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title=\"Wikipedija:Kriteriji značajnosti po temama\"\u003ERaspravite \u003Cb\u003Erazradu po temama kriterija\u003C/b\u003E o značajnosti!\u003C/a\u003E \u003C/td\u003E\n\u003C/tr\u003E\n\u003C/tbody\u003E\u003C/table\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Mrežno mjesto"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Sadržaj" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Početak</div> </a> </li> <li id="toc-Svojstva" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Svojstva"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Svojstva</span> </div> </a> <button aria-controls="toc-Svojstva-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>Sadržaj cjeline Svojstva</span> </button> <ul id="toc-Svojstva-sublist" class="vector-toc-list"> <li id="toc-Usmjerenost" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Usmjerenost"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Usmjerenost</span> </div> </a> <ul id="toc-Usmjerenost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eulerova_karakteristika" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eulerova_karakteristika"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Eulerova karakteristika</span> </div> </a> <ul id="toc-Eulerova_karakteristika-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometrija_i_topologija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometrija_i_topologija"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Geometrija i topologija</span> </div> </a> <button aria-controls="toc-Geometrija_i_topologija-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>Sadržaj cjeline Geometrija i topologija</span> </button> <ul id="toc-Geometrija_i_topologija-sublist" class="vector-toc-list"> <li id="toc-Topologija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topologija"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Topologija</span> </div> </a> <ul id="toc-Topologija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Računalna_grafika" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Računalna_grafika"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Računalna grafika</span> </div> </a> <ul id="toc-Računalna_grafika-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Slični_objekti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Slični_objekti"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Slični objekti</span> </div> </a> <ul id="toc-Slični_objekti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primjena" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primjena"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Primjena</span> </div> </a> <ul id="toc-Primjena-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vanjske_poveznice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vanjske_poveznice"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Vanjske poveznice</span> </div> </a> <ul id="toc-Vanjske_poveznice-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="Sadržaj" 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="Prikaz sadržaja stranice" > <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">Prikaz sadržaja stranice</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">Möbiusova vrpca</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="Idi na druge jezične varijante članka. Dostupan je na 62 jezika" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-62" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">62 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%B1%D9%8A%D8%B7_%D9%85%D9%88%D8%A8%D9%8A%D9%88%D8%B3" title="شريط موبيوس – arapski" lang="ar" hreflang="ar" data-title="شريط موبيوس" data-language-autonym="العربية" data-language-local-name="arapski" 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/M%C3%B6bius_lenti" title="Möbius lenti – azerbajdžanski" lang="az" hreflang="az" data-title="Möbius lenti" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%82%D1%83%D0%B6%D0%BA%D0%B0_%D0%9C%D1%91%D0%B1%D1%96%D1%83%D1%81%D0%B0" title="Стужка Мёбіуса – bjeloruski" lang="be" hreflang="be" data-title="Стужка Мёбіуса" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D1%81%D1%82_%D0%BD%D0%B0_%D0%9C%D1%8C%D0%BE%D0%B1%D0%B8%D1%83%D1%81" title="Лист на Мьобиус – bugarski" lang="bg" hreflang="bg" data-title="Лист на Мьобиус" data-language-autonym="Български" data-language-local-name="bugarski" 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/Cinta_de_M%C3%B6bius" title="Cinta de Möbius – katalonski" lang="ca" hreflang="ca" data-title="Cinta de Möbius" data-language-autonym="Català" data-language-local-name="katalonski" 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/M%C3%B6biova_p%C3%A1ska" title="Möbiova páska – češki" lang="cs" hreflang="cs" data-title="Möbiova páska" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D1%91%D0%B1%D0%B8%D1%83%D1%81_%D1%85%C4%83%D0%B9%C4%83%D0%B2%C4%95" title="Мёбиус хăйăвĕ – čuvaški" lang="cv" hreflang="cv" data-title="Мёбиус хăйăвĕ" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Stribyn_M%C3%B6bius" title="Stribyn Möbius – velški" lang="cy" hreflang="cy" data-title="Stribyn Möbius" data-language-autonym="Cymraeg" data-language-local-name="velški" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/M%C3%B6biusb%C3%A5nd" title="Möbiusbånd – danski" lang="da" hreflang="da" data-title="Möbiusbånd" data-language-autonym="Dansk" data-language-local-name="danski" 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/M%C3%B6biusband" title="Möbiusband – njemački" lang="de" hreflang="de" data-title="Möbiusband" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CF%89%CF%81%CE%AF%CE%B4%CE%B1_%CF%84%CE%BF%CF%85_%CE%9C%CE%AD%CE%BC%CF%80%CE%B9%CE%BF%CF%85%CF%82" title="Λωρίδα του Μέμπιους – grčki" lang="el" hreflang="el" data-title="Λωρίδα του Μέμπιους" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="dobri članak"><a href="https://en.wikipedia.org/wiki/M%C3%B6bius_strip" title="Möbius strip – engleski" lang="en" hreflang="en" data-title="Möbius strip" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rubando_de_M%C3%B6bius" title="Rubando de Möbius – esperanto" lang="eo" hreflang="eo" data-title="Rubando de Möbius" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Banda_de_M%C3%B6bius" title="Banda de Möbius – španjolski" lang="es" hreflang="es" data-title="Banda de Möbius" data-language-autonym="Español" data-language-local-name="španjolski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/M%C3%B6biuse_leht" title="Möbiuse leht – estonski" lang="et" hreflang="et" data-title="Möbiuse leht" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Moebius_banda" title="Moebius banda – baskijski" lang="eu" hreflang="eu" data-title="Moebius banda" data-language-autonym="Euskara" data-language-local-name="baskijski" 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/%D9%86%D9%88%D8%A7%D8%B1_%D9%85%D9%88%D8%A8%DB%8C%D9%88%D8%B3" title="نوار موبیوس – perzijski" lang="fa" hreflang="fa" data-title="نوار موبیوس" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/M%C3%B6biuksen_nauha" title="Möbiuksen nauha – finski" lang="fi" hreflang="fi" data-title="Möbiuksen nauha" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ruban_de_M%C3%B6bius" title="Ruban de Möbius – francuski" lang="fr" hreflang="fr" data-title="Ruban de Möbius" data-language-autonym="Français" data-language-local-name="francuski" 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/M%C3%B6biusb%C3%A2n" title="Möbiusbân – zapadnofrizijski" lang="fy" hreflang="fy" data-title="Möbiusbân" data-language-autonym="Frysk" data-language-local-name="zapadnofrizijski" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Stiall_M%C3%B6bius" title="Stiall Möbius – irski" lang="ga" hreflang="ga" data-title="Stiall Möbius" data-language-autonym="Gaeilge" data-language-local-name="irski" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Banda_de_M%C3%B6bius" title="Banda de Möbius – galicijski" lang="gl" hreflang="gl" data-title="Banda de Möbius" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%91%D7%A2%D7%AA_%D7%9E%D7%91%D7%99%D7%95%D7%A1" title="טבעת מביוס – hebrejski" lang="he" hreflang="he" data-title="טבעת מביוס" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C3%B6bius-szalag" title="Möbius-szalag – mađarski" lang="hu" hreflang="hu" data-title="Möbius-szalag" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%B5%D5%B8%D5%A2%D5%AB%D5%B8%D6%82%D5%BD%D5%AB_%D5%A9%D5%A5%D6%80%D5%A9" title="Մյոբիուսի թերթ – armenski" lang="hy" hreflang="hy" data-title="Մյոբիուսի թերթ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Banda_de_M%C3%B6bius" title="Banda de Möbius – interlingua" lang="ia" hreflang="ia" data-title="Banda de Möbius" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pita_M%C3%B6bius" title="Pita Möbius – indonezijski" lang="id" hreflang="id" data-title="Pita Möbius" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Mobius-strio" title="Mobius-strio – ido" lang="io" hreflang="io" data-title="Mobius-strio" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Nastro_di_M%C3%B6bius" title="Nastro di Möbius – talijanski" lang="it" hreflang="it" data-title="Nastro di Möbius" data-language-autonym="Italiano" data-language-local-name="talijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A1%E3%83%93%E3%82%A6%E3%82%B9%E3%81%AE%E5%B8%AF" title="メビウスの帯 – japanski" lang="ja" hreflang="ja" data-title="メビウスの帯" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/dasri_pe_la_.mobius." title="dasri pe la .mobius. – lojban" lang="jbo" hreflang="jbo" data-title="dasri pe la .mobius." data-language-autonym="La .lojban." data-language-local-name="lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B1%D0%B8%D1%83%D1%81_%D0%B6%D0%B0%D0%BF%D1%8B%D1%80%D0%B0%D2%93%D1%8B" title="Мбиус жапырағы – kazaški" lang="kk" hreflang="kk" data-title="Мбиус жапырағы" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AB%BC%EB%B9%84%EC%9A%B0%EC%8A%A4%EC%9D%98_%EB%9D%A0" title="뫼비우스의 띠 – korejski" lang="ko" hreflang="ko" data-title="뫼비우스의 띠" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Taenia_Moebii" title="Taenia Moebii – latinski" lang="la" hreflang="la" data-title="Taenia Moebii" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/M%C3%B6biusschleef" title="Möbiusschleef – luksemburški" lang="lb" hreflang="lb" data-title="Möbiusschleef" data-language-autonym="Lëtzebuergesch" data-language-local-name="luksemburški" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/M%C4%93biusa_lente" title="Mēbiusa lente – latvijski" lang="lv" hreflang="lv" data-title="Mēbiusa lente" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jalur_M%C3%B6bius" title="Jalur Möbius – malajski" lang="ms" hreflang="ms" data-title="Jalur Möbius" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – nizozemski" lang="nl" hreflang="nl" data-title="Möbiusband" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – norveški nynorsk" lang="nn" hreflang="nn" data-title="Möbiusband" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/M%C3%B6bius-b%C3%A5nd" title="Möbius-bånd – norveški bokmål" lang="nb" hreflang="nb" data-title="Möbius-bånd" data-language-autonym="Norsk bokmål" data-language-local-name="norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Mobius-bende" title="Mobius-bende – Novial" lang="nov" hreflang="nov" data-title="Mobius-bende" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wst%C4%99ga_M%C3%B6biusa" title="Wstęga Möbiusa – poljski" lang="pl" hreflang="pl" data-title="Wstęga Möbiusa" data-language-autonym="Polski" data-language-local-name="poljski" 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/Fita_de_M%C3%B6bius" title="Fita de Möbius – portugalski" lang="pt" hreflang="pt" data-title="Fita de Möbius" data-language-autonym="Português" data-language-local-name="portugalski" 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/Banda_lui_M%C3%B6bius" title="Banda lui Möbius – rumunjski" lang="ro" hreflang="ro" data-title="Banda lui Möbius" data-language-autonym="Română" data-language-local-name="rumunjski" 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%9B%D0%B5%D0%BD%D1%82%D0%B0_%D0%9C%D1%91%D0%B1%D0%B8%D1%83%D1%81%D0%B0" title="Лента Мёбиуса – ruski" lang="ru" hreflang="ru" data-title="Лента Мёбиуса" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Nastru_di_M%C3%B6bius" title="Nastru di Möbius – sicilijski" lang="scn" hreflang="scn" data-title="Nastru di Möbius" data-language-autonym="Sicilianu" data-language-local-name="sicilijski" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/M%C3%B6bius_strip" title="Möbius strip – Simple English" lang="en-simple" hreflang="en-simple" data-title="Möbius strip" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/M%C3%B6biov_list" title="Möbiov list – slovački" lang="sk" hreflang="sk" data-title="Möbiov list" data-language-autonym="Slovenčina" data-language-local-name="slovački" 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/M%C3%B6biusov_trak" title="Möbiusov trak – slovenski" lang="sl" hreflang="sl" data-title="Möbiusov trak" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B1%D0%B8%D1%98%D1%83%D1%81%D0%BE%D0%B2%D0%B0_%D1%82%D1%80%D0%B0%D0%BA%D0%B0" title="Мебијусова трака – srpski" lang="sr" hreflang="sr" data-title="Мебијусова трака" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%B6biusband" title="Möbiusband – švedski" lang="sv" hreflang="sv" data-title="Möbiusband" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Faborka_M%C3%B6biusa" title="Faborka Möbiusa – Silesian" lang="szl" hreflang="szl" data-title="Faborka Möbiusa" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%8B%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B8%E0%AF%8D_%E0%AE%A8%E0%AE%BE%E0%AE%9F%E0%AE%BE" title="மோபியஸ் நாடா – tamilski" lang="ta" hreflang="ta" data-title="மோபியஸ் நாடா" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%96%E0%B8%9A%E0%B9%80%E0%B8%A1%E0%B8%AD%E0%B8%9A%E0%B8%B4%E0%B8%AD%E0%B8%B8%E0%B8%AA" title="แถบเมอบิอุส – tajlandski" lang="th" hreflang="th" data-title="แถบเมอบิอุส" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/M%C3%B6bius_%C5%9Feridi" title="Möbius şeridi – turski" lang="tr" hreflang="tr" data-title="Möbius şeridi" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D1%96%D1%87%D0%BA%D0%B0_%D0%9C%D0%B5%D0%B1%D1%96%D1%83%D1%81%D0%B0" title="Стрічка Мебіуса – ukrajinski" lang="uk" hreflang="uk" data-title="Стрічка Мебіуса" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%B7t_Mobius" title="Mặt Mobius – vijetnamski" lang="vi" hreflang="vi" data-title="Mặt Mobius" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带 – wu kineski" lang="wuu" hreflang="wuu" data-title="莫比乌斯带" data-language-autonym="吴语" data-language-local-name="wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9C%D3%A9%D0%B1%D0%B8%D1%83%D1%81%D0%B8%D0%BD_%D0%BA%D2%AF%D1%81%D0%BC" title="Мөбиусин күсм – kalmyk" lang="xal" hreflang="xal" data-title="Мөбиусин күсм" data-language-autonym="Хальмг" data-language-local-name="kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带 – kineski" lang="zh" hreflang="zh" data-title="莫比乌斯带" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%8E%AB%E6%AF%94%E7%83%8F%E6%96%AF%E5%B8%B6" title="莫比烏斯帶 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="莫比烏斯帶" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%8E%AB%E6%AF%94%E7%83%8F%E6%96%AF%E5%B8%B6" title="莫比烏斯帶 – kantonski" lang="yue" hreflang="yue" data-title="莫比烏斯帶" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q226843#sitelinks-wikipedia" title="Poveznice na druge jezike" class="wbc-editpage">Uredi poveznice</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Imenski prostori"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/M%C3%B6biusova_vrpca" title="Pogledaj sadržaj [c]" accesskey="c"><span>Stranica</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Razgovor:M%C3%B6biusova_vrpca&action=edit&redlink=1" rel="discussion" class="new" title="Razgovorna stranica (stranica ne postoji) [t]" accesskey="t"><span>Razgovor</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Promijeni jezičnu varijantu" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">hrvatski</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Pogledi"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/M%C3%B6biusova_vrpca"><span>Čitaj</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit" title="Uredite ovu stranicu [v]" accesskey="v"><span>Uredi</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit" title="Uredite izvorni kôd ove stranice [e]" accesskey="e"><span>Uredi kôd</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=history" title="Ranije izmjene na ovoj stranici [h]" accesskey="h"><span>Vidi povijest</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Pomagala"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Pomagala" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Pomagala</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > 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class="box-Wikipedizirati messagebox plainlinks" style="width:min(50em, 96%); margin: 0 auto; border-width: 1px 1px 1px 10px; border-style: solid; border-color: #2371CB; padding: 0.5em 1em;"> <tbody><tr style="width: 100%;"> <td style="width: 50px;"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Datoteka:Ryanscontribs.svg" class="mw-file-description" title="Möbiusova vrpca"><img alt="Möbiusova vrpca" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Ryanscontribs.svg/40px-Ryanscontribs.svg.png" decoding="async" width="40" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Ryanscontribs.svg/60px-Ryanscontribs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Ryanscontribs.svg/80px-Ryanscontribs.svg.png 2x" data-file-width="210" data-file-height="204" /></a><figcaption>Möbiusova vrpca</figcaption></figure> </td> <td style="width: calc(100% - 50px);">Ovaj članak ili odjeljak <b>nije wikipediziran</b>. (<a href="/w/index.php?title=Razgovor:M%C3%B6biusova_vrpca&action=edit&redlink=1" class="new" title="Razgovor:Möbiusova vrpca (stranica ne postoji)">Rasprava</a>)<small style="display: block; margin-top: 5px;">Tekst članka treba <a href="/wiki/Wikipedija:Kako_ure%C4%91ivati_stranicu" title="Wikipedija:Kako uređivati stranicu">preurediti u wikitekst</a>. Potom uklonite ovaj predložak.</small> </td></tr></tbody></table> <table class="box-Nedostaju_izvori messagebox plainlinks" style="width:min(50em, 96%); margin: 0 auto; border-width: 1px 1px 1px 10px; border-style: solid; border-color: #2371CB; padding: 0.5em 1em;"> <tbody><tr style="width: 100%;"> <td style="width: 50px;"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Datoteka:Question_book-4.svg" class="mw-file-description" title="Möbiusova vrpca"><img alt="Möbiusova vrpca" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/40px-Question_book-4.svg.png" decoding="async" width="40" height="31" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/60px-Question_book-4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/80px-Question_book-4.svg.png 2x" data-file-width="262" data-file-height="204" /></a><figcaption>Möbiusova vrpca</figcaption></figure> </td> <td style="width: calc(100% - 50px);">Ovaj članak ili dio članka <b>nije pokriven <a href="/wiki/Wikipedija:Vjerodostojni_izvori" title="Wikipedija:Vjerodostojni izvori">izvorima</a></b>.<small style="display: block; margin-top: 5px;">Pomozite Wikipediji <a href="/wiki/Wikipedija:Navo%C4%91enje_izvora" title="Wikipedija:Navođenje izvora">navođenjem odgovarajućih knjiga, članaka u časopisima ili internetskih stranica</a>.</small> </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:M%C3%B6bius_strip_3D_red.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/M%C3%B6bius_strip_3D_red.png/220px-M%C3%B6bius_strip_3D_red.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/M%C3%B6bius_strip_3D_red.png/330px-M%C3%B6bius_strip_3D_red.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/M%C3%B6bius_strip_3D_red.png/440px-M%C3%B6bius_strip_3D_red.png 2x" data-file-width="1280" data-file-height="720" /></a><figcaption>Möbiusova vrpca</figcaption></figure> <p><b>Möbiusova vrpca</b> (<a href="/wiki/Engleski_jezik" title="Engleski jezik">eng.</a> <i>Möbius band</i>, <a href="/wiki/Njema%C4%8Dki_jezik" title="Njemački jezik">njem.</a> <i>Möbiusband</i>), <b>Möbiusova petlja</b> ili <b>Möbiusova ploha</b><sup id="cite_ref-he_1-0" class="reference"><a href="#cite_note-he-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> jednostrana je <a href="/wiki/Povr%C5%A1ina" title="Površina">površina</a> (<a href="/wiki/Ploha_(geometrija)" title="Ploha (geometrija)">ploha</a>) nastala zakretanjem jedne stranice pravokutne vrpce za 180 stupnjeva i njezinim prianjanjem sa suprotnom stranicom. Obilaskom po vrpci mijenja se smjer okomice u suprotan.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Drugim riječima, to je jednostrana ploha s jednom neprekidnom orijentacijom.<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> Otkrili su je njemački matematičari <a href="/w/index.php?title=August_Ferdinand_M%C3%B6bius&action=edit&redlink=1" class="new" title="August Ferdinand Möbius (stranica ne postoji)">August Ferdinand Möbius</a> i <a href="/w/index.php?title=Johan_Benedikt_Listing&action=edit&redlink=1" class="new" title="Johan Benedikt Listing (stranica ne postoji)">Johan Benedikt Listing</a> 1858. godine, neovisno jedan o drugomu, no prozvana je po Möbiusu.<sup id="cite_ref-he_1-1" class="reference"><a href="#cite_note-he-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:August_Ferdinand_M%C3%B6bius.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/August_Ferdinand_M%C3%B6bius.jpg/220px-August_Ferdinand_M%C3%B6bius.jpg" decoding="async" width="220" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/August_Ferdinand_M%C3%B6bius.jpg/330px-August_Ferdinand_M%C3%B6bius.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/August_Ferdinand_M%C3%B6bius.jpg/440px-August_Ferdinand_M%C3%B6bius.jpg 2x" data-file-width="900" data-file-height="1100" /></a><figcaption>August Ferdinand Möbius</figcaption></figure> <p>Möbiusova vrpca nije površina jedinstvene veličine i oblika. Naime, matematičari smatraju Möbiusovom vrpcom bilo koju površinu koja je <a href="/w/index.php?title=Homeomorfizam&action=edit&redlink=1" class="new" title="Homeomorfizam (stranica ne postoji)">homeomorfna</a> s njom. Njezina granica je jednostavna zatvorena krivulja, odnosno homeomorfna je <a href="/wiki/Kru%C5%BEnica" title="Kružnica">kružnici</a>. To omogućava razne geometrijske inačice Möbiusove vrpce, gdje svaka ima određenu veličinu i oblik. </p><p>Poluokret u smjeru kazaljke na satu daje drukčiju vrstu vrpce u odnosu na poluokret u suprotnome smjeru. To znači da je kao objekt u euklidskome prostoru Möbiusova vrpca <a href="/w/index.php?title=Kiralnost_(matematika)&action=edit&redlink=1" class="new" title="Kiralnost (matematika) (stranica ne postoji)">kiralan</a> objekt pozitivne ili negativne orijentacije. Postoji beskonačno mnogo topološki različitih utapanja istoga topološkog prostora u trodimenzijski prostor, što znači da se i Möbiusova vrpca može stvoriti na više načina, primjerice uvrtanjem vrpce neparan broj puta ili njenim vezivanjem u čvor i uvrtanjem (prije spajanja krajeva). </p><p>Izvrnuti papirni model Möbiusove vrpce površina je <a href="/w/index.php?title=Gaussova_zakrivljenost&action=edit&redlink=1" class="new" title="Gaussova zakrivljenost (stranica ne postoji)">Gaussove zakrivljenosti</a> nula. Sustav diferencijalno-algebarskih jednadžbi koji opisuje modele ove vrste objavljen je 2007. godine zajedno s rješenjem.<sup>[<i><a href="/wiki/Wikipedija:Navo%C4%91enje_izvora" title="Wikipedija:Navođenje izvora">nedostaje izvor</a></i>]</sup> </p><p><a href="/w/index.php?title=Eulerova_karakteristika&action=edit&redlink=1" class="new" title="Eulerova karakteristika (stranica ne postoji)">Eulerova karakteristika</a> Möbiusove vrpce jednaka je nuli. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Svojstva">Svojstva</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=1" title="Uredi odlomak: Svojstva" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=1" title="Uredi kôd odjeljka Svojstva"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Mobius_Half_final.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Mobius_Half_final.gif/220px-Mobius_Half_final.gif" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Mobius_Half_final.gif/330px-Mobius_Half_final.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Mobius_Half_final.gif/440px-Mobius_Half_final.gif 2x" data-file-width="634" data-file-height="422" /></a><figcaption>Rezanje Möbiusove vrpce po sredini</figcaption></figure><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Trecina.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trecina.gif/220px-Trecina.gif" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trecina.gif/330px-Trecina.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trecina.gif/440px-Trecina.gif 2x" data-file-width="690" data-file-height="388" /></a><figcaption>Rezanje vrpce po trećini širine</figcaption></figure><p>Möbiusova vrpca ima nekoliko zanimljivih svojstava. </p><p>Kao primjer se često navodi <a href="/wiki/Mrav" class="mw-redirect" title="Mrav">mrav</a> koji hoda duž nje. Nakon jednoga obilaska naći će se sa suprotne strane svoje početne točke, a nakon dva, u svojoj početnoj točki. </p><p>Rezanjem Möbiusove vrpce po sredini dobiva se jedna duža vrpca s dva puna obrta, a ne dvije odvojene vrpce kao što bi se očekivalo. Dobivena vrpca nije Möbiusova. </p><p>S druge strane, ako se Möbiusova vrpca ne reže po sredini, već na razdaljini od jedne trećine svoje širine do ruba, rezultat sječenja bit će dvije vrpce: kraća, koja jest Möbiusova, i duža, koja sadrži dva puna obrta (i koja nije Möbiusova i koja bi se inače dobila rezanjem početne vrpce na dva dijela).<sup>[<i><a href="/wiki/Wikipedija:Navo%C4%91enje_izvora" title="Wikipedija:Navođenje izvora">nedostaje izvor</a></i>]</sup> </p><p>Uopćeno, prepolavljanjem trake koja ima 2n-1 poluobrta, dobiva se traka s 2n punih obrta. Presijecanjem trake koja ima 2n poluobrta, dobivaju se dvije iste takve vrpce, međusobno uvijene n puta. Na primjer, za n = 2, nakon rezanja će jedna vrpca biti dva puta omotana oko druge. Za n = 1 dobit će se dvije karike lanca. </p><p>Dodavanjem još obrta i spajanjem krajeva dobivaju se figure koje se nazivaju <a href="/w/index.php?title=Paradromski_prsteni&action=edit&redlink=1" class="new" title="Paradromski prsteni (stranica ne postoji)">paradromski prsteni</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Usmjerenost">Usmjerenost</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=2" title="Uredi odlomak: Usmjerenost" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=2" title="Uredi kôd odjeljka Usmjerenost"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Surface_of_M%C3%B6bius_strip.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Surface_of_M%C3%B6bius_strip.gif/220px-Surface_of_M%C3%B6bius_strip.gif" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Surface_of_M%C3%B6bius_strip.gif/330px-Surface_of_M%C3%B6bius_strip.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Surface_of_M%C3%B6bius_strip.gif/440px-Surface_of_M%C3%B6bius_strip.gif 2x" data-file-width="629" data-file-height="454" /></a><figcaption>Obilazak okomice po vrpci</figcaption></figure> <p>Površina je usmjerena ako za proizvoljnu jednostavnu zatvorenu krivulju na toj površini i bilo koju točku na toj krivulji vrijedi sljedeće: <a href="/wiki/Vektor" title="Vektor">normalni vektor</a> u toj točki neprekidnim kretanjem duž krivulje vraća se u svoj početni položaj bez promjene smjera. </p><p>Neusmjerenost Möbiusove vrpce može se dokazati i ako se ona predstavi <a href="/wiki/Poliedar" title="Poliedar">poliedarski</a>. Tada je tablica povezanosti tjemena, rub i plohe ovoga modela: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\{0,1,2,3,4,5\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\{0,1,2,3,4,5\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d820a937bf4130a39e5cc803c3e8ece5c64fd901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.204ex; height:2.843ex;" alt="{\displaystyle T=\{0,1,2,3,4,5\}}"></span> </p><p><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 I=\{03,32,21,10,24,45,53,40,15\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>03</mn> <mo>,</mo> <mn>32</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>24</mn> <mo>,</mo> <mn>45</mn> <mo>,</mo> <mn>53</mn> <mo>,</mo> <mn>40</mn> <mo>,</mo> <mn>15</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\{03,32,21,10,24,45,53,40,15\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d7fb67dfc4be6ef4a44995f9c0da19a325c113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.791ex; height:2.843ex;" alt="{\displaystyle I=\{03,32,21,10,24,45,53,40,15\}}"></span> </p><p><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 p_{0}=\langle 0,3,2,1\rangle ,\ p_{1}=\langle 3,2,4,5\rangle ,\ p_{2}=\langle 4,0,1,5\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>4</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}=\langle 0,3,2,1\rangle ,\ p_{1}=\langle 3,2,4,5\rangle ,\ p_{2}=\langle 4,0,1,5\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76fb2089b16512d5b6256af612511e4ffd6ca901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:47.968ex; height:2.843ex;" alt="{\displaystyle p_{0}=\langle 0,3,2,1\rangle ,\ p_{1}=\langle 3,2,4,5\rangle ,\ p_{2}=\langle 4,0,1,5\rangle }"></span> </p><p>Tu T predstavlja skup tjemena, I skup rubova, a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01084a31964201514f3e6bd0136989e11ea6e58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:2.009ex;" alt="{\displaystyle p_{k}}"></span> su plohe. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Poliedarski.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Poliedarski.png/220px-Poliedarski.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Poliedarski.png/330px-Poliedarski.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Poliedarski.png/440px-Poliedarski.png 2x" data-file-width="1440" data-file-height="1080" /></a><figcaption>Poliedarski prikaz Möbiusove vrpce s trima plohama</figcaption></figure> <p>Poliedarska površina je usmjerena ako se može uskladiti orijentacija susjednih plohi, odnosno, kada svake dvije susjedne plohe induciraju suprotnu orijentaciju zajedničkoga ruba. </p><p>Usmjerenja (orijentacije) Möbiusove vrpce ne mogu se uskladiti, što se može dokazati promatrajući najjednostavniji poliedarski model Möbiusove vrpce. Pritom se uočava kako su sve tri plohe susjedne. To znači da orijentacija svake plohe mora biti usklađena s ostalim dvjema. Ako se uskladi orijentacija <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 p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> s <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 p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span>, dobiva se: <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 p_{1}=\langle 5,4,2,3\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>5</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}=\langle 5,4,2,3\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d89ff26bc9b6751952174c050c2a859167d572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.972ex; height:2.843ex;" alt="{\displaystyle p_{1}=\langle 5,4,2,3\rangle }"></span>. Tada je <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 p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f1b08d7d69712872e051c2b33fdfa9f5d42319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{2}}"></span> s <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 p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> usklađeno, ali <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 p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f1b08d7d69712872e051c2b33fdfa9f5d42319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{2}}"></span> s <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 p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span> nije. Promjenom orijentacije <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 p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span>, ona više neće biti usklađena s <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 p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Eulerova_karakteristika">Eulerova karakteristika</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=3" title="Uredi odlomak: Eulerova karakteristika" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=3" title="Uredi kôd odjeljka Eulerova karakteristika"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Formula" title="Formula">Formula</a> za izračunavanje Eulerove karakteristike poliedarske površine je sljedeća: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi =T-I+P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>I</mi> <mo>+</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi =T-I+P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec756e4983fc9a0aef83f79825cff4f474e04b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.788ex; height:2.509ex;" alt="{\displaystyle \chi =T-I+P}"></span> </p><p>Tu je T broj tjemena, I broj rubova, a P broj ploha.<sup>[<span style="color:#22c"><i>što s tim?</i></span>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Geometrija_i_topologija">Geometrija i topologija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=4" title="Uredi odlomak: Geometrija i topologija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=4" title="Uredi kôd odjeljka Geometrija i topologija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Moebius_strip.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/220px-Moebius_strip.svg.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/330px-Moebius_strip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/440px-Moebius_strip.svg.png 2x" data-file-width="712" data-file-height="640" /></a><figcaption>Parametarski prikaz Möbiusove vrpce</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:M%C3%B6biusStripAsSquare.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/M%C3%B6biusStripAsSquare.svg/220px-M%C3%B6biusStripAsSquare.svg.png" decoding="async" width="220" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/M%C3%B6biusStripAsSquare.svg/330px-M%C3%B6biusStripAsSquare.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/M%C3%B6biusStripAsSquare.svg/440px-M%C3%B6biusStripAsSquare.svg.png 2x" data-file-width="512" data-file-height="542" /></a><figcaption>Da bi se napravila Möbiusova vrpca od <a href="/wiki/Kvadrat" title="Kvadrat">kvadrata</a>, treba spojiti stranice tako da se strelice poklope</figcaption></figure> <p>Jedan način za predstavljanje Möbiusove vrpce kao <a href="/wiki/Podskup" title="Podskup">podskupa</a> <b>R</b>3 je putem <a href="/w/index.php?title=Parametrizacija&action=edit&redlink=1" class="new" title="Parametrizacija (stranica ne postoji)">parametrizacije</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b432c59b99f10f24616c9afded1298521e26e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.772ex; height:4.843ex;" alt="{\displaystyle x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u}"></span> </p><p><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 y(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f242e20f236aa09c7145157f030b1e1735a78c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.342ex; height:4.843ex;" alt="{\displaystyle y(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u}"></span> </p><p><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 z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73755c808e0bea6a76c166dcba8ccc2cac777393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.281ex; height:4.676ex;" alt="{\displaystyle z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}}"></span> </p><p>Tu je <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 0\leq u<2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>u</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq u<2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8227e96f13d7d97a7dbe7a215b83f5ec070ce25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.184ex; height:2.343ex;" alt="{\displaystyle 0\leq u<2\pi }"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq v\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>v</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq v\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78efe5b7c254d5806ccaf1ba44f98b7a975ab91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.458ex; height:2.343ex;" alt="{\displaystyle -1\leq v\leq 1}"></span>. Ovo daje Möbiusovu vrpcu širine 1, čija središnja kružnica ima <a href="/wiki/Polumjer" class="mw-redirect" title="Polumjer">polumjer</a> 1, leži u ravnini Oxy i središte joj je u koordinatnome početku. Parametar <i>u</i> kreće se duž vrpce, dok <i>v</i> ide od jednoga ruba do drugog. U <a href="/wiki/Cilindri%C4%8Dni_koordinatni_sustav" title="Cilindrični koordinatni sustav">cilindričnim koordinatama</a> Möbiusova vrpca može se predstaviti pomoću jednadžbe <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 \log(r)\sin \left({\frac {1}{2}}\theta \right)=z\cos \left({\frac {1}{2}}\theta \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(r)\sin \left({\frac {1}{2}}\theta \right)=z\cos \left({\frac {1}{2}}\theta \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a46432ae0ae3d73c2ea74c7e9e226aba8592681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.424ex; height:6.176ex;" alt="{\displaystyle \log(r)\sin \left({\frac {1}{2}}\theta \right)=z\cos \left({\frac {1}{2}}\theta \right).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Topologija">Topologija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=5" title="Uredi odlomak: Topologija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=5" title="Uredi kôd odjeljka Topologija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Topologija" title="Topologija">Topološki</a>, Möbiusova vrpca može se odrediti kao <a href="/wiki/Kvadrat" title="Kvadrat">kvadrat</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]\times [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]\times [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f35a051af39d8299688d7c4a63e39ee5f95c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.145ex; height:2.843ex;" alt="{\displaystyle [0,1]\times [0,1]}"></span> s identifikacijom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,0)~(1-x,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,0)~(1-x,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30eda4f55147aa658bafc5b10194755cdb4e5a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.254ex; height:2.843ex;" alt="{\displaystyle (x,0)~(1-x,1)}"></span> za <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 0\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30810e06ad49f3a837bd2193d4392eda1f74e7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.852ex; height:2.343ex;" alt="{\displaystyle 0\leq x\leq 1}"></span>. Möbiusova vrpca na ovaj je način predstavljena kao površina s povezanom granicom. Neusmjerena je i uzima se kao savršen primjer topološke osobine neorijentabilnosti iz sljedećih razloga: </p> <ul><li>Ne postoje neorijentabilne <a href="/wiki/Mnogostrukost" title="Mnogostrukost">mnogostrukosti</a> dimenzije manje od dva.</li> <li>Möbiusova vrpca je površina koja predstavlja <a href="/wiki/Topolo%C5%A1ki_prostor" title="Topološki prostor">topološki potprostor</a> svake neorijentabilne površine, iz čega slijedi da je dana površina neorijentabilna ako i samo ako sadrži Möbiusovu vrpcu kao svoj potprostor.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Računalna_grafika"><span id="Ra.C4.8Dunalna_grafika"></span>Računalna grafika</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=6" title="Uredi odlomak: Računalna grafika" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=6" title="Uredi kôd odjeljka Računalna grafika"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Creating_a_Mobius_Strip_from_a_rectangle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Creating_a_Mobius_Strip_from_a_rectangle.gif/220px-Creating_a_Mobius_Strip_from_a_rectangle.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Creating_a_Mobius_Strip_from_a_rectangle.gif/330px-Creating_a_Mobius_Strip_from_a_rectangle.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Creating_a_Mobius_Strip_from_a_rectangle.gif/440px-Creating_a_Mobius_Strip_from_a_rectangle.gif 2x" data-file-width="598" data-file-height="448" /></a><figcaption>Stvaranje Möbiusove vrpce svijanjem <a href="/wiki/Pravokutnik" title="Pravokutnik">pravokutnika</a></figcaption></figure> <p>Često se koristi u računalnoj grafici ili softverskim paketima za modeliranje. </p><p>Na primjer, u programu <a href="/w/index.php?title=Autodesk_3ds_Max&action=edit&redlink=1" class="new" title="Autodesk 3ds Max (stranica ne postoji)">3D Studio Max</a>, Möbiusova vrpca dobiva se iscrtavanjem kvadrata (<i>plane</i>) i primjenom modifikatora <i>twist</i> i <i>bend</i> za po 180º i 360º. </p> <div class="mw-heading mw-heading2"><h2 id="Slični_objekti"><span id="Sli.C4.8Dni_objekti"></span>Slični objekti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=7" title="Uredi odlomak: Slični objekti" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=7" title="Uredi kôd odjeljka Slični objekti"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Mobius_to_Klein.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Mobius_to_Klein.gif/150px-Mobius_to_Klein.gif" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/87/Mobius_to_Klein.gif 1.5x" data-file-width="180" data-file-height="180" /></a><figcaption>Slaganje Kleinove boce od dviju Möbiusovih vrpci</figcaption></figure> <p>Möbiusova je vrpca usko povezana s <a href="/w/index.php?title=Kleinova_boca&action=edit&redlink=1" class="new" title="Kleinova boca (stranica ne postoji)">Kleinovom bocom</a>. Boca se može dobiti spajanjem dviju Möbiusovih vrpca po njihovim granicama. Međutim, ovo se u trodimenzijskome <a href="/w/index.php?title=Euklidov_prostor&action=edit&redlink=1" class="new" title="Euklidov prostor (stranica ne postoji)">euklidskom prostoru</a> ne može postići bez samopresjeka. </p><p>Još jedan sličan objekt je <a href="/w/index.php?title=Realna_projekcijska_ravnina&action=edit&redlink=1" class="new" title="Realna projekcijska ravnina (stranica ne postoji)">realna projekcijska ravnina</a> koja se dobiva lijepljenjem Möbiusove vrpce i diska po njihovim rubovima. Također, ne može biti ostvarena u trodimenzijskome prostoru bez samopresjeka. </p><p>U <a href="/wiki/Teorija_grafova" title="Teorija grafova">teoriji grafova</a>, <a href="/w/index.php?title=M%C3%B6biusove_ljestve&action=edit&redlink=1" class="new" title="Möbiusove ljestve (stranica ne postoji)">Möbiusove ljestve</a> su <a href="/wiki/Stupanj_(teorija_grafova)#Regularan_graf" title="Stupanj (teorija grafova)">3-regularan graf</a> s 2n čvorova. Ovakvi grafovi nose ovo ime jer u sebi (osim u slučaju kada je broj čvorova šest, tj. n=3) sadrže točno n ciklusa dužine 4 čijim se spajanjem po zajedničkim rubovima dobiva Möbiusova vrpca. </p> <div class="mw-heading mw-heading2"><h2 id="Primjena">Primjena</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=8" title="Uredi odlomak: Primjena" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=8" title="Uredi kôd odjeljka Primjena"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Möbiusova vrpca, zbog svojih osobina, nalazi brojne primjene u raznim područjima. Koristi se u <a href="/wiki/Fizika" title="Fizika">fizici</a> i <a href="/wiki/Elektrotehnika" title="Elektrotehnika">elektrotehnici</a>. U elektrotehnici se uporabljuje u proizvodnji niskoomskih neinduktivnih <a href="/wiki/Otpornik" title="Otpornik">otpornika</a>, posebice u visokofrekvencijskim i impulsnim uređajima, sustava <a href="/wiki/Kondenzator" class="mw-disambig" title="Kondenzator">kondenzatora</a> u visokofrekvencijskim sklopovima, mikrovalnih rezonatora i <a href="/wiki/Filtar" class="mw-disambig" title="Filtar">filtara</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> U <a href="/wiki/Tvornica" title="Tvornica">tvornicama</a> se koristi kao <a href="/wiki/Teku%C4%87a_vrpca" title="Tekuća vrpca">tekuća vrpca</a>. Na taj način je svaki dio vrpce opterećen istom <a href="/wiki/Te%C5%BEina" title="Težina">težinom</a>, te je ona dugotrajnija. Ista ideja se primjenjivala i u industriji kaseta (kako bi se udvostručilo vrijeme trajanja snimki).<sup id="cite_ref-he_1-2" class="reference"><a href="#cite_note-he-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Upotrebljava se i u proizvodnji <a href="/wiki/Pisa%C4%8D" title="Pisač">pisača</a> i pisaćih strojeva. </p><p><a href="/w/index.php?title=M%C3%B6biusov_otpornik&action=edit&redlink=1" class="new" title="Möbiusov otpornik (stranica ne postoji)">Möbiusov otpornik</a> element je električnoga kruga koji ima svojstvo poništavanja vlastite indukcijske reaktante. <a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Nikola Tesla</a> je početkom dvadesetoga stoljeća patentirao sličnu tehnologiju, koju je namjeravao koristiti u svojemu sustavu za svjetski bežični prijenos energije.<sup>[<i><a href="/wiki/Wikipedija:Navo%C4%91enje_izvora" title="Wikipedija:Navođenje izvora">nedostaje izvor</a></i>]</sup> </p><p>Nadahnuće je za veliki broj umjetničkih djela. Nalazi se ispred ulaza u Muzej američke povijesti u Washingtonu. Nalazila se na brazilskoj <a href="/wiki/Po%C5%A1tanska_marka" title="Poštanska marka">poštanskoj markici</a> 1967. godine, a na belgijskoj 1969. godine. Möbiusova vrpca usvojena je kao međunarodni znak za <a href="/wiki/Oporaba" class="mw-redirect" title="Oporaba">oporabu</a>. Na logotipu je <a href="/wiki/Google_Drive" title="Google Drive">Googleovog Drivea</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=9" title="Uredi odlomak: Izvori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=9" title="Uredi kôd odjeljka Izvori"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r6541845">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-he-1"><span class="mw-cite-backlink">↑ <a href="#cite_ref-he_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-he_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-he_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <a rel="nofollow" class="external text" href="https://www.enciklopedija.hr/natuknica.aspx?ID=69990">Möbiusova ploha</a> <i>enciklopedija.hr</i>. <a href="/wiki/Hrvatska_enciklopedija_(LZMK)" title="Hrvatska enciklopedija (LZMK)">Hrvatska enciklopedija</a>. LZMK. Zagreb. </span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"> <a rel="nofollow" class="external text" href="https://tl.lzmk.hr/clanak/4087">Möbiusova vrpca</a> <i>tl.lzmk.hr</i>. Tehnički leksikon, LZMK: Zagreb. </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"> <a rel="nofollow" class="external text" href="http://lavica.fesb.unist.hr/matematika3/predavanja/node20.html">Plošni integral vektorskog polja</a> <i>lavica.fesb.unist.hr</i>. FESB Split. </span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r7012897">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"»""«"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style><cite id="CITEREFVujević2007" class="citation journal cs1">Vujević, Dušan. 2007. <a rel="nofollow" class="external text" href="https://hrcak.srce.hr/clanak/31708">Primjena Möbiusove vrpce u elektrotehnici</a>. <i>Journal of Energy: Energija</i>. <b>56</b> (6): 700-711.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Energy%3A+Energija&rft.atitle=Primjena+M%C3%B6biusove+vrpce+u+elektrotehnici&rft.volume=56&rft.issue=6&rft.pages=700-711.&rft.date=2007&rft.aulast=Vujevi%C4%87&rft.aufirst=Du%C5%A1an&rft_id=https%3A%2F%2Fhrcak.srce.hr%2Fclanak%2F31708&rfr_id=info%3Asid%2Fhr.wikipedia.org%3AM%C3%B6biusova+vrpca" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Vanjske_poveznice">Vanjske poveznice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&veaction=edit&section=10" title="Uredi odlomak: Vanjske poveznice" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=M%C3%B6biusova_vrpca&action=edit&section=10" title="Uredi kôd odjeljka Vanjske poveznice"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <table border="0" class="noprint" style="clear: right; border: solid var(--border-color-base, #a2a9b1) 1px; margin: 0 0 1em 1em; font-size: 90%; background: var(--background-color-neutral-subtle, #f8f9fa); color: inherit; width: 250px; padding: 4px; spacing: 0px; text-align: left; float: right;"> <tbody><tr> <td><figure class="mw-halign-none" typeof="mw:File"><span title="Logotip Zajedničkog poslužitelja"><img alt="Logotip Zajedničkog poslužitelja" src="//upload.wikimedia.org/wikipedia/commons/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/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span><figcaption>Logotip Zajedničkog poslužitelja</figcaption></figure> </td> <td><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Glavna_stranica?uselang=hr">Zajednički poslužitelj</a> ima još gradiva o temi <i><b><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Moebius_strip?uselang=hr">Möbiusova vrpca</a></b></i></span> </td></tr></tbody></table> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/MoebiusStrip.html">MathWorld : Moebius Strip</a> <span style="cursor:help; 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