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href="#Constructions"> <div class="vector-toc-text"> 3 Constructions </div> </a> <button aria-controls="toc-Constructions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Constructions subsection </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Sweeping_a_line_segment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sweeping_a_line_segment"> <div class="vector-toc-text"> 3.1 Sweeping a line segment </div> </a> <ul id="toc-Sweeping_a_line_segment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polyhedral_surfaces_and_flat_foldings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polyhedral_surfaces_and_flat_foldings"> <div class="vector-toc-text"> 3.2 Polyhedral surfaces and flat foldings </div> </a> <ul id="toc-Polyhedral_surfaces_and_flat_foldings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Smoothly_embedded_rectangles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Smoothly_embedded_rectangles"> <div class="vector-toc-text"> 3.3 Smoothly embedded rectangles </div> </a> <ul id="toc-Smoothly_embedded_rectangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Making_the_boundary_circular" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Making_the_boundary_circular"> <div class="vector-toc-text"> 3.4 Making the boundary circular </div> </a> <ul id="toc-Making_the_boundary_circular-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surfaces_of_constant_curvature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surfaces_of_constant_curvature"> <div class="vector-toc-text"> 3.5 Surfaces of constant curvature </div> </a> <ul id="toc-Surfaces_of_constant_curvature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spaces_of_lines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spaces_of_lines"> <div class="vector-toc-text"> 3.6 Spaces of lines </div> </a> <ul id="toc-Spaces_of_lines-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 4 Applications </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> 5 In popular culture </div> </a> <ul id="toc-In_popular_culture-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"> 6 See also </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"> 7 Notes </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"> 8 References </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"> 9 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </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">Möbius strip</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 62 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-62" aria-hidden="true" > 62 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%B1%D9%8A%D8%B7_%D9%85%D9%88%D8%A8%D9%8A%D9%88%D8%B3" title="شريط موبيوس – Arabic" lang="ar" hreflang="ar" data-title="شريط موبيوس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C3%B6bius_lenti" title="Möbius lenti – Azerbaijani" lang="az" hreflang="az" data-title="Möbius lenti" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%82%D1%83%D0%B6%D0%BA%D0%B0_%D0%9C%D1%91%D0%B1%D1%96%D1%83%D1%81%D0%B0" title="Стужка Мёбіуса – Belarusian" lang="be" hreflang="be" data-title="Стужка Мёбіуса" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D1%81%D1%82_%D0%BD%D0%B0_%D0%9C%D1%8C%D0%BE%D0%B1%D0%B8%D1%83%D1%81" title="Лист на Мьобиус – Bulgarian" lang="bg" hreflang="bg" data-title="Лист на Мьобиус" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cinta_de_M%C3%B6bius" title="Cinta de Möbius – Catalan" lang="ca" hreflang="ca" data-title="Cinta de Möbius" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D1%91%D0%B1%D0%B8%D1%83%D1%81_%D1%85%C4%83%D0%B9%C4%83%D0%B2%C4%95" title="Мёбиус хăйăвĕ – Chuvash" lang="cv" hreflang="cv" data-title="Мёбиус хăйăвĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/M%C3%B6biova_p%C3%A1ska" title="Möbiova páska – Czech" lang="cs" hreflang="cs" data-title="Möbiova páska" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Stribyn_M%C3%B6bius" title="Stribyn Möbius – Welsh" lang="cy" hreflang="cy" data-title="Stribyn Möbius" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/M%C3%B6biusb%C3%A5nd" title="Möbiusbånd – Danish" lang="da" hreflang="da" data-title="Möbiusbånd" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – German" lang="de" hreflang="de" data-title="Möbiusband" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/M%C3%B6biuse_leht" title="Möbiuse leht – Estonian" lang="et" hreflang="et" data-title="Möbiuse leht" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CF%89%CF%81%CE%AF%CE%B4%CE%B1_%CF%84%CE%BF%CF%85_%CE%9C%CE%AD%CE%BC%CF%80%CE%B9%CE%BF%CF%85%CF%82" title="Λωρίδα του Μέμπιους – Greek" lang="el" hreflang="el" data-title="Λωρίδα του Μέμπιους" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Banda_de_M%C3%B6bius" title="Banda de Möbius – Spanish" lang="es" hreflang="es" data-title="Banda de Möbius" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rubando_de_M%C3%B6bius" title="Rubando de Möbius – Esperanto" lang="eo" hreflang="eo" data-title="Rubando de Möbius" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Moebius_banda" title="Moebius banda – Basque" lang="eu" hreflang="eu" data-title="Moebius banda" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%88%D8%A7%D8%B1_%D9%85%D9%88%D8%A8%DB%8C%D9%88%D8%B3" title="نوار موبیوس – Persian" lang="fa" hreflang="fa" data-title="نوار موبیوس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ruban_de_M%C3%B6bius" title="Ruban de Möbius – French" lang="fr" hreflang="fr" data-title="Ruban de Möbius" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/M%C3%B6biusb%C3%A2n" title="Möbiusbân – Western Frisian" lang="fy" hreflang="fy" data-title="Möbiusbân" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Stiall_M%C3%B6bius" title="Stiall Möbius – Irish" lang="ga" hreflang="ga" data-title="Stiall Möbius" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Banda_de_M%C3%B6bius" title="Banda de Möbius – Galician" lang="gl" hreflang="gl" data-title="Banda de Möbius" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9C%D3%A9%D0%B1%D0%B8%D1%83%D1%81%D0%B8%D0%BD_%D0%BA%D2%AF%D1%81%D0%BC" title="Мөбиусин күсм – Kalmyk" lang="xal" hreflang="xal" data-title="Мөбиусин күсм" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AB%BC%EB%B9%84%EC%9A%B0%EC%8A%A4%EC%9D%98_%EB%9D%A0" title="뫼비우스의 띠 – Korean" lang="ko" hreflang="ko" data-title="뫼비우스의 띠" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%B5%D5%B8%D5%A2%D5%AB%D5%B8%D6%82%D5%BD%D5%AB_%D5%A9%D5%A5%D6%80%D5%A9" title="Մյոբիուսի թերթ – Armenian" lang="hy" hreflang="hy" data-title="Մյոբիուսի թերթ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/M%C3%B6biusova_vrpca" title="Möbiusova vrpca – Croatian" lang="hr" hreflang="hr" data-title="Möbiusova vrpca" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Mobius-strio" title="Mobius-strio – Ido" lang="io" hreflang="io" data-title="Mobius-strio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pita_M%C3%B6bius" title="Pita Möbius – Indonesian" lang="id" hreflang="id" data-title="Pita Möbius" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Banda_de_M%C3%B6bius" title="Banda de Möbius – Interlingua" lang="ia" hreflang="ia" data-title="Banda de Möbius" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Nastro_di_M%C3%B6bius" title="Nastro di Möbius – Italian" lang="it" hreflang="it" data-title="Nastro di Möbius" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%91%D7%A2%D7%AA_%D7%9E%D7%91%D7%99%D7%95%D7%A1" title="טבעת מביוס – Hebrew" lang="he" hreflang="he" data-title="טבעת מביוס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B1%D0%B8%D1%83%D1%81_%D0%B6%D0%B0%D0%BF%D1%8B%D1%80%D0%B0%D2%93%D1%8B" title="Мбиус жапырағы – Kazakh" lang="kk" hreflang="kk" data-title="Мбиус жапырағы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Taenia_Moebii" title="Taenia Moebii – Latin" lang="la" hreflang="la" data-title="Taenia Moebii" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/M%C4%93biusa_lente" title="Mēbiusa lente – Latvian" lang="lv" hreflang="lv" data-title="Mēbiusa lente" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/M%C3%B6biusschleef" title="Möbiusschleef – Luxembourgish" lang="lb" hreflang="lb" data-title="Möbiusschleef" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/dasri_pe_la_.mobius." title="dasri pe la .mobius. – Lojban" lang="jbo" hreflang="jbo" data-title="dasri pe la .mobius." data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C3%B6bius-szalag" title="Möbius-szalag – Hungarian" lang="hu" hreflang="hu" data-title="Möbius-szalag" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jalur_M%C3%B6bius" title="Jalur Möbius – Malay" lang="ms" hreflang="ms" data-title="Jalur Möbius" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – Dutch" lang="nl" hreflang="nl" data-title="Möbiusband" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A1%E3%83%93%E3%82%A6%E3%82%B9%E3%81%AE%E5%B8%AF" title="メビウスの帯 – Japanese" lang="ja" hreflang="ja" data-title="メビウスの帯" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/M%C3%B6bius-b%C3%A5nd" title="Möbius-bånd – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Möbius-bånd" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Möbiusband" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Mobius-bende" title="Mobius-bende – Novial" lang="nov" hreflang="nov" data-title="Mobius-bende" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wst%C4%99ga_M%C3%B6biusa" title="Wstęga Möbiusa – Polish" lang="pl" hreflang="pl" data-title="Wstęga Möbiusa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fita_de_M%C3%B6bius" title="Fita de Möbius – Portuguese" lang="pt" hreflang="pt" data-title="Fita de Möbius" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Banda_lui_M%C3%B6bius" title="Banda lui Möbius – Romanian" lang="ro" hreflang="ro" data-title="Banda lui Möbius" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B5%D0%BD%D1%82%D0%B0_%D0%9C%D1%91%D0%B1%D0%B8%D1%83%D1%81%D0%B0" title="Лента Мёбиуса – Russian" lang="ru" hreflang="ru" data-title="Лента Мёбиуса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Nastru_di_M%C3%B6bius" title="Nastru di Möbius – Sicilian" lang="scn" hreflang="scn" data-title="Nastru di Möbius" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/M%C3%B6bius_strip" title="Möbius strip – Simple English" lang="en-simple" hreflang="en-simple" data-title="Möbius strip" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/M%C3%B6biov_list" title="Möbiov list – Slovak" lang="sk" hreflang="sk" data-title="Möbiov list" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/M%C3%B6biusov_trak" title="Möbiusov trak – Slovenian" lang="sl" hreflang="sl" data-title="Möbiusov trak" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Faborka_M%C3%B6biusa" title="Faborka Möbiusa – Silesian" lang="szl" hreflang="szl" data-title="Faborka Möbiusa" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B1%D0%B8%D1%98%D1%83%D1%81%D0%BE%D0%B2%D0%B0_%D1%82%D1%80%D0%B0%D0%BA%D0%B0" title="Мебијусова трака – Serbian" lang="sr" hreflang="sr" data-title="Мебијусова трака" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/M%C3%B6biuksen_nauha" title="Möbiuksen nauha – Finnish" lang="fi" hreflang="fi" data-title="Möbiuksen nauha" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – Swedish" lang="sv" hreflang="sv" data-title="Möbiusband" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%8B%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B8%E0%AF%8D_%E0%AE%A8%E0%AE%BE%E0%AE%9F%E0%AE%BE" title="மோபியஸ் நாடா – Tamil" lang="ta" hreflang="ta" data-title="மோபியஸ் நாடா" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%96%E0%B8%9A%E0%B9%80%E0%B8%A1%E0%B8%AD%E0%B8%9A%E0%B8%B4%E0%B8%AD%E0%B8%B8%E0%B8%AA" title="แถบเมอบิอุส – Thai" lang="th" hreflang="th" data-title="แถบเมอบิอุส" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/M%C3%B6bius_%C5%9Feridi" title="Möbius şeridi – Turkish" lang="tr" hreflang="tr" data-title="Möbius şeridi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D1%96%D1%87%D0%BA%D0%B0_%D0%9C%D0%B5%D0%B1%D1%96%D1%83%D1%81%D0%B0" title="Стрічка Мебіуса – Ukrainian" lang="uk" hreflang="uk" data-title="Стрічка Мебіуса" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-good-star" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Good_articles*" title="This is a good article. Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Non-orientable surface with one edge</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:M%C3%B6bius_Strip.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/M%C3%B6bius_Strip.jpg/280px-M%C3%B6bius_Strip.jpg" decoding="async" width="280" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/M%C3%B6bius_Strip.jpg/420px-M%C3%B6bius_Strip.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/M%C3%B6bius_Strip.jpg/560px-M%C3%B6bius_Strip.jpg 2x" data-file-width="2000" data-file-height="1240" /></a><figcaption>A Möbius strip made with paper and adhesive tape</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a Möbius strip, Möbius band, or Möbius loop<a href="#cite_note-2">[a]</a> is a <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a> that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by <a href="/wiki/Johann_Benedict_Listing" title="Johann Benedict Listing">Johann Benedict Listing</a> and <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a> in 1858, but it had already appeared in <a href="/wiki/Ancient_Rome" title="Ancient Rome">Roman</a> mosaics from the third century <a href="/wiki/Common_Era" title="Common Era">CE</a>. The Möbius strip is a <a href="/wiki/Orientability" title="Orientability">non-orientable</a> surface, meaning that within it one cannot consistently distinguish <a href="/wiki/Clockwise" title="Clockwise">clockwise</a> from counterclockwise turns. Every non-orientable surface contains a Möbius strip. As an abstract <a href="/wiki/Topological_space" title="Topological space">topological space</a>, the Möbius strip can be embedded into three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knotted</a> centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">topologically equivalent</a>. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a single <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary curve</a>. Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a <a href="/wiki/Ruled_surface" title="Ruled surface">ruled surface</a> by a line segment rotating in a rotating plane, with or without self-crossings. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a <a href="/wiki/Developable_surface" title="Developable surface">developable surface</a> or be <a href="/wiki/Mathematics_of_paper_folding#Flat_folding" title="Mathematics of paper folding">folded flat</a>; the flattened Möbius strips include the <a href="/wiki/Trihexaflexagon" class="mw-redirect" title="Trihexaflexagon">trihexaflexagon</a>. The Sudanese Möbius strip is a <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surface</a> in a <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a>, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary. A Möbius strip without its boundary, called an open Möbius strip, can form <a href="/wiki/Uniformization_theorem" title="Uniformization theorem">surfaces of constant curvature</a>. Certain highly symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip. The many applications of Möbius strips include <a href="/wiki/Belt_(mechanical)" title="Belt (mechanical)">mechanical belts</a> that wear evenly on both sides, dual-track <a href="/wiki/Roller_coaster" title="Roller coaster">roller coasters</a> whose carriages alternate between the two tracks, and <a href="/wiki/World_map" title="World map">world maps</a> printed so that <a href="/wiki/Antipodes" title="Antipodes">antipodes</a> appear opposite each other. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in <a href="/wiki/Social_choice_theory" title="Social choice theory">social choice theory</a>. In popular culture, Möbius strips appear in artworks by <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>, <a href="/wiki/Max_Bill" title="Max Bill">Max Bill</a>, and others, and in the design of the <a href="/wiki/Recycling_symbol" title="Recycling symbol">recycling symbol</a>. Many architectural concepts have been inspired by the Möbius strip, including the building design for the <a href="/wiki/NASCAR_Hall_of_Fame" title="NASCAR Hall of Fame">NASCAR Hall of Fame</a>. Performers including <a href="/wiki/Harry_Blackstone_Sr." title="Harry Blackstone Sr.">Harry Blackstone Sr.</a> and <a href="/wiki/Thomas_Nelson_Downs" class="mw-redirect" title="Thomas Nelson Downs">Thomas Nelson Downs</a> have based stage magic tricks on the properties of the Möbius strip. The <a href="/wiki/Canon_(music)" title="Canon (music)">canons</a> of <a href="/wiki/J._S._Bach" class="mw-redirect" title="J. S. Bach">J. S. Bach</a> have been analyzed using Möbius strips. Many works of <a href="/wiki/Speculative_fiction" title="Speculative fiction">speculative fiction</a> feature Möbius strips; more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=1" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/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 img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:472px;max-width:472px"><div class="trow"><div class="tsingle" style="width:274px;max-width:274px"><div class="thumbimage" style="height:248px;overflow:hidden"><a href="/wiki/File:Aion_mosaic_Glyptothek_Munich_W504.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Aion_mosaic_Glyptothek_Munich_W504.jpg/272px-Aion_mosaic_Glyptothek_Munich_W504.jpg" decoding="async" width="272" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Aion_mosaic_Glyptothek_Munich_W504.jpg/408px-Aion_mosaic_Glyptothek_Munich_W504.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Aion_mosaic_Glyptothek_Munich_W504.jpg/544px-Aion_mosaic_Glyptothek_Munich_W504.jpg 2x" data-file-width="2120" data-file-height="1938" /></a></div><div class="thumbcaption">Mosaic from ancient <a href="/wiki/Sentinum" title="Sentinum">Sentinum</a> depicting <a href="/wiki/Aion_(deity)" title="Aion (deity)">Aion</a> holding a Möbius strip</div></div><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:248px;overflow:hidden"><a href="/wiki/File:Al-Jazari_Automata_1205.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Al-Jazari_Automata_1205.jpg/192px-Al-Jazari_Automata_1205.jpg" decoding="async" width="192" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Al-Jazari_Automata_1205.jpg/288px-Al-Jazari_Automata_1205.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/9/96/Al-Jazari_Automata_1205.jpg 2x" data-file-width="292" data-file-height="379" /></a></div><div class="thumbcaption"><a href="/wiki/Chain_pump" title="Chain pump">Chain pump</a> with a Möbius drive chain, by <a href="/wiki/Ismail_al-Jazari" title="Ismail al-Jazari">Ismail al-Jazari</a> (1206)</div></div></div></div></div> The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians <a href="/wiki/Johann_Benedict_Listing" title="Johann Benedict Listing">Johann Benedict Listing</a> and <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a> in 1858.<a href="#cite_note-pickover-3">[2]</a> However, it had been known long before, both as a physical object and in artistic depictions; in particular, it can be seen in several Roman mosaics from the third century CE.<a href="#cite_note-roman-4">[3]</a><a href="#cite_note-ancient-5">[4]</a> In many cases these merely depict coiled ribbons as boundaries. When the number of coils is odd, these ribbons are Möbius strips, but for an even number of coils they are topologically equivalent to <a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">untwisted rings</a>. Therefore, whether the ribbon is a Möbius strip may be coincidental, rather than a deliberate choice. In at least one case, a ribbon with different colors on different sides was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did not match up.<a href="#cite_note-roman-4">[3]</a> Another mosaic from the town of <a href="/wiki/Sentinum" title="Sentinum">Sentinum</a> (depicted) shows the <a href="/wiki/Zodiac" title="Zodiac">zodiac</a>, held by the god <a href="/wiki/Aion_(deity)" title="Aion (deity)">Aion</a>, as a band with only a single twist. There is no clear evidence that the one-sidedness of this visual representation of celestial time was intentional; it could have been chosen merely as a way to make all of the signs of the zodiac appear on the visible side of the strip. Some other ancient depictions of the <a href="/wiki/Ourobouros" class="mw-redirect" title="Ourobouros">ourobouros</a> or of <a href="/wiki/Lemniscate" title="Lemniscate">figure-eight</a>-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is unclear.<a href="#cite_note-ancient-5">[4]</a> Independently of the mathematical tradition, machinists have long known that <a href="/wiki/Belt_(mechanical)" title="Belt (mechanical)">mechanical belts</a> wear half as quickly when they form Möbius strips, because they use the entire surface of the belt rather than only the inner surface of an untwisted belt.<a href="#cite_note-roman-4">[3]</a> Additionally, such a belt may be less prone to curling from side to side. An early written description of this technique dates to 1871, which is after the first mathematical publications regarding the Möbius strip. Much earlier, an image of a <a href="/wiki/Chain_pump" title="Chain pump">chain pump</a> in a work of <a href="/wiki/Ismail_al-Jazari" title="Ismail al-Jazari">Ismail al-Jazari</a> from 1206 depicts a Möbius strip configuration for its drive chain.<a href="#cite_note-ancient-5">[4]</a> Another use of this surface was made by seamstresses in Paris (at an unspecified date): they initiated novices by requiring them to stitch a Möbius strip as a collar onto a garment.<a href="#cite_note-roman-4">[3]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=2" title="Edit section: Properties">edit</a>]</div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Fiddler_crab_mobius_strip.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Fiddler_crab_mobius_strip.gif/170px-Fiddler_crab_mobius_strip.gif" decoding="async" width="170" height="262" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Fiddler_crab_mobius_strip.gif/255px-Fiddler_crab_mobius_strip.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b7/Fiddler_crab_mobius_strip.gif 2x" data-file-width="317" data-file-height="489" /></a><figcaption>A 2D object traversing once around the Möbius strip returns in mirrored form</figcaption></figure> The Möbius strip has several curious properties. It is a <a href="/wiki/Orientability" title="Orientability">non-orientable surface</a>: if an asymmetric two-dimensional object slides one time around the strip, it returns to its starting position as its mirror image. In particular, a curved arrow pointing clockwise (↻) would return as an arrow pointing counterclockwise (↺), implying that, within the Möbius strip, it is impossible to consistently define what it means to be clockwise or counterclockwise. It is the simplest non-orientable surface: any other surface is non-orientable if and only if it has a Möbius strip as a subset.<a href="#cite_note-chirality-6">[5]</a> Relatedly, when embedded into <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, the Möbius strip has only one side. A three-dimensional object that slides one time around the surface of the strip is not mirrored, but instead returns to the same point of the strip on what appears locally to be its other side, showing that both positions are really part of a single side. This behavior is different from familiar <a href="/wiki/Orientable_surface" class="mw-redirect" title="Orientable surface">orientable surfaces</a> in three dimensions such as those modeled by flat sheets of paper, cylindrical drinking straws, or hollow balls, for which one side of the surface is not connected to the other.<a href="#cite_note-FOOTNOTEPickover20058–9-7">[6]</a> However, this is a property of its embedding into space rather than an intrinsic property of the Möbius strip itself: there exist other topological spaces in which the Möbius strip can be embedded so that it has two sides.<a href="#cite_note-woll-8">[7]</a> For instance, if the front and back faces of a cube are glued to each other with a left-right mirror reflection, the result is a three-dimensional topological space (the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of a Möbius strip with an interval) in which the top and bottom halves of the cube can be separated from each other by a two-sided Möbius strip.<a href="#cite_note-10">[b]</a> In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed an <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable set</a> of <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> copies into three-dimensional space, only a countable number of Möbius strips can be simultaneously embedded.<a href="#cite_note-frolkina-11">[9]</a><a href="#cite_note-defy-12">[10]</a><a href="#cite_note-melikhov-13">[11]</a> A path along the edge of a Möbius strip, traced until it returns to its starting point on the edge, includes all boundary points of the Möbius strip in a single continuous curve. For a Möbius strip formed by gluing and twisting a rectangle, it has twice the length of the centerline of the strip. In this sense, the Möbius strip is different from an untwisted ring and like a circular disk in having only one boundary.<a href="#cite_note-FOOTNOTEPickover20058–9-7">[6]</a> A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a> object with right- or left-handedness.<a href="#cite_note-FOOTNOTEPickover200552-14">[12]</a> Möbius strips with odd numbers of half-twists greater than one, or that are knotted before gluing, are distinct as embedded subsets of three-dimensional space, even though they are all equivalent as two-dimensional topological surfaces.<a href="#cite_note-FOOTNOTEPickover200512-15">[13]</a> More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine the same knot and they have the same number of twists as each other.<a href="#cite_note-kyle-16">[14]</a> With an even number of twists, however, one obtains a different topological surface, called the <a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a>.<a href="#cite_note-FOOTNOTEPickover200511-17">[15]</a> The Möbius strip can be continuously transformed into its centerline, by making it narrower while fixing the points on the centerline. This transformation is an example of a <a href="/wiki/Deformation_retraction" class="mw-redirect" title="Deformation retraction">deformation retraction</a>, and its existence means that the Möbius strip has many of the same properties as its centerline, which is topologically a circle. In particular, its <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> is the same as the fundamental group of a circle, an <a href="/wiki/Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a>. Therefore, paths on the Möbius strip that start and end at the same point can be distinguished topologically (up to <a href="/wiki/Homotopy" title="Homotopy">homotopy</a>) only by the number of times they loop around the strip.<a href="#cite_note-massey-18">[16]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:472px;max-width:472px"><div class="trow"><div class="tsingle" style="width:240px;max-width:240px"><div class="thumbimage" style="height:151px;overflow:hidden"><a href="/wiki/File:Moebiusband-1s.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Moebiusband-1s.svg/238px-Moebiusband-1s.svg.png" decoding="async" width="238" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Moebiusband-1s.svg/357px-Moebiusband-1s.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Moebiusband-1s.svg/476px-Moebiusband-1s.svg.png 2x" data-file-width="444" data-file-height="283" /></a></div><div class="thumbcaption">Cutting the centerline produces a double-length two-sided (non-Möbius) strip</div></div><div class="tsingle" style="width:228px;max-width:228px"><div class="thumbimage" style="height:151px;overflow:hidden"><a href="/wiki/File:Moebiusband-2s.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Moebiusband-2s.svg/226px-Moebiusband-2s.svg.png" decoding="async" width="226" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Moebiusband-2s.svg/339px-Moebiusband-2s.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Moebiusband-2s.svg/452px-Moebiusband-2s.svg.png 2x" data-file-width="463" data-file-height="310" /></a></div><div class="thumbcaption">A single off-center cut produces a Möbius strip (purple) linked with a double-length two-sided strip</div></div></div></div></div> Cutting a Möbius strip along the centerline with a pair of scissors yields one long strip with four half-twists in it (relative to an untwisted annulus or cylinder) rather than two separate strips. Two of the half-twists come from the fact that this thinner strip goes two times through the half-twist in the original Möbius strip, and the other two come from the way the two halves of the thinner strip wrap around each other. The result is not a Möbius strip, but instead is topologically equivalent to a cylinder. Cutting this double-twisted strip again along its centerline produces two linked double-twisted strips. If, instead, a Möbius strip is cut lengthwise, a third of the way across its width, it produces two linked strips. One of the two is a central, thinner, Möbius strip, while the other has two half-twists.<a href="#cite_note-FOOTNOTEPickover20058–9-7">[6]</a> These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called paradromic rings.<a href="#cite_note-rouseball-19">[17]</a><a href="#cite_note-paradromic-20">[18]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:472px;max-width:472px"><div class="trow"><div class="tsingle" style="width:234px;max-width:234px"><div class="thumbimage" style="height:174px;overflow:hidden"><a href="/wiki/File:Tietze-Moebius.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Tietze-Moebius.svg/232px-Tietze-Moebius.svg.png" decoding="async" width="232" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Tietze-Moebius.svg/348px-Tietze-Moebius.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Tietze-Moebius.svg/464px-Tietze-Moebius.svg.png 2x" data-file-width="735" data-file-height="552" /></a></div><div class="thumbcaption">Subdivision into six mutually-adjacent regions, bounded by <a href="/wiki/Tietze%27s_graph" title="Tietze's graph">Tietze's graph</a></div></div><div class="tsingle" style="width:234px;max-width:234px"><div class="thumbimage" style="height:174px;overflow:hidden"><a href="/wiki/File:3_utilities_problem_moebius.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/3_utilities_problem_moebius.svg/232px-3_utilities_problem_moebius.svg.png" decoding="async" width="232" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/3_utilities_problem_moebius.svg/348px-3_utilities_problem_moebius.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/3_utilities_problem_moebius.svg/464px-3_utilities_problem_moebius.svg.png 2x" data-file-width="512" data-file-height="384" /></a></div><div class="thumbcaption">Solution to the <a href="/wiki/Three_utilities_problem" title="Three utilities problem">three utilities problem</a> on a Möbius strip</div></div></div></div></div> The Möbius strip can be cut into six mutually-adjacent regions, showing that maps on the surface of the Möbius strip can sometimes require six colors, in contrast to the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> for the plane.<a href="#cite_note-tietze-21">[19]</a> Six colors are always enough. This result is part of the <a href="/wiki/Ringel%E2%80%93Youngs_theorem" class="mw-redirect" title="Ringel–Youngs theorem">Ringel–Youngs theorem</a>, which states how many colors each topological surface needs.<a href="#cite_note-ringel-youngs-22">[20]</a> The edges and vertices of these six regions form <a href="/wiki/Tietze%27s_graph" title="Tietze's graph">Tietze's graph</a>, which is a <a href="/wiki/Dual_graph" title="Dual graph">dual graph</a> on this surface for the six-vertex <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a> but cannot be <a href="/wiki/Planar_graph" title="Planar graph">drawn without crossings on a plane</a>. Another family of graphs that can be <a href="/wiki/Graph_embedding" title="Graph embedding">embedded</a> on the Möbius strip, but not on the plane, are the <a href="/wiki/M%C3%B6bius_ladder" title="Möbius ladder">Möbius ladders</a>, the boundaries of subdivisions of the Möbius strip into rectangles meeting end-to-end.<a href="#cite_note-jab-rad-saz-23">[21]</a> These include the utility graph, a six-vertex <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite graph</a> whose embedding into the Möbius strip shows that, unlike in the plane, the <a href="/wiki/Three_utilities_problem" title="Three utilities problem">three utilities problem</a> can be solved on a transparent Möbius strip.<a href="#cite_note-larsen-24">[22]</a> The <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of the Möbius strip is <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a>, meaning that for any subdivision of the strip by vertices and edges into regions, the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> of vertices, edges, and regions satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-E+F=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V-E+F=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3ec2cf77a129f5cef68b1f6f88b84f400762f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.245ex; height:2.343ex;" alt="{\displaystyle V-E+F=0}">. For instance, Tietze's graph has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}"> vertices, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 18}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>18</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b85947866bcf8f0368c690f27b785fb20cdd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 18}"> edges, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"> regions; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12-18+6=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>−</mo> <mn>18</mn> <mo>+</mo> <mn>6</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12-18+6=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd0cd2c4e5a33f060e305809f6d37d7295ef7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.754ex; height:2.343ex;" alt="{\displaystyle 12-18+6=0}">.<a href="#cite_note-tietze-21">[19]</a> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=3" title="Edit section: Constructions">edit</a>]</div> There are many different ways of defining geometric surfaces with the topology of the Möbius strip, yielding realizations with additional geometric properties. <div class="mw-heading mw-heading3"><h3 id="Sweeping_a_line_segment">Sweeping a line segment</h3>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=4" title="Edit section: Sweeping a line segment">edit</a>]</div> <div class="thumb tright"> <div class="thumbinner" style="width: 187px;"> <div class="thumbimage" style="width: 185px; height: 150px; overflow: hidden;"> <div style="position: relative; top: -115px; left: -105px; width: 400px"><div class="noresize"><a href="/wiki/File:Mobius_strip.gif" class="mw-file-description"><img alt="A Möbius strip swept out by a rotating line segment in a rotating plane" src="//upload.wikimedia.org/wikipedia/commons/7/73/Mobius_strip.gif" decoding="async" width="400" height="379" class="mw-file-element" data-file-width="392" data-file-height="371" /></a></div></div> </div> <div class="thumbcaption"> <div class="magnify"><a href="/wiki/File:Mobius_strip.gif" title="File:Mobius strip.gif"> </a></div>A Möbius strip swept out by a rotating line segment in a rotating plane </div> </div> </div> <div class="thumb tright"> <div class="thumbinner" style="width: 242px;"> <div class="thumbimage" style="width: 240px; height: 240px; overflow: hidden;"> <div style="position: relative; top: -60px; left: -60px; width: 360px"><div class="noresize"><a href="/wiki/File:Plucker%27s_conoid_(n%3D2).gif" class="mw-file-description"><img alt="Plücker's conoid swept out by a different motion of a line segment" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Plucker%27s_conoid_%28n%3D2%29.gif/360px-Plucker%27s_conoid_%28n%3D2%29.gif" decoding="async" width="360" height="396" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d7/Plucker%27s_conoid_%28n%3D2%29.gif 1.5x" data-file-width="394" data-file-height="433" /></a></div></div> </div> <div class="thumbcaption"> <div class="magnify"><a href="/wiki/File:Plucker%27s_conoid_(n%3D2).gif" title="File:Plucker's conoid (n=2).gif"> </a></div><a href="/wiki/Pl%C3%BCcker%27s_conoid" title="Plücker's conoid">Plücker's conoid</a> swept out by a different motion of a line segment </div> </div> </div> One way to embed the Möbius strip in three-dimensional Euclidean space is to sweep it out by a line segment rotating in a plane, which in turn rotates around one of its lines.<a href="#cite_note-maschke-25">[23]</a> For the swept surface to meet up with itself after a half-twist, the line segment should rotate around its center at half the angular velocity of the plane's rotation. This can be described as a <a href="/wiki/Parametric_surface" title="Parametric surface">parametric surface</a> defined by equations for the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of its points, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{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="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>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{2}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b53cbccc386f8f5ca10eb9b1223ac8eee95c31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:30.523ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{2}}\\\end{aligned}}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq u<2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>u</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq u<2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8227e96f13d7d97a7dbe7a215b83f5ec070ce25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.184ex; height:2.343ex;" alt="{\displaystyle 0\leq u<2\pi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq v\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>v</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq v\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78efe5b7c254d5806ccaf1ba44f98b7a975ab91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.458ex; height:2.343ex;" alt="{\displaystyle -1\leq v\leq 1}">, where one parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> describes the rotation angle of the plane around its central axis and the other parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> describes the position of a point along the rotating line segment. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}">-plane and is centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c6fd55d5621fd95ca93549660fbb355fd9bd22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,0)}">.<a href="#cite_note-parameterization-26">[24]</a> The same method can produce Möbius strips with any odd number of half-twists, by rotating the segment more quickly in its plane. The rotating segment sweeps out a circular disk in the plane that it rotates within, and the Möbius strip that it generates forms a slice through the <a href="/wiki/Solid_torus" title="Solid torus">solid torus</a> swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains connected.<a href="#cite_note-split-tori-27">[25]</a> A line or line segment swept in a different motion, rotating in a horizontal plane around the origin as it moves up and down, forms <a href="/wiki/Pl%C3%BCcker%27s_conoid" title="Plücker's conoid">Plücker's conoid</a> or cylindroid, an algebraic <a href="/wiki/Ruled_surface" title="Ruled surface">ruled surface</a> in the form of a self-crossing Möbius strip.<a href="#cite_note-francis-28">[26]</a> It has applications in the design of <a href="/wiki/Gear" title="Gear">gears</a>.<a href="#cite_note-dooner-seirig-29">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Polyhedral_surfaces_and_flat_foldings">Polyhedral surfaces and flat foldings</h3>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=5" title="Edit section: Polyhedral surfaces and flat foldings">edit</a>]</div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Flexagon.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Flexagon.gif/130px-Flexagon.gif" decoding="async" width="130" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Flexagon.gif/195px-Flexagon.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a1/Flexagon.gif 2x" data-file-width="240" data-file-height="248" /></a><figcaption>Trihexaflexagon being flexed</figcaption></figure> A strip of paper can form a <a href="/wiki/Mathematics_of_paper_folding#Flat_folding" title="Mathematics of paper folding">flattened</a> Möbius strip in the plane by folding it at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c42292485b447b7f627a7accd90d5b439c11d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 60^{\circ }}"> angles so that its center line lies along an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>, and attaching the ends. The shortest strip for which this is possible consists of three equilateral triangles, folded at the edges where two triangles meet. Its <a href="/wiki/Aspect_ratio" title="Aspect ratio">aspect ratio</a> – the ratio of the strip's length<a href="#cite_note-30">[c]</a> to its width – is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}\approx 1.73}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.73</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}\approx 1.73}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b339ab1e263eb16e37516f58c367e2210afb5c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.331ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}\approx 1.73}">, and the same folding method works for any larger aspect ratio.<a href="#cite_note-barr-31">[28]</a><a href="#cite_note-fuchs-tabachnikov-32">[29]</a> For a strip of nine equilateral triangles, the result is a <a href="/wiki/Trihexaflexagon" class="mw-redirect" title="Trihexaflexagon">trihexaflexagon</a>, which can be flexed to reveal different parts of its surface.<a href="#cite_note-pook-33">[30]</a> For strips too short to apply this method directly, one can first "accordion fold" the strip in its wide direction back and forth using an even number of folds. With two folds, for example, a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}"> strip would become a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times {\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times {\tfrac {1}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf25dcdef83f6c2f9a97989f28f58de45b44033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.661ex; height:3.676ex;" alt="{\displaystyle 1\times {\tfrac {1}{3}}}"> folded strip whose <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross section</a> is in the shape of an 'N' and would remain an 'N' after a half-twist. The narrower accordion-folded strip can then be folded and joined in the same way that a longer strip would be.<a href="#cite_note-barr-31">[28]</a><a href="#cite_note-fuchs-tabachnikov-32">[29]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:191px;max-width:191px"><div class="thumbimage" style="height:188px;overflow:hidden"><a href="/wiki/File:5-vertex_polyhedral_M%C3%B6bius_strip.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/5-vertex_polyhedral_M%C3%B6bius_strip.svg/189px-5-vertex_polyhedral_M%C3%B6bius_strip.svg.png" decoding="async" width="189" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/5-vertex_polyhedral_M%C3%B6bius_strip.svg/284px-5-vertex_polyhedral_M%C3%B6bius_strip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/5-vertex_polyhedral_M%C3%B6bius_strip.svg/378px-5-vertex_polyhedral_M%C3%B6bius_strip.svg.png 2x" data-file-width="512" data-file-height="512" /></a></div></div><div class="tsingle" style="width:197px;max-width:197px"><div class="thumbimage" style="height:188px;overflow:hidden"><a href="/wiki/File:Pentagonal_M%C3%B6bius_strip.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Pentagonal_M%C3%B6bius_strip.svg/195px-Pentagonal_M%C3%B6bius_strip.svg.png" decoding="async" width="195" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Pentagonal_M%C3%B6bius_strip.svg/293px-Pentagonal_M%C3%B6bius_strip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Pentagonal_M%C3%B6bius_strip.svg/390px-Pentagonal_M%C3%B6bius_strip.svg.png 2x" data-file-width="522" data-file-height="504" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Five-vertex polyhedral and flat-folded Möbius strips</div></div></div></div> The Möbius strip can also be embedded as a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedral surface</a> in space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than the <a href="/wiki/Cylinder" title="Cylinder">cylinder</a>, which requires six triangles and six vertices, even when represented more abstractly as a <a href="/wiki/Abstract_simplicial_complex" title="Abstract simplicial complex">simplicial complex</a>.<a href="#cite_note-9vertex-34">[31]</a><a href="#cite_note-35">[d]</a> A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of a <a href="/wiki/5-cell" title="5-cell">four-dimensional regular simplex</a>. This four-dimensional polyhedral Möbius strip is the only tight Möbius strip, one that is fully four-dimensional and for which all cuts by <a href="/wiki/Hyperplane" title="Hyperplane">hyperplanes</a> separate it into two parts that are topologically equivalent to disks or circles.<a href="#cite_note-kuiper-36">[32]</a> Other polyhedral embeddings of Möbius strips include one with four convex <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilaterals</a> as faces, another with three non-convex quadrilateral faces,<a href="#cite_note-szilassi-37">[33]</a> and one using the vertices and center point of a regular <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, with a triangular boundary.<a href="#cite_note-tuckerman-38">[34]</a> Every abstract triangulation of the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its faces;<a href="#cite_note-bon-nak-39">[35]</a> an example is the six-vertex projective plane obtained by adding one vertex to the five-vertex Möbius strip, connected by triangles to each of its boundary edges.<a href="#cite_note-9vertex-34">[31]</a> However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral surface.<a href="#cite_note-brehm-40">[36]</a> To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the triangulation.<a href="#cite_note-nak-tsu-41">[37]</a> <div class="mw-heading mw-heading3"><h3 id="Smoothly_embedded_rectangles">Smoothly embedded rectangles</h3>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=6" title="Edit section: Smoothly embedded rectangles">edit</a>]</div> A rectangular Möbius strip, made by attaching the ends of a paper rectangle, can be embedded smoothly into three-dimensional space whenever its aspect ratio is greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}\approx 1.73}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.73</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}\approx 1.73}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b339ab1e263eb16e37516f58c367e2210afb5c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.331ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}\approx 1.73}">, the same ratio as for the flat-folded equilateral-triangle version of the Möbius strip.<a href="#cite_note-sadowsky-translation-42">[38]</a> This flat triangular embedding can lift to a smooth<a href="#cite_note-44">[e]</a> embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of the planes.<a href="#cite_note-sadowsky-translation-42">[38]</a> Mathematically, a smoothly embedded sheet of paper can be modeled as a <a href="/wiki/Developable_surface" title="Developable surface">developable surface</a>, that can bend but cannot stretch.<a href="#cite_note-bartels-hornung-43">[39]</a><a href="#cite_note-starostin-vdh-45">[40]</a> As its aspect ratio decreases toward <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}">, all smooth embeddings seem to approach the same triangular form.<a href="#cite_note-darkside-46">[41]</a> The lengthwise folds of an accordion-folded flat Möbius strip prevent it from forming a three-dimensional embedding in which the layers are separated from each other and bend smoothly without crumpling or stretching away from the folds.<a href="#cite_note-fuchs-tabachnikov-32">[29]</a> Instead, unlike in the flat-folded case, there is a lower limit to the aspect ratio of smooth rectangular Möbius strips. Their aspect ratio cannot be less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2\approx 1.57}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>≈</mo> <mn>1.57</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2\approx 1.57}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a36ff9e1888aa40c1aded554c531ed6c74ac50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.89ex; height:2.843ex;" alt="{\displaystyle \pi /2\approx 1.57}">, even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this bound.<a href="#cite_note-fuchs-tabachnikov-32">[29]</a><a href="#cite_note-halpern-weaver-47">[42]</a> Without self-intersections, the aspect ratio must be at least<a href="#cite_note-schwartz-48">[43]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{3}}{\sqrt {3+2{\sqrt {3}}}}\approx 1.695.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </msqrt> </mrow> <mo>≈</mo> <mn>1.695.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}{\sqrt {3+2{\sqrt {3}}}}\approx 1.695.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e1500db8ca0318d54d02088938ab098970637c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.628ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}{\sqrt {3+2{\sqrt {3}}}}\approx 1.695.}"> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div>Unsolved problem in mathematics:</div> <div class="unsolved-body">Can a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12\times 7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>×</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\times 7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e04a403a64170e2656d90bffc99293633e36c06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.328ex; height:2.176ex;" alt="{\displaystyle 12\times 7}"> paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in space? <a href="#cite_note-49">[f]</a></div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> For aspect ratios between this bound and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}">, it has been an open problem whether smooth embeddings, without self-intersection, exist.<a href="#cite_note-fuchs-tabachnikov-32">[29]</a><a href="#cite_note-halpern-weaver-47">[42]</a><a href="#cite_note-schwartz-48">[43]</a> In 2023, <a href="/wiki/Richard_Schwartz_(mathematician)" title="Richard Schwartz (mathematician)">Richard Schwartz</a> announced a proof that they do not exist, but this result still awaits peer review and publication.<a href="#cite_note-optimal-50">[44]</a><a href="#cite_note-crowell-51">[45]</a> If the requirement of smoothness is relaxed to allow <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a> surfaces, the <a href="/wiki/Nash%E2%80%93Kuiper_theorem" class="mw-redirect" title="Nash–Kuiper theorem">Nash–Kuiper theorem</a> implies that any two opposite edges of any rectangle can be glued to form an embedded Möbius strip, no matter how small the aspect ratio becomes.<a href="#cite_note-52">[g]</a> The limiting case, a surface obtained from an infinite strip of the plane between two parallel lines, glued with the opposite orientation to each other, is called the unbounded Möbius strip or the real <a href="/wiki/Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">tautological line bundle</a>.<a href="#cite_note-dundas-53">[46]</a> Although it has no smooth closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclidean space.<a href="#cite_note-blanusa-54">[47]</a> The minimum-energy shape of a smooth Möbius strip glued from a rectangle does not have a known analytic description, but can be calculated numerically, and has been the subject of much study in <a href="/wiki/Plate_theory" title="Plate theory">plate theory</a> since the initial work on this subject in 1930 by <a href="/wiki/Michael_Sadowsky" title="Michael Sadowsky">Michael Sadowsky</a>.<a href="#cite_note-bartels-hornung-43">[39]</a><a href="#cite_note-starostin-vdh-45">[40]</a> It is also possible to find <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surfaces</a> that contain rectangular developable Möbius strips.<a href="#cite_note-wunderlich-55">[48]</a><a href="#cite_note-schwarz-56">[49]</a> <div class="mw-heading mw-heading3"><h3 id="Making_the_boundary_circular">Making the boundary circular</h3>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=7" title="Edit section: Making the boundary circular">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:472px;max-width:472px"><div class="trow"><div class="tsingle" style="width:232px;max-width:232px"><div class="thumbimage" style="height:230px;overflow:hidden"><a href="/wiki/File:Mobius_to_Klein.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/8/87/Mobius_to_Klein.gif" decoding="async" width="230" height="230" class="mw-file-element" data-file-width="180" data-file-height="180" /></a></div><div class="thumbcaption">Gluing two Möbius strips to form a Klein bottle</div></div><div class="tsingle" style="width:236px;max-width:236px"><div class="thumbimage" style="height:230px;overflow:hidden"><a href="/wiki/File:MobiusStrip-02.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/MobiusStrip-02.png/234px-MobiusStrip-02.png" decoding="async" width="234" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/80/MobiusStrip-02.png 1.5x" data-file-width="276" data-file-height="272" /></a></div><div class="thumbcaption">A projection of the Sudanese Möbius strip</div></div></div></div></div> The edge, or <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a>, of a Möbius strip is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">topologically equivalent</a> to a <a href="/wiki/Circle" title="Circle">circle</a>. In common forms of the Möbius strip, it has a different shape from a circle, but it is <a href="/wiki/Unknot" title="Unknot">unknotted</a>, and therefore the whole strip can be stretched without crossing itself to make the edge perfectly circular.<a href="#cite_note-hilbert-cohn-vossen-57">[50]</a> One such example is based on the topology of the <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a>, a one-sided surface with no boundary that cannot be embedded into three-dimensional space, but can be <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersed</a> (allowing the surface to cross itself in certain restricted ways). A Klein bottle is the surface that results when two Möbius strips are glued together edge-to-edge, and – reversing that process – a Klein bottle can be sliced along a carefully chosen cut to produce two Möbius strips.<a href="#cite_note-spivak-58">[51]</a> For a form of the Klein bottle known as Lawson's Klein bottle, the curve along which it is sliced can be made circular, resulting in Möbius strips with circular edges.<a href="#cite_note-ddg-59">[52]</a> Lawson's Klein bottle is a self-crossing <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surface</a> in the <a href="/wiki/Unit_hypersphere" class="mw-redirect" title="Unit hypersphere">unit hypersphere</a> of 4-dimensional space, the set of points of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>,</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb68f484bde24a37cd86edde261fd61b8fee96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.652ex; height:2.843ex;" alt="{\displaystyle (\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \theta <\pi ,0\leq \phi <2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo><</mo> <mi>π</mi> <mo>,</mo> <mn>0</mn> <mo>≤</mo> <mi>ϕ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \theta <\pi ,0\leq \phi <2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f89774a6b4869ce2be7e624e745213384651de88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.055ex; height:2.509ex;" alt="{\displaystyle 0\leq \theta <\pi ,0\leq \phi <2\pi }">.<a href="#cite_note-lawson-60">[53]</a> Half of this Klein bottle, the subset with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \phi <\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>ϕ</mi> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \phi <\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14866553885d9726ee86a3f073e17663e095dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.077ex; height:2.509ex;" alt="{\displaystyle 0\leq \phi <\pi }">, gives a Möbius strip embedded in the hypersphere as a minimal surface with a <a href="/wiki/Great_circle" title="Great circle">great circle</a> as its boundary.<a href="#cite_note-schleimer-segerman-61">[54]</a> This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the 1970s.<a href="#cite_note-sudanese-62">[55]</a> Geometrically Lawson's Klein bottle can be constructed by sweeping a great circle through a great-circular motion in the 3-sphere, and the Sudanese Möbius strip is obtained by sweeping a semicircle instead of a circle, or equivalently by slicing the Klein bottle along a circle that is perpendicular to all of the swept circles.<a href="#cite_note-ddg-59">[52]</a><a href="#cite_note-franzoni-63">[56]</a> <a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its boundary.<a href="#cite_note-ddg-59">[52]</a> The most symmetric projection is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles, but produces an unbounded embedding with the projection point removed from its centerline.<a href="#cite_note-schleimer-segerman-61">[54]</a> Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978098671ba9316b57d09845e071c44f221ed52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.78ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (2)}">, the group of symmetries of a circle.<a href="#cite_note-lawson-60">[53]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cross-cap_level_sets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Cross-cap_level_sets.svg/180px-Cross-cap_level_sets.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Cross-cap_level_sets.svg/270px-Cross-cap_level_sets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Cross-cap_level_sets.svg/360px-Cross-cap_level_sets.svg.png 2x" data-file-width="306" data-file-height="306" /></a><figcaption>Schematic depiction of a cross-cap with an open bottom, showing its <a href="/wiki/Level_set" title="Level set">level sets</a>. This surface crosses itself along the vertical line segment.</figcaption></figure> The Sudanese Möbius strip extends on all sides of its boundary circle, unavoidably if the surface is to avoid crossing itself. Another form of the Möbius strip, called the cross-cap or crosscap, also has a circular boundary, but otherwise stays on only one side of the plane of this circle,<a href="#cite_note-huggett-jordan-64">[57]</a> making it more convenient for attaching onto circular holes in other surfaces. In order to do so, it crosses itself. It can be formed by removing a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> from the top of a hemisphere, orienting the edges of the quadrilateral in alternating directions, and then gluing opposite pairs of these edges consistently with this orientation.<a href="#cite_note-flapan-65">[58]</a> The two parts of the surface formed by the two glued pairs of edges cross each other with a <a href="/wiki/Pinch_point_(mathematics)" title="Pinch point (mathematics)">pinch point</a> like that of a <a href="/wiki/Whitney_umbrella" title="Whitney umbrella">Whitney umbrella</a> at each end of the crossing segment,<a href="#cite_note-richeson-66">[59]</a> the same topological structure seen in Plücker's conoid.<a href="#cite_note-francis-28">[26]</a> <div class="mw-heading mw-heading3"><h3 id="Surfaces_of_constant_curvature">Surfaces of constant curvature</h3>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=8" title="Edit section: Surfaces of constant curvature">edit</a>]</div> The open Möbius strip is the <a href="/wiki/Relative_interior" title="Relative interior">relative interior</a> of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a> of constant positive, negative, or zero <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>. The cases of negative and zero curvature form geodesically complete surfaces, which means that all <a href="/wiki/Geodesic" title="Geodesic">geodesics</a> ("straight lines" on the surface) may be extended indefinitely in either direction. <dl><dt>Zero curvature</dt> <dd>An open strip with zero curvature may be constructed by gluing the opposite sides of a plane strip between two parallel lines, described above as the tautological line bundle.<a href="#cite_note-dundas-53">[46]</a> The resulting metric makes the open Möbius strip into a (geodesically) complete flat surface (i.e., having zero Gaussian curvature everywhere). This is the unique metric on the Möbius strip, up to uniform scaling, that is both flat and complete. It is the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> of a plane by a <a href="/wiki/Glide_reflection" title="Glide reflection">glide reflection</a>, and (together with the plane, <a href="/wiki/Cylinder" title="Cylinder">cylinder</a>, <a href="/wiki/Torus" title="Torus">torus</a>, and <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a>) is one of only five two-dimensional complete <a href="/wiki/Flat_manifold" title="Flat manifold">flat manifolds</a>.<a href="#cite_note-godinho-natario-67">[60]</a></dd> <dt>Negative curvature</dt> <dd>The open Möbius strip also admits complete metrics of constant negative curvature. One way to see this is to begin with the <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">upper half plane (Poincaré) model</a> of the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic plane</a>, a geometry of constant curvature whose lines are represented in the model by semicircles that meet the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-axis at right angles. Take the subset of the upper half-plane between any two nested semicircles, and identify the outer semicircle with the left-right reversal of the inner semicircle. The result is topologically a complete and non-compact Möbius strip with constant negative curvature. It is a "nonstandard" complete hyperbolic surface in the sense that it contains a complete hyperbolic <a href="/wiki/Half-plane" class="mw-redirect" title="Half-plane">half-plane</a> (actually two, on opposite sides of the axis of glide-reflection), and is one of only 13 nonstandard surfaces.<a href="#cite_note-cantwell-conlon-68">[61]</a> Again, this can be understood as the quotient of the hyperbolic plane by a glide reflection.<a href="#cite_note-stillwell-69">[62]</a></dd> <dt>Positive curvature</dt> <dd>A Möbius strip of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the <a href="/wiki/Real_projective_plane" title="Real projective plane">projective plane</a>.<a href="#cite_note-godinho-natario-67">[60]</a> However, in a sense it is only one point away from being a complete surface, as the open Möbius strip is homeomorphic to the <a href="/wiki/Once-punctured" class="mw-redirect" title="Once-punctured">once-punctured</a> projective plane, the surface obtained by removing any one point from the projective plane.<a href="#cite_note-seifert-threlfall-70">[63]</a></dd></dl> The <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surfaces</a> are described as having constant zero <a href="/wiki/Mean_curvature" title="Mean curvature">mean curvature</a> instead of constant Gaussian curvature. The Sudanese Möbius strip was constructed as a minimal surface bounded by a great circle in a 3-sphere, but there is also a unique complete (boundaryless) minimal surface immersed in Euclidean space that has the topology of an open Möbius strip. It is called the Meeks Möbius strip,<a href="#cite_note-lopez-martin-71">[64]</a> after its 1982 description by <a href="/wiki/William_Hamilton_Meeks,_III" title="William Hamilton Meeks, III">William Hamilton Meeks, III</a>.<a href="#cite_note-meeks-72">[65]</a> Although globally unstable as a minimal surface, small patches of it, bounded by non-contractible curves within the surface, can form stable embedded Möbius strips as minimal surfaces.<a href="#cite_note-systolic-73">[66]</a> Both the Meeks Möbius strip, and every higher-dimensional minimal surface with the topology of the Möbius strip, can be constructed using solutions to the <a href="/wiki/Bj%C3%B6rling_problem" title="Björling problem">Björling problem</a>, which defines a minimal surface uniquely from its boundary curve and tangent planes along this curve.<a href="#cite_note-bjorling-74">[67]</a> <div class="mw-heading mw-heading3"><h3 id="Spaces_of_lines">Spaces of lines</h3>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=9" title="Edit section: Spaces of lines">edit</a>]</div> The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines is <a href="/wiki/Diffeomorphic" class="mw-redirect" title="Diffeomorphic">topologically equivalent</a> to the open Möbius strip.<a href="#cite_note-parker-75">[68]</a> One way to see this is to extend the Euclidean plane to the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a> by adding one more line, the <a href="/wiki/Line_at_infinity" title="Line at infinity">line at infinity</a>. By <a href="/wiki/Projective_duality" class="mw-redirect" title="Projective duality">projective duality</a> the space of lines in the projective plane is equivalent to its space of points, the projective plane itself. Removing the line at infinity, to produce the space of Euclidean lines, punctures this space of projective lines.<a href="#cite_note-bickel-76">[69]</a> Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbius strip.<a href="#cite_note-seifert-threlfall-70">[63]</a> The space of lines in the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a> can be parameterized by <a href="/wiki/Unordered_pair" title="Unordered pair">unordered pairs</a> of distinct points on a circle, the pairs of points at infinity of each line. This space, again, has the topology of an open Möbius strip.<a href="#cite_note-mangahas-77">[70]</a> These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>, and the symmetries of hyperbolic lines include the <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>.<a href="#cite_note-ramirez-seade-78">[71]</a> The affine transformations and Möbius transformations both form 6-dimensional <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>, topological spaces having a compatible <a href="/wiki/Symmetry_group" title="Symmetry group">algebraic structure</a> describing the composition of symmetries.<a href="#cite_note-fomenko-kunii-79">[72]</a><a href="#cite_note-isham-80">[73]</a> Because every line in the plane is symmetric to every other line, the open Möbius strip is a <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous space</a>, a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are called <a href="/wiki/Solvmanifold" title="Solvmanifold">solvmanifolds</a>, and the Möbius strip can be used as a <a href="/wiki/Counterexample" title="Counterexample">counterexample</a>, showing that not every solvmanifold is a <a href="/wiki/Nilmanifold" title="Nilmanifold">nilmanifold</a>, and that not every solvmanifold can be factored into a <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of a <a href="/wiki/Compact_space" title="Compact space">compact</a> solvmanifold with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">. These symmetries also provide another way to construct the Möbius strip itself, as a group model of these Lie groups. A group model consists of a Lie group and a <a href="/wiki/Stabilizer_subgroup" class="mw-redirect" title="Stabilizer subgroup">stabilizer subgroup</a> of its action; contracting the <a href="/wiki/Coset" title="Coset">cosets</a> of the subgroup to points produces a space with the same topology as the underlying homogenous space. In the case of the symmetries of Euclidean lines, the stabilizer of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-axis consists of all symmetries that take the axis to itself. Each line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> corresponds to a coset, the set of symmetries that map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-axis. Therefore, the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a>, a space that has one point per coset and inherits its topology from the space of symmetries, is the same as the space of lines, and is again an open Möbius strip.<a href="#cite_note-gor-oni-vin-81">[74]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=10" title="Edit section: Applications">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:M%C3%B6bius_resistor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/M%C3%B6bius_resistor.svg/180px-M%C3%B6bius_resistor.svg.png" decoding="async" width="180" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/M%C3%B6bius_resistor.svg/270px-M%C3%B6bius_resistor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/M%C3%B6bius_resistor.svg/360px-M%C3%B6bius_resistor.svg.png 2x" data-file-width="500" data-file-height="450" /></a><figcaption>Electrical flow in a <a href="/wiki/M%C3%B6bius_resistor" title="Möbius resistor">Möbius resistor</a></figcaption></figure> Beyond the already-discussed applications of Möbius strips to the design of mechanical belts that wear evenly on their entire surface, and of the Plücker conoid to the design of gears, other applications of Möbius strips include: <ul><li><a href="/wiki/Graphene" title="Graphene">Graphene</a> ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism<a href="#cite_note-graphene-82">[75]</a></li> <li><a href="/wiki/M%C3%B6bius_aromaticity" title="Möbius aromaticity">Möbius aromaticity</a>, a property of <a href="/wiki/Organic_chemical" class="mw-redirect" title="Organic chemical">organic chemicals</a> whose molecular structure forms a cycle, with <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular orbitals</a> aligned along the cycle in the pattern of a Möbius strip<a href="#cite_note-aromaticity1-83">[76]</a><a href="#cite_note-aromaticity2-84">[77]</a></li> <li>The <a href="/wiki/M%C3%B6bius_resistor" title="Möbius resistor">Möbius resistor</a>, a strip of conductive material covering the single side of a <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> Möbius strip, in a way that cancels its own <a href="/wiki/Self-inductance" class="mw-redirect" title="Self-inductance">self-inductance</a><a href="#cite_note-resistor-85">[78]</a><a href="#cite_note-FOOTNOTEPickover200545–46-86">[79]</a></li> <li><a href="/wiki/Resonator" title="Resonator">Resonators</a> with a compact design and a resonant frequency that is half that of identically constructed linear coils<a href="#cite_note-resonator-87">[80]</a><a href="#cite_note-resonator2-88">[81]</a></li> <li><a href="/wiki/Polarization_(waves)" title="Polarization (waves)">Polarization</a> patterns in light emerging from a <a href="/wiki/Q-plate" title="Q-plate">q-plate</a><a href="#cite_note-polarization-89">[82]</a></li> <li>A proof of the impossibility of continuous, anonymous, and unanimous two-party aggregation rules in <a href="/wiki/Social_choice_theory" title="Social choice theory">social choice theory</a><a href="#cite_note-can-ind-90">[83]</a></li> <li><a href="/wiki/M%C3%B6bius_loop_roller_coaster" class="mw-redirect" title="Möbius loop roller coaster">Möbius loop roller coasters</a>, a form of dual-tracked roller coaster in which the two tracks spiral around each other an odd number of times, so that the carriages return to the other track than the one they started on<a href="#cite_note-coaster1-91">[84]</a><a href="#cite_note-coaster2-92">[85]</a></li> <li><a href="/wiki/World_map" title="World map">World maps</a> projected onto a Möbius strip with the convenient properties that there are no east–west boundaries, and that the <a href="/wiki/Antipodes" title="Antipodes">antipode</a> of any point on the map can be found on the other printed side of the surface at the same point of the Möbius strip<a href="#cite_note-maps1-93">[86]</a><a href="#cite_note-maps2-94">[87]</a></li></ul> Scientists have also studied the energetics of <a href="/wiki/Soap_film" title="Soap film">soap films</a> shaped as Möbius strips,<a href="#cite_note-courant-95">[88]</a><a href="#cite_note-soap-96">[89]</a> the <a href="/wiki/Chemical_synthesis" title="Chemical synthesis">chemical synthesis</a> of <a href="/wiki/Molecule" title="Molecule">molecules</a> with a Möbius strip shape,<a href="#cite_note-synthesis-97">[90]</a><a href="#cite_note-FOOTNOTEPickover200552–58-98">[91]</a> and the formation of larger <a href="/wiki/Nanoscale" class="mw-redirect" title="Nanoscale">nanoscale</a> Möbius strips using <a href="/wiki/DNA_origami" title="DNA origami">DNA origami</a>.<a href="#cite_note-dna-99">[92]</a> <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=11" title="Edit section: In popular culture">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg/220px-Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg/330px-Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg/440px-Middelheim_Max_Bill_Eindeloze_kronkel_1956_03_Cropped.jpg 2x" data-file-width="1449" data-file-height="1449" /></a><figcaption>Endless Twist, <a href="/wiki/Max_Bill" title="Max Bill">Max Bill</a>, 1956, from the <a href="/wiki/Middelheim_Open_Air_Sculpture_Museum" title="Middelheim Open Air Sculpture Museum">Middelheim Open Air Sculpture Museum</a></figcaption></figure> Two-dimensional artworks featuring the Möbius strip include an untitled 1947 painting by <a href="/wiki/Corrado_Cagli" title="Corrado Cagli">Corrado Cagli</a> (memorialized in a poem by <a href="/wiki/Charles_Olson" title="Charles Olson">Charles Olson</a>),<a href="#cite_note-emmer-100">[93]</a><a href="#cite_note-olson-101">[94]</a> and two prints by <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>: Möbius Band I (1961), depicting three folded <a href="/wiki/Flatfish" title="Flatfish">flatfish</a> biting each others' tails; and Möbius Band II (1963), depicting ants crawling around a <a href="/wiki/Lemniscate" title="Lemniscate">lemniscate</a>-shaped Möbius strip.<a href="#cite_note-escher1-102">[95]</a><a href="#cite_note-escher2-103">[96]</a> It is also a popular subject of <a href="/wiki/Mathematical_sculpture" title="Mathematical sculpture">mathematical sculpture</a>, including works by <a href="/wiki/Max_Bill" title="Max Bill">Max Bill</a> (Endless Ribbon, 1953), <a href="/wiki/Jos%C3%A9_de_Rivera" title="José de Rivera">José de Rivera</a> (<a href="/wiki/Infinity_(de_Rivera)" title="Infinity (de Rivera)">Infinity</a>, 1967), and <a href="/wiki/Sebasti%C3%A1n_(sculptor)" title="Sebastián (sculptor)">Sebastián</a>.<a href="#cite_note-emmer-100">[93]</a> A <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil-knotted</a> Möbius strip was used in <a href="/wiki/John_Robinson_(sculptor)" title="John Robinson (sculptor)">John Robinson</a>'s Immortality (1982).<a href="#cite_note-FOOTNOTEPickover200513-104">[97]</a> <a href="/wiki/Charles_O._Perry" title="Charles O. Perry">Charles O. Perry</a>'s <a href="/wiki/Continuum_(sculpture)" title="Continuum (sculpture)">Continuum</a> (1976) is one of several pieces by Perry exploring variations of the Möbius strip.<a href="#cite_note-brecher-105">[98]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage" style="height:141px;overflow:hidden"><a href="/wiki/File:Recycle001.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Recycle001.svg/150px-Recycle001.svg.png" decoding="async" width="150" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Recycle001.svg/225px-Recycle001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Recycle001.svg/300px-Recycle001.svg.png 2x" data-file-width="777" data-file-height="733" /></a></div><div class="thumbcaption"><a href="/wiki/Recycling_symbol" title="Recycling symbol">Recycling symbol</a></div></div><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:141px;overflow:hidden"><a href="/wiki/File:Logo_of_Google_Drive_(2012-2014).svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Logo_of_Google_Drive_%282012-2014%29.svg/142px-Logo_of_Google_Drive_%282012-2014%29.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Logo_of_Google_Drive_%282012-2014%29.svg/213px-Logo_of_Google_Drive_%282012-2014%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Logo_of_Google_Drive_%282012-2014%29.svg/284px-Logo_of_Google_Drive_%282012-2014%29.svg.png 2x" data-file-width="640" data-file-height="640" /></a></div><div class="thumbcaption"><a href="/wiki/Google_Drive" title="Google Drive">Google Drive</a> logo (2012–2014)</div></div><div class="tsingle" style="width:90px;max-width:90px"><div class="thumbimage" style="height:141px;overflow:hidden"><a href="/wiki/File:Stamp_of_Brazil_-_1967_-_Colnect_263101_-_Mobius_Symbol.jpeg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Stamp_of_Brazil_-_1967_-_Colnect_263101_-_Mobius_Symbol.jpeg/88px-Stamp_of_Brazil_-_1967_-_Colnect_263101_-_Mobius_Symbol.jpeg" decoding="async" width="88" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Stamp_of_Brazil_-_1967_-_Colnect_263101_-_Mobius_Symbol.jpeg/132px-Stamp_of_Brazil_-_1967_-_Colnect_263101_-_Mobius_Symbol.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1b/Stamp_of_Brazil_-_1967_-_Colnect_263101_-_Mobius_Symbol.jpeg 2x" data-file-width="140" data-file-height="225" /></a></div><div class="thumbcaption"><a href="/wiki/Instituto_Nacional_de_Matem%C3%A1tica_Pura_e_Aplicada" title="Instituto Nacional de Matemática Pura e Aplicada">IMPA</a> logo on stamp</div></div></div></div></div> Because of their easily recognized form, Möbius strips are a common element of <a href="/wiki/Graphic_design" title="Graphic design">graphic design</a>.<a href="#cite_note-FOOTNOTEPickover200513-104">[97]</a> The familiar <a href="/wiki/Recycling_symbol" title="Recycling symbol">three-arrow logo</a> for <a href="/wiki/Recycling" title="Recycling">recycling</a>, designed in 1970, is based on the smooth triangular form of the Möbius strip,<a href="#cite_note-peterson-106">[99]</a> as was the logo for the environmentally-themed <a href="/wiki/Expo_%2774" title="Expo '74">Expo '74</a>.<a href="#cite_note-expo74-107">[100]</a> Some variations of the recycling symbol use a different embedding with three half-twists instead of one,<a href="#cite_note-peterson-106">[99]</a> and the original version of the <a href="/wiki/Google_Drive" title="Google Drive">Google Drive</a> logo used a flat-folded three-twist Möbius strip, as have other similar designs.<a href="#cite_note-gdrive-108">[101]</a> The Brazilian <a href="/wiki/Instituto_Nacional_de_Matem%C3%A1tica_Pura_e_Aplicada" title="Instituto Nacional de Matemática Pura e Aplicada">Instituto Nacional de Matemática Pura e Aplicada</a> (IMPA) uses a stylized smooth Möbius strip as their logo, and has a matching large sculpture of a Möbius strip on display in their building.<a href="#cite_note-impa-109">[102]</a> The Möbius strip has also featured in the artwork for <a href="/wiki/Postage_stamp" title="Postage stamp">postage stamps</a> from countries including Brazil, Belgium, the Netherlands, and Switzerland.<a href="#cite_note-FOOTNOTEPickover2005156–157-110">[103]</a><a href="#cite_note-briefmarken-111">[104]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NASCAR_Hall_of_Fame_(7553589908).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/NASCAR_Hall_of_Fame_%287553589908%29.jpg/170px-NASCAR_Hall_of_Fame_%287553589908%29.jpg" decoding="async" width="170" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/NASCAR_Hall_of_Fame_%287553589908%29.jpg/255px-NASCAR_Hall_of_Fame_%287553589908%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/NASCAR_Hall_of_Fame_%287553589908%29.jpg/340px-NASCAR_Hall_of_Fame_%287553589908%29.jpg 2x" data-file-width="4000" data-file-height="3000" /></a><figcaption><a href="/wiki/NASCAR_Hall_of_Fame" title="NASCAR Hall of Fame">NASCAR Hall of Fame</a> entrance</figcaption></figure> Möbius strips have been a frequent inspiration for the architectural design of buildings and bridges. However, many of these are projects or conceptual designs rather than constructed objects, or stretch their interpretation of the Möbius strip beyond its recognizability as a mathematical form or a functional part of the architecture.<a href="#cite_note-architecture-112">[105]</a><a href="#cite_note-bridges-113">[106]</a> An example is the <a href="/wiki/National_Library_of_Kazakhstan" title="National Library of Kazakhstan">National Library of Kazakhstan</a>, for which a building was planned in the shape of a thickened Möbius strip but refinished with a different design after the original architects pulled out of the project.<a href="#cite_note-kazakh-114">[107]</a> One notable building incorporating a Möbius strip is the <a href="/wiki/NASCAR_Hall_of_Fame" title="NASCAR Hall of Fame">NASCAR Hall of Fame</a>, which is surrounded by a large twisted ribbon of stainless steel acting as a façade and canopy, and evoking the curved shapes of racing tracks.<a href="#cite_note-nascar-115">[108]</a> On a smaller scale, Moebius Chair (2006) by <a href="/wiki/Pedro_Reyes_(artist)" title="Pedro Reyes (artist)">Pedro Reyes</a> is a <a href="/wiki/Courting_bench" class="mw-redirect" title="Courting bench">courting bench</a> whose base and sides have the form of a Möbius strip.<a href="#cite_note-reyes-116">[109]</a> As a form of <a href="/wiki/Mathematics_and_fiber_arts" title="Mathematics and fiber arts">mathematics and fiber arts</a>, <a href="/wiki/Scarf" title="Scarf">scarves</a> have been <a href="/wiki/Knitting" title="Knitting">knit</a> into Möbius strips since the work of <a href="/wiki/Elizabeth_Zimmermann" title="Elizabeth Zimmermann">Elizabeth Zimmermann</a> in the early 1980s.<a href="#cite_note-zimmermann-117">[110]</a> In <a href="/wiki/Food_styling" class="mw-redirect" title="Food styling">food styling</a>, Möbius strips have been used for slicing <a href="/wiki/Bagel" title="Bagel">bagels</a>,<a href="#cite_note-bagel-118">[111]</a> making loops out of <a href="/wiki/Bacon" title="Bacon">bacon</a>,<a href="#cite_note-bacon-119">[112]</a> and creating new shapes for <a href="/wiki/Pasta" title="Pasta">pasta</a>.<a href="#cite_note-pasta-120">[113]</a> Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked in <a href="/wiki/Speculative_fiction" title="Speculative fiction">speculative fiction</a> as the basis for a <a href="/wiki/Time_loop" title="Time loop">time loop</a> into which unwary victims may become trapped. Examples of this trope include <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a>'s "No-Sided Professor" (1946), <a href="/wiki/Armin_Joseph_Deutsch" title="Armin Joseph Deutsch">Armin Joseph Deutsch</a>'s "<a href="/wiki/A_Subway_Named_Mobius" title="A Subway Named Mobius">A Subway Named Mobius</a>" (1950) and the film <a href="/wiki/Moebius_(1996_film)" title="Moebius (1996 film)">Moebius</a> (1996) based on it. An entire world shaped like a Möbius strip is the setting of <a href="/wiki/Arthur_C._Clarke" title="Arthur C. Clarke">Arthur C. Clarke</a>'s "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever inventions in multiple stories of <a href="/wiki/William_Hazlett_Upson" title="William Hazlett Upson">William Hazlett Upson</a> from the 1940s.<a href="#cite_note-FOOTNOTEPickover2005174–177-121">[114]</a> Other works of fiction have been analyzed as having a Möbius strip–like structure, in which elements of the plot repeat with a twist; these include <a href="/wiki/Marcel_Proust" title="Marcel Proust">Marcel Proust</a>'s <a href="/wiki/In_Search_of_Lost_Time" title="In Search of Lost Time">In Search of Lost Time</a> (1913–1927), <a href="/wiki/Luigi_Pirandello" title="Luigi Pirandello">Luigi Pirandello</a>'s <a href="/wiki/Six_Characters_in_Search_of_an_Author" title="Six Characters in Search of an Author">Six Characters in Search of an Author</a> (1921), <a href="/wiki/Frank_Capra" title="Frank Capra">Frank Capra</a>'s <a href="/wiki/It%27s_a_Wonderful_Life" title="It's a Wonderful Life">It's a Wonderful Life</a> (1946), <a href="/wiki/John_Barth" title="John Barth">John Barth</a>'s <a href="/wiki/Lost_in_the_Funhouse" title="Lost in the Funhouse">Lost in the Funhouse</a> (1968), <a href="/wiki/Samuel_R._Delany" title="Samuel R. Delany">Samuel R. Delany</a>'s <a href="/wiki/Dhalgren" title="Dhalgren">Dhalgren</a> (1975) and the film <a href="/wiki/Donnie_Darko" title="Donnie Darko">Donnie Darko</a> (2001).<a href="#cite_note-FOOTNOTEPickover2005179–187-122">[115]</a> One of the <a href="/wiki/Canon_(music)" title="Canon (music)">musical canons</a> by <a href="/wiki/J._S._Bach" class="mw-redirect" title="J. S. Bach">J. S. Bach</a>, the fifth of 14 canons (<a href="/wiki/BWV_1087" class="mw-redirect" title="BWV 1087">BWV 1087</a>) discovered in 1974 in Bach's copy of the <a href="/wiki/Goldberg_Variations" title="Goldberg Variations">Goldberg Variations</a>, features a <a href="/wiki/Glide_reflection" title="Glide reflection">glide-reflect</a> symmetry in which each voice in the canon repeats, with <a href="/wiki/Inversion_(music)" title="Inversion (music)">inverted notes</a>, the same motif from two measures earlier. Because of this symmetry, this canon can be thought of as having its score written on a Möbius strip.<a href="#cite_note-phillips-123">[116]</a><a href="#cite_note-124">[h]</a> In <a href="/wiki/Music_theory" title="Music theory">music theory</a>, tones that differ by an octave are generally considered to be equivalent notes, and the space of possible notes forms a circle, the <a href="/wiki/Chromatic_circle" title="Chromatic circle">chromatic circle</a>. Because the Möbius strip is the <a href="/wiki/Configuration_space_(mathematics)" title="Configuration space (mathematics)">configuration space</a> of two unordered points on a circle, the space of all <a href="/wiki/Dyad_(music)" title="Dyad (music)">two-note chords</a> takes the shape of a Möbius strip. This conception, and generalizations to more points, is a significant <a href="/wiki/Orbifold#Music_theory" title="Orbifold">application of orbifolds to music theory</a>.<a href="#cite_note-music-125">[117]</a><a href="#cite_note-chords-126">[118]</a> Modern musical groups taking their name from the Möbius strip include American electronic rock trio <a href="/wiki/Mobius_Band_(band)" title="Mobius Band (band)">Mobius Band</a><a href="#cite_note-bandband-127">[119]</a> and Norwegian progressive rock band <a href="/wiki/Ring_Van_M%C3%B6bius" title="Ring Van Möbius">Ring Van Möbius</a>.<a href="#cite_note-ringvan-128">[120]</a> Möbius strips and their properties have been used in the design of <a href="/wiki/Magic_(illusion)" title="Magic (illusion)">stage magic</a>. One such trick, known as the Afghan bands, uses the fact that the Möbius strip remains in one piece as a single strip when cut lengthwise. It originated in the 1880s, and was very popular in the first half of the twentieth century. Many versions of this trick exist and have been performed by famous illusionists such as <a href="/wiki/Harry_Blackstone_Sr." title="Harry Blackstone Sr.">Harry Blackstone Sr.</a> and <a href="/wiki/Thomas_Nelson_Downs" class="mw-redirect" title="Thomas Nelson Downs">Thomas Nelson Downs</a>.<a href="#cite_note-magic-129">[121]</a><a href="#cite_note-gardner-130">[122]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=12" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/M%C3%B6bius_counter" class="mw-redirect" title="Möbius counter">Möbius counter</a>, a shift register whose output bit is complemented before being fed back into the input bit</li> <li><a href="/wiki/Penrose_triangle" title="Penrose triangle">Penrose triangle</a>, an impossible figure whose boundary appears to wrap around it in a Möbius strip</li> <li><a href="/wiki/Ribbon_theory" class="mw-redirect" title="Ribbon theory">Ribbon theory</a>, the mathematical theory of infinitesimally thin strips that follow knotted space curves</li> <li><a href="/wiki/Smale%E2%80%93Williams_attractor" class="mw-redirect" title="Smale–Williams attractor">Smale–Williams attractor</a>, a fractal formed by repeatedly thickening a space curve to a Möbius strip and then replacing it with the boundary edge</li> <li><a href="/wiki/Umbilic_torus" title="Umbilic torus">Umbilic torus</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=13" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Pronounced <style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><a href="/wiki/American_English" title="American English">US</a>: <a href="/wiki/Help:IPA/English" title="Help:IPA/English">/ˈmoʊbiəs, ˈmeɪ-/</a> <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key">MOH-bee-əs, MAY-</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1177148991"><a href="/wiki/British_English" title="British English">UK</a>: <a href="/wiki/Help:IPA/English" title="Help:IPA/English">/ˈmɜːbiəs/</a>;<a href="#cite_note-1">[1]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1177148991">German: <a href="/wiki/Help:IPA/Standard_German" title="Help:IPA/Standard German">[ˈmøːbi̯ʊs]</a>. As is common for words containing an <a href="/wiki/Umlaut_(diacritic)" title="Umlaut (diacritic)">umlaut</a>, it is also often spelled Mobius or Moebius. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Essentially this example, but for a <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> rather than a Möbius strip, is given by <a href="#CITEREFBlackett1982">Blackett (1982)</a>.<a href="#cite_note-blackett-9">[8]</a> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> The flat-folded Möbius strip formed from three equilateral triangles does not come from an <a href="/wiki/Abstract_simplicial_complex" title="Abstract simplicial complex">abstract simplicial complex</a>, because all three triangles share the same three vertices, while abstract simplicial complexes require each triangle to have a different set of vertices. </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> This piecewise planar and cylindrical embedding has <a href="/wiki/Smoothness" title="Smoothness">smoothness</a> class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd6a5946b7e916352b0afc557f992328bac85e8" 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^{2}}">, and can be approximated arbitrarily accurately by <a href="/wiki/Infinitely_differentiable" class="mw-redirect" title="Infinitely differentiable">infinitely differentiable</a> (class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971ed05871d69309df32efdfd2020128c9cf69d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}">) embeddings.<a href="#cite_note-bartels-hornung-43">[39]</a> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> 12/7 is the simplest rational number in the range of aspect ratios, between 1.695 and 1.73, for which the existence of a smooth embedding is unknown. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> These surfaces have smoothness class <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><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}}">. For a more fine-grained analysis of the smoothness assumptions that force an embedding to be developable versus the assumptions under which the <a href="/wiki/Nash%E2%80%93Kuiper_theorem" class="mw-redirect" title="Nash–Kuiper theorem">Nash–Kuiper theorem</a> allows arbitrarily flexible embeddings, see remarks by <a href="#CITEREFBartelsHornung2015">Bartels & Hornung (2015)</a>, p. 116, following Theorem 2.2.<a href="#cite_note-bartels-hornung-43">[39]</a> </li> <li id="cite_note-124"><a href="#cite_ref-124">^</a> Möbius strips have also been used to analyze many other canons by Bach and others, but in most of these cases other looping surfaces such as a cylinder could have been used equally well.<a href="#cite_note-phillips-123">[116]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=14" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWells2008" class="citation book cs1"><a href="/wiki/John_C._Wells" title="John C. Wells">Wells, John C.</a> (2008). Longman Pronunciation Dictionary (3rd ed.). Longman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4058-8118-0" title="Special:BookSources/978-1-4058-8118-0"><bdi>978-1-4058-8118-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Longman+Pronunciation+Dictionary&rft.edition=3rd&rft.pub=Longman&rft.date=2008&rft.isbn=978-1-4058-8118-0&rft.aulast=Wells&rft.aufirst=John+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-pickover-3"><a href="#cite_ref-pickover_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickover2005" class="citation book cs1"><a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Pickover, Clifford A.</a> (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/mbiusstripdrau00pick">The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology</a>. Thunder's Mouth Press. pp. 28–29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56025-826-1" title="Special:BookSources/978-1-56025-826-1"><bdi>978-1-56025-826-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+M%C3%B6bius+Strip%3A+Dr.+August+M%C3%B6bius%27s+Marvelous+Band+in+Mathematics%2C+Games%2C+Literature%2C+Art%2C+Technology%2C+and+Cosmology&rft.pages=28-29&rft.pub=Thunder%27s+Mouth+Press&rft.date=2005&rft.isbn=978-1-56025-826-1&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmbiusstripdrau00pick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-roman-4">^ <a href="#cite_ref-roman_4-0">a</a> <a href="#cite_ref-roman_4-1">b</a> <a href="#cite_ref-roman_4-2">c</a> <a href="#cite_ref-roman_4-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarison1973" class="citation journal cs1">Larison, Lorraine L. (1973). "The Möbius band in Roman mosaics". <a href="/wiki/American_Scientist" title="American Scientist">American Scientist</a>. 61 (5): 544–547. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973AmSci..61..544L">1973AmSci..61..544L</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27843983">27843983</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=The+M%C3%B6bius+band+in+Roman+mosaics&rft.volume=61&rft.issue=5&rft.pages=544-547&rft.date=1973&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27843983%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1973AmSci..61..544L&rft.aulast=Larison&rft.aufirst=Lorraine+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-ancient-5">^ <a href="#cite_ref-ancient_5-0">a</a> <a href="#cite_ref-ancient_5-1">b</a> <a href="#cite_ref-ancient_5-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartwrightGonzález2016" class="citation journal cs1"><a href="/wiki/Julyan_Cartwright" title="Julyan Cartwright">Cartwright, Julyan H. E.</a>; González, Diego L. (2016). "Möbius strips before Möbius: topological hints in ancient representations". <a href="/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a>. 38 (2): 69–76. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1609.07779">1609.07779</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016arXiv160907779C">2016arXiv160907779C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-016-9631-8">10.1007/s00283-016-9631-8</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=3507121">3507121</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119587191">119587191</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=M%C3%B6bius+strips+before+M%C3%B6bius%3A+topological+hints+in+ancient+representations&rft.volume=38&rft.issue=2&rft.pages=69-76&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119587191%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2016arXiv160907779C&rft_id=info%3Aarxiv%2F1609.07779&rft_id=info%3Adoi%2F10.1007%2Fs00283-016-9631-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3507121%23id-name%3DMR&rft.aulast=Cartwright&rft.aufirst=Julyan+H.+E.&rft.au=Gonz%C3%A1lez%2C+Diego+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-chirality-6"><a href="#cite_ref-chirality_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlapan2000" class="citation book cs1"><a href="/wiki/Erica_Flapan" title="Erica Flapan">Flapan, Erica</a> (2000). <a href="/wiki/When_Topology_Meets_Chemistry" title="When Topology Meets Chemistry">When Topology Meets Chemistry: A Topological Look at Molecular Chirality</a>. Outlooks. Washington, DC: Mathematical Association of America. pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ytjLCgAAQBAJ&pg=PA82">82–83</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511626272">10.1017/CBO9780511626272</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66254-0" title="Special:BookSources/0-521-66254-0"><bdi>0-521-66254-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1781912">1781912</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=When+Topology+Meets+Chemistry%3A+A+Topological+Look+at+Molecular+Chirality&rft.place=Washington%2C+DC&rft.series=Outlooks&rft.pages=82-83&rft.pub=Mathematical+Association+of+America&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1781912%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9780511626272&rft.isbn=0-521-66254-0&rft.aulast=Flapan&rft.aufirst=Erica&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-FOOTNOTEPickover20058–9-7">^ <a href="#cite_ref-FOOTNOTEPickover20058–9_7-0">a</a> <a href="#cite_ref-FOOTNOTEPickover20058–9_7-1">b</a> <a href="#cite_ref-FOOTNOTEPickover20058–9_7-2">c</a> <a href="#CITEREFPickover2005">Pickover (2005)</a>, pp. 8–9. </li> <li id="cite_note-woll-8"><a href="#cite_ref-woll_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoll1971" class="citation journal cs1">Woll, John W. Jr. (Spring 1971). 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Washington, DC: Mathematical Association of America. pp. 289–293. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-516-X" title="Special:BookSources/0-88385-516-X"><bdi>0-88385-516-X</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=1303141">1303141</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Misunderstanding+my+mazy+mazes+may+make+me+miserable&rft.btitle=Proceedings+of+the+Eug%C3%A8ne+Strens+Memorial+Conference+on+Recreational+Mathematics+and+its+History+held+at+the+University+of+Calgary%2C+Calgary%2C+Alberta%2C+August+1986&rft.place=Washington%2C+DC&rft.series=MAA+Spectrum&rft.pages=289-293&rft.pub=Mathematical+Association+of+America&rft.date=1994&rft.isbn=0-88385-516-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1303141%23id-name%3DMR&rft.aulast=Larsen&rft.aufirst=Mogens+Esrom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988">. See <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FsH2DwAAQBAJ&pg=PA292">Figure 7, p. 292</a>. </li> <li id="cite_note-maschke-25"><a href="#cite_ref-maschke_25-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaschke1900" class="citation journal cs1"><a href="/wiki/Heinrich_Maschke" title="Heinrich Maschke">Maschke, Heinrich</a> (1900). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1986401">"Note on the unilateral surface of Moebius"</a>. <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>. 1 (1): 39. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1986401">10.2307/1986401</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1986401">1986401</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=1500522">1500522</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Note+on+the+unilateral+surface+of+Moebius&rft.volume=1&rft.issue=1&rft.pages=39&rft.date=1900&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1500522%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1986401%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1986401&rft.aulast=Maschke&rft.aufirst=Heinrich&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1986401&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-parameterization-26"><a href="#cite_ref-parameterization_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJunghenn2015" class="citation book cs1">Junghenn, Hugo D. 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Séquin">Séquin, Carlo H.</a> (2005). <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2005/bridges2005-211.html">"Splitting tori, knots, and Moebius bands"</a>. In Sarhangi, Reza; Moody, Robert V. (eds.). Renaissance Banff: Mathematics, Music, Art, Culture. 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"Lecture 14: Paper Möbius band". <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160424020238/http://www.math.psu.edu/tabachni/Books/taba.pdf">Mathematical Omnibus: Thirty Lectures on Classic Mathematics</a> (PDF). Providence, Rhode Island: American Mathematical Society. pp. 199–206. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fmbk%2F046">10.1090/mbk/046</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4316-1" title="Special:BookSources/978-0-8218-4316-1"><bdi>978-0-8218-4316-1</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=2350979">2350979</a>. 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"Complete minimal surfaces in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e57c690f890937838c10ba57853ff21bf30ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}">". <a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. Second Series. 92 (3): 335–374. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970625">10.2307/1970625</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970625">1970625</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=0270280">0270280</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Complete+minimal+surfaces+in+MATH+RENDER+ERROR&rft.volume=92&rft.issue=3&rft.pages=335-374&rft.date=1970&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D270280%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970625%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1970625&rft.aulast=Lawson&rft.aufirst=H.+Blaine+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> See Section 7, pp. 350–353, where the Klein bottle is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{1,2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54a50358663cdf0840a19535445510c79077f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.35ex; height:2.343ex;" alt="{\displaystyle \tau _{1,2}}">. </li> <li id="cite_note-schleimer-segerman-61">^ <a href="#cite_ref-schleimer-segerman_61-0">a</a> <a href="#cite_ref-schleimer-segerman_61-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchleimerSegerman2012" class="citation conference cs1">Schleimer, Saul; Segerman, Henry (2012). <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2012/bridges2012-103.html">"Sculptures in S3"</a>. 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Notices of the American Mathematical Society. 59 (8): 1076–1082. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti880">10.1090/noti880</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=2985809">2985809</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=The+Klein+bottle%3A+variations+on+a+theme&rft.volume=59&rft.issue=8&rft.pages=1076-1082&rft.date=2012&rft_id=info%3Adoi%2F10.1090%2Fnoti880&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2985809%23id-name%3DMR&rft.aulast=Franzoni&rft.aufirst=Gregorio&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fnoti880&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-huggett-jordan-64"><a href="#cite_ref-huggett-jordan_64-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggettJordan2009" class="citation book cs1">Huggett, Stephen; Jordan, David (2009). A Topological Aperitif (Revised ed.). Springer-Verlag. p. 57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84800-912-7" title="Special:BookSources/978-1-84800-912-7"><bdi>978-1-84800-912-7</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=2483686">2483686</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Topological+Aperitif&rft.pages=57&rft.edition=Revised&rft.pub=Springer-Verlag&rft.date=2009&rft.isbn=978-1-84800-912-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2483686%23id-name%3DMR&rft.aulast=Huggett&rft.aufirst=Stephen&rft.au=Jordan%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-flapan-65"><a href="#cite_ref-flapan_65-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlapan2016" class="citation book cs1"><a href="/wiki/Erica_Flapan" title="Erica Flapan">Flapan, Erica</a> (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q4RbCwAAQBAJ&pg=PA99">Knots, Molecules, and the Universe: An Introduction to Topology</a>. 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Retrieved 2022-03-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Spaces+of+geodesics&rft.btitle=Differential+Geometry+Workshop+on+Spaces+of+Geometry+%28Guanajuato%2C+1992%29&rft.series=Aportaciones+Mat.+Notas+Investigaci%C3%B3n&rft.pages=67-79&rft.pub=Soc.+Mat.+Mexicana%2C+M%C3%A9xico&rft.date=1993&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1304924%23id-name%3DMR&rft.aulast=Parker&rft.aufirst=Phillip+E.&rft_id=http%3A%2F%2Fwww.math.wichita.edu%2F~pparker%2Fresearch%2Fsog%2Fsog1.zip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>) </li> <li id="cite_note-bickel-76"><a href="#cite_ref-bickel_76-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBickel1999" class="citation journal cs1">Bickel, Holger (1999). "Duality in stable planes and related closure and kernel operations". Journal of Geometry. 64 (1–2): 8–15. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01229209">10.1007/BF01229209</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=1675956">1675956</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122209943">122209943</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geometry&rft.atitle=Duality+in+stable+planes+and+related+closure+and+kernel+operations&rft.volume=64&rft.issue=1%E2%80%932&rft.pages=8-15&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1675956%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122209943%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01229209&rft.aulast=Bickel&rft.aufirst=Holger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-mangahas-77"><a href="#cite_ref-mangahas_77-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMangahas2017" class="citation book cs1">Mangahas, Johanna (July 2017). "Office Hour Five: The Ping-Pong Lemma". In Clay, Matt; Margalit, Dan (eds.). Office Hours with a Geometric Group Theorist. Princeton University Press. pp. 85–105. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400885398">10.1515/9781400885398</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781400885398" title="Special:BookSources/9781400885398"><bdi>9781400885398</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Office+Hour+Five%3A+The+Ping-Pong+Lemma&rft.btitle=Office+Hours+with+a+Geometric+Group+Theorist&rft.pages=85-105&rft.pub=Princeton+University+Press&rft.date=2017-07&rft_id=info%3Adoi%2F10.1515%2F9781400885398&rft.isbn=9781400885398&rft.aulast=Mangahas&rft.aufirst=Johanna&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> See in particular Project 7, pp. 104–105. </li> <li id="cite_note-ramirez-seade-78"><a href="#cite_ref-ramirez-seade_78-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamírez_GalarzaSeade2007" class="citation book cs1">Ramírez Galarza, Ana Irene; Seade, José (2007). Introduction to Classical Geometries. Basel: Birkhäuser Verlag. pp. 83–88, 157–163. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-7517-1" title="Special:BookSources/978-3-7643-7517-1"><bdi>978-3-7643-7517-1</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=2305055">2305055</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Classical+Geometries&rft.place=Basel&rft.pages=83-88%2C+157-163&rft.pub=Birkh%C3%A4user+Verlag&rft.date=2007&rft.isbn=978-3-7643-7517-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2305055%23id-name%3DMR&rft.aulast=Ram%C3%ADrez+Galarza&rft.aufirst=Ana+Irene&rft.au=Seade%2C+Jos%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-fomenko-kunii-79"><a href="#cite_ref-fomenko-kunii_79-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFomenkoKunii2013" class="citation book cs1"><a href="/wiki/Anatoly_Fomenko" title="Anatoly Fomenko">Fomenko, Anatolij T.</a>; Kunii, Tosiyasu L. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8bn0CAAAQBAJ&pg=PA269">Topological Modeling for Visualization</a>. Springer. p. 269. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9784431669562" title="Special:BookSources/9784431669562"><bdi>9784431669562</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Modeling+for+Visualization&rft.pages=269&rft.pub=Springer&rft.date=2013&rft.isbn=9784431669562&rft.aulast=Fomenko&rft.aufirst=Anatolij+T.&rft.au=Kunii%2C+Tosiyasu+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8bn0CAAAQBAJ%26pg%3DPA269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-isham-80"><a href="#cite_ref-isham_80-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsham1999" class="citation book cs1">Isham, Chris J. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8bn0CAAAQBAJ&pg=PA269">Modern Differential Geometry for Physicists</a>. World Scientific lecture notes in physics. Vol. 61 (2nd ed.). World Scientific. p. 269. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-02-3555-0" title="Special:BookSources/981-02-3555-0"><bdi>981-02-3555-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1698234">1698234</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Differential+Geometry+for+Physicists&rft.series=World+Scientific+lecture+notes+in+physics&rft.pages=269&rft.edition=2nd&rft.pub=World+Scientific&rft.date=1999&rft.isbn=981-02-3555-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1698234%23id-name%3DMR&rft.aulast=Isham&rft.aufirst=Chris+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8bn0CAAAQBAJ%26pg%3DPA269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-gor-oni-vin-81"><a href="#cite_ref-gor-oni-vin_81-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGorbatsevichOnishchikVinberg1993" class="citation book cs1">Gorbatsevich, V. V.; Onishchik, A. L.; Vinberg, È. B. (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9UCOjHdD_hgC&pg=PA164">Lie groups and Lie algebras I: Foundations of Lie Theory; Lie Transformation Groups</a>. Encyclopaedia of Mathematical Sciences. Vol. 20. Springer-Verlag, Berlin. pp. 164–166. <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%2F978-3-642-57999-8">10.1007/978-3-642-57999-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-18697-2" title="Special:BookSources/3-540-18697-2"><bdi>3-540-18697-2</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=1306737">1306737</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+groups+and+Lie+algebras+I%3A+Foundations+of+Lie+Theory%3B+Lie+Transformation+Groups&rft.series=Encyclopaedia+of+Mathematical+Sciences&rft.pages=164-166&rft.pub=Springer-Verlag%2C+Berlin&rft.date=1993&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1306737%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-57999-8&rft.isbn=3-540-18697-2&rft.aulast=Gorbatsevich&rft.aufirst=V.+V.&rft.au=Onishchik%2C+A.+L.&rft.au=Vinberg%2C+%C3%88.+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9UCOjHdD_hgC%26pg%3DPA164&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-graphene-82"><a href="#cite_ref-graphene_82-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYamashiroShimoiHarigayaWakabayashi2004" class="citation journal cs1">Yamashiro, Atsushi; Shimoi, Yukihiro; Harigaya, Kikuo; Wakabayashi, Katsunori (2004). "Novel electronic states in graphene ribbons: competing spin and charge orders". Physica E. 22 (1–3): 688–691. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0309636">cond-mat/0309636</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004PhyE...22..688Y">2004PhyE...22..688Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physe.2003.12.100">10.1016/j.physe.2003.12.100</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17102453">17102453</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+E&rft.atitle=Novel+electronic+states+in+graphene+ribbons%3A+competing+spin+and+charge+orders&rft.volume=22&rft.issue=1%E2%80%933&rft.pages=688-691&rft.date=2004&rft_id=info%3Aarxiv%2Fcond-mat%2F0309636&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17102453%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physe.2003.12.100&rft_id=info%3Abibcode%2F2004PhyE...22..688Y&rft.aulast=Yamashiro&rft.aufirst=Atsushi&rft.au=Shimoi%2C+Yukihiro&rft.au=Harigaya%2C+Kikuo&rft.au=Wakabayashi%2C+Katsunori&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-aromaticity1-83"><a href="#cite_ref-aromaticity1_83-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRzepa2005" class="citation journal cs1">Rzepa, Henry S. (September 2005). "Möbius aromaticity and delocalization". Chemical Reviews. 105 (10): 3697–3715. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fcr030092l">10.1021/cr030092l</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16218564">16218564</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Reviews&rft.atitle=M%C3%B6bius+aromaticity+and+delocalization&rft.volume=105&rft.issue=10&rft.pages=3697-3715&rft.date=2005-09&rft_id=info%3Adoi%2F10.1021%2Fcr030092l&rft_id=info%3Apmid%2F16218564&rft.aulast=Rzepa&rft.aufirst=Henry+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-aromaticity2-84"><a href="#cite_ref-aromaticity2_84-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoonOsukaKim2009" class="citation journal cs1">Yoon, Zin Seok; Osuka, Atsuhiro; Kim, Dongho (May 2009). "Möbius aromaticity and antiaromaticity in expanded porphyrins". Nature Chemistry. 1 (2): 113–122. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NatCh...1..113Y">2009NatCh...1..113Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnchem.172">10.1038/nchem.172</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21378823">21378823</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Chemistry&rft.atitle=M%C3%B6bius+aromaticity+and+antiaromaticity+in+expanded+porphyrins&rft.volume=1&rft.issue=2&rft.pages=113-122&rft.date=2009-05&rft_id=info%3Apmid%2F21378823&rft_id=info%3Adoi%2F10.1038%2Fnchem.172&rft_id=info%3Abibcode%2F2009NatCh...1..113Y&rft.aulast=Yoon&rft.aufirst=Zin+Seok&rft.au=Osuka%2C+Atsuhiro&rft.au=Kim%2C+Dongho&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-resistor-85"><a href="#cite_ref-resistor_85-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation magazine cs1"><a rel="nofollow" class="external text" href="https://content.time.com/time/subscriber/article/0,33009,876181,00.html">"Making resistors with math"</a>. <a href="/wiki/Time_(magazine)" title="Time (magazine)">Time</a>. 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The Verge.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Verge&rft.atitle=How+to+make+a+mathematically-endless+strip+of+bacon&rft.date=2014-09-05&rft.aulast=Miller&rft.aufirst=Ross&rft_id=https%3A%2F%2Fwww.theverge.com%2F2014%2F9%2F5%2F6110563%2Fmobius-bacon-recipe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-pasta-120"><a href="#cite_ref-pasta_120-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChang2012" class="citation news cs1">Chang, Kenneth (January 9, 2012). <a rel="nofollow" class="external text" href="https://www.nytimes.com/2012/01/10/science/pasta-inspires-scientists-to-use-their-noodle.html">"Pasta Graduates From Alphabet Soup to Advanced Geometry"</a>. <a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=Pasta+Graduates+From+Alphabet+Soup+to+Advanced+Geometry&rft.date=2012-01-09&rft.aulast=Chang&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2012%2F01%2F10%2Fscience%2Fpasta-inspires-scientists-to-use-their-noodle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-FOOTNOTEPickover2005174–177-121"><a href="#cite_ref-FOOTNOTEPickover2005174–177_121-0">^</a> <a href="#CITEREFPickover2005">Pickover (2005)</a>, pp. 174–177. </li> <li id="cite_note-FOOTNOTEPickover2005179–187-122"><a href="#cite_ref-FOOTNOTEPickover2005179–187_122-0">^</a> <a href="#CITEREFPickover2005">Pickover (2005)</a>, pp. 179–187. </li> <li id="cite_note-phillips-123">^ <a href="#cite_ref-phillips_123-0">a</a> <a href="#cite_ref-phillips_123-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhillips2016" class="citation magazine cs1">Phillips, Tony (November 25, 2016). <a rel="nofollow" class="external text" href="https://plus.maths.org/content/topology-music-m-bius-strip">"Bach and the musical Möbius strip"</a>. <a href="/wiki/Plus_Magazine" title="Plus Magazine">Plus Magazine</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Plus+Magazine&rft.atitle=Bach+and+the+musical+M%C3%B6bius+strip&rft.date=2016-11-25&rft.aulast=Phillips&rft.aufirst=Tony&rft_id=https%3A%2F%2Fplus.maths.org%2Fcontent%2Ftopology-music-m-bius-strip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> Reprinted from an American Mathematical Society Feature Column. </li> <li id="cite_note-music-125"><a href="#cite_ref-music_125-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoskowitz2008" class="citation web cs1"><a href="/wiki/Clara_Moskowitz" title="Clara Moskowitz">Moskowitz, Clara</a> (May 6, 2008). <a rel="nofollow" class="external text" href="https://www.livescience.com/strangenews/080507-math-music.html">"Music reduced to beautiful math"</a>. <a href="/wiki/Live_Science" title="Live Science">Live Science</a>. Retrieved 2022-03-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Live+Science&rft.atitle=Music+reduced+to+beautiful+math&rft.date=2008-05-06&rft.aulast=Moskowitz&rft.aufirst=Clara&rft_id=https%3A%2F%2Fwww.livescience.com%2Fstrangenews%2F080507-math-music.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-chords-126"><a href="#cite_ref-chords_126-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTymoczko2006" class="citation journal cs1"><a href="/wiki/Dmitri_Tymoczko" title="Dmitri Tymoczko">Tymoczko, Dmitri</a> (July 7, 2006). <a rel="nofollow" class="external text" href="http://dmitri.mycpanel.princeton.edu/voiceleading.pdf">"The geometry of musical chords"</a> (PDF). <a href="/wiki/Science_(journal)" title="Science (journal)">Science</a>. 313 (5783): 72–4. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006Sci...313...72T">2006Sci...313...72T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1126287">10.1126/science.1126287</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3846592">3846592</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16825563">16825563</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2877171">2877171</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=The+geometry+of+musical+chords&rft.volume=313&rft.issue=5783&rft.pages=72-4&rft.date=2006-07-07&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2877171%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2006Sci...313...72T&rft_id=info%3Apmid%2F16825563&rft_id=info%3Adoi%2F10.1126%2Fscience.1126287&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3846592%23id-name%3DJSTOR&rft.aulast=Tymoczko&rft.aufirst=Dmitri&rft_id=http%3A%2F%2Fdmitri.mycpanel.princeton.edu%2Fvoiceleading.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-bandband-127"><a href="#cite_ref-bandband_127-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParks2007" class="citation magazine cs1">Parks, Andrew (August 30, 2007). <a rel="nofollow" class="external text" href="https://magnetmagazine.com/2007/08/30/mobius-band-friendly-fire/">"Mobius Band: Friendly Fire"</a>. <a href="/wiki/Magnet_(magazine)" title="Magnet (magazine)">Magnet</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Magnet&rft.atitle=Mobius+Band%3A+Friendly+Fire&rft.date=2007-08-30&rft.aulast=Parks&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fmagnetmagazine.com%2F2007%2F08%2F30%2Fmobius-band-friendly-fire%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-ringvan-128"><a href="#cite_ref-ringvan_128-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawson2021" class="citation magazine cs1">Lawson, Dom (February 9, 2021). <a rel="nofollow" class="external text" href="https://www.pressreader.com/uk/prog/20210209/281535113669133">"Ring Van Möbius"</a>. <a href="/wiki/Prog_(magazine)" title="Prog (magazine)">Prog</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Prog&rft.atitle=Ring+Van+M%C3%B6bius&rft.date=2021-02-09&rft.aulast=Lawson&rft.aufirst=Dom&rft_id=https%3A%2F%2Fwww.pressreader.com%2Fuk%2Fprog%2F20210209%2F281535113669133&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-magic-129"><a href="#cite_ref-magic_129-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrevos2018" class="citation book cs1">Prevos, Peter (2018). <a rel="nofollow" class="external text" href="https://magicperspectives.net/afghan-bands/">The Möbius Strip in Magic: A Treatise on the Afghan Bands</a>. Kangaroo Flat: Third Hemisphere.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+M%C3%B6bius+Strip+in+Magic%3A+A+Treatise+on+the+Afghan+Bands&rft.place=Kangaroo+Flat&rft.pub=Third+Hemisphere&rft.date=2018&rft.aulast=Prevos&rft.aufirst=Peter&rft_id=https%3A%2F%2Fmagicperspectives.net%2Fafghan-bands%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> <li id="cite_note-gardner-130"><a href="#cite_ref-gardner_130-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1956" class="citation book cs1"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (1956). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WkS4BQAAQBAJ&pg=PA70">"The Afghan Bands"</a>. Mathematics, Magic and Mystery. New York: Dover Books. pp. 70–73.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Afghan+Bands&rft.btitle=Mathematics%2C+Magic+and+Mystery&rft.place=New+York&rft.pages=70-73&rft.pub=Dover+Books&rft.date=1956&rft.aulast=Gardner&rft.aufirst=Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWkS4BQAAQBAJ%26pg%3DPA70&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=M%C3%B6bius_strip&action=edit&section=15" title="Edit section: External links">edit</a>]</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"><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" /></div> <div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/M%C3%B6bius_strip" class="extiw" title="wiktionary:Möbius strip">Möbius strip</a> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><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/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Moebius_Strip" class="extiw" title="commons:Moebius Strip">Moebius Strip</a> at Wikimedia Commons</li> <li><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/MoebiusStrip.html">"Möbius Strip"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</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=M%C3%B6bius+Strip&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMoebiusStrip.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AM%C3%B6bius+strip" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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.navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Compact_topological_surfaces" title="Template:Compact topological surfaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Compact_topological_surfaces" title="Template talk:Compact 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_3D" 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 href="/wiki/Torus" title="Torus">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 class="mw-selflink selflink">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 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 rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematics_of_paper_folding" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematics_of_paper_folding" title="Template:Mathematics of paper folding"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematics_of_paper_folding" title="Template talk:Mathematics of paper folding"><abbr title="Discuss this 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theorem">Kawasaki's theorem</a></li> <li><a href="/wiki/Maekawa%27s_theorem" title="Maekawa's theorem">Maekawa's theorem</a></li> <li><a href="/wiki/Map_folding" title="Map folding">Map folding</a></li> <li><a href="/wiki/Napkin_folding_problem" title="Napkin folding problem">Napkin folding problem</a></li> <li><a href="/wiki/Pureland_origami" title="Pureland origami">Pureland origami</a></li> <li><a href="/wiki/Yoshizawa%E2%80%93Randlett_system" title="Yoshizawa–Randlett system">Yoshizawa–Randlett system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Strip folding</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/Dragon_curve" title="Dragon curve">Dragon curve</a></li> <li><a href="/wiki/Flexagon" title="Flexagon">Flexagon</a></li> <li><a class="mw-selflink selflink">Möbius strip</a></li> <li><a href="/wiki/Regular_paperfolding_sequence" title="Regular paperfolding sequence">Regular paperfolding sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">3d structures</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/Miura_fold" title="Miura fold">Miura fold</a></li> <li><a href="/wiki/Modular_origami" title="Modular origami">Modular origami</a></li> <li><a href="/wiki/Paper_bag_problem" title="Paper bag problem">Paper bag problem</a></li> <li><a href="/wiki/Rigid_origami" title="Rigid origami">Rigid origami</a></li> <li><a href="/wiki/Schwarz_lantern" title="Schwarz lantern">Schwarz lantern</a></li> <li><a href="/wiki/Sonobe" title="Sonobe">Sonobe</a></li> <li><a href="/wiki/Yoshimura_buckling" title="Yoshimura buckling">Yoshimura buckling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Polyhedra</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/Alexandrov%27s_uniqueness_theorem" title="Alexandrov's uniqueness theorem">Alexandrov's uniqueness theorem</a></li> <li><a href="/wiki/Flexible_polyhedron" title="Flexible polyhedron">Flexible polyhedron</a> (<a href="/wiki/Bricard_octahedron" title="Bricard octahedron">Bricard octahedron</a>, <a href="/wiki/Steffen%27s_polyhedron" title="Steffen's polyhedron">Steffen's polyhedron</a>)</li> <li><a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">Net</a> <ul><li><a href="/wiki/Blooming_(geometry)" title="Blooming (geometry)">Blooming</a></li> <li><a href="/wiki/Common_net" title="Common net">Common net</a></li> <li><a href="/wiki/Source_unfolding" title="Source unfolding">Source unfolding</a></li> <li><a href="/wiki/Star_unfolding" title="Star unfolding">Star unfolding</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</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/Fold-and-cut_theorem" title="Fold-and-cut theorem">Fold-and-cut theorem</a></li> <li><a href="/wiki/Lill%27s_method" title="Lill's method">Lill's method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Publications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geometric_Exercises_in_Paper_Folding" title="Geometric Exercises in Paper Folding">Geometric Exercises in Paper Folding</a></li> <li><a href="/wiki/Geometric_Folding_Algorithms" title="Geometric Folding Algorithms">Geometric Folding Algorithms</a></li> <li><a href="/wiki/Geometric_Origami" title="Geometric Origami">Geometric Origami</a></li> <li><a href="/wiki/A_History_of_Folding_in_Mathematics" title="A History of Folding in Mathematics">A History of Folding in Mathematics</a></li> <li><a href="/wiki/Origami_Polyhedra_Design" title="Origami Polyhedra Design">Origami Polyhedra Design</a></li> <li><a href="/wiki/Origamics" title="Origamics">Origamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Roger_C._Alperin" title="Roger C. Alperin">Roger C. Alperin</a></li> <li><a href="/wiki/Margherita_Piazzola_Beloch" title="Margherita Piazzola Beloch">Margherita Piazzola Beloch</a></li> <li><a href="/wiki/Yan_Chen_(mechanical_engineer)" title="Yan Chen (mechanical engineer)">Yan Chen</a></li> <li><a href="/wiki/Robert_Connelly" title="Robert Connelly">Robert Connelly</a></li> <li><a href="/wiki/Erik_Demaine" title="Erik Demaine">Erik Demaine</a></li> <li><a href="/wiki/Martin_Demaine" title="Martin Demaine">Martin Demaine</a></li> <li><a href="/wiki/Rona_Gurkewitz" title="Rona Gurkewitz">Rona Gurkewitz</a></li> <li><a href="/wiki/David_A._Huffman" title="David A. Huffman">David A. Huffman</a></li> <li><a href="/wiki/Tom_Hull_(mathematician)" title="Tom Hull (mathematician)">Tom Hull</a></li> <li><a href="/wiki/K%C3%B4di_Husimi" title="Kôdi Husimi">Kôdi Husimi</a></li> <li><a href="/wiki/Humiaki_Huzita" title="Humiaki Huzita">Humiaki Huzita</a></li> <li><a href="/wiki/Toshikazu_Kawasaki" title="Toshikazu Kawasaki">Toshikazu Kawasaki</a></li> <li><a href="/wiki/Robert_J._Lang" title="Robert J. Lang">Robert J. Lang</a></li> <li><a href="/wiki/Anna_Lubiw" title="Anna Lubiw">Anna Lubiw</a></li> <li><a href="/wiki/Jun_Maekawa" title="Jun Maekawa">Jun Maekawa</a></li> <li><a href="/wiki/K%C5%8Dry%C5%8D_Miura" title="Kōryō Miura">Kōryō Miura</a></li> <li><a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">Joseph O'Rourke</a></li> <li><a href="/wiki/Tomohiro_Tachi" title="Tomohiro Tachi">Tomohiro Tachi</a></li> <li><a href="/wiki/Eve_Torrence" title="Eve Torrence">Eve Torrence</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <a href="https://www.wikidata.org/wiki/Q226843#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></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 rel="nofollow" class="external text" href="https://d-nb.info/gnd/1106880307">Germany</a></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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