Klein bottle - Wikipedia

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class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dissection" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dissection"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Dissection</span> </div> </a> <ul id="toc-Dissection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple-closed_curves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Simple-closed_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Simple-closed curves</span> </div> </a> <ul id="toc-Simple-closed_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parametrization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parametrization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Parametrization</span> </div> </a> <button aria-controls="toc-Parametrization-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Parametrization subsection</span> </button> <ul id="toc-Parametrization-sublist" class="vector-toc-list"> <li id="toc-The_figure_8_immersion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_figure_8_immersion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>The figure 8 immersion</span> </div> </a> <ul id="toc-The_figure_8_immersion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4-D_non-intersecting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#4-D_non-intersecting"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>4-D non-intersecting</span> </div> </a> <ul id="toc-4-D_non-intersecting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3D_pinched_torus_/_4D_Möbius_tube" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#3D_pinched_torus_/_4D_Möbius_tube"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>3D pinched torus / 4D Möbius tube</span> </div> </a> <ul id="toc-3D_pinched_torus_/_4D_Möbius_tube-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bottle_shape" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bottle_shape"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Bottle shape</span> </div> </a> <ul id="toc-Bottle_shape-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Homotopy_classes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Homotopy_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Homotopy classes</span> </div> </a> <ul id="toc-Homotopy_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Klein_surface" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Klein_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Klein surface</span> </div> </a> <ul id="toc-Klein_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Klein bottle</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 44 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-44" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">44 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Klein-bottel" title="Klein-bottel – Afrikaans" lang="af" hreflang="af" data-title="Klein-bottel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%B1%D9%88%D8%B1%D8%A9_%D9%83%D9%84%D8%A7%D9%8A%D9%86" title="قارورة كلاين – Arabic" lang="ar" hreflang="ar" data-title="قارورة كلاين" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kleyn_%C5%9F%C3%BC%C5%9F%C9%99si" title="Kleyn şüşəsi – Azerbaijani" lang="az" hreflang="az" data-title="Kleyn şüşəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D1%83%D1%82%D0%B8%D0%BB%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%9A%D0%BB%D0%B0%D0%B9%D0%BD" title="Бутилка на Клайн – Bulgarian" lang="bg" hreflang="bg" data-title="Бутилка на Клайн" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ampolla_de_Klein" title="Ampolla de Klein – Catalan" lang="ca" hreflang="ca" data-title="Ampolla de Klein" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kleinova_l%C3%A1hev" title="Kleinova láhev – Czech" lang="cs" hreflang="cs" data-title="Kleinova láhev" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kleinflaske" title="Kleinflaske – Danish" lang="da" hreflang="da" data-title="Kleinflaske" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kleinsche_Flasche" title="Kleinsche Flasche – German" lang="de" hreflang="de" data-title="Kleinsche Flasche" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kleini_pudel" title="Kleini pudel – Estonian" lang="et" hreflang="et" data-title="Kleini pudel" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CE%B9%CE%AC%CE%BB%CE%B7_%CF%84%CE%BF%CF%85_%CE%9A%CE%BB%CE%AC%CE%B9%CE%BD" title="Φιάλη του Κλάιν – Greek" lang="el" hreflang="el" data-title="Φιάλη του Κλάιν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Botella_de_Klein" title="Botella de Klein – Spanish" lang="es" hreflang="es" data-title="Botella de Klein" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Botelo_de_Klein" title="Botelo de Klein – Esperanto" lang="eo" hreflang="eo" data-title="Botelo de Klein" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Klein_botila" title="Klein botila – Basque" lang="eu" hreflang="eu" data-title="Klein botila" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%B7%D8%B1%DB%8C_%DA%A9%D9%84%D8%A7%DB%8C%D9%86" title="بطری کلاین – Persian" lang="fa" hreflang="fa" data-title="بطری کلاین" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bouteille_de_Klein" title="Bouteille de Klein – French" lang="fr" hreflang="fr" data-title="Bouteille de Klein" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kleinflesse" title="Kleinflesse – Western Frisian" lang="fy" hreflang="fy" data-title="Kleinflesse" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Botella_de_Klein" title="Botella de Klein – Galician" lang="gl" hreflang="gl" data-title="Botella de Klein" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%81%B4%EB%9D%BC%EC%9D%B8_%EB%B3%91" title="클라인 병 – Korean" lang="ko" hreflang="ko" data-title="클라인 병" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%AC%D5%A1%D5%B5%D5%B6%D5%AB_%D5%B7%D5%AB%D5%B7" title="Կլայնի շիշ – Armenian" lang="hy" hreflang="hy" data-title="Կլայնի շիշ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Klein-botelo" title="Klein-botelo – Ido" lang="io" hreflang="io" data-title="Klein-botelo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Klein-flaska" title="Klein-flaska – Icelandic" lang="is" hreflang="is" data-title="Klein-flaska" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Bottiglia_di_Klein" title="Bottiglia di Klein – Italian" lang="it" hreflang="it" data-title="Bottiglia di Klein" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%91%D7%A7%D7%91%D7%95%D7%A7_%D7%A7%D7%9C%D7%99%D7%99%D7%9F" title="בקבוק קליין – Hebrew" lang="he" hreflang="he" data-title="בקבוק קליין" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kleina_pudele" title="Kleina pudele – Latvian" lang="lv" hreflang="lv" data-title="Kleina pudele" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Klein-Fl%C3%A4sch" title="Klein-Fläsch – Luxembourgish" lang="lb" hreflang="lb" data-title="Klein-Fläsch" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kleino_butelis" title="Kleino butelis – Lithuanian" lang="lt" hreflang="lt" data-title="Kleino butelis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Klein-f%C3%A9le_palack" title="Klein-féle palack – Hungarian" lang="hu" hreflang="hu" data-title="Klein-féle palack" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kleinse_fles" title="Kleinse fles – Dutch" lang="nl" hreflang="nl" data-title="Kleinse fles" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AF%E3%83%A9%E3%82%A4%E3%83%B3%E3%81%AE%E5%A3%BA" title="クラインの壺 – Japanese" lang="ja" hreflang="ja" data-title="クラインの壺" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Klein-botle" title="Klein-botle – Novial" lang="nov" hreflang="nov" data-title="Klein-botle" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Butelka_Kleina" title="Butelka Kleina – Polish" lang="pl" hreflang="pl" data-title="Butelka Kleina" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Garrafa_de_Klein" title="Garrafa de Klein – Portuguese" lang="pt" hreflang="pt" data-title="Garrafa de Klein" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sticla_lui_Klein" title="Sticla lui Klein – Romanian" lang="ro" hreflang="ro" data-title="Sticla lui Klein" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D1%83%D1%82%D1%8B%D0%BB%D0%BA%D0%B0_%D0%9A%D0%BB%D0%B5%D0%B9%D0%BD%D0%B0" title="Бутылка Клейна – Russian" lang="ru" hreflang="ru" data-title="Бутылка Клейна" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Klein_bottle" title="Klein bottle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Klein bottle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kleinova_f%C4%BEa%C5%A1a" title="Kleinova fľaša – Slovak" lang="sk" hreflang="sk" data-title="Kleinova fľaša" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kleinova_steklenica" title="Kleinova steklenica – Slovenian" lang="sl" hreflang="sl" data-title="Kleinova steklenica" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kleinin_pullo" title="Kleinin pullo – Finnish" lang="fi" hreflang="fi" data-title="Kleinin pullo" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kleinflaska" title="Kleinflaska – Swedish" lang="sv" hreflang="sv" data-title="Kleinflaska" data-language-autonym="Svenska" data-language-local-name="Swedish" 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Non-orientable mathematical surface</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Klein_bottle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/170px-Klein_bottle.svg.png" decoding="async" width="170" height="326" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/255px-Klein_bottle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/340px-Klein_bottle.svg.png 2x" data-file-width="250" data-file-height="480" /></a><figcaption>A two-dimensional representation of the Klein bottle <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersed</a> in three-dimensional space</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Klein bottle</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'k' in 'kind'">k</span><span title="'l' in 'lie'">l</span><span title="/aɪ/: 'i' in 'tide'">aɪ</span><span title="'n' in 'nigh'">n</span></span>/</a></span></span>) is an example of a <a href="/wiki/Orientability" title="Orientability">non-orientable</a> <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a>; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> <a href="/wiki/Manifold" title="Manifold">manifold</a> on which one cannot define a <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal vector</a> at each point that varies <a href="/wiki/Continuous_function" title="Continuous function">continuously</a> over the whole manifold. Other related non-orientable surfaces include the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> and the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>. While a Möbius strip is a surface with a <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a>, a Klein bottle has no boundary. For comparison, a <a href="/wiki/Sphere" title="Sphere">sphere</a> is an orientable surface with no boundary. </p><p>The Klein bottle was first described in 1882 by the mathematician <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>.<sup id="cite_ref-FOOTNOTEStillwell1993651.2.3_The_Klein_Bottle_1-0" class="reference"><a href="#cite_note-FOOTNOTEStillwell1993651.2.3_The_Klein_Bottle-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=1" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following square is a <a href="/wiki/Fundamental_polygon" title="Fundamental polygon">fundamental polygon</a> of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Klein_Bottle_Folding_1.svg/150px-Klein_Bottle_Folding_1.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Klein_Bottle_Folding_1.svg/225px-Klein_Bottle_Folding_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Klein_Bottle_Folding_1.svg/300px-Klein_Bottle_Folding_1.svg.png 2x" data-file-width="150" data-file-height="150" /></a></span></dd></dl> <p>To construct the Klein bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, one would pass one end through the side of the cylinder. This creates a curve of self-intersection; this is thus an <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersion</a> of the Klein bottle in the <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Klein_Bottle_Folding_1.svg/120px-Klein_Bottle_Folding_1.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Klein_Bottle_Folding_1.svg/180px-Klein_Bottle_Folding_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Klein_Bottle_Folding_1.svg/240px-Klein_Bottle_Folding_1.svg.png 2x" data-file-width="150" data-file-height="150" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Klein_Bottle_Folding_2.svg/60px-Klein_Bottle_Folding_2.svg.png" decoding="async" width="60" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Klein_Bottle_Folding_2.svg/90px-Klein_Bottle_Folding_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Klein_Bottle_Folding_2.svg/120px-Klein_Bottle_Folding_2.svg.png 2x" data-file-width="75" data-file-height="150" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Klein_Bottle_Folding_3.svg/90px-Klein_Bottle_Folding_3.svg.png" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Klein_Bottle_Folding_3.svg/135px-Klein_Bottle_Folding_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Klein_Bottle_Folding_3.svg/180px-Klein_Bottle_Folding_3.svg.png 2x" data-file-width="225" data-file-height="300" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Klein_Bottle_Folding_4.svg/90px-Klein_Bottle_Folding_4.svg.png" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Klein_Bottle_Folding_4.svg/135px-Klein_Bottle_Folding_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Klein_Bottle_Folding_4.svg/180px-Klein_Bottle_Folding_4.svg.png 2x" data-file-width="225" data-file-height="300" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Klein_Bottle_Folding_5.svg/90px-Klein_Bottle_Folding_5.svg.png" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Klein_Bottle_Folding_5.svg/135px-Klein_Bottle_Folding_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Klein_Bottle_Folding_5.svg/180px-Klein_Bottle_Folding_5.svg.png 2x" data-file-width="225" data-file-height="300" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Klein_Bottle_Folding_6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Klein_Bottle_Folding_6.svg/90px-Klein_Bottle_Folding_6.svg.png" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Klein_Bottle_Folding_6.svg/135px-Klein_Bottle_Folding_6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Klein_Bottle_Folding_6.svg/180px-Klein_Bottle_Folding_6.svg.png 2x" data-file-width="225" data-file-height="300" /></a></span></div> <div class="gallerytext"></div> </li> </ul> <p>This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no <i>boundary</i>, where the surface stops abruptly, and it is <a href="/wiki/Orientability" title="Orientability">non-orientable</a>, as reflected in the one-sidedness of the immersion. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Science_Museum_London_1110529_nevit.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Science_Museum_London_1110529_nevit.jpg/150px-Science_Museum_London_1110529_nevit.jpg" decoding="async" width="150" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Science_Museum_London_1110529_nevit.jpg/225px-Science_Museum_London_1110529_nevit.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Science_Museum_London_1110529_nevit.jpg/300px-Science_Museum_London_1110529_nevit.jpg 2x" data-file-width="2304" data-file-height="3072" /></a><figcaption>Immersed Klein bottles in the <a href="/wiki/Science_Museum_(London)" class="mw-redirect" title="Science Museum (London)">Science Museum in London</a></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Acme_klein_bottle.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Acme_klein_bottle.jpg/150px-Acme_klein_bottle.jpg" decoding="async" width="150" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Acme_klein_bottle.jpg/225px-Acme_klein_bottle.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Acme_klein_bottle.jpg/300px-Acme_klein_bottle.jpg 2x" data-file-width="1216" data-file-height="1660" /></a><figcaption>A hand-blown Klein Bottle</figcaption></figure> <p>The common physical model of a Klein bottle is a similar construction. The <a href="/wiki/Science_Museum_(London)" class="mw-redirect" title="Science Museum (London)">Science Museum in London</a> has a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three-dimensional space. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand off the plane.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-0" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Klein_bottle_time_evolution_in_xyzt-space.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Klein_bottle_time_evolution_in_xyzt-space.gif/220px-Klein_bottle_time_evolution_in_xyzt-space.gif" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Klein_bottle_time_evolution_in_xyzt-space.gif/330px-Klein_bottle_time_evolution_in_xyzt-space.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/2/24/Klein_bottle_time_evolution_in_xyzt-space.gif 2x" data-file-width="384" data-file-height="192" /></a><figcaption><a href="/wiki/Time_evolution" title="Time evolution">Time evolution</a> of a Klein figure in <i>xyzt</i>-space</figcaption></figure> <p>Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed in <i>xyzt</i>-space. The accompanying illustration ("Time evolution...") shows one useful evolution of the figure. At <span class="nowrap"><i>t</i> = 0</span> the wall sprouts from a bud somewhere near the "intersection" point. After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like the <a href="/wiki/Cheshire_Cat" title="Cheshire Cat">Cheshire Cat</a> but leaving its ever-expanding smile behind. By the time the growth front gets to where the bud had been, there is nothing there to intersect and the growth completes without piercing existing structure. The 4-figure as defined cannot exist in 3-space but is easily understood in 4-space.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-1" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>More formally, the Klein bottle is the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> described as the <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> [0,1] × [0,1] with sides identified by the relations <span class="nowrap">(0, <i>y</i>) ~ (1, <i>y</i>)</span> for <span class="nowrap">0 ≤ <i>y</i> ≤ 1</span> and <span class="nowrap">(<i>x</i>, 0) ~ (1 − <i>x</i>, 1)</span> for <span class="nowrap">0 ≤ <i>x</i> ≤ 1</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a>, the Klein bottle is a two-dimensional <a href="/wiki/Manifold" title="Manifold">manifold</a> which is not <a href="/wiki/Orientability" title="Orientability">orientable</a>. Unlike the Möbius strip, it is a <i>closed</i> manifold, meaning it is a <a href="/wiki/Compact_space" title="Compact space">compact</a> manifold without boundary. While the Möbius strip can be embedded in three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup>3</sup>, the Klein bottle cannot. It can be embedded in <b>R</b><sup>4</sup>, however.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-2" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Continuing this sequence, for example creating a 3-manifold which cannot be embedded in <b>R</b><sup>4</sup> but can be in <b>R</b><sup>5</sup>, is possible; in this case, connecting two ends of a <a href="/wiki/Spherinder" title="Spherinder">spherinder</a> to each other in the same manner as the two ends of a cylinder for a Klein bottle, creates a figure, referred to as a "spherinder Klein bottle", that cannot fully be embedded in <b>R</b><sup>4</sup>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The Klein bottle can be seen as a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> over the <a href="/wiki/Circle" title="Circle">circle</a> <i>S</i><sup>1</sup>, with fibre <i>S</i><sup>1</sup>, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be <i>E</i>, the total space, while the base space <i>B</i> is given by the unit interval in <i>y</i>, modulo <i>1~0</i>. The projection π:<i>E</i>→<i>B</i> is then given by <span class="nowrap">π([<i>x</i>, <i>y</i>]) = [<i>y</i>]</span>. </p><p>The Klein bottle can be constructed (in a four dimensional space, because in three dimensional space it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips, as described in the following <a href="/wiki/Limerick_(poetry)" title="Limerick (poetry)">limerick</a> by <a href="/wiki/Leo_Moser" title="Leo Moser">Leo Moser</a>:<sup id="cite_ref-Darling2004_6-0" class="reference"><a href="#cite_note-Darling2004-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><div class="poem"> <p>A mathematician named <a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a><br /> Thought the Möbius band was divine.<br />      Said he: "If you glue<br />      The edges of two,<br /> You'll get a weird bottle like mine." </p> </div></blockquote> <p>The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a <a href="/wiki/CW_complex" title="CW complex">CW complex</a> structure with one 0-cell <i>P</i>, two 1-cells <i>C</i><sub>1</sub>, <i>C</i><sub>2</sub> and one 2-cell <i>D</i>. Its <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> is therefore <span class="nowrap">1 − 2 + 1 = 0</span>. The boundary homomorphism is given by <span class="nowrap">∂<i>D</i> = 2<i>C</i><sub>1</sub></span> and <span class="nowrap">∂<i>C</i><sub>1</sub> = ∂<i>C</i><sub>2</sub> = 0</span>, yielding the <a href="/wiki/Cellular_homology" title="Cellular homology">homology groups</a> of the Klein bottle <i>K</i> to be <span class="nowrap">H<sub>0</sub>(<i>K</i>, <b>Z</b>) = <b>Z</b></span>, <span class="nowrap">H<sub>1</sub>(<i>K</i>, <b>Z</b>) = <b>Z</b>×(<b>Z</b>/2<b>Z</b>)</span> and <span class="nowrap">H<sub><i>n</i></sub>(<i>K</i>, <b>Z</b>) = 0</span> for <span class="nowrap"><i>n</i> > 1</span>. </p><p>There is a 2-1 <a href="/wiki/Covering_map" class="mw-redirect" title="Covering map">covering map</a> from the <a href="/wiki/Torus" title="Torus">torus</a> to the Klein bottle, because two copies of the <a href="/wiki/Fundamental_region" class="mw-redirect" title="Fundamental region">fundamental region</a> of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. The <a href="/wiki/Universal_cover" class="mw-redirect" title="Universal cover">universal cover</a> of both the torus and the Klein bottle is the plane <b>R</b><sup>2</sup>. </p><p>The <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of the Klein bottle can be determined as the <a href="/wiki/Deck_transformation#Deck_transformation_group,_regular_covers" class="mw-redirect" title="Deck transformation">group of deck transformations</a> of the universal cover and has the <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> <span class="nowrap"><span class="nowrap">⟨<i>a</i>, <i>b</i> | <i>ab</i> = <i>b</i><sup>−1</sup><i>a</i>⟩</span></span>. It follows that it is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \rtimes \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⋊</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \rtimes \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f26abecdd1e8a79a088aa6aa2f85e5d9933090be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \rtimes \mathbb {Z} }"></span>, the only nontrivial semidirect product of the additive group of integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> with itself. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Klein_bottle_colouring.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Klein_bottle_colouring.svg/170px-Klein_bottle_colouring.svg.png" decoding="async" width="170" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Klein_bottle_colouring.svg/255px-Klein_bottle_colouring.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Klein_bottle_colouring.svg/340px-Klein_bottle_colouring.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>A 6-colored Klein bottle, the only exception to the Heawood conjecture</figcaption></figure> <p>Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the <a href="/wiki/Heawood_conjecture" title="Heawood conjecture">Heawood conjecture</a>, a generalization of the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a>, which would require seven. </p><p>A Klein bottle is homeomorphic to the <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> of two <a href="/wiki/Projective_plane" title="Projective plane">projective planes</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> It is also homeomorphic to a sphere plus two <a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">cross-caps</a>. </p><p>When embedded in Euclidean space, the Klein bottle is one-sided. However, there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dissection">Dissection</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=3" title="Edit section: Dissection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:KleinBottle-cut.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/KleinBottle-cut.svg/150px-KleinBottle-cut.svg.png" decoding="async" width="150" height="321" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/KleinBottle-cut.svg/225px-KleinBottle-cut.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/KleinBottle-cut.svg/300px-KleinBottle-cut.svg.png 2x" data-file-width="217" data-file-height="465" /></a><figcaption>Dissecting the Klein bottle results in Möbius strips.</figcaption></figure> <p>Dissecting a Klein bottle into halves along its <a href="/wiki/Plane_of_symmetry" class="mw-redirect" title="Plane of symmetry">plane of symmetry</a> results in two mirror image <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strips</a>, i.e. one with a left-handed half-twist and the other with a right-handed half-twist (one of these is pictured on the right). Remember that the intersection pictured is not really there.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Simple-closed_curves">Simple-closed curves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=4" title="Edit section: Simple-closed curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One description of the types of simple-closed curves that may appear on the surface of the Klein bottle is given by the use of the first homology group of the Klein bottle calculated with integer coefficients. This group is isomorphic to <b>Z</b>×<b>Z</b><sub>2</sub>. Up to reversal of orientation, the only homology classes which contain simple-closed curves are as follows: (0,0), (1,0), (1,1), (2,0), (0,1). Up to reversal of the orientation of a simple closed curve, if it lies within one of the two cross-caps that make up the Klein bottle, then it is in homology class (1,0) or (1,1); if it cuts the Klein bottle into two Möbius strips, then it is in homology class (2,0); if it cuts the Klein bottle into an annulus, then it is in homology class (0,1); and if bounds a disk, then it is in homology class (0,0).<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-3" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Parametrization">Parametrization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=5" title="Edit section: Parametrization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:KleinBottle-Figure8-01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/KleinBottle-Figure8-01.svg/220px-KleinBottle-Figure8-01.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/KleinBottle-Figure8-01.svg/330px-KleinBottle-Figure8-01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/KleinBottle-Figure8-01.svg/440px-KleinBottle-Figure8-01.svg.png 2x" data-file-width="676" data-file-height="542" /></a><figcaption>The "figure 8" immersion of the Klein bottle.</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Kleinbagel_cross_section.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Kleinbagel_cross_section.png/220px-Kleinbagel_cross_section.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/9e/Kleinbagel_cross_section.png 1.5x" data-file-width="300" data-file-height="301" /></a><figcaption>Klein bagel cross section, showing a figure eight curve (the <a href="/wiki/Lemniscate_of_Gerono" title="Lemniscate of Gerono">lemniscate of Gerono</a>).</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="The_figure_8_immersion">The figure 8 immersion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=6" title="Edit section: The figure 8 immersion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To make the "figure 8" or "bagel" <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersion</a> of the Klein bottle, one can start with a <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> and curl it to bring the edge to the midline; since there is only one edge, it will meet itself there, passing through the midline. It has a particularly simple parametrization as a "figure-8" torus with a half-twist:<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-4" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=\left(r+\cos {\frac {\theta }{2}}\sin v-\sin {\frac {\theta }{2}}\sin 2v\right)\cos \theta \\y&=\left(r+\cos {\frac {\theta }{2}}\sin v-\sin {\frac {\theta }{2}}\sin 2v\right)\sin \theta \\z&=\sin {\frac {\theta }{2}}\sin v+\cos {\frac {\theta }{2}}\sin 2v\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> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>v</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=\left(r+\cos {\frac {\theta }{2}}\sin v-\sin {\frac {\theta }{2}}\sin 2v\right)\cos \theta \\y&=\left(r+\cos {\frac {\theta }{2}}\sin v-\sin {\frac {\theta }{2}}\sin 2v\right)\sin \theta \\z&=\sin {\frac {\theta }{2}}\sin v+\cos {\frac {\theta }{2}}\sin 2v\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d68a5596bd253a7121aeaefd55eec778b88a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:41.721ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}x&=\left(r+\cos {\frac {\theta }{2}}\sin v-\sin {\frac {\theta }{2}}\sin 2v\right)\cos \theta \\y&=\left(r+\cos {\frac {\theta }{2}}\sin v-\sin {\frac {\theta }{2}}\sin 2v\right)\sin \theta \\z&=\sin {\frac {\theta }{2}}\sin v+\cos {\frac {\theta }{2}}\sin 2v\end{aligned}}}"></span></dd></dl> <p>for 0 ≤ <i>θ</i> < 2π, 0 ≤ <i>v</i> < 2π and <i>r</i> > 2. </p><p>In this immersion, the self-intersection circle (where sin(<i>v</i>) is zero) is a geometric <a href="/wiki/Circle" title="Circle">circle</a> in the <i>xy</i> plane. The positive constant <i>r</i> is the radius of this circle. The parameter <i>θ</i> gives the angle in the <i>xy</i> plane as well as the rotation of the figure 8, and <i>v</i> specifies the position around the 8-shaped cross section. With the above parametrization the cross section is a 2:1 <a href="/wiki/Lissajous_curve" title="Lissajous curve">Lissajous curve</a>. </p> <div class="mw-heading mw-heading3"><h3 id="4-D_non-intersecting">4-D non-intersecting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=7" title="Edit section: 4-D non-intersecting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A non-intersecting 4-D parametrization can be modeled after that of the <a href="/wiki/Flat_torus#Flat_torus" class="mw-redirect" title="Flat torus">flat torus</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=R\left(\cos {\frac {\theta }{2}}\cos v-\sin {\frac {\theta }{2}}\sin 2v\right)\\y&=R\left(\sin {\frac {\theta }{2}}\cos v+\cos {\frac {\theta }{2}}\sin 2v\right)\\z&=P\cos \theta \left(1+\varepsilon \sin v\right)\\w&=P\sin \theta \left(1+{\varepsilon }\sin v\right)\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> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>v</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>v</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>w</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=R\left(\cos {\frac {\theta }{2}}\cos v-\sin {\frac {\theta }{2}}\sin 2v\right)\\y&=R\left(\sin {\frac {\theta }{2}}\cos v+\cos {\frac {\theta }{2}}\sin 2v\right)\\z&=P\cos \theta \left(1+\varepsilon \sin v\right)\\w&=P\sin \theta \left(1+{\varepsilon }\sin v\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db23614907802c1e80163602c5cf36fedfc6f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:35.598ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}x&=R\left(\cos {\frac {\theta }{2}}\cos v-\sin {\frac {\theta }{2}}\sin 2v\right)\\y&=R\left(\sin {\frac {\theta }{2}}\cos v+\cos {\frac {\theta }{2}}\sin 2v\right)\\z&=P\cos \theta \left(1+\varepsilon \sin v\right)\\w&=P\sin \theta \left(1+{\varepsilon }\sin v\right)\end{aligned}}}"></span></dd></dl> <p>where <i>R</i> and <i>P</i> are constants that determine aspect ratio, <i>θ</i> and <i>v</i> are similar to as defined above. <i>v</i> determines the position around the figure-8 as well as the position in the x-y plane. <i>θ</i> determines the rotational angle of the figure-8 as well and the position around the z-w plane. <i>ε</i> is any small constant and <i>ε</i> sin<i>v</i> is a small <i>v</i> dependent bump in <i>z-w</i> space to avoid self intersection. The <i>v</i> bump causes the self intersecting 2-D/planar figure-8 to spread out into a 3-D stylized "potato chip" or saddle shape in the x-y-w and x-y-z space viewed edge on. When <i>ε=0</i> the self intersection is a circle in the z-w plane <0, 0, cos<i>θ</i>, sin<i>θ</i>>.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-5" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="3D_pinched_torus_/_4D_Möbius_tube"><span id="3D_pinched_torus_.2F_4D_M.C3.B6bius_tube"></span>3D pinched torus / 4D Möbius tube</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=8" title="Edit section: 3D pinched torus / 4D Möbius tube"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Pinched_Torus_Klein_bottle.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pinched_Torus_Klein_bottle.jpg/220px-Pinched_Torus_Klein_bottle.jpg" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pinched_Torus_Klein_bottle.jpg/330px-Pinched_Torus_Klein_bottle.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pinched_Torus_Klein_bottle.jpg/440px-Pinched_Torus_Klein_bottle.jpg 2x" data-file-width="819" data-file-height="757" /></a><figcaption>The pinched torus immersion of the Klein bottle.</figcaption></figure> <p>The pinched torus is perhaps the simplest parametrization of the klein bottle in both three and four dimensions. It's a torus that, in three dimensions, flattens and passes through itself on one side. Unfortunately, in three dimensions this parametrization has two <a href="/wiki/Pinch_point_(mathematics)" title="Pinch point (mathematics)">pinch points</a>, which makes it undesirable for some applications. In four dimensions the <i>z</i> amplitude rotates into the <i>w</i> amplitude and there are no self intersections or pinch points.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-6" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \cos \left({\frac {\varphi }{2}}\right)\\w(\theta ,\varphi )&=r\sin \theta \sin \left({\frac {\varphi }{2}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>w</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \cos \left({\frac {\varphi }{2}}\right)\\w(\theta ,\varphi )&=r\sin \theta \sin \left({\frac {\varphi }{2}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa65a057b04a6372d1029f5ac2f7c9ab08cd05d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:28.811ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \cos \left({\frac {\varphi }{2}}\right)\\w(\theta ,\varphi )&=r\sin \theta \sin \left({\frac {\varphi }{2}}\right)\end{aligned}}}"></span></dd></dl> <p>One can view this as a tube or cylinder that wraps around, as in a torus, but its circular cross section flips over in four dimensions, presenting its "backside" as it reconnects, just as a Möbius strip cross section rotates before it reconnects. The 3D orthogonal projection of this is the pinched torus shown above. Just as a Möbius strip is a subset of a solid torus, the Möbius tube is a subset of a toroidally closed <a href="/wiki/Spherinder" title="Spherinder">spherinder</a> (solid <a href="/w/index.php?title=Spheritorus&action=edit&redlink=1" class="new" title="Spheritorus (page does not exist)">spheritorus</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Bottle_shape">Bottle shape</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=9" title="Edit section: Bottle shape"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Klein_bottle_translucent.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Klein_bottle_translucent.png/220px-Klein_bottle_translucent.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Klein_bottle_translucent.png/330px-Klein_bottle_translucent.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Klein_bottle_translucent.png/440px-Klein_bottle_translucent.png 2x" data-file-width="565" data-file-height="367" /></a><figcaption>Klein Bottle with slight transparency</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x(u,v)=-&{\frac {2}{15}}\cos u\left(3\cos {v}-30\sin {u}+90\cos ^{4}{u}\sin {u}\right.-\\&\left.60\cos ^{6}{u}\sin {u}+5\cos {u}\cos {v}\sin {u}\right)\\[3pt]y(u,v)=-&{\frac {1}{15}}\sin u\left(3\cos {v}-3\cos ^{2}{u}\cos {v}-48\cos ^{4}{u}\cos {v}+48\cos ^{6}{u}\cos {v}\right.-\\&60\sin {u}+5\cos {u}\cos {v}\sin {u}-5\cos ^{3}{u}\cos {v}\sin {u}-\\&\left.80\cos ^{5}{u}\cos {v}\sin {u}+80\cos ^{7}{u}\cos {v}\sin {u}\right)\\[3pt]z(u,v)=&{\frac {2}{15}}\left(3+5\cos {u}\sin {u}\right)\sin {v}\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 0.6em 0.3em 0.3em 0.6em 0.3em" 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> <mo>=</mo> <mo>−</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo>−</mo> <mn>30</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>+</mo> <mn>90</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> <mo>−</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mn>60</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>+</mo> <mn>5</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo>−</mo> <mn>3</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo>−</mo> <mn>48</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo>+</mo> <mn>48</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> <mo>−</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>60</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>+</mo> <mn>5</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>−</mo> <mn>5</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>−</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mn>80</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>+</mo> <mn>80</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(u,v)=-&{\frac {2}{15}}\cos u\left(3\cos {v}-30\sin {u}+90\cos ^{4}{u}\sin {u}\right.-\\&\left.60\cos ^{6}{u}\sin {u}+5\cos {u}\cos {v}\sin {u}\right)\\[3pt]y(u,v)=-&{\frac {1}{15}}\sin u\left(3\cos {v}-3\cos ^{2}{u}\cos {v}-48\cos ^{4}{u}\cos {v}+48\cos ^{6}{u}\cos {v}\right.-\\&60\sin {u}+5\cos {u}\cos {v}\sin {u}-5\cos ^{3}{u}\cos {v}\sin {u}-\\&\left.80\cos ^{5}{u}\cos {v}\sin {u}+80\cos ^{7}{u}\cos {v}\sin {u}\right)\\[3pt]z(u,v)=&{\frac {2}{15}}\left(3+5\cos {u}\sin {u}\right)\sin {v}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5d921d40ba9c619980533c34235fd121d6e557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.338ex; width:78.411ex; height:27.843ex;" alt="{\displaystyle {\begin{aligned}x(u,v)=-&{\frac {2}{15}}\cos u\left(3\cos {v}-30\sin {u}+90\cos ^{4}{u}\sin {u}\right.-\\&\left.60\cos ^{6}{u}\sin {u}+5\cos {u}\cos {v}\sin {u}\right)\\[3pt]y(u,v)=-&{\frac {1}{15}}\sin u\left(3\cos {v}-3\cos ^{2}{u}\cos {v}-48\cos ^{4}{u}\cos {v}+48\cos ^{6}{u}\cos {v}\right.-\\&60\sin {u}+5\cos {u}\cos {v}\sin {u}-5\cos ^{3}{u}\cos {v}\sin {u}-\\&\left.80\cos ^{5}{u}\cos {v}\sin {u}+80\cos ^{7}{u}\cos {v}\sin {u}\right)\\[3pt]z(u,v)=&{\frac {2}{15}}\left(3+5\cos {u}\sin {u}\right)\sin {v}\end{aligned}}}"></span></dd></dl> <p>for 0 ≤ <i>u</i> < π and 0 ≤ <i>v</i> < 2π.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-7" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Homotopy_classes">Homotopy classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=10" title="Edit section: Homotopy classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Regular 3D immersions of the Klein bottle fall into three <a href="/wiki/Regular_homotopy" title="Regular homotopy">regular homotopy</a> classes.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The three are represented by: </p> <ul><li>the "traditional" Klein bottle;</li> <li>the left-handed figure-8 Klein bottle;</li> <li>the right-handed figure-8 Klein bottle.</li></ul> <p>The traditional Klein bottle immersion is <a href="/wiki/Chirality" title="Chirality">achiral</a>. The figure-8 immersion is chiral. (The pinched torus immersion above is not regular, as it has pinch points, so it is not relevant to this section.) </p><p>If the traditional Klein bottle is cut in its plane of symmetry it breaks into two Möbius strips of opposite chirality. A figure-8 Klein bottle can be cut into two Möbius strips of the <i>same</i> chirality, and cannot be regularly deformed into its mirror image.<sup id="cite_ref-FOOTNOTEAllingGreenleaf1969_4-8" class="reference"><a href="#cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=11" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The generalization of the Klein bottle to higher <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> is given in the article on the <a href="/wiki/Fundamental_polygon" title="Fundamental polygon">fundamental polygon</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In another order of ideas, constructing <a href="/wiki/3-manifold" title="3-manifold">3-manifolds</a>, it is known that a <a href="/wiki/Solid_Klein_bottle" title="Solid Klein bottle">solid Klein bottle</a> is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of a <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> and a closed interval. The <i>solid Klein bottle</i> is the non-orientable version of the <b>solid torus</b>, equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{2}\times S^{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{2}\times S^{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c74f77b5679d871289f6a39c8b65065c68d209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.042ex; height:2.676ex;" alt="{\displaystyle D^{2}\times S^{1}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Klein_surface">Klein surface</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=12" title="Edit section: Klein surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>Klein surface</b> is, as for <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, a surface with an atlas allowing the <a href="/wiki/Transition_map" class="mw-redirect" title="Transition map">transition maps</a> to be composed using <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. One can obtain the so-called <a href="/wiki/Dianalytic_structure" class="mw-redirect" title="Dianalytic structure">dianalytic structure</a> of the space and has only one side.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic topology</a></li> <li><a href="/wiki/Alice_universe" class="mw-redirect" title="Alice universe">Alice universe</a></li> <li><a href="/wiki/Systoles_of_surfaces#Klein_bottle" title="Systoles of surfaces">Bavard's Klein bottle systolic inequality</a></li> <li><a href="/wiki/Boy%27s_surface" title="Boy's surface">Boy's surface</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=15" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEStillwell1993651.2.3_The_Klein_Bottle-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStillwell1993651.2.3_The_Klein_Bottle_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStillwell1993">Stillwell 1993</a>, p. 65, 1.2.3 The Klein Bottle.</span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeeks2020" class="citation book cs1">Weeks, Jeffrey (2020). <a rel="nofollow" class="external text" href="https://www.crcpress.com/The-Shape-of-Space/Weeks/p/book/9781138061217"><i>The Shape of Space, 3rd Edn</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1138061217" title="Special:BookSources/978-1138061217"><bdi>978-1138061217</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+Shape+of+Space%2C+3rd+Edn.&rft.pub=CRC+Press&rft.date=2020&rft.isbn=978-1138061217&rft.aulast=Weeks&rft.aufirst=Jeffrey&rft_id=https%3A%2F%2Fwww.crcpress.com%2FThe-Shape-of-Space%2FWeeks%2Fp%2Fbook%2F9781138061217&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061128155852/http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp">"Strange Surfaces: New Ideas"</a>. Science Museum London. Archived from <a rel="nofollow" class="external text" href="http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp">the original</a> on 2006-11-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Strange+Surfaces%3A+New+Ideas&rft.pub=Science+Museum+London&rft_id=http%3A%2F%2Fwww.sciencemuseum.org.uk%2Fon-line%2Fsurfaces%2Fnew.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEAllingGreenleaf1969-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAllingGreenleaf1969_4-8"><sup><i><b>i</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAllingGreenleaf1969">Alling & Greenleaf 1969</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="/wiki/Marc_ten_Bosch" class="mw-redirect" title="Marc ten Bosch">Marc ten Bosch</a> - <a rel="nofollow" class="external free" href="https://marctenbosch.com/news/2021/12/4d-toys-version-1-7-klein-bottles/">https://marctenbosch.com/news/2021/12/4d-toys-version-1-7-klein-bottles/</a></span> </li> <li id="cite_note-Darling2004-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Darling2004_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Darling2004" class="citation book cs1">David Darling (11 August 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nnpChqstvg0C&q=get+a+weird+bottle+like+mine&pg=PA176"><i>The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes</i></a>. John Wiley & Sons. p. 176. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-27047-8" title="Special:BookSources/978-0-471-27047-8"><bdi>978-0-471-27047-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+Book+of+Mathematics%3A+From+Abracadabra+to+Zeno%27s+Paradoxes&rft.pages=176&rft.pub=John+Wiley+%26+Sons&rft.date=2004-08-11&rft.isbn=978-0-471-27047-8&rft.au=David+Darling&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnnpChqstvg0C%26q%3Dget%2Ba%2Bweird%2Bbottle%2Blike%2Bmine%26pg%3DPA176&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShick2007" class="citation book cs1">Shick, Paul (2007). <i>Topology: Point-Set and Geometric</i>. Wiley-Interscience. pp. 191–192. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470096055" title="Special:BookSources/9780470096055"><bdi>9780470096055</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology%3A+Point-Set+and+Geometric&rft.pages=191-192&rft.pub=Wiley-Interscience&rft.date=2007&rft.isbn=9780470096055&rft.aulast=Shick&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=I3ZlhxaT_Ko">Cutting a Klein Bottle in Half – Numberphile on YouTube</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSéquin2013" class="citation journal cs1">Séquin, Carlo H (1 June 2013). "On the number of Klein bottle types". <i>Journal of Mathematics and the Arts</i>. <b>7</b> (2): 51–63. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.637.4811">10.1.1.637.4811</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F17513472.2013.795883">10.1080/17513472.2013.795883</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:16444067">16444067</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+Mathematics+and+the+Arts&rft.atitle=On+the+number+of+Klein+bottle+types&rft.volume=7&rft.issue=2&rft.pages=51-63&rft.date=2013-06-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.637.4811%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16444067%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F17513472.2013.795883&rft.aulast=S%C3%A9quin&rft.aufirst=Carlo+H&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDay2014" class="citation web cs1">Day, Adam (17 February 2014). <a rel="nofollow" class="external text" href="https://cqgplus.com/2014/02/17/quantum-gravity-on-a-klein-bottle/">"Quantum gravity on a Klein bottle"</a>. <i>CQG+</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CQG%2B&rft.atitle=Quantum+gravity+on+a+Klein+bottle&rft.date=2014-02-17&rft.aulast=Day&rft.aufirst=Adam&rft_id=https%3A%2F%2Fcqgplus.com%2F2014%2F02%2F17%2Fquantum-gravity-on-a-klein-bottle%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBitetto2020" class="citation book cs1">Bitetto, Dr Marco (2020-02-14). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K4DQDwAAQBAJ&dq=A+Klein+surface+is%2C+as+for+Riemann+surfaces%2C+a+surface+with+an+atlas+allowing+the+transition+maps+to+be+composed+using+complex+conjugation.+One+can+obtain+the+so-called+dianalytic+structure+of+the+space&pg=PA222"><i>Hyperspatial Dynamics</i></a>. Dr. Marco A. V. Bitetto.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hyperspatial+Dynamics&rft.pub=Dr.+Marco+A.+V.+Bitetto&rft.date=2020-02-14&rft.aulast=Bitetto&rft.aufirst=Dr+Marco&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK4DQDwAAQBAJ%26dq%3DA%2BKlein%2Bsurface%2Bis%252C%2Bas%2Bfor%2BRiemann%2Bsurfaces%252C%2Ba%2Bsurface%2Bwith%2Ban%2Batlas%2Ballowing%2Bthe%2Btransition%2Bmaps%2Bto%2Bbe%2Bcomposed%2Busing%2Bcomplex%2Bconjugation.%2BOne%2Bcan%2Bobtain%2Bthe%2Bso-called%2Bdianalytic%2Bstructure%2Bof%2Bthe%2Bspace%26pg%3DPA222&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=16" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><i>This article incorporates material from Klein bottle on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i></li> <li><span class="citation mathworld" id="Reference-Mathworld-Klein_Bottle"><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/KleinBottle.html">"Klein Bottle"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Klein+Bottle&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FKleinBottle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAllingGreenleaf1969" class="citation journal cs1">Alling, Norman; Greenleaf, Newcomb (1969). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1969-12332-3">"Klein surfaces and real algebraic function fields"</a>. <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>. <b>75</b> (4): 627–888. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1969-12332-3">10.1090/S0002-9904-1969-12332-3</a></span>. <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=0251213">0251213</a>. <a href="/wiki/Project_Euclid" title="Project Euclid">PE</a> <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.jdg/euclid.bams/1183530665">euclid.bams/1183530665</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Klein+surfaces+and+real+algebraic+function+fields&rft.volume=75&rft.issue=4&rft.pages=627-888&rft.date=1969&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1969-12332-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0251213%23id-name%3DMR&rft.aulast=Alling&rft.aufirst=Norman&rft.au=Greenleaf%2C+Newcomb&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1969-12332-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span> (A classical on the theory of Klein surfaces)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell1993" class="citation book cs1"><a href="/wiki/John_Stillwell" title="John Stillwell">Stillwell, John</a> (1993). <i>Classical Topology and Combinatorial Group Theory</i> (2nd ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97970-0" title="Special:BookSources/0-387-97970-0"><bdi>0-387-97970-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Topology+and+Combinatorial+Group+Theory&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=0-387-97970-0&rft.aulast=Stillwell&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKlein+bottle" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_bottle&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Klein_bottle" class="extiw" title="commons:Category:Klein bottle">Klein bottle</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://plus.maths.org/content/os/issue26/features/mathart/index">Imaging Maths - The Klein Bottle</a></li> <li><a rel="nofollow" class="external text" href="http://www.kleinbottle.com/meter_tall_klein_bottle.html">The biggest Klein bottle in all the world</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=E8rifKlq5hc">Klein Bottle animation: produced for a topology seminar at the Leibniz University Hannover.</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=sRTKSzAOBr4&fmt=22">Klein Bottle animation from 2010 including a car ride through the bottle and the original description by Felix Klein: produced at the Free University Berlin.</a></li> <li><a rel="nofollow" class="external text" href="https://archive.today/20130713133627/https://github.com/danfuzz/xscreensaver/blob/master/hacks/glx/klein.man">Klein Bottle</a>, <a href="/wiki/XScreenSaver" title="XScreenSaver">XScreenSaver</a> "hack". A screensaver for <a href="/wiki/X_Window_System" title="X Window System">X 11</a> and <a href="/wiki/OS_X" class="mw-redirect" title="OS X">OS X</a> featuring an animated Klein Bottle.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Compact_topological_surfaces_and_their_immersions_in_3D" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .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 class="mw-selflink selflink">Klein bottle</a> (genus 2)</li> <li><a href="/wiki/Dyck%27s_surface" class="mw-redirect" title="Dyck's surface">Dyck's surface</a> (genus 3) ...</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With boundary</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">Disk</a> <ul><li>Semisphere</li></ul></li> <li>Ribbon <ul><li><a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">Annulus</a></li> <li><a href="/wiki/Cylinder" title="Cylinder">Cylinder</a></li></ul></li> <li><a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> <ul><li><a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">Cross-cap</a></li></ul></li> <li><a href="/wiki/Pair_of_pants_(mathematics)" title="Pair of pants (mathematics)">Sphere with three holes</a> ...</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />notions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Connected_space" title="Connected space">Connectedness</a></li> <li><a href="/wiki/Compact_space" title="Compact space">Compactness</a></li> <li><a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">Triangulatedness</a> or <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">smoothness</a></li> <li><a href="/wiki/Orientability" title="Orientability">Orientability</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Characteristics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Number of <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> components</li> <li><a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">Genus</a></li> <li><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Connected_sum" title="Connected sum">Connected sum</a></li> <li>Making a hole</li> <li>Gluing a <a href="/wiki/Handle_decomposition" title="Handle decomposition">handle</a></li> <li>Gluing a <a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">cross-cap</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary)" 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:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></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/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</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/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</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/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</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; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Klein_bottle&oldid=1257844402">https://en.wikipedia.org/w/index.php?title=Klein_bottle&oldid=1257844402</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Geometric_topology" title="Category:Geometric topology">Geometric topology</a></li><li><a href="/wiki/Category:Manifolds" 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Klein bottle - Wikipedia