short).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Ring_Torus_to_Degenerate_Torus_%28Short%29.gif/220px-Ring_Torus_to_Degenerate_Torus_%28Short%29.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/8f/Ring_Torus_to_Degenerate_Torus_%28Short%29.gif 1.5x" data-file-width="240" data-file-height="180" /></a><figcaption>As the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally <a href="/wiki/Degeneracy_(mathematics)" title="Degeneracy (mathematics)">degenerates</a> into a double-covered <a href="/wiki/Sphere" title="Sphere">sphere</a>.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Torus_cycles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Torus_cycles.svg/220px-Torus_cycles.svg.png" decoding="async" width="220" height="335" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Torus_cycles.svg/330px-Torus_cycles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Torus_cycles.svg/440px-Torus_cycles.svg.png 2x" data-file-width="512" data-file-height="780" /></a><figcaption>A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>torus</b> (<abbr title="plural">pl.</abbr>: <b>tori</b> or <b>toruses</b>) is a <a href="/wiki/Surface_of_revolution" title="Surface of revolution">surface of revolution</a> generated by revolving a <a href="/wiki/Circle" title="Circle">circle</a> in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> one full revolution about an axis that is <a href="/wiki/Coplanarity" title="Coplanarity">coplanar</a> with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a <b>donut</b> or <b>doughnut</b>. </p><p>If the <a href="/wiki/Axis_of_revolution" class="mw-redirect" title="Axis of revolution">axis of revolution</a> does not touch the circle, the surface has a ring shape and is called a <b>torus of revolution</b>, also known as a <b>ring torus</b>. If the axis of revolution is <a href="/wiki/Tangent" title="Tangent">tangent</a> to the circle, the surface is a <b>horn torus</b>. If the axis of revolution passes twice through the circle, the surface is a <b><a href="/wiki/Lemon_(geometry)" title="Lemon (geometry)">spindle torus</a></b> (or <i>self-crossing torus</i> or <i>self-intersecting torus</i>). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered <a href="/wiki/Sphere" title="Sphere">sphere</a>. If the revolved curve is not a circle, the surface is called a <i><a href="/wiki/Toroid" title="Toroid">toroid</a></i>, as in a square toroid. </p><p>Real-world objects that approximate a torus of revolution include <a href="/wiki/Swim_ring" title="Swim ring">swim rings</a>, <a href="/wiki/Inner_tube" title="Inner tube">inner tubes</a> and <a href="/wiki/Ringette_ring" class="mw-redirect" title="Ringette ring">ringette rings</a>. </p><p>A torus should not be confused with a <i><a href="/wiki/Solid_torus" title="Solid torus">solid torus</a></i>, which is formed by rotating a <a href="/wiki/Disk_(geometry)" class="mw-redirect" title="Disk (geometry)">disk</a>, rather than a circle, around an axis. A solid torus is a torus plus the <a href="/wiki/Volume" title="Volume">volume</a> inside the torus. Real-world objects that approximate a <i>solid torus</i> include <a href="/wiki/O-ring" title="O-ring">O-rings</a>, non-inflatable <a href="/wiki/Lifebuoy" title="Lifebuoy">lifebuoys</a>, ring <a href="/wiki/Doughnut" title="Doughnut">doughnuts</a>, and <a href="/wiki/Bagel" title="Bagel">bagels</a>. </p><p>In <a href="/wiki/Topology" title="Topology">topology</a>, a ring torus is <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> to the <a href="/wiki/Product_topology" title="Product topology">Cartesian product</a> of two <a href="/wiki/Circle" title="Circle">circles</a>: <span class="texhtml"><i>S</i><sup>1</sup> × <i>S</i><sup>1</sup></span>, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, but another way to do this is the Cartesian product of the <a href="/wiki/Embedding" title="Embedding">embedding</a> of <span class="texhtml"><i>S</i><sup>1</sup></span> in the plane with itself. This produces a geometric object called the <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a>, a surface in <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-space</a>. </p><p>In the field of <a href="/wiki/Topology" title="Topology">topology</a>, a torus is any topological space that is homeomorphic to a torus.<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> The surface of a coffee cup and a doughnut are both topological tori with <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> one. </p><p>An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="https://en.wiktionary.org/wiki/torus" class="extiw" title="wikt:torus">Torus</a></i> is a Latin word for "a round, swelling, elevation, protuberance". </p> <div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=2" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:222px;max-width:222px"><div class="trow"><div class="theader">Bottom-halves and<br />vertical cross-sections</div></div><div class="trow"><div class="tsingle" style="width:220px;max-width:220px"><div class="thumbimage" style="height:272px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Standard_torus-ring.png" class="mw-file-description"><img alt="ring" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Standard_torus-ring.png/218px-Standard_torus-ring.png" decoding="async" width="218" height="273" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Standard_torus-ring.png/327px-Standard_torus-ring.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Standard_torus-ring.png/436px-Standard_torus-ring.png 2x" data-file-width="1024" data-file-height="1280" /></a></span></div><div class="thumbcaption"><span class="texhtml"><i>R</i> > <i>r</i></span>: ring torus or anchor ring</div></div></div><div class="trow"><div class="tsingle" style="width:220px;max-width:220px"><div class="thumbimage" style="height:272px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Standard_torus-horn.png" class="mw-file-description"><img alt="horn" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Standard_torus-horn.png/218px-Standard_torus-horn.png" decoding="async" width="218" height="273" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Standard_torus-horn.png/327px-Standard_torus-horn.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Standard_torus-horn.png/436px-Standard_torus-horn.png 2x" data-file-width="1024" data-file-height="1280" /></a></span></div><div class="thumbcaption"><span class="texhtml"><i>R</i>=<i>r</i></span>: horn torus</div></div></div><div class="trow"><div class="tsingle" style="width:220px;max-width:220px"><div class="thumbimage" style="height:272px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Standard_torus-spindle.png" class="mw-file-description"><img alt="spindle" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Standard_torus-spindle.png/218px-Standard_torus-spindle.png" decoding="async" width="218" height="273" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Standard_torus-spindle.png/327px-Standard_torus-spindle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Standard_torus-spindle.png/436px-Standard_torus-spindle.png 2x" data-file-width="1024" data-file-height="1280" /></a></span></div><div class="thumbcaption"><span class="texhtml"><i>R</i> < <i>r</i></span>: self-intersecting spindle torus</div></div></div></div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Toroidal_coord.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/250px-Toroidal_coord.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/330px-Toroidal_coord.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/500px-Toroidal_coord.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>Poloidal direction (red arrow) and toroidal direction (blue arrow)</figcaption></figure> <p>A torus of revolution in 3-space can be <a href="/wiki/Parametric_equation" title="Parametric equation">parametrized</a> as:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\sin \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\sin \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\cos \theta \\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\sin \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\sin \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\cos \theta \\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18b1a6a8ce733d6dd748f37ebcfe97f9b4d4854" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.222ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\sin \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\sin \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\cos \theta \\\end{aligned}}}" /></span> using angular coordinates <span class="texhtml"><i>θ</i></span>, <span class="texhtml"><i>φ</i> ∈ [0, 2π)</span>, representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the <i>major radius</i> <span class="texhtml"><i>R</i></span> is the distance from the center of the tube to the center of the torus and the <i>minor radius</i> <span class="texhtml"><i>r</i></span> is the radius of the tube.<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> </p><p>The ratio <span class="texhtml"><i>R</i>/<i>r</i></span> is called the <i><a href="/wiki/Aspect_ratio" title="Aspect ratio">aspect ratio</a></i> of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2. </p><p>An <a href="/wiki/Implicit_function" title="Implicit function">implicit</a> equation in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> for a torus radially symmetric about the z-<a href="/wiki/Coordinate_axis" class="mw-redirect" title="Coordinate axis">axis</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>R</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bed37611ee14e52c7fc94fb7753b9a8d1331edb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.384ex; height:3.676ex;" alt="{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.}" /></span> </p><p>Algebraically eliminating the <a href="/wiki/Square_root" title="Square root">square root</a> gives a <a href="/wiki/Quartic_equation" title="Quartic equation">quartic equation</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45915d1998a49b426ecffd22205920cf86772ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.279ex; height:3.843ex;" alt="{\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).}" /></span> </p><p>The three classes of standard tori correspond to the three possible aspect ratios between <span class="texhtml mvar" style="font-style:italic;">R</span> and <span class="texhtml mvar" style="font-style:italic;">r</span>: </p> <ul><li>When <span class="texhtml"><i>R</i> > <i>r</i></span>, the surface will be the familiar ring torus or anchor ring.</li> <li><span class="texhtml"><i>R</i> = <i>r</i></span> corresponds to the horn torus, which in effect is a torus with no "hole".</li> <li><span class="texhtml"><i>R</i> < <i>r</i></span> describes the self-intersecting spindle torus; its inner shell is a <i><a href="/wiki/Lemon_(geometry)" title="Lemon (geometry)">lemon</a></i> and its outer shell is an <i><a href="/wiki/Apple_(geometry)" class="mw-redirect" title="Apple (geometry)">apple</a></i>.</li> <li>When <span class="texhtml"><i>R</i> = 0</span>, the torus degenerates to the sphere radius <span class="texhtml"><i>r</i></span>.</li> <li>When <span class="texhtml"><i>r</i> = 0</span>, the torus degenerates to the circle radius <span class="texhtml"><i>R</i></span>.</li></ul> <p>When <span class="texhtml"><i>R</i> ≥ <i>r</i></span>, the <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>R</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4162026c6785270428ced3c583affaffceb3883" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.738ex; height:3.676ex;" alt="{\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}}" /></span> of this torus is <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> (and, hence, homeomorphic) to a <a href="/wiki/Cartesian_product" title="Cartesian product">product</a> of a <a href="/wiki/Disk_(geometry)" class="mw-redirect" title="Disk (geometry)">Euclidean open disk</a> and a circle. The <a href="/wiki/Volume" title="Volume">volume</a> of this solid torus and the <a href="/wiki/Surface_area" title="Surface area">surface area</a> of its torus are easily computed using <a href="/wiki/Pappus%27s_centroid_theorem" title="Pappus's centroid theorem">Pappus's centroid theorem</a>, giving:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mi>r</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759188c729542a0e2cac84f62573b2d5745054c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.82ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}}" /></span> </p><p>These formulas are the same as for a cylinder of length <span class="texhtml">2π<i>R</i></span> and radius <span class="texhtml mvar" style="font-style:italic;">r</span>, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. </p><p>Expressing the surface area and the volume by the distance <span class="texhtml mvar" style="font-style:italic;">p</span> of an outermost point on the surface of the torus to the center, and the distance <span class="texhtml mvar" style="font-style:italic;">q</span> of an innermost point to the center (so that <span class="texhtml"><i>R</i> = <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>p</i> + <i>q</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and <span class="texhtml"><i>r</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>p</i> − <i>q</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>), yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daad86f31332f0d59870358e78e2b38488e1340f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:52.313ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}}" /></span> </p><p>As a torus is the product of two circles, a modified version of the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a> is sometimes used. In traditional spherical coordinates there are three measures, <span class="texhtml mvar" style="font-style:italic;">R</span>, the distance from the center of the coordinate system, and <span class="texhtml mvar" style="font-style:italic;">θ</span> and <span class="texhtml mvar" style="font-style:italic;">φ</span>, angles measured from the center point. </p><p>As a torus has, effectively, two center points, the centerpoints of the angles are moved; <span class="texhtml mvar" style="font-style:italic;">φ</span> measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of <span class="texhtml mvar" style="font-style:italic;">θ</span> is moved to the center of <span class="texhtml mvar" style="font-style:italic;">r</span>, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".<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> </p><p>In modern use, <a href="/wiki/Toroidal_and_poloidal" class="mw-redirect" title="Toroidal and poloidal">toroidal and poloidal</a> are more commonly used to discuss <a href="/wiki/Magnetic_confinement_fusion" title="Magnetic confinement fusion">magnetic confinement fusion</a> devices. </p> <div class="mw-heading mw-heading2"><h2 id="Topology">Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=3" title="Edit section: Topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">November 2015</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><a href="/wiki/Topology" title="Topology">Topologically</a>, a torus is a <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> defined as the <a href="/wiki/Product_topology" title="Product topology">product</a> of two <a href="/wiki/Circle" title="Circle">circles</a>: <span class="texhtml"><i>S</i><sup>1</sup> × <i>S</i><sup>1</sup></span>. This can be viewed as lying in <a href="/wiki/Complex_coordinate_space" title="Complex coordinate space"><span class="texhtml"><b>C</b><sup>2</sup></span></a> and is a subset of the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> <span class="texhtml"><i>S</i><sup>3</sup></span> of radius <span class="texhtml">√2</span>. This topological torus is also often called the <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In fact, <span class="texhtml"><i>S</i><sup>3</sup></span> is <a href="/wiki/Foliation" title="Foliation">filled out</a> by a family of nested tori in this manner (with two degenerate circles), a fact that is important in the study of <span class="texhtml"><i>S</i><sup>3</sup></span> as a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> over <span class="texhtml"><i>S</i><sup>2</sup></span> (the <a href="/wiki/Hopf_bundle" class="mw-redirect" title="Hopf bundle">Hopf bundle</a>). </p><p>The surface described above, given the <a href="/wiki/Relative_topology" class="mw-redirect" title="Relative topology">relative topology</a> from <a href="/wiki/Real_coordinate_space" title="Real coordinate space"><span class="texhtml"><b>R</b><sup>3</sup></span></a>, is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographically projecting</a> the topological torus into <span class="texhtml"><b>R</b><sup>3</sup></span> from the north pole of <span class="texhtml"><i>S</i><sup>3</sup></span>. </p><p>The torus can also be described as a <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient</a> of the <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a> under the identifications </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\sim (x+1,y)\sim (x,y+1),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\sim (x+1,y)\sim (x,y+1),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/414dca7e234140267cda58f81f46f8aaae995ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.222ex; height:2.843ex;" alt="{\displaystyle (x,y)\sim (x+1,y)\sim (x,y+1),\,}" /></span></dd></dl> <p>or, equivalently, as the quotient of the <a href="/wiki/Unit_square" title="Unit square">unit square</a> by pasting the opposite edges together, described as a <a href="/wiki/Fundamental_polygon" title="Fundamental polygon">fundamental polygon</a> <span class="texhtml"><i>ABA</i><sup>−1</sup><i>B</i><sup>−1</sup></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Inside-out_torus_(animated,_small).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/ba/Inside-out_torus_%28animated%2C_small%29.gif" decoding="async" width="170" height="170" class="mw-file-element" data-file-width="170" data-file-height="170" /></a><figcaption>Turning a punctured torus inside-out</figcaption></figure> <p>The <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of the torus is just the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of the fundamental group of the circle with itself: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(T^{2})=\pi _{1}(S^{1})\times \pi _{1}(S^{1})\cong \mathrm {Z} \times \mathrm {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(T^{2})=\pi _{1}(S^{1})\times \pi _{1}(S^{1})\cong \mathrm {Z} \times \mathrm {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b117c5b61554a943285e27368218593b7131b939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.857ex; height:3.176ex;" alt="{\displaystyle \pi _{1}(T^{2})=\pi _{1}(S^{1})\times \pi _{1}(S^{1})\cong \mathrm {Z} \times \mathrm {Z} .}" /></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Intuitively speaking, this means that a <a href="/wiki/Loop_(topology)" title="Loop (topology)">closed path</a> that circles the torus's "hole" (say, a circle that traces out a particular latitude) and then circles the torus's "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. </p><p>If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation. </p><p>The first <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology group</a> of the torus is <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to the fundamental group (this follows from <a href="/wiki/Hurewicz_theorem" title="Hurewicz theorem">Hurewicz theorem</a> since the fundamental group is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Two-sheeted_cover">Two-sheeted cover</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=4" title="Edit section: Two-sheeted cover"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 2-torus is a twofold branched cover of the 2-sphere, with four <a href="/wiki/Ramification_point" class="mw-redirect" title="Ramification point">ramification points</a>. Every <a href="/wiki/Conformal_structure" class="mw-redirect" title="Conformal structure">conformal structure</a> on the 2-torus can be represented as such a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the <a href="/wiki/Weierstrass_point" title="Weierstrass point">Weierstrass points</a>. In fact, the conformal type of the torus is determined by the <a href="/wiki/Cross-ratio" title="Cross-ratio">cross-ratio</a> of the four points. </p> <div class="mw-heading mw-heading2"><h2 id="n-dimensional_torus"><i>n</i>-dimensional torus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=5" title="Edit section: n-dimensional torus"><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:Clifford-torus.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Clifford-torus.gif/220px-Clifford-torus.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6f/Clifford-torus.gif 1.5x" data-file-width="255" data-file-height="255" /></a><figcaption>A stereographic projection of a <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a> in four dimensions performing a simple rotation through the <i>xz</i>-plane</figcaption></figure> <p>The torus has a generalization to higher dimensions, the <em><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="''n''-dimensional_torus"></span><span id="Finite_dimensional_torus"></span><span class="vanchor-text"><i>n</i>-dimensional torus</span></span></em>, often called the <em><span class="texhtml"><i>n</i></span>-torus</em> or <em>hypertorus</em> for short. (This is the more typical meaning of the term "<span class="texhtml"><i>n</i></span>-torus", the other referring to <span class="texhtml"><i>n</i></span> holes or of genus <span class="texhtml"><i>n</i></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>) Just as the ordinary torus is topologically the product space of two circles, the <span class="texhtml"><i>n</i></span>-dimensional torus is <i>topologically equivalent to</i> the product of <span class="texhtml"><i>n</i></span> circles. That is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29edd1206dcbcc6a4fa65013ec54992d610290f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:20.24ex; height:6.009ex;" alt="{\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}.}" /></span></dd></dl> <p>The standard 1-torus is just the circle: <span class="texhtml"><i>T</i><sup>1</sup> = <i>S</i><sup>1</sup></span>. The torus discussed above is the standard 2-torus, <span class="texhtml"><i>T</i><sup>2</sup></span>. And similar to the 2-torus, the <span class="texhtml"><i>n</i></span>-torus, <span class="texhtml"><i>T</i><sup><i>n</i></sup></span> can be described as a quotient of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> under integral shifts in any coordinate. That is, the <i>n</i>-torus is <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> modulo the <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of the integer <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> <span class="texhtml"><b>Z</b><sup><i>n</i></sup></span> (with the action being taken as vector addition). Equivalently, the <span class="texhtml"><i>n</i></span>-torus is obtained from the <span class="texhtml"><i>n</i></span>-dimensional <a href="/wiki/Hypercube" title="Hypercube">hypercube</a> by gluing the opposite faces together. </p><p>An <span class="texhtml"><i>n</i></span>-torus in this sense is an example of an <i>n-</i>dimensional <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Manifold" title="Manifold">manifold</a>. It is also an example of a compact <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. This follows from the fact that the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> is a compact abelian Lie group (when identified with the unit <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. </p><p>Toroidal groups play an important part in the theory of <a href="/wiki/Compact_Lie_group" class="mw-redirect" title="Compact Lie group">compact Lie groups</a>. This is due in part to the fact that in any compact Lie group <span class="texhtml"><i>G</i></span> one can always find a <a href="/wiki/Maximal_torus" title="Maximal torus">maximal torus</a>; that is, a closed <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> which is a torus of the largest possible dimension. Such maximal tori <span class="texhtml"><i>T</i></span> have a controlling role to play in theory of connected <span class="texhtml"><i>G</i></span>. Toroidal groups are examples of <a href="/wiki/Protorus" title="Protorus">protori</a>, which (like tori) are compact connected abelian groups, which are not required to be <a href="/wiki/Manifold" title="Manifold">manifolds</a>. </p><p><a href="/wiki/Automorphism" title="Automorphism">Automorphisms</a> of <span class="texhtml"><i>T</i></span> are easily constructed from automorphisms of the lattice <span class="texhtml"><b>Z</b><sup><i>n</i></sup></span>, which are classified by <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> <a href="/wiki/Integral_matrices" class="mw-redirect" title="Integral matrices">integral matrices</a> of size <span class="texhtml"><i>n</i></span> with an integral inverse; these are just the integral matrices with determinant <span class="texhtml">±1</span>. Making them act on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> in the usual way, one has the typical <i>toral automorphism</i> on the quotient. </p><p>The <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of an <i>n</i>-torus is a <a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian group</a> of rank <span class="texhtml"><i>n</i></span>. The <span class="texhtml"><i>k</i></span>th <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology group</a> of an <span class="texhtml"><i>n</i></span>-torus is a free abelian group of rank <i>n</i> <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">choose</a> <span class="texhtml"><i>k</i></span>. It follows that the <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of the <span class="texhtml"><i>n</i></span>-torus is <span class="texhtml">0</span> for all <span class="texhtml"><i>n</i></span>. The <a href="/wiki/Cohomology_ring" title="Cohomology ring">cohomology ring</a> <i>H</i><sup>•</sup>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a5182df16f38b52918786915cbaa047bb46a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.938ex; height:2.343ex;" alt="{\displaystyle T^{n}}" /></span>, <b>Z</b>) can be identified with the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> over the <span class="texhtml"><b>Z</b></span>-<a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> <span class="texhtml"><b>Z</b><sup><i>n</i></sup></span> whose generators are the duals of the <span class="texhtml"><i>n</i></span> nontrivial cycles. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quasitoric_manifold" title="Quasitoric manifold">Quasitoric manifold</a></div> <div class="mw-heading mw-heading3"><h3 id="Configuration_space">Configuration space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=6" title="Edit section: Configuration space"><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:Moebius_Surface_1_Display_Small.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/71/Moebius_Surface_1_Display_Small.png" decoding="async" width="180" height="140" class="mw-file-element" data-file-width="180" data-file-height="140" /></a><figcaption>The configuration space of 2 not necessarily distinct points on the circle is the <a href="/wiki/Orbifold" title="Orbifold">orbifold</a> quotient of the 2-torus, <span class="texhtml"><i>T</i><sup>2</sup> / <i>S</i><sub>2</sub></span>, which is the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a>.</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Neo-Riemannian_Tonnetz.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Neo-Riemannian_Tonnetz.svg/220px-Neo-Riemannian_Tonnetz.svg.png" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Neo-Riemannian_Tonnetz.svg/330px-Neo-Riemannian_Tonnetz.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Neo-Riemannian_Tonnetz.svg/440px-Neo-Riemannian_Tonnetz.svg.png 2x" data-file-width="2149" data-file-height="1104" /></a><figcaption>The <i><a href="/wiki/Tonnetz" title="Tonnetz">Tonnetz</a></i> is an example of a torus in music theory.<br /><small>The Tonnetz is only truly a torus if <a href="/wiki/Enharmonic_equivalence" title="Enharmonic equivalence">enharmonic equivalence</a> is assumed, so that the <span class="nowrap">(F♯-A♯)</span> segment of the right edge of the repeated parallelogram is identified with the <span class="nowrap">(G♭-B♭)</span> segment of the left edge.</small></figcaption></figure> <p>As the <span class="texhtml"><i>n</i></span>-torus is the <span class="texhtml"><i>n</i></span>-fold product of the circle, the <span class="texhtml"><i>n</i></span>-torus is the <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> of <span class="texhtml"><i>n</i></span> ordered, not necessarily distinct points on the circle. Symbolically, <span class="texhtml"><i>T</i><sup><i>n</i></sup> = (<i>S</i><sup>1</sup>)<sup><i>n</i></sup></span>. The configuration space of <i>unordered</i>, not necessarily distinct points is accordingly the <a href="/wiki/Orbifold" title="Orbifold">orbifold</a> <span class="texhtml"><i>T</i><sup><i>n</i></sup> / <i>S</i><sup><i>n</i></sup></span>, which is the quotient of the torus by the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> on <span class="texhtml"><i>n</i></span> letters (by permuting the coordinates). </p><p>For <span class="texhtml"><i>n</i> = 2</span>, the quotient is the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a>, the edge corresponding to the orbifold points where the two coordinates coincide. For <span class="texhtml"><i>n</i> = 3</span> this quotient may be described as a solid torus with cross-section an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>, with a <a href="/wiki/Dehn_twist" title="Dehn twist">twist</a>; equivalently, as a <a href="/wiki/Triangular_prism" title="Triangular prism">triangular prism</a> whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical. </p><p>These orbifolds have found significant <a href="/wiki/Orbifold#Music_theory" title="Orbifold">applications to music theory</a> in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model <a href="/wiki/Triad_(music)" title="Triad (music)">musical triads</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Flat_torus">Flat torus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=7" title="Edit section: Flat torus"><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:Torus_from_rectangle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Torus_from_rectangle.gif/250px-Torus_from_rectangle.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Torus_from_rectangle.gif/330px-Torus_from_rectangle.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/60/Torus_from_rectangle.gif 2x" data-file-width="400" data-file-height="300" /></a><figcaption>In three dimensions, one can bend a rectangle into a torus, but doing this necessarily affects the distances measured along the surface, as seen by the distortion of the checkered pattern.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Duocylinder_ridge_animated.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7e/Duocylinder_ridge_animated.gif" decoding="async" width="162" height="156" class="mw-file-element" data-file-width="162" data-file-height="156" /></a><figcaption>Seen in <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>, a 4D <i>flat torus</i> can be projected into 3-dimensions and rotated on a fixed axis.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Toroidal_monohedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Toroidal_monohedron.png/220px-Toroidal_monohedron.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Toroidal_monohedron.png/330px-Toroidal_monohedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Toroidal_monohedron.png/440px-Toroidal_monohedron.png 2x" data-file-width="1409" data-file-height="896" /></a><figcaption>The simplest tiling of a flat torus is <a href="/wiki/Regular_map_(graph_theory)#Toroidal_polyhedra" title="Regular map (graph theory)">{4,4}<sub>1,0</sub></a>, constructed on the surface of a <a href="/wiki/Duocylinder" title="Duocylinder">duocylinder</a> with 1 vertex, 2 orthogonal edges, and one square face. It is seen here stereographically projected into 3-space as a torus.</figcaption></figure> <p>A flat torus is a torus with the metric inherited from its representation as the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient</a>, <span class="texhtml"><b>R</b><sup>2</sup> / <b>L</b></span>, where <span class="texhtml"><b>L</b></span> is a discrete subgroup of <span class="texhtml"><b>R</b><sup>2</sup></span> isomorphic to <span class="texhtml"><b>Z</b><sup>2</sup></span>. This gives the quotient the structure of a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when <span class="texhtml"><b>L</b> = <b>Z</b><sup>2</sup></span>: <span class="texhtml"><b>R</b><sup>2</sup> / <b>Z</b><sup>2</sup></span>, which can also be described as the <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a> under the identifications <span class="texhtml">(<i>x</i>, <i>y</i>) ~ (<i>x</i> + 1, <i>y</i>) ~ (<i>x</i>, <i>y</i> + 1)</span>. This particular flat torus (and any uniformly scaled version of it) is known as the "square" flat torus. </p><p>This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). </p><p>A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mi>cos</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> <mi>R</mi> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> <mi>P</mi> <mi>cos</mi> <mo>⁡</mo> <mi>v</mi> <mo>,</mo> <mi>P</mi> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db7687001d7c02328716c2cb696c75ae4e5f8b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.122ex; height:2.843ex;" alt="{\displaystyle (x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)}" /></span></dd></dl> <p>where <i>R</i> and <i>P</i> are positive constants determining the aspect ratio. It is <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> to a regular torus but not <a href="/wiki/Isometry" title="Isometry">isometric</a>. It can not be <a href="/wiki/Analytic_function" title="Analytic function">analytically</a> embedded (<a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> of class <span class="texhtml"><i>C<sup>k</sup></i>, 2 ≤ <i>k</i> ≤ ∞</span>) into Euclidean 3-space. <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Mapping</a> it into <i>3</i>-space requires one to stretch it, in which case it looks like a regular torus. For example, in the following map: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)=((R+P\sin v)\cos u,(R+P\sin v)\sin u,P\cos v).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>P</mi> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>P</mi> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> <mi>P</mi> <mi>cos</mi> <mo>⁡</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)=((R+P\sin v)\cos u,(R+P\sin v)\sin u,P\cos v).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/020560be6ce8f841c385fe15edf98ad2b6747b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.839ex; height:2.843ex;" alt="{\displaystyle (x,y,z)=((R+P\sin v)\cos u,(R+P\sin v)\sin u,P\cos v).}" /></span></dd></dl> <p>If <span class="texhtml"><i>R</i></span> and <span class="texhtml"><i>P</i></span> in the above flat torus parametrization form a unit vector <span class="texhtml">(<i>R</i>, <i>P</i>) = (cos(<i>η</i>), sin(<i>η</i>))</span> then <i>u</i>, <i>v</i>, and <span class="texhtml">0 < <i>η</i> < π/2</span> parameterize the unit 3-sphere as <a href="/wiki/3-sphere#Hopf_coordinates" title="3-sphere">Hopf coordinates</a>. In particular, for certain very specific choices of a square flat torus in the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> <i>S</i><sup>3</sup>, where <span class="texhtml"><i>η</i> = π/4</span> above, the torus will partition the 3-sphere into two <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> solid tori subsets with the aforesaid flat torus surface as their common <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a>. One example is the torus <span class="texhtml"><i>T</i></span> defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\left\{(x,y,z,w)\in S^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\left\{(x,y,z,w)\in S^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07eb65b12a9ef37f78e3f231b41d67cdaed225ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.708ex; height:6.176ex;" alt="{\displaystyle T=\left\{(x,y,z,w)\in S^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.}" /></span></dd></dl> <p>Other tori in <span class="texhtml"><i>S</i><sup>3</sup></span> having this partitioning property include the square tori of the form <span class="texhtml"><i>Q</i> ⋅ <i>T</i></span>, where <span class="texhtml"><i>Q</i></span> is a rotation of 4-dimensional space <span class="texhtml"><b>R</b><sup>4</sup></span>, or in other words <span class="texhtml"><i>Q</i></span> is a member of the Lie group <span class="texhtml">SO(4)</span>. </p><p>It is known that there exists no <span class="texhtml"><i>C</i><sup>2</sup></span> (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the <a href="/wiki/Nash_embedding_theorem" class="mw-redirect" title="Nash embedding theorem">Nash-Kuiper theorem</a>, which was proven in the 1950s, an isometric <i>C</i><sup>1</sup> embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Flat_torus_Havea_embedding.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Flat_torus_Havea_embedding.png/250px-Flat_torus_Havea_embedding.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Flat_torus_Havea_embedding.png/330px-Flat_torus_Havea_embedding.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Flat_torus_Havea_embedding.png/500px-Flat_torus_Havea_embedding.png 2x" data-file-width="1920" data-file-height="1080" /></a><figcaption><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" /></span> isometric embedding of a flat torus in <span class="texhtml"><b>R</b><sup>3</sup></span>, with corrugations</figcaption></figure> <p>In April 2012, an explicit <i>C</i><sup>1</sup> (continuously differentiable) isometric embedding of a flat torus into 3-dimensional Euclidean space <span class="texhtml"><b>R</b><sup>3</sup></span> was found.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a <a href="/wiki/Fractal" title="Fractal">fractal</a> as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">surface normals</a>, yielding a so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths".<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics. </p> <div class="mw-heading mw-heading3"><h3 id="Conformal_classification_of_flat_tori">Conformal classification of flat tori</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=8" title="Edit section: Conformal classification of flat tori"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the study of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The <a href="/wiki/Uniformization_theorem" title="Uniformization theorem">Uniformization theorem</a> guarantees that every Riemann surface is <a href="/wiki/Conformal_map" title="Conformal map">conformally equivalent</a> to one that has constant <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>. In the case of a torus, the constant curvature must be zero. Then one defines the "<a href="/wiki/Moduli_space" title="Moduli space">moduli space</a>" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space <i>M</i> may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle π and the other has total angle 2π/3. </p><p><i>M</i> may be turned into a compact space <i>M*</i> – topologically equivalent to a sphere – by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with <i>three</i> points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, <i>M*</i> may be constructed by glueing together two congruent <a href="/wiki/Geodesic_triangle" class="mw-redirect" title="Geodesic triangle">geodesic triangles</a> in the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a> along their (identical) boundaries, where each triangle has angles of <span class="texhtml">π/2</span>, <span class="texhtml">π/3</span>, and <span class="texhtml">0</span>. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the <a href="/wiki/Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a> shows that the area of each triangle can be calculated as <span class="texhtml">π − (π/2 + π/3 + 0) = π/6</span>, so it follows that the compactified moduli space <i>M*</i> has area equal to <span class="texhtml">π/3</span>. </p><p>The other two cusps occur at the points corresponding in <i>M*</i> to (a) the square torus (total angle <span class="texhtml">π</span>) and (b) the hexagonal torus (total angle <span class="texhtml">2π/3</span>). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation. </p> <div class="mw-heading mw-heading2"><h2 id="Genus_g_surface">Genus <i>g</i> surface</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=9" title="Edit section: Genus g surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Genus_g_surface" title="Genus g surface">Genus g surface</a></div> <p>In the theory of <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a> there is a more general family of objects, the "<a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a>" <span class="texhtml"><i>g</i></span> surfaces. A genus <span class="texhtml"><i>g</i></span> surface is the <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> of <span class="texhtml"><i>g</i></span> two-tori. (And so the torus itself is the surface of genus 1.) To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. (That is, merge the two boundary circles so they become just one circle.) To form the connected sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus <span class="texhtml"><i>g</i></span> surface resembles the surface of <span class="texhtml"><i>g</i></span> doughnuts stuck together side by side, or a <a href="/wiki/Sphere" title="Sphere">2-sphere</a> with <span class="texhtml"><i>g</i></span> handles attached. </p><p>As examples, a genus zero surface (without boundary) is the <a href="/wiki/Sphere" title="Sphere">two-sphere</a> while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called <span class="texhtml"><i>n</i></span>-holed tori (or, rarely, <span class="texhtml"><i>n</i></span>-fold tori). The terms <a href="/wiki/Double_torus" class="mw-redirect" title="Double torus">double torus</a> and <a href="/wiki/Triple_torus" class="mw-redirect" title="Triple torus">triple torus</a> are also occasionally used. </p><p>The <a href="/wiki/Classification_theorem" title="Classification theorem">classification theorem</a> for surfaces states that every <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Connected_space" title="Connected space">connected</a> surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real <a href="/wiki/Projective_plane" title="Projective plane">projective planes</a>. </p> <table class="wikitable"> <tbody><tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Double_torus_illustration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Double_torus_illustration.png/160px-Double_torus_illustration.png" decoding="async" width="160" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Double_torus_illustration.png/240px-Double_torus_illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Double_torus_illustration.png/320px-Double_torus_illustration.png 2x" data-file-width="985" data-file-height="1077" /></a></span><br /><a href="/wiki/Double_torus" class="mw-redirect" title="Double torus">genus two</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Triple_torus_illustration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Triple_torus_illustration.png/250px-Triple_torus_illustration.png" decoding="async" width="240" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Triple_torus_illustration.png/500px-Triple_torus_illustration.png 1.5x" data-file-width="2204" data-file-height="1550" /></a></span><br /><a href="/wiki/Triple_torus" class="mw-redirect" title="Triple torus">genus three</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Toroidal_polyhedra">Toroidal polyhedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=10" title="Edit section: Toroidal polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Toroidal_polyhedron" title="Toroidal polyhedron">Toroidal polyhedron</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hexagonal_torus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hexagonal_torus.svg/220px-Hexagonal_torus.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hexagonal_torus.svg/330px-Hexagonal_torus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hexagonal_torus.svg/440px-Hexagonal_torus.svg.png 2x" data-file-width="830" data-file-height="576" /></a><figcaption>A <a href="/wiki/Toroidal_polyhedron" title="Toroidal polyhedron">toroidal polyhedron</a> with <span class="nowrap">6 × 4 = 24</span> <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> faces</figcaption></figure> <p><a href="/wiki/Polyhedron" title="Polyhedron">Polyhedra</a> with the topological type of a torus are called toroidal polyhedra, and have <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> <span class="texhtml"><i>V</i> − <i>E</i> + <i>F</i> = 0</span>. For any number of holes, the formula generalizes to <span class="texhtml"><i>V</i> − <i>E</i> + <i>F</i> = 2 − 2<i>N</i></span>, where <span class="texhtml"><i>N</i></span> is the number of holes. </p><p>The term "toroidal polyhedron" is also used for higher-genus polyhedra and for <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersions</a> of toroidal polyhedra. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444" /><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Torus&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">April 2010</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Automorphisms">Automorphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=11" title="Edit section: Automorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Homeomorphism_group" title="Homeomorphism group">homeomorphism group</a> (or the subgroup of diffeomorphisms) of the torus is studied in <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a>. Its <a href="/wiki/Mapping_class_group" title="Mapping class group">mapping class group</a> (the connected components of the homeomorphism group) is surjective onto the group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbf {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbf {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0629f244178ed70d3088c7c93f75a2f5ce2a1af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.149ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbf {Z} )}" /></span> of invertible integer matrices, which can be realized as linear maps on the universal covering space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" /></span> that preserve the standard lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f958aa7a83ea03c9641cbcbdb83dc7a36db14972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.343ex;" alt="{\displaystyle \mathbf {Z} ^{n}}" /></span> (this corresponds to integer coefficients) and thus descend to the quotient. </p><p>At the level of <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> and <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a>, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a>, as these are all naturally isomorphic; also the first <a href="/wiki/Cohomology_group" class="mw-redirect" title="Cohomology group">cohomology group</a> generates the <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> algebra: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {MCG} _{\operatorname {Ho} }(T^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbf {Z} ^{n})=\operatorname {GL} (n,\mathbf {Z} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>MCG</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Ho</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Aut</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Aut</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {MCG} _{\operatorname {Ho} }(T^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbf {Z} ^{n})=\operatorname {GL} (n,\mathbf {Z} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/051fd513bd3eb206471919a811323e8cf1e99e41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.279ex; height:2.843ex;" alt="{\displaystyle \operatorname {MCG} _{\operatorname {Ho} }(T^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbf {Z} ^{n})=\operatorname {GL} (n,\mathbf {Z} ).}" /></span></dd></dl> <p>Since the torus is an <a href="/wiki/Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a> <span class="texhtml"><i>K</i>(<i>G</i>, 1)</span>, its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism. </p><p>Thus the <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of the mapping class group splits (an identification of the torus as the quotient of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" /></span> gives a splitting, via the linear maps, as above): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\to \operatorname {Homeo} _{0}(T^{n})\to \operatorname {Homeo} (T^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(T^{n})\to 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <msub> <mi>Homeo</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Homeo</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>MCG</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>TOP</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\to \operatorname {Homeo} _{0}(T^{n})\to \operatorname {Homeo} (T^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(T^{n})\to 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6570699f62332b1b779c8a1d36a331785c08be53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.248ex; height:2.843ex;" alt="{\displaystyle 1\to \operatorname {Homeo} _{0}(T^{n})\to \operatorname {Homeo} (T^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(T^{n})\to 1.}" /></span></dd></dl> <p>The mapping class group of higher genus surfaces is much more complicated, and an area of active research. </p> <div class="mw-heading mw-heading2"><h2 id="Coloring_a_torus">Coloring a torus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=12" title="Edit section: Coloring a torus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The torus's <a href="/wiki/Heawood_number" title="Heawood number">Heawood number</a> is seven, meaning every graph that can be <a href="/wiki/Toroidal_graph" title="Toroidal graph">embedded on the torus</a> has a <a href="/wiki/Chromatic_number" class="mw-redirect" title="Chromatic number">chromatic number</a> of at most seven. (Since the <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {K_{7}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="sans-serif">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="sans-serif">7</mn> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {K_{7}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/249d86a709d4d226aeec84e5797927e7546127ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.667ex; height:2.509ex;" alt="{\displaystyle {\mathsf {K_{7}}}}" /></span> can be embedded on the torus, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ({\mathsf {K_{7}}})=7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="sans-serif">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="sans-serif">7</mn> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ({\mathsf {K_{7}}})=7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28407f3455e445b5646ef864db8499110883ddb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.193ex; height:2.843ex;" alt="{\displaystyle \chi ({\mathsf {K_{7}}})=7}" /></span>, the upper bound is tight.) Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color. (Contrast with the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> for the <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>.) </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Projection_color_torus.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Projection_color_torus.png/480px-Projection_color_torus.png" decoding="async" width="480" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/37/Projection_color_torus.png 1.5x" data-file-width="523" data-file-height="166" /></a><figcaption>This construction shows the torus divided into seven regions, every one of which touches every other, meaning each must be assigned a unique color.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="de_Bruijn_torus">de Bruijn torus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=13" title="Edit section: de Bruijn torus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/De_Bruijn_torus" title="De Bruijn torus">de Bruijn torus</a></div> <figure typeof="mw:File/Thumb"><span class="mw-3d-wrapper" data-label="3D"><a href="http://viewstl.com/classic/?embedded&url=http://upload.wikimedia.org/wikipedia/commons/1/1e/De_bruijn_torus_3x3.stl&bgcolor=black" rel="nofollow"><img resource="/wiki/File:De_bruijn_torus_3x3.stl" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/De_bruijn_torus_3x3.stl/250px-De_bruijn_torus_3x3.stl.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/De_bruijn_torus_3x3.stl/500px-De_bruijn_torus_3x3.stl.png 1.5x" data-file-width="5120" data-file-height="2880" /></a></span><figcaption><a href="/wiki/STL_(file_format)" title="STL (file format)">STL</a> model of de Bruijn torus <span class="nowrap">(16,32;3,3)<sub>2</sub></span> with 1s as panels and 0s as holes in the mesh – with consistent orientation, every 3×3 matrix appears exactly once</figcaption></figure> <p>In <a href="/wiki/Combinatorics" title="Combinatorics">combinatorial</a> mathematics, a <i>de Bruijn torus</i> is an <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">array</a> of symbols from an alphabet (often just 0 and 1) that contains every <span class="texhtml"><i>m</i></span>-by-<span class="texhtml"><i>n</i></span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the <a href="/wiki/De_Bruijn_sequence" title="De Bruijn sequence">De Bruijn sequence</a>, which can be considered a special case where <span class="texhtml"><i>n</i></span> is 1 (one dimension). </p> <div class="mw-heading mw-heading2"><h2 id="Cutting_a_torus">Cutting a torus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=14" title="Edit section: Cutting a torus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A solid torus of revolution can be cut by <i>n</i> (> 0) planes into at most </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d78de8aef833bd423f390ab995e42fca7f6cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.845ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)}" /></span></dd></dl> <p>parts.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> (This assumes the pieces may not be rearranged but must remain in place for all cuts.) </p><p>The first 11 numbers of parts, for <span class="texhtml">0 ≤ <i>n</i> ≤ 10</span> (including the case of <span class="texhtml"><i>n</i> = 0</span>, not covered by the above formulas), are as follows: </p> <dl><dd>1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (sequence <span class="nowrap external"><a href="//oeis.org/A003600" class="extiw" title="oeis:A003600">A003600</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <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=Torus&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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style="column-width: 22em;"> <ul><li><a href="/wiki/3-torus" title="3-torus">3-torus</a></li> <li><a href="/wiki/Algebraic_torus" title="Algebraic torus">Algebraic torus</a></li> <li><a href="/wiki/Angenent_torus" title="Angenent torus">Angenent torus</a></li> <li><a href="/wiki/Annulus_(geometry)" class="mw-redirect" title="Annulus (geometry)">Annulus (geometry)</a></li> <li><a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a></li> <li><a href="/wiki/Complex_torus" title="Complex torus">Complex torus</a></li> <li><a href="/wiki/Dupin_cyclide" title="Dupin cyclide">Dupin cyclide</a></li> <li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li> <li><a href="/wiki/Irrational_winding_of_a_torus" class="mw-redirect" title="Irrational winding of a torus">Irrational winding of a torus</a></li> <li><a href="/wiki/Joint_European_Torus" title="Joint European Torus">Joint European Torus</a></li> <li><a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a></li> <li><a href="/wiki/Loewner%27s_torus_inequality" title="Loewner's torus inequality">Loewner's torus inequality</a></li> <li><a href="/wiki/Maximal_torus" title="Maximal torus">Maximal torus</a></li> <li><a href="/wiki/Period_lattice" class="mw-redirect" title="Period lattice">Period lattice</a></li> <li><a href="/wiki/Real_projective_plane" title="Real projective plane">Real projective plane</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Spiric_section" title="Spiric section">Spiric section</a></li> <li><a href="/wiki/Surface_(topology)" title="Surface (topology)">Surface (topology)</a></li> <li><a href="/wiki/Toric_lens" title="Toric lens">Toric lens</a></li> <li><a href="/wiki/Toric_section" title="Toric section">Toric section</a></li> <li><a href="/wiki/Toric_variety" title="Toric variety">Toric variety</a></li> <li><a href="/wiki/Toroid" title="Toroid">Toroid</a></li> <li><a href="/wiki/Toroidal_and_poloidal" class="mw-redirect" title="Toroidal and poloidal">Toroidal and poloidal</a></li> <li><a href="/wiki/Torus-based_cryptography" title="Torus-based cryptography">Torus-based cryptography</a></li> <li><a href="/wiki/Torus_knot" title="Torus knot">Torus knot</a></li> <li><a href="/wiki/Umbilic_torus" title="Umbilic torus">Umbilic torus</a></li> <li><a href="/wiki/Villarceau_circles" title="Villarceau circles">Villarceau circles</a></li></ul></div> <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=Torus&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Nociones de Geometría Analítica y Álgebra Lineal</i>, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation 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.cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-970-10-6596-9" title="Special:BookSources/978-970-10-6596-9">978-970-10-6596-9</a>, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish</li> <li>Allen Hatcher. <a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html"><i>Algebraic Topology</i></a>. Cambridge University Press, 2002. <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/0-521-79540-0" title="Special:BookSources/0-521-79540-0">0-521-79540-0</a>.</li> <li>V. V. Nikulin, I. R. Shafarevich. <i>Geometries and Groups</i>. Springer, 1987. <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/3-540-15281-4" title="Special:BookSources/3-540-15281-4">3-540-15281-4</a>, <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/978-3-540-15281-1" title="Special:BookSources/978-3-540-15281-1">978-3-540-15281-1</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.mathcurve.com/surfaces/tore/tore.shtml">"Tore (notion géométrique)" at <i>Encyclopédie des Formes Mathématiques Remarquables</i></a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGallierXu2013" class="citation book cs1"><a href="/wiki/Jean_Gallier" title="Jean Gallier">Gallier, Jean</a>; <a href="/wiki/Dianna_Xu" title="Dianna Xu">Xu, Dianna</a> (2013). <a href="/wiki/A_Guide_to_the_Classification_Theorem_for_Compact_Surfaces" title="A Guide to the Classification Theorem for Compact Surfaces"><i>A Guide to the Classification Theorem for Compact Surfaces</i></a>. Geometry and Computing. Vol. 9. Springer, Heidelberg. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-34364-3">10.1007/978-3-642-34364-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-34363-6" title="Special:BookSources/978-3-642-34363-6"><bdi>978-3-642-34363-6</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3026641">3026641</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+the+Classification+Theorem+for+Compact+Surfaces&rft.series=Geometry+and+Computing&rft.pub=Springer%2C+Heidelberg&rft.date=2013&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3026641%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-34364-3&rft.isbn=978-3-642-34363-6&rft.aulast=Gallier&rft.aufirst=Jean&rft.au=Xu%2C+Dianna&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.geom.uiuc.edu/zoo/toptype/torus/standard/eqns.html">"Equations for the Standard Torus"</a>. 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Torus.html">"Torus"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Torus&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTorus.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dictionary.oed.com/cgi/entry/50183023?single=1&query_type=word&queryword=poloidal&first=1&max_to_show=10">"poloidal"</a>. <i>Oxford English Dictionary Online</i>. 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Archived from <a rel="nofollow" class="external text" href="http://www2.cnrs.fr/en/2027.htm">the original</a> on 5 July 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">21 July</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematics%3A+first-ever+image+of+a+flat+torus+in+3D+%E2%80%93+CNRS+Web+site+%E2%80%93+CNRS&rft_id=http%3A%2F%2Fwww2.cnrs.fr%2Fen%2F2027.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120618084643/http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html">"Flat tori finally visualized!"</a>. Math.univ-lyon1.fr. 18 April 2012. Archived from <a rel="nofollow" class="external text" href="http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html">the original</a> on 18 June 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">21 July</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Flat+tori+finally+visualized%21&rft.pub=Math.univ-lyon1.fr&rft.date=2012-04-18&rft_id=http%3A%2F%2Fmath.univ-lyon1.fr%2F~borrelli%2FHevea%2FPresse%2Findex-en.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHoang2016" class="citation web cs1">Hoang, Lê Nguyên (2016). <a rel="nofollow" class="external text" href="http://www.science4all.org/article/flat-torus/">"The Tortuous Geometry of the Flat Torus"</a>. <i>Science4All</i><span class="reference-accessdate">. Retrieved <span class="nowrap">1 November</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Science4All&rft.atitle=The+Tortuous+Geometry+of+the+Flat+Torus&rft.date=2016&rft.aulast=Hoang&rft.aufirst=L%C3%AA+Nguy%C3%AAn&rft_id=http%3A%2F%2Fwww.science4all.org%2Farticle%2Fflat-torus%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Torus_Cutting"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TorusCutting.html">"Torus Cutting"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Torus+Cutting&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTorusCutting.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/torus" class="extiw" title="wiktionary:Special:Search/torus">torus</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-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/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a> has media related to:<br /> <a href="https://commons.wikimedia.org/wiki/Torus" class="extiw" title="commons:Torus"><span style="font-style:italic; font-weight:bold;">Torus</span></a> (<a href="https://commons.wikimedia.org/wiki/Category:Torus" class="extiw" title="commons:Category:Torus">category</a>)</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/shortcut.shtml">Creation of a torus</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.dr-mikes-maths.com/4d-torus.html">"4D torus"</a> Fly-through cross-sections of a four-dimensional torus</li> <li><a rel="nofollow" class="external text" href="http://www.visumap.net/index.aspx?p=Resources/RpmOverview">"Relational Perspective Map"</a> Visualizing high dimensional data with flat torus</li> <li><a rel="nofollow" class="external text" href="http://tofique.fatehi.us/Mathematics/Polydoes/polydoes.html">Polydoes, doughnut-shaped polygons</a></li> <li>Archived at <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/3_VydFQmtZ8">Ghostarchive</a> and the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140128170125/http://www.youtube.com/watch?v=3_VydFQmtZ8&gl=US&hl=en">Wayback Machine</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSéquin2014" class="citation web cs1"><a href="/wiki/Carlo_H._S%C3%A9quin" title="Carlo H. Séquin">Séquin, Carlo H</a> (27 January 2014). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=3_VydFQmtZ8">"Topology of a Twisted Torus – Numberphile"</a> <span class="cs1-format">(video)</span>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Topology+of+a+Twisted+Torus+%E2%80%93+Numberphile&rft.pub=Brady+Haran&rft.date=2014-01-27&rft.aulast=S%C3%A9quin&rft.aufirst=Carlo+H&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D3_VydFQmtZ8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAnders_Sandberg2014" class="citation web cs1">Anders Sandberg (4 February 2014). <a rel="nofollow" class="external text" href="http://www.aleph.se/andart/archives/2014/02/torusearth.html">"Torus Earth"</a><span class="reference-accessdate">. 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topological surfaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Compact_topological_surfaces" title="Special:EditPage/Template:Compact topological surfaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Compact_topological_surfaces_and_their_immersions_in_3D102" style="font-size:114%;margin:0 4em">Compact topological surfaces and their immersions in 3D</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Without boundary</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Orientable</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/Sphere" title="Sphere">Sphere</a> (genus 0)</li> <li><a class="mw-selflink selflink">Torus</a> (genus 1)</li> <li>Number 8 (genus 2)</li> <li>Pretzel (genus 3) ...</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-orientable</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/Real_projective_plane" title="Real projective plane">Real projective plane</a> <ul><li>genus 1; <a href="/wiki/Boy%27s_surface" title="Boy's surface">Boy's surface</a></li> <li><a href="/wiki/Roman_surface" title="Roman surface">Roman surface</a></li></ul></li> <li><a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> (genus 2)</li> <li><a href="/wiki/Dyck%27s_surface" class="mw-redirect" title="Dyck's surface">Dyck's surface</a> (genus 3) ...</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With boundary</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">Disk</a> <ul><li>Semisphere</li></ul></li> <li>Ribbon <ul><li><a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">Annulus</a></li> <li><a href="/wiki/Cylinder" title="Cylinder">Cylinder</a></li></ul></li> <li><a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> <ul><li><a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">Cross-cap</a></li></ul></li> <li><a href="/wiki/Pair_of_pants_(mathematics)" title="Pair of pants (mathematics)">Sphere with three holes</a> ...</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />notions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Properties</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/Connected_space" title="Connected space">Connectedness</a></li> <li><a href="/wiki/Compact_space" title="Compact space">Compactness</a></li> <li><a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">Triangulatedness</a> or <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">smoothness</a></li> <li><a href="/wiki/Orientability" title="Orientability">Orientability</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Characteristics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Number of <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> components</li> <li><a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">Genus</a></li> <li><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operations</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/Connected_sum" title="Connected sum">Connected sum</a></li> <li>Making a hole</li> <li>Gluing a <a href="/wiki/Handle_decomposition" title="Handle decomposition">handle</a></li> <li>Gluing a <a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">cross-cap</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li></ul> </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 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