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<div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <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="目录" 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">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button 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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#高维度的環面"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>高维度的環面</span> </div> </a> <ul id="toc-高维度的環面-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-环面着色" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#环面着色"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>环面着色</span> </div> </a> <ul id="toc-环面着色-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参見"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>参見</span> </div> </a> <ul id="toc-参見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-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="目录" 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="开关目录" > <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">开关目录</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">环面</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="前往另一种语言写成的文章。68种语言可用" > <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-68" 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">68种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Torus" title="Torus – 南非荷兰语" lang="af" hreflang="af" data-title="Torus" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D8%B1%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="طارة (رياضيات) – 阿拉伯语" lang="ar" hreflang="ar" data-title="طارة (رياضيات)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" 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/Tor_(h%C9%99nd%C9%99si_fiqur)" title="Tor (həndəsi fiqur) – 阿塞拜疆语" lang="az" hreflang="az" data-title="Tor (həndəsi fiqur)" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F)" title="Тор (геаметрыя) – 白俄罗斯语" lang="be" hreflang="be" data-title="Тор (геаметрыя)" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Тор (геометрия) – 保加利亚语" lang="bg" hreflang="bg" data-title="Тор (геометрия)" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Torus" title="Torus – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Torus" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tor_(geometria)" title="Tor (geometria) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Tor (geometria)" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Toru_(giumitria)" title="Toru (giumitria) – 科西嘉语" lang="co" hreflang="co" data-title="Toru (giumitria)" data-language-autonym="Corsu" data-language-local-name="科西嘉语" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Torus" title="Torus – 捷克语" lang="cs" hreflang="cs" data-title="Torus" data-language-autonym="Čeština" data-language-local-name="捷克语" 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%A2%D0%BE%D1%80_(%C3%A7%D0%B8%D0%B9)" title="Тор (çий) – 楚瓦什语" lang="cv" hreflang="cv" data-title="Тор (çий)" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Torus" title="Torus – 丹麦语" lang="da" hreflang="da" data-title="Torus" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Torus" title="Torus – 德语" lang="de" hreflang="de" data-title="Torus" data-language-autonym="Deutsch" data-language-local-name="德语" 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%A4%CF%8C%CF%81%CE%BF%CF%82" title="Τόρος – 希腊语" lang="el" hreflang="el" data-title="Τόρος" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Torus" title="Torus – 英语" lang="en" hreflang="en" data-title="Torus" data-language-autonym="English" data-language-local-name="英语" 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/Toro_(geometrio)" title="Toro (geometrio) – 世界语" lang="eo" hreflang="eo" data-title="Toro (geometrio)" data-language-autonym="Esperanto" data-language-local-name="世界语" 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/Toro_(geometr%C3%ADa)" title="Toro (geometría) – 西班牙语" lang="es" hreflang="es" data-title="Toro (geometría)" data-language-autonym="Español" data-language-local-name="西班牙语" 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/Toor" title="Toor – 爱沙尼亚语" lang="et" hreflang="et" data-title="Toor" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" 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/Toru" title="Toru – 巴斯克语" lang="eu" hreflang="eu" data-title="Toru" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%A8%D8%B1%D9%87" title="چنبره – 波斯语" lang="fa" hreflang="fa" data-title="چنبره" data-language-autonym="فارسی" data-language-local-name="波斯语" 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/Torus" title="Torus – 芬兰语" lang="fi" hreflang="fi" data-title="Torus" data-language-autonym="Suomi" data-language-local-name="芬兰语" 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/Tore" title="Tore – 法语" lang="fr" hreflang="fr" data-title="Tore" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Ring_(Geometrii)" title="Ring (Geometrii) – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Ring (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/T%C3%B3ras" title="Tóras – 爱尔兰语" lang="ga" hreflang="ga" data-title="Tóras" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Toro_(xeometr%C3%ADa_e_topolox%C3%ADa)" title="Toro (xeometría e topoloxía) – 加利西亚语" lang="gl" hreflang="gl" data-title="Toro (xeometría e topoloxía)" data-language-autonym="Galego" data-language-local-name="加利西亚语" 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%95%D7%A8%D7%95%D7%A1" title="טורוס – 希伯来语" lang="he" hreflang="he" data-title="טורוס" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9F%E0%A5%89%E0%A4%B0%E0%A4%B8" title="टॉरस – 印地语" lang="hi" hreflang="hi" data-title="टॉरस" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Torus" title="Torus – 克罗地亚语" lang="hr" hreflang="hr" data-title="Torus" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B3rusz" title="Tórusz – 匈牙利语" lang="hu" hreflang="hu" data-title="Tórusz" data-language-autonym="Magyar" data-language-local-name="匈牙利语" 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%8F%D5%B8%D6%80_(%D5%B4%D5%A1%D5%AF%D5%A5%D6%80%D6%87%D5%B8%D6%82%D5%B5%D5%A9)" title="Տոր (մակերևույթ) – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Տոր (մակերևույթ)" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" 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/Toro" title="Toro – 国际语" lang="ia" hreflang="ia" data-title="Toro" data-language-autonym="Interlingua" data-language-local-name="国际语" 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/Torus" title="Torus – 印度尼西亚语" lang="id" hreflang="id" data-title="Torus" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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/Toro" title="Toro – 伊多语" lang="io" hreflang="io" data-title="Toro" data-language-autonym="Ido" data-language-local-name="伊多语" 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/Toro_(geometria)" title="Toro (geometria) – 意大利语" lang="it" hreflang="it" data-title="Toro (geometria)" data-language-autonym="Italiano" data-language-local-name="意大利语" 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%88%E3%83%BC%E3%83%A9%E3%82%B9" title="トーラス – 日语" lang="ja" hreflang="ja" data-title="トーラス" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98_(%E1%83%92%E1%83%94%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90)" title="ტორი (გეომეტრია) – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="ტორი (გეომეტრია)" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Mbiriga_(mathabu)" title="Mbiriga (mathabu) – 吉库尤语" lang="ki" hreflang="ki" data-title="Mbiriga (mathabu)" data-language-autonym="Gĩkũyũ" data-language-local-name="吉库尤语" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%ED%99%98%EB%A9%B4" title="원환면 – 韩语" lang="ko" hreflang="ko" data-title="원환면" data-language-autonym="한국어" data-language-local-name="韩语" 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/Torus_(mathematica)" title="Torus (mathematica) – 拉丁语" lang="la" hreflang="la" data-title="Torus (mathematica)" data-language-autonym="Latina" data-language-local-name="拉丁语" 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/Torus" title="Torus – 卢森堡语" lang="lb" hreflang="lb" data-title="Torus" data-language-autonym="Lëtzebuergesch" data-language-local-name="卢森堡语" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Toras_(geometrija)" title="Toras (geometrija) – 立陶宛语" lang="lt" hreflang="lt" data-title="Toras (geometrija)" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Tors_(%C4%A3eometrija)" title="Tors (ģeometrija) – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Tors (ģeometrija)" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Тор (геометрија) – 马其顿语" lang="mk" hreflang="mk" data-title="Тор (геометрија)" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9F%E0%B5%8B%E0%B4%B1%E0%B4%B8%E0%B5%8D" title="ടോറസ് – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="ടോറസ്" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Torus" title="Torus – 马来语" lang="ms" hreflang="ms" data-title="Torus" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Torus" title="Torus – 荷兰语" lang="nl" hreflang="nl" data-title="Torus" data-language-autonym="Nederlands" data-language-local-name="荷兰语" 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/Torus" title="Torus – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Torus" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" 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/Torus" title="Torus – 书面挪威语" lang="nb" hreflang="nb" data-title="Torus" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/T%C3%B2r_(geometria)" title="Tòr (geometria) – 奥克语" lang="oc" hreflang="oc" data-title="Tòr (geometria)" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Torus_(matematyka)" title="Torus (matematyka) – 波兰语" lang="pl" hreflang="pl" data-title="Torus (matematyka)" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Toro_(topologia)" title="Toro (topologia) – 葡萄牙语" lang="pt" hreflang="pt" data-title="Toro (topologia)" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Tor" title="Tor – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Tor" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%BF%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D1%8C)" title="Тор (поверхность) – 俄语" lang="ru" hreflang="ru" data-title="Тор (поверхность)" data-language-autonym="Русский" data-language-local-name="俄语" 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/Toru_(giometr%C3%ACa)" title="Toru (giometrìa) – 西西里语" lang="scn" hreflang="scn" data-title="Toru (giometrìa)" data-language-autonym="Sicilianu" data-language-local-name="西西里语" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Torus" title="Torus – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Torus" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Torus" title="Torus – Simple English" lang="en-simple" hreflang="en-simple" data-title="Torus" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Torus_(geometria)" title="Torus (geometria) – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Torus (geometria)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Torus" title="Torus – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Torus" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Torusi" title="Torusi – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Torusi" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80%D1%83%D1%81" title="Торус – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Торус" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" 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/Torus" title="Torus – 瑞典语" lang="sv" hreflang="sv" data-title="Torus" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%B0%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="உருள்வளையம் – 泰米尔语" lang="ta" hreflang="ta" data-title="உருள்வளையம்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%AD%E0%B8%A3%E0%B8%B1%E0%B8%AA_(%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95)" title="ทอรัส (เรขาคณิต) – 泰语" lang="th" hreflang="th" data-title="ทอรัส (เรขาคณิต)" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Simit_(geometri)" title="Simit (geometri) – 土耳其语" lang="tr" hreflang="tr" data-title="Simit (geometri)" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F)" title="Тор (геометрія) – 乌克兰语" lang="uk" hreflang="uk" data-title="Тор (геометрія)" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tor_(jism)" title="Tor (jism) – 乌兹别克语" lang="uz" hreflang="uz" data-title="Tor (jism)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_xuy%E1%BA%BFn" title="Hình xuyến – 越南语" lang="vi" hreflang="vi" data-title="Hình xuyến" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8E%AF%E9%9D%A2" title="环面 – 吴语" lang="wuu" hreflang="wuu" data-title="环面" data-language-autonym="吴语" data-language-local-name="吴语" 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%A2%D0%BE%D1%80" title="Тор – 卡尔梅克语" lang="xal" hreflang="xal" data-title="Тор" data-language-autonym="Хальмг" data-language-local-name="卡尔梅克语" class="interlanguage-link-target"><span>Хальмг</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/Q12510#sitelinks-wikipedia" title="编辑跨语言链接" 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><div id="noteTA-fc6b83e0" class="noteTA"><div class="noteTA-local"><div data-noteta-code="zh-hant:甜甜圈;zh-hans:甜甜圈;zh-hk:冬甩;"></div></div></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Torus.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/80/Torus.jpg" decoding="async" width="300" height="168" class="mw-file-element" data-file-width="300" data-file-height="168" /></a><figcaption>一个环面。</figcaption></figure> <p>在<a href="/wiki/%E5%87%A0%E4%BD%95" class="mw-redirect" title="几何">几何</a>上，一个<b>环面</b>是一个<a href="/wiki/%E6%89%8B%E9%95%AF" title="手镯">手镯</a>形状的<a href="/wiki/%E6%97%8B%E8%BD%89%E6%9B%B2%E9%9D%A2" title="旋轉曲面">旋转曲面</a>，由一个<a href="/wiki/%E5%9C%86" title="圆">圆</a>绕一个和该圆<a href="/wiki/%E5%85%B1%E9%9D%A2" title="共面">共面</a>的一个轴回转所生成。<a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a>可以视为环面的特殊情况，也就是旋转轴是该圆的<a href="/wiki/%E7%9B%B4%E5%BE%84" title="直径">直径</a>时。若转轴和圆不相交，圆面中间有一个洞，就像一个手镯、<a href="/wiki/%E7%94%9C%E7%94%9C%E5%9C%88" title="甜甜圈">甜甜圈</a>、<a href="/wiki/%E5%91%BC%E5%95%A6%E5%9C%88" class="mw-redirect" title="呼啦圈">呼啦圈</a>，或者一个充了气的<a href="/wiki/%E8%BD%AE%E8%83%8E" class="mw-redirect" title="轮胎">轮胎</a>。另一个情况，也就是轴是圆的一根<a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a>的时候，就产生一个挤扁了的球面，就像一个圆的座垫那样。英文<i>Torus</i>曾是<a href="/wiki/%E6%8B%89%E4%B8%81%E6%96%87" class="mw-redirect" title="拉丁文">拉丁文</a>的这种形状的座垫。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="几何"><span id=".E5.87.A0.E4.BD.95"></span>几何</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2&action=edit&section=1" title="编辑章节：几何"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆环面可以参数式地定义为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(u,v)=(R+r\cos {v})\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> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(u,v)=(R+r\cos {v})\cos {u}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab9470ae59c91d22ae36a83a92715cb1155e0d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.806ex; height:2.843ex;" alt="{\displaystyle x(u,v)=(R+r\cos {v})\cos {u}\,}"></span></dd> <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 y(u,v)=(R+r\cos {v})\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> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(u,v)=(R+r\cos {v})\sin {u}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8c4ed3a5ead836a3674244d8bb479d3de473bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.376ex; height:2.843ex;" alt="{\displaystyle y(u,v)=(R+r\cos {v})\sin {u}\,}"></span></dd> <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 z(u,v)=r\sin {v}\,}"> <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> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(u,v)=r\sin {v}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4a814d2a4afc823c9bc0f95ffa8772ba0ecb58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.68ex; height:2.843ex;" alt="{\displaystyle z(u,v)=r\sin {v}\,}"></span></dd></dl> <p>其中 </p> <dl><dd><i>u</i>, <i>v</i> ∈ [0, 2π],</dd> <dd><i>R</i>是管子的中心到画面的中心的距离，</dd> <dd><i>r</i>是圆管的半径。</dd></dl> <p><a href="/wiki/%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E7%B3%BB" class="mw-redirect" title="直角坐标系">直角坐标系</a>中的关于z-<a href="/wiki/%E5%9D%90%E6%A0%87%E8%BD%B4" class="mw-redirect" title="坐标轴">轴</a>方位角对称的环面方程是 </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 \left(R-{\sqrt {x^{2}+y^{2}}}\right)^{2}+z^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(R-{\sqrt {x^{2}+y^{2}}}\right)^{2}+z^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e523d2fae8d455dbff39ee3357a3a59a897a4bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.029ex; height:6.509ex;" alt="{\displaystyle \left(R-{\sqrt {x^{2}+y^{2}}}\right)^{2}+z^{2}=r^{2}}"></span></dd></dl> <p>该圆环面的<a href="/wiki/%E8%A1%A8%E9%9D%A2%E7%A7%AF" class="mw-redirect" title="表面积">表面积</a>和内部<a href="/wiki/%E4%BD%93%E7%A7%AF" title="体积">体积</a>如下 </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 A=4\pi ^{2}Rr=\left(2\pi r\right)\left(2\pi R\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mi>r</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi ^{2}Rr=\left(2\pi r\right)\left(2\pi R\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f899e6e086fe8074cf1d274f8998655ec2abdc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.498ex; height:3.176ex;" alt="{\displaystyle A=4\pi ^{2}Rr=\left(2\pi r\right)\left(2\pi R\right)\,}"></span></dd> <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 V=2\pi ^{2}Rr^{2}=\left(\pi r^{2}\right)\left(2\pi R\right).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=2\pi ^{2}Rr^{2}=\left(\pi r^{2}\right)\left(2\pi R\right).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c958abea3c70c6107cad679110edaf880f8ebc09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.843ex; height:3.343ex;" alt="{\displaystyle V=2\pi ^{2}Rr^{2}=\left(\pi r^{2}\right)\left(2\pi R\right).\,}"></span></dd></dl> <p>根据更一般的定义，环面的<a href="/wiki/%E7%94%9F%E6%88%90%E5%85%83" class="mw-redirect mw-disambig" title="生成元">生成元</a>不必是圆，而可以是<a href="/wiki/%E6%A4%AD%E5%9C%86" title="椭圆">椭圆</a>或任何<a href="/wiki/%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF" title="圆锥曲线">圆锥曲线</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="拓扑"><span id=".E6.8B.93.E6.89.91"></span>拓扑</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2&action=edit&section=2" title="编辑章节：拓扑"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Torus_cycles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Torus_cycles.png/220px-Torus_cycles.png" decoding="async" width="220" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/54/Torus_cycles.png 1.5x" data-file-width="229" data-file-height="311" /></a><figcaption>一个环面是两个圆的乘积。</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Inside-out_torus_(animated,_small).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/ba/Inside-out_torus_%28animated%2C_small%29.gif" decoding="async" width="170" height="170" class="mw-file-element" data-file-width="170" data-file-height="170" /></a><figcaption>将一个有小洞的环面翻转。</figcaption></figure> <p><a href="/wiki/%E6%8B%93%E6%89%91%E5%AD%A6" title="拓扑学">拓扑学</a>上，一个<b>环面</b>是一个定义为两个<a href="/wiki/%E5%9C%86" title="圆">圆</a>的<a href="/wiki/%E7%A7%AF%E6%8B%93%E6%89%91" class="mw-redirect" title="积拓扑">积</a>的闭合<a href="/wiki/%E6%9B%B2%E9%9D%A2" title="曲面">曲面</a>：<i>S</i><sup>1</sup> × <i>S</i><sup>1</sup>。上述曲面，若采用<b>R</b><sup>3</sup>诱导的<a href="/wiki/%E7%9B%B8%E5%AF%B9%E6%8B%93%E6%89%91" class="mw-redirect" title="相对拓扑">相对拓扑</a>，则<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>于一个拓扑环面，只要它不和自己的轴相交。 </p><p>该环面也可用<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%B9%B3%E9%9D%A2" class="mw-redirect" title="欧几里得平面">欧几里得平面</a>的一个<a href="/wiki/%E5%95%86%E7%A9%BA%E9%97%B4" title="商空间">商空间</a>来表述，这是通过如下的等价关系来完成的 </p> <dl><dd>(<i>x</i>,<i>y</i>) ~ (<i>x</i>+1,<i>y</i>) ~ (<i>x</i>,<i>y</i>+1)</dd></dl> <p>或者等价地说，作为<a href="/wiki/%E5%8D%95%E4%BD%8D%E6%AD%A3%E6%96%B9%E5%BD%A2" title="单位正方形">单位正方形</a>将对边粘合的商空间，表述为<a href="/wiki/%E5%9F%BA%E6%9C%AC%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="基本多边形">基本多边形</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 ABA^{-1}B^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABA^{-1}B^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6caa444b9e9549d0ae674d9805da32d35f3993f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.68ex; height:2.676ex;" alt="{\displaystyle ABA^{-1}B^{-1}}"></span>。 </p><p>环面的<a href="/wiki/%E5%9F%BA%E6%9C%AC%E7%BE%A4" title="基本群">基本群</a>是圆的基本群和自身的<a href="/wiki/%E7%9B%B4%E7%A7%AF" title="直积">直积</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 \pi _{1}(\mathbb {T} ^{2})=\pi _{1}(\mathbb {S} ^{1})\times \pi _{1}(\mathbb {S} ^{1})\cong \mathbb {Z} \times \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≅</mo> <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 \pi _{1}(\mathbb {T} ^{2})=\pi _{1}(\mathbb {S} ^{1})\times \pi _{1}(\mathbb {S} ^{1})\cong \mathbb {Z} \times \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ea9c6df7d9a41faa49fbd8c062f11d6bac80ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.842ex; height:3.176ex;" alt="{\displaystyle \pi _{1}(\mathbb {T} ^{2})=\pi _{1}(\mathbb {S} ^{1})\times \pi _{1}(\mathbb {S} ^{1})\cong \mathbb {Z} \times \mathbb {Z} }"></span></dd></dl> <p>直观地讲，这意味着一个先绕着环面的“洞”（譬如，沿着某个纬度方向的圆）然后绕着环面“实体”（譬如，沿着特定经度方向的圆）的闭<a href="/wiki/%E8%B7%AF%E5%BE%84_(%E6%8B%93%E6%89%91%E5%AD%A6)" class="mw-redirect" title="路径 (拓扑学)">路径</a>可以变形成为先绕实体后绕空心的路径。所以，严格的经度方向和严格的纬度方向的路径是可交换的。这可以想象成为两个鞋带互相穿过然后解开再系上。 </p><p>环面的第一<a href="/wiki/%E5%90%8C%E8%B0%83%E7%BE%A4" class="mw-redirect" title="同调群">同调群</a>和基本群<a href="/wiki/%E5%90%8C%E6%9E%84" title="同构">同构</a>（因为基本群是<a href="/wiki/%E4%BA%A4%E6%8D%A2%E7%BE%A4" class="mw-redirect" title="交换群">交换群</a>）。 </p> <div class="mw-heading mw-heading2"><h2 id="高维度的環面"><span id=".E9.AB.98.E7.BB.B4.E5.BA.A6.E7.9A.84.E7.92.B0.E9.9D.A2"></span>高维度的環面</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2&action=edit&section=3" title="编辑章节：高维度的環面"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>环面很容易推广到任意维。<b><i>n</i>维标准环面</b>可以定义为 <i>n</i> 个标准圆的乘积： </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 \mathbb {T} ^{n}=\mathbb {S} ^{1}\times \mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ^{n}=\mathbb {S} ^{1}\times \mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9cc4f1ca69e92eb44e5e0a5d54057d77890515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.152ex; height:2.676ex;" alt="{\displaystyle \mathbb {T} ^{n}=\mathbb {S} ^{1}\times \mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}}"></span></dd></dl> <p>上面所述的环面就是 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 维环面。 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> 维环面就是圆。 <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 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span> 维环面很难描述。和 <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> 维环面一样， <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环面可以表述为 <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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> 在各个坐标方向整数平移下的商空间。也即， <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环面是 <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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> 模(modulo)整数<a href="/wiki/%E6%A0%BC%E7%82%B9" class="mw-redirect" title="格点">格点</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 \mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}"></span> 的<a href="/wiki/%E7%BE%A4%E4%BD%9C%E7%94%A8" title="群作用">群作用</a>（该作用就是向量和）。等价地说，<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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环面是 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维<a href="/wiki/%E7%AB%8B%E6%96%B9%E4%BD%93" class="mw-redirect" title="立方体">立方体</a>把相对的面两两粘合起来得到的空间。 </p><p><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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环是 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维<a href="/wiki/%E7%B4%A7%E8%87%B4%E7%A9%BA%E9%97%B4" class="mw-redirect" title="紧致空间">紧致</a><a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a>的一个例子。它也是紧致<a href="/wiki/%E5%8F%AF%E4%BA%A4%E6%8D%A2" class="mw-redirect" title="可交换">可交换</a><a href="/wiki/%E6%9D%8E%E7%BE%A4" title="李群">李群</a>的一个例子。这是因为<a href="/wiki/%E5%8D%95%E4%BD%8D%E5%9C%86" title="单位圆">单位圆</a>是一个紧致可交换李群（如果把它作为定义了乘法的单位长度<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>来看）。环面上的群乘法可以定义为各坐标分别相乘。 </p><p>环面群在紧致李群理论中有重要的作用。部分原因在于所有紧致李群中总是存在一个<a href="/wiki/%E6%A5%B5%E5%A4%A7%E7%92%B0%E9%9D%A2" title="極大環面">极大环面</a>；也就是最大可能维度的闭<a href="/wiki/%E5%AD%90%E7%BE%A4" title="子群">子群</a>环面。 </p><p><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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环面的<a href="/wiki/%E5%9F%BA%E6%9C%AC%E7%BE%A4" title="基本群">基本群</a>是一个<i>n</i>阶<a href="/w/index.php?title=%E8%87%AA%E7%94%B1%E5%8F%AF%E4%BA%A4%E6%8D%A2%E7%BE%A4&action=edit&redlink=1" class="new" title="自由可交换群（页面不存在）">自由可交换群</a>。 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环面的 <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span></i> 阶<a href="/wiki/%E5%90%8C%E8%B0%83%E7%BE%A4" class="mw-redirect" title="同调群">同调群</a>是 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> <a href="/wiki/%E4%BA%8C%E9%A1%B9%E5%BC%8F%E7%B3%BB%E6%95%B0" class="mw-redirect" title="二项式系数">取</a> <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span></i> 阶的自由可交换群。因此可以推出 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 维环面的<a href="/wiki/%E6%AC%A7%E6%8B%89%E7%A4%BA%E6%80%A7%E6%95%B0" title="欧拉示性数">欧拉示性数</a> 是0。<a href="/wiki/%E4%B8%8A%E5%90%8C%E8%B0%83%E7%8E%AF" title="上同调环">上同调环</a><i>H</i><sup>·</sup>(<b>T</b><sup><i>n</i></sup>,<b>Z</b>)可以等同为<b>Z</b>-<a href="/wiki/%E6%A8%A1" title="模">模</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 \mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}"></span>上的<a href="/wiki/%E5%A4%96%E4%BB%A3%E6%95%B0" title="外代数">外代数</a>，其生成元为 <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> 非平凡闭链的对偶。 </p> <div class="mw-heading mw-heading2"><h2 id="环面着色"><span id=".E7.8E.AF.E9.9D.A2.E7.9D.80.E8.89.B2"></span>环面着色</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2&action=edit&section=4" title="编辑章节：环面着色"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果把环面分成若干区域，那么总是可以用最多7种颜色来着色，使得每对相邻区域有不同的颜色。（这和<a href="/wiki/%E5%9B%9B%E8%89%B2%E9%97%AE%E9%A2%98" class="mw-redirect" title="四色问题">四色问题</a>不同。）在下面的例子中，环面被分为7个区域，两两相邻，说明7色是必须的：<span typeof="mw:File"><a href="/wiki/File:Torus-with-seven-colours.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Torus-with-seven-colours.png/200px-Torus-with-seven-colours.png" decoding="async" width="200" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Torus-with-seven-colours.png/300px-Torus-with-seven-colours.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Torus-with-seven-colours.png/400px-Torus-with-seven-colours.png 2x" data-file-width="991" data-file-height="450" /></a></span> </p> <div class="mw-heading mw-heading2"><h2 id="参見"><span id=".E5.8F.82.E8.A6.8B"></span>参見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2&action=edit&section=5" title="编辑章节：参見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%E4%BB%A3%E6%95%B0%E7%8E%AF%E9%9D%A2&action=edit&redlink=1" class="new" title="代数环面（页面不存在）">代数环面</a></li> <li><a href="/wiki/%E5%9C%86%E7%8E%AF" class="mw-redirect" title="圆环">圆环</a></li> <li><a href="/wiki/%E7%92%B0%E5%9C%88" title="環圈">環圈</a></li> <li><a href="/wiki/%E6%A4%AD%E5%9C%86%E6%9B%B2%E7%BA%BF" title="椭圆曲线">椭圆曲线</a></li> <li><a href="/wiki/%E6%A5%B5%E5%A4%A7%E7%92%B0%E9%9D%A2" title="極大環面">极大环面</a></li> <li><a href="/w/index.php?title=%E5%91%A8%E6%9C%9F%E6%A0%BC&action=edit&redlink=1" class="new" title="周期格（页面不存在）">周期格</a></li> <li><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a></li> <li><a href="/wiki/%E6%9B%B2%E9%9D%A2" title="曲面">曲面</a></li> <li><a href="/w/index.php?title=%E7%8E%AF%E4%BD%93&action=edit&redlink=1" class="new" title="环体（页面不存在）">环体</a></li> <li><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2_(%E5%8E%9F%E5%AD%90%E7%89%A9%E7%90%86)&action=edit&redlink=1" class="new" title="环面 (原子物理)（页面不存在）">环面 (原子物理)</a></li> <li><a href="/w/index.php?title=Villarceau%E5%9C%86&action=edit&redlink=1" class="new" title="Villarceau圆（页面不存在）">Villarceau圆</a></li> <li><a href="/wiki/%E7%92%B0%E9%9D%A2%E6%9B%B2%E7%B7%9A" title="環面曲線">环面曲线</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%8E%AF%E9%9D%A2&action=edit&section=6" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/shortcut.shtml#torus">环面的建立</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20060419190041/http://www.cut-the-knot.org/shortcut.shtml#torus">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）位于网站</li> <li><a rel="nofollow" class="external text" href="http://www.mathsisfun.com/geometry/torus.html">更多环的图像</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20060517195808/http://www.mathsisfun.com/geometry/torus.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） <i>(位于 <a rel="nofollow" class="external text" href="http://www.mathsisfun.com/">趣味数学</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20060428075122/http://www.mathsisfun.com/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）)</i></li> <li>Eric W. Weisstein. "环面." 位于MathWorld--Wolfram网络资源。 <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/Torus.html">http://mathworld.wolfram.com/Torus.html</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20060604035419/http://mathworld.wolfram.com/Torus.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061011195647/http://www.bubblerings.com/bubblerings/media.cfm">气泡圈的图像和视频</a> 位于David Whiteis的<a rel="nofollow" class="external text" href="http://www.bubblerings.com">BubbleRings.com</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.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 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