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class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc 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class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 49 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-49" 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">49 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Zweidimensional" title="Zweidimensional – Alemannic" lang="gsw" hreflang="gsw" data-title="Zweidimensional" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A_%D8%A7%D9%84%D8%A3%D8%A8%D8%B9%D8%A7%D8%AF" title="فضاء ثنائي الأبعاد – Arabic" lang="ar" hreflang="ar" data-title="فضاء ثنائي الأبعاد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_bidimensional" title="Espaciu bidimensional – Asturian" lang="ast" hreflang="ast" data-title="Espaciu bidimensional" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%8D%E0%A6%AC%E0%A6%BF%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="দ্বিমাত্রিক ক্ষেত্র – Bangla" lang="bn" hreflang="bn" data-title="দ্বিমাত্রিক ক্ষেত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/2_diminsi" title="2 diminsi – Banjar" lang="bjn" hreflang="bjn" data-title="2 diminsi" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D0%BA%D0%B5_%D2%AF%D0%BB%D1%81%D3%99%D0%BC%D0%BB%D0%B5_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%A1" title="Ике үлсәмле арауыҡ – Bashkir" lang="ba" hreflang="ba" data-title="Ике үлсәмле арауыҡ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B2%D1%83%D0%BC%D0%B5%D1%80%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Двумерно пространство – Bulgarian" lang="bg" hreflang="bg" data-title="Двумерно пространство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_bidimensional" title="Espai bidimensional – Catalan" lang="ca" hreflang="ca" data-title="Espai bidimensional" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BA_%D0%B2%D0%B8%C3%A7%D0%B5%D0%BB%D0%BB%C4%95_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Ик виçеллĕ уçлăх – Chuvash" lang="cv" hreflang="cv" data-title="Ик виçеллĕ уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/2D" title="2D – Czech" lang="cs" hreflang="cs" data-title="2D" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_dau_ddimensiwn" title="Gofod dau ddimensiwn – Welsh" lang="cy" hreflang="cy" data-title="Gofod dau ddimensiwn" data-language-autonym="Cymraeg" data-language-local-name="Welsh" 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/2D" title="2D – Danish" lang="da" hreflang="da" data-title="2D" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/2D" title="2D – German" lang="de" hreflang="de" data-title="2D" data-language-autonym="Deutsch" data-language-local-name="German" 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%94%CE%B9%CF%83%CE%B4%CE%B9%CE%AC%CF%83%CF%84%CE%B1%CF%84%CE%BF%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Δισδιάστατος χώρος – Greek" lang="el" hreflang="el" data-title="Δισδιάστατος χώρος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Bidimensional" title="Bidimensional – Spanish" lang="es" hreflang="es" data-title="Bidimensional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bidimentsional" title="Bidimentsional – Basque" lang="eu" hreflang="eu" data-title="Bidimentsional" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D8%AF%D9%88%D8%A8%D8%B9%D8%AF%DB%8C" title="فضای دوبعدی – Persian" lang="fa" hreflang="fa" data-title="فضای دوبعدی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Deux_dimensions" title="Deux dimensions – French" lang="fr" hreflang="fr" data-title="Deux dimensions" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_bidimensional" title="Espazo bidimensional – Galician" lang="gl" hreflang="gl" data-title="Espazo bidimensional" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/2%EC%B0%A8%EC%9B%90" title="2차원 – Korean" lang="ko" hreflang="ko" data-title="2차원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D5%B9%D5%A1%D6%83_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Երկչափ տարածություն – Armenian" lang="hy" hreflang="hy" data-title="Երկչափ տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF-%E0%A4%B5%E0%A4%BF%E0%A4%AE_%E0%A4%B8%E0%A4%AE%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%BF" title="द्वि-विम समष्टि – Hindi" lang="hi" hreflang="hi" data-title="द्वि-विम समष्टि" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Dvodimenzionalni_prostor" title="Dvodimenzionalni prostor – Croatian" lang="hr" hreflang="hr" data-title="Dvodimenzionalni prostor" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_dimensi_2" title="Ruang dimensi 2 – Indonesian" lang="id" hreflang="id" data-title="Ruang dimensi 2" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-2D_(Uhlobo_lokumila)" title="I-2D (Uhlobo lokumila) – Xhosa" lang="xh" hreflang="xh" data-title="I-2D (Uhlobo lokumila)" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Bidimensionalit%C3%A0" title="Bidimensionalità – Italian" lang="it" hreflang="it" data-title="Bidimensionalità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%93%D7%95-%D7%9E%D7%9E%D7%93%D7%99" title="מרחב דו-ממדי – Hebrew" lang="he" hreflang="he" data-title="מרחב דו-ממדי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Dua_dimensi" title="Dua dimensi – Malay" lang="ms" hreflang="ms" data-title="Dua dimensi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tweedimensionaal" title="Tweedimensionaal – Dutch" lang="nl" hreflang="nl" data-title="Tweedimensionaal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/2%E6%AC%A1%E5%85%83" title="2次元 – Japanese" lang="ja" hreflang="ja" data-title="2次元" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Todimensjonal" title="Todimensjonal – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Todimensjonal" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_dwuwymiarowa" title="Przestrzeń dwuwymiarowa – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń dwuwymiarowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_bidimensional" title="Espaço bidimensional – Portuguese" lang="pt" hreflang="pt" data-title="Espaço bidimensional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_bidimensional" title="Spațiu bidimensional – Romanian" lang="ro" hreflang="ro" data-title="Spațiu bidimensional" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B2%D1%83%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Двумерное пространство – Russian" lang="ru" hreflang="ru" data-title="Двумерное пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Plani_(matematik%C3%AB)" title="Plani (matematikë) – Albanian" lang="sq" hreflang="sq" data-title="Plani (matematikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/2D" title="2D – Simple English" lang="en-simple" hreflang="en-simple" data-title="2D" 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/2D" title="2D – Slovak" lang="sk" hreflang="sk" data-title="2D" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Dvodimenzionalni_prostor" title="Dvodimenzionalni prostor – Serbian" lang="sr" hreflang="sr" data-title="Dvodimenzionalni prostor" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kaksiulotteisuus" title="Kaksiulotteisuus – Finnish" lang="fi" hreflang="fi" data-title="Kaksiulotteisuus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tv%C3%A5dimensionellt_rum" title="Tvådimensionellt rum – Swedish" lang="sv" hreflang="sv" data-title="Tvådimensionellt rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B0%E0%AE%B3%E0%AE%B5%E0%AF%81_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="ஈரளவு வெளி – Tamil" lang="ta" hreflang="ta" data-title="ஈரளவு வெளி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9B%E0%B8%A3%E0%B8%B4%E0%B8%A0%E0%B8%B9%E0%B8%A1%E0%B8%B4%E0%B8%AA%E0%B8%AD%E0%B8%87%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%B4" title="ปริภูมิสองมิติ – Thai" lang="th" hreflang="th" data-title="ปริภูมิสองมิติ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0ki_boyutlu_uzay" title="İki boyutlu uzay – Turkish" lang="tr" hreflang="tr" data-title="İki boyutlu uzay" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B2%D0%BE%D0%B2%D0%B8%D0%BC%D1%96%D1%80%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Двовимірний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Двовимірний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/2D" title="2D – Venetian" lang="vec" hreflang="vec" data-title="2D" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_hai_chi%E1%BB%81u" title="Không gian hai chiều – Vietnamese" lang="vi" hreflang="vi" data-title="Không gian hai chiều" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BA%8C%E7%B6%AD" title="二維 – Cantonese" lang="yue" hreflang="yue" data-title="二維" 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.mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Plane_(disambiguation)" class="mw-redirect mw-disambig" title="Plane (disambiguation)">Plane (disambiguation)</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>plane</b> is a <a href="/wiki/Two-dimensional_space" title="Two-dimensional space">two-dimensional space</a> or <a href="/wiki/Flat_space" class="mw-redirect" title="Flat space">flat</a> <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> that extends indefinitely. A plane is the two-dimensional analogue of a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> (zero dimensions), a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> (one dimension) and <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>. When working exclusively in two-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, the definite article is used, so <i>the</i> <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> refers to the whole space. </p><p>Several notions of a plane may be defined. The Euclidean plane follows <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, and in particular the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. A <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The <a href="/wiki/Elliptic_plane" class="mw-redirect" title="Elliptic plane">elliptic plane</a> may be further defined by adding a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> to the real projective plane. One may also conceive of a <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a>, which obeys <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> and has a negative <a href="/wiki/Curvature" title="Curvature">curvature</a>. </p><p>Abstractly, one may forget all structure except the topology, producing the topological plane, which is homeomorphic to an <a href="/wiki/Open_disk" class="mw-redirect" title="Open disk">open disk</a>. Viewing the plane as an <a href="/wiki/Affine_space" title="Affine space">affine space</a> produces the <a href="/wiki/Affine_plane" title="Affine plane">affine plane</a>, which lacks a notion of distance but preserves the notion of <a href="/wiki/Collinearity" title="Collinearity">collinearity</a>. Conversely, in adding more structure, one may view the plane as a <a href="/wiki/1-dimensional" class="mw-redirect" title="1-dimensional">1-dimensional</a> <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a>, called the <a href="/wiki/Complex_line" title="Complex line">complex line</a>. </p><p>Many fundamental tasks in mathematics, <a href="/wiki/Geometry" title="Geometry">geometry</a>, <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, and <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphing</a> are performed in a two-dimensional or <i>planar</i> space.<sup id="cite_ref-Janich_Zook_1992_p._50_1-0" class="reference"><a href="#cite_note-Janich_Zook_1992_p._50-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Euclidean_plane">Euclidean plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=1" title="Edit section: Euclidean plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Euclidean_plane&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian-coordinate-system.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/300px-Cartesian-coordinate-system.svg.png" decoding="async" width="300" height="297" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/450px-Cartesian-coordinate-system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/600px-Cartesian-coordinate-system.svg.png 2x" data-file-width="661" data-file-height="654" /></a><figcaption>Bi-dimensional <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> is a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> of <a href="/wiki/Two-dimensional_space" title="Two-dimensional space">dimension two</a>, denoted <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 {\textbf {E}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">E</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {E}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8667dea0c8ec83f2ed680240ba70d6e1c1bcd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.811ex; height:2.676ex;" alt="{\displaystyle {\textbf {E}}^{2}}"></span> or <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 {E} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c2c8e9a82f057e1a8f354057896167fe621734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {E} ^{2}}"></span>. It is a <a href="/wiki/Geometric_space" class="mw-redirect" title="Geometric space">geometric space</a> in which two <a href="/wiki/Real_number" title="Real number">real numbers</a> are required to determine the <a href="/wiki/Position_(geometry)" title="Position (geometry)">position</a> of each <a href="/wiki/Point_(mathematics)" class="mw-redirect" title="Point (mathematics)">point</a>. It is an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, which includes in particular the concept of <a href="/wiki/Parallel_lines" class="mw-redirect" title="Parallel lines">parallel lines</a>. It has also <a href="/wiki/Measurement" title="Measurement">metrical properties</a> induced by a <a href="/wiki/Euclidean_distance" title="Euclidean distance">distance</a>, which allows to define <a href="/wiki/Circle" title="Circle">circles</a>, and <a href="/wiki/Angle" title="Angle">angle measurement</a>. </p><p>A Euclidean plane with a chosen <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> is called a <i><a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a></i>. </p> The set <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} ^{2}}"> <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"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> of the ordered pairs of real numbers (the <a href="/wiki/Real_coordinate_plane" class="mw-redirect" title="Real coordinate plane">real coordinate plane</a>), equipped with the <a href="/wiki/Dot_product" title="Dot product">dot product</a>, is often called <i>the</i> Euclidean plane or <i>standard Euclidean plane</i>, since every Euclidean plane is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to it.</div></div> <div class="mw-heading mw-heading3"><h3 id="Embedding_in_three-dimensional_space">Embedding in three-dimensional space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=2" title="Edit section: Embedding in three-dimensional space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Euclidean_planes_in_three-dimensional_space" title="Euclidean planes in three-dimensional space">Euclidean planes in three-dimensional space</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Euclidean_planes_in_three-dimensional_space&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plane_equation_qtl3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Plane_equation_qtl3.svg/220px-Plane_equation_qtl3.svg.png" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Plane_equation_qtl3.svg/330px-Plane_equation_qtl3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Plane_equation_qtl3.svg/440px-Plane_equation_qtl3.svg.png 2x" data-file-width="470" data-file-height="310" /></a><figcaption>Plane equation in normal form</figcaption></figure> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, a <a href="/wiki/Euclidean_planes_in_three-dimensional_space" title="Euclidean planes in three-dimensional space">plane</a> is a <a href="/wiki/Flat_space" class="mw-redirect" title="Flat space">flat</a> two-<a href="/wiki/Dimension" title="Dimension">dimensional</a> <a href="/wiki/Surface_(geometry)" class="mw-redirect" title="Surface (geometry)">surface</a> that extends indefinitely. <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean planes</a> often arise as <a href="/wiki/Euclidean_subspace" class="mw-redirect" title="Euclidean subspace">subspaces</a> of <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</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 {R} ^{3}}"> <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"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. </p> While a pair of real numbers <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} ^{2}}"> <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"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their <a href="/wiki/Embedding" title="Embedding">embedding</a> in the <a href="/wiki/Ambient_space" class="mw-redirect" title="Ambient space">ambient space</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 {R} ^{3}}"> <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"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Elliptic_plane">Elliptic plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=3" title="Edit section: Elliptic plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Elliptic_geometry#Elliptic_plane" title="Elliptic geometry">Elliptic geometry § Elliptic plane</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Elliptic_geometry&action=edit#Elliptic_plane">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>The elliptic plane is the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a> provided with a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>. <a href="/wiki/Kepler" class="mw-redirect" title="Kepler">Kepler</a> and <a href="/wiki/Desargues" class="mw-redirect" title="Desargues">Desargues</a> used the <a href="/wiki/Gnomonic_projection" title="Gnomonic projection">gnomonic projection</a> to relate a plane σ to points on a <a href="/wiki/Sphere" title="Sphere">hemisphere</a> tangent to it. With <i>O</i> the center of the hemisphere, a point <i>P</i> in σ determines a line <i>OP</i> intersecting the hemisphere, and any line <i>L</i> ⊂ σ determines a plane <i>OL</i> which intersects the hemisphere in half of a <a href="/wiki/Great_circle" title="Great circle">great circle</a>. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of <i>σ</i> corresponds to this plane; instead a <a href="/wiki/Line_at_infinity" title="Line at infinity">line at infinity</a> is appended to <i>σ</i>. As any line in this extension of σ corresponds to a plane through <i>O</i>, and since any pair of such planes intersects in a line through <i>O</i>, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.<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> </p> Given <i>P</i> and <i>Q</i> in <i>σ</i>, the elliptic distance between them is the measure of the angle <i>POQ</i>, usually taken in radians. <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> initiated the study of elliptic geometry when he wrote "On the definition of distance".<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><sup class="reference nowrap"><span title="Page / location: 82">: 82 </span></sup> This venture into abstraction in geometry was followed by <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> and <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> leading to <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> and <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Projective_plane">Projective plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=4" title="Edit section: Projective plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Projective_plane" title="Projective plane">Projective plane</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Projective_plane&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Finite_projective_planes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Finite_projective_planes.svg/170px-Finite_projective_planes.svg.png" decoding="async" width="170" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Finite_projective_planes.svg/255px-Finite_projective_planes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Finite_projective_planes.svg/340px-Finite_projective_planes.svg.png 2x" data-file-width="512" data-file-height="819" /></a><figcaption>Drawings of the finite projective planes of orders 2 (the <a href="/wiki/Fano_plane" title="Fano plane">Fano plane</a>) and 3, in grid layout, showing a method of creating such drawings for prime orders</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Railroad-Tracks-Perspective.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Railroad-Tracks-Perspective.jpg/170px-Railroad-Tracks-Perspective.jpg" decoding="async" width="170" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Railroad-Tracks-Perspective.jpg/255px-Railroad-Tracks-Perspective.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Railroad-Tracks-Perspective.jpg/340px-Railroad-Tracks-Perspective.jpg 2x" data-file-width="1536" data-file-height="2048" /></a><figcaption>These parallel lines appear to intersect in the <a href="/wiki/Vanishing_point" title="Vanishing point">vanishing point</a> "at infinity". In a projective plane this is actually true.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> is a geometric structure that extends the concept of a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus <i>any</i> two distinct lines in a projective plane intersect at exactly one point. </p><p>Renaissance artists, in developing the techniques of drawing in <a href="/wiki/Perspective_(graphical)#Renaissance" title="Perspective (graphical)">perspective</a>, laid the groundwork for this mathematical topic. The archetypical example is the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>, also known as the extended Euclidean plane.<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> This example, in slightly different guises, is important in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="/wiki/Topology" title="Topology">topology</a> and <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> where it may be denoted variously by <span class="nowrap">PG(2, R)</span>, RP<sup>2</sup>, or P<sub>2</sub>(R), among other notations. There are many other projective planes, both infinite, such as the <a href="/wiki/Complex_projective_plane" title="Complex projective plane">complex projective plane</a>, and finite, such as the <a href="/wiki/Fano_plane" title="Fano plane">Fano plane</a>. </p> A projective plane is a 2-dimensional <a href="/wiki/Projective_space" title="Projective space">projective space</a>. Not all projective planes can be <a href="/wiki/Embedding" title="Embedding">embedded</a> in 3-dimensional projective spaces; such embeddability is a consequence of a property known as <a href="/wiki/Desargues%27_theorem" class="mw-redirect" title="Desargues' theorem">Desargues' theorem</a>, not shared by all projective planes.</div></div> <div class="mw-heading mw-heading2"><h2 id="Further_generalizations">Further generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=5" title="Edit section: Further generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to its familiar <a href="/wiki/Geometric" class="mw-redirect" title="Geometric">geometric</a> structure, with <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a> that are <a href="/wiki/Isometry" title="Isometry">isometries</a> with respect to the usual inner product, the plane may be viewed at various other levels of <a href="/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)">abstraction</a>. Each level of abstraction corresponds to a specific <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. </p><p>At one extreme, all geometrical and <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> concepts may be dropped to leave the <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> plane, which may be thought of as an idealized <a href="/wiki/Homotopy" title="Homotopy">homotopically</a> trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a> (or 2-manifolds) classified in <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional topology</a>. Isomorphisms of the topological plane are all <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Bijection" title="Bijection">bijections</a>. The topological plane is the natural context for the branch of <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> that deals with <a href="/wiki/Planar_graphs" class="mw-redirect" title="Planar graphs">planar graphs</a>, and results such as the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a>. </p><p>The plane may also be viewed as an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but <a href="/wiki/Collinearity" title="Collinearity">collinearity</a> and ratios of distances on any line are preserved. </p><p><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> views a plane as a 2-dimensional real <a href="/wiki/Manifold" title="Manifold">manifold</a>, a topological plane which is provided with a <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a>. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> or <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability. </p><p>In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> and the major area of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>. The complex field has only two isomorphisms that leave the real line fixed, the identity and <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">conjugation</a>. </p><p>In the same way as in the real case, the plane may also be viewed as the simplest, <a href="/wiki/One-dimensional" class="mw-redirect" title="One-dimensional">one-dimensional</a> (in terms of <a href="/wiki/Complex_dimension" title="Complex dimension">complex dimension</a>, over the complex numbers) <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a>, sometimes called the <a href="/wiki/Complex_line" title="Complex line">complex line</a>. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all <a href="/wiki/Conformal_map" title="Conformal map">conformal</a> bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation. </p><p>In addition, the Euclidean geometry (which has zero <a href="/wiki/Curvature" title="Curvature">curvature</a> everywhere) is not the only geometry that the plane may have. The plane may be given a <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a> by using the <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>. This can be thought of as placing a sphere tangent to the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point. This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature. </p><p>Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic plane</a>. The latter possibility finds an application in the theory of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a <a href="/wiki/Timelike" class="mw-redirect" title="Timelike">timelike</a> <a href="/wiki/Hypersurface" title="Hypersurface">hypersurface</a> in three-dimensional <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Topological_and_differential_geometric_notions">Topological and differential geometric notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=6" title="Edit section: Topological and differential geometric notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/One-point_compactification" class="mw-redirect" title="One-point compactification">one-point compactification</a> of the plane is homeomorphic to a <a href="/wiki/Sphere" title="Sphere">sphere</a> (see <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a <a href="/wiki/Manifold" title="Manifold">manifold</a> referred to as the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a> or the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a> <a href="/wiki/Projective_line" title="Projective line">projective line</a>. The projection from the Euclidean plane to a sphere without a point is a <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> and even a <a href="/wiki/Conformal_map" title="Conformal map">conformal map</a>. </p><p>The plane itself is homeomorphic (and diffeomorphic) to an open <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a>. For the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic plane</a> such diffeomorphism is conformal, but for the Euclidean plane it is not. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Affine_plane" title="Affine plane">Affine plane</a></li> <li><a href="/wiki/Half-plane" class="mw-redirect" title="Half-plane">Half-plane</a></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic geometry</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plane_(mathematics)&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Janich_Zook_1992_p._50-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Janich_Zook_1992_p._50_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJanichZook1992" class="citation book cs1">Janich, P.; Zook, D. (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0DJ5Fq35NYQC&pg=PA50"><i>Euclid's Heritage. Is Space Three-Dimensional?</i></a>. The Western Ontario Series in Philosophy of Science. Springer Netherlands. p. 50. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-2025-8" title="Special:BookSources/978-0-7923-2025-8"><bdi>978-0-7923-2025-8</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euclid%27s+Heritage.+Is+Space+Three-Dimensional%3F&rft.series=The+Western+Ontario+Series+in+Philosophy+of+Science&rft.pages=50&rft.pub=Springer+Netherlands&rft.date=1992&rft.isbn=978-0-7923-2025-8&rft.aulast=Janich&rft.aufirst=P.&rft.au=Zook%2C+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0DJ5Fq35NYQC%26pg%3DPA50&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlane+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">H. S. M. Coxeter</a> (1965) Introduction to Geometry, page 92</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCayley1859" class="citation cs2"><a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley, Arthur</a> (1859), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1432432">"A sixth memoir upon quantics"</a>, <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society_of_London" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London">Philosophical Transactions of the Royal Society of London</a></i>, <b>149</b>: 61–90, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1859.0004">10.1098/rstl.1859.0004</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0080-4614">0080-4614</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/108690">108690</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=A+sixth+memoir+upon+quantics&rft.volume=149&rft.pages=61-90&rft.date=1859&rft.issn=0080-4614&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F108690%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frstl.1859.0004&rft.aulast=Cayley&rft.aufirst=Arthur&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1432432&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlane+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">The phrases "projective plane", "extended affine plane" and "extended Euclidean plane" may be distinguished according to whether the line at infinity is regarded as special (in the so-called "projective" plane it is not, in the "extended" planes it is) and to whether Euclidean metric is regarded as meaningful (in the projective and affine planes it is not). Similarly for projective or extended spaces of other dimensions.</span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Plane_(mathematics)&oldid=1250022913">https://en.wikipedia.org/w/index.php?title=Plane_(mathematics)&oldid=1250022913</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Geometry" title="Category:Geometry">Geometry</a></li><li><a href="/wiki/Category:Surfaces" title="Category:Surfaces">Surfaces</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_with_excerpts" title="Category:Articles with excerpts">Articles with excerpts</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 October 2024, at 01:49<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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