Oktonion – Wikipedia

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id="toc-Fano-planet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fano-planet"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fano-planet</span> </div> </a> <ul id="toc-Fano-planet-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Se_også" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Se også</span> </div> </a> <ul id="toc-Se_også-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eksterne_lenker" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eksterne_lenker"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Eksterne lenker</span> </div> </a> <ul id="toc-Eksterne_lenker-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="Innhold" 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="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Oktonion</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="Gå til en artikkel på et annet språk. Tilgjengelig på 33 språk" > <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-33" 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">33 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Oktonion" title="Oktonion – svensk" lang="sv" hreflang="sv" data-title="Oktonion" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AB%D9%85%D8%A7%D9%86%D9%8A_%D9%85%D8%B1%D9%83%D8%A8" title="عدد ثماني مركب – arabisk" lang="ar" hreflang="ar" data-title="عدد ثماني مركب" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Octoni%C3%B3" title="Octonió – katalansk" lang="ca" hreflang="ca" data-title="Octonió" data-language-autonym="Català" data-language-local-name="katalansk" 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%9A%D1%8D%D0%BB%D0%B8_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8" title="Кэли алгебри – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Кэли алгебри" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" 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/Oktonion" title="Oktonion – tsjekkisk" lang="cs" hreflang="cs" data-title="Oktonion" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Oktave_(Mathematik)" title="Oktave (Mathematik) – tysk" lang="de" hreflang="de" data-title="Oktave (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="tysk" 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%9F%CE%BA%CF%84%CF%8C%CE%BD%CE%B9%CE%BF" title="Οκτόνιο – gresk" lang="el" hreflang="el" data-title="Οκτόνιο" data-language-autonym="Ελληνικά" data-language-local-name="gresk" 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/Octonion" title="Octonion – engelsk" lang="en" hreflang="en" data-title="Octonion" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Octoni%C3%B3n" title="Octonión – spansk" lang="es" hreflang="es" data-title="Octonión" data-language-autonym="Español" data-language-local-name="spansk" 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/Oktonioi" title="Oktonioi – baskisk" lang="eu" hreflang="eu" data-title="Oktonioi" data-language-autonym="Euskara" data-language-local-name="baskisk" 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%87%D8%B4%D8%AA%DA%AF%D8%A7%D9%86%E2%80%8C%D9%87%D8%A7" title="هشتگان‌ها – persisk" lang="fa" hreflang="fa" data-title="هشتگان‌ها" data-language-autonym="فارسی" data-language-local-name="persisk" 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/Octonion" title="Octonion – fransk" lang="fr" hreflang="fr" data-title="Octonion" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8C%94%EC%9B%90%EC%88%98" title="팔원수 – koreansk" lang="ko" hreflang="ko" data-title="팔원수" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Oktonion" title="Oktonion – indonesisk" lang="id" hreflang="id" data-title="Oktonion" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Octonion" title="Octonion – interlingua" lang="ia" hreflang="ia" data-title="Octonion" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ottetto_(matematica)" title="Ottetto (matematica) – italiensk" lang="it" hreflang="it" data-title="Ottetto (matematica)" data-language-autonym="Italiano" data-language-local-name="italiensk" 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%90%D7%9C%D7%92%D7%91%D7%A8%D7%AA_%D7%94%D7%90%D7%95%D7%A7%D7%98%D7%95%D7%A0%D7%99%D7%95%D7%A0%D7%99%D7%9D_%D7%A9%D7%9C_%D7%A7%D7%99%D7%99%D7%9C%D7%99" title="אלגברת האוקטוניונים של קיילי – hebraisk" lang="he" hreflang="he" data-title="אלגברת האוקטוניונים של קיילי" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_octonus" title="Numerus octonus – latin" lang="la" hreflang="la" data-title="Numerus octonus" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Okt%C3%B3ni%C3%B3k" title="Októniók – ungarsk" lang="hu" hreflang="hu" data-title="Októniók" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Octonion" title="Octonion – nederlandsk" lang="nl" hreflang="nl" data-title="Octonion" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" 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/%E5%85%AB%E5%85%83%E6%95%B0" title="八元数 – japansk" lang="ja" hreflang="ja" data-title="八元数" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Oktawy_Cayleya" title="Oktawy Cayleya – polsk" lang="pl" hreflang="pl" data-title="Oktawy Cayleya" data-language-autonym="Polski" data-language-local-name="polsk" 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/Octoni%C3%A3o" title="Octonião – portugisisk" lang="pt" hreflang="pt" data-title="Octonião" data-language-autonym="Português" data-language-local-name="portugisisk" 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/Octonion" title="Octonion – rumensk" lang="ro" hreflang="ro" data-title="Octonion" data-language-autonym="Română" data-language-local-name="rumensk" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%9A%D1%8D%D0%BB%D0%B8" title="Алгебра Кэли – russisk" lang="ru" hreflang="ru" data-title="Алгебра Кэли" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Octonion" title="Octonion – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Octonion" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Oktonion" title="Oktonion – slovensk" lang="sl" hreflang="sl" data-title="Oktonion" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%AD%E0%B8%81%E0%B9%82%E0%B8%97%E0%B9%80%E0%B8%99%E0%B8%B5%E0%B8%A2%E0%B8%99" title="ออกโทเนียน – thai" lang="th" hreflang="th" data-title="ออกโทเนียน" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%BA%D1%82%D0%BE%D0%BD%D1%96%D0%BE%D0%BD" title="Октоніон – ukrainsk" lang="uk" hreflang="uk" data-title="Октоніон" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Octonion" title="Octonion – vietnamesisk" lang="vi" hreflang="vi" data-title="Octonion" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%85%AB%E5%85%83%E6%95%B8" title="八元數 – klassisk kinesisk" lang="lzh" hreflang="lzh" data-title="八元數" data-language-autonym="文言" data-language-local-name="klassisk kinesisk" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%AB%E5%85%83%E6%95%B8" title="八元數 – kantonesisk" lang="yue" hreflang="yue" data-title="八元數" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%AB%E5%85%83%E6%95%B0" title="八元数 – kinesisk" lang="zh" hreflang="zh" data-title="八元数" data-language-autonym="中文" data-language-local-name="kinesisk" class="interlanguage-link-target"><span>中文</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/Q743418#sitelinks-wikipedia" title="Rediger lenker til artikkelen på andre språk" class="wbc-editpage">Rediger lenker</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Navnerom"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Oktonion" title="Vis innholdssiden [c]" accesskey="c"><span>Artikkel</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskusjon:Oktonion&action=edit&redlink=1" rel="discussion" class="new" title="Diskusjon om innholdssiden (ikke skrevet ennå) [t]" accesskey="t"><span>Diskusjon</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Bytt språkvariant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">norsk bokmål</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Visninger"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Oktonion"><span>Les</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Oktonion&veaction=edit" title="Rediger siden [v]" accesskey="v"><span>Rediger</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Oktonion&action=edit" title="Rediger kildekoden for denne siden [e]" accesskey="e"><span>Rediger kilde</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Oktonion&action=history" title="Tidligere sideversjoner av denne siden [h]" accesskey="h"><span>Vis historikk</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sideverktøy"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Verktøy" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Verktøy</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Verktøy</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skjul</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Flere alternativer" > <div class="vector-menu-heading"> Handlinger </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Oktonion"><span>Les</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Oktonion&veaction=edit" title="Rediger siden [v]" accesskey="v"><span>Rediger</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Oktonion&action=edit" title="Rediger kildekoden for denne siden [e]" accesskey="e"><span>Rediger kilde</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Oktonion&action=history"><span>Vis historikk</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generelt </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Spesial:Lenker_hit/Oktonion" title="Liste over alle wikisider som lenker hit [j]" accesskey="j"><span>Lenker hit</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Spesial:Relaterte_endringer/Oktonion" rel="nofollow" title="Siste endringer i sider som blir lenket fra denne siden [k]" accesskey="k"><span>Relaterte endringer</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=no" title="Last opp filer [u]" accesskey="u"><span>Last opp fil</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Spesial:Spesialsider" title="Liste over alle spesialsider [q]" accesskey="q"><span>Spesialsider</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Oktonion&oldid=22061304" title="Permanent lenke til denne versjonen av siden"><span>Permanent lenke</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Oktonion&action=info" title="Mer informasjon om denne siden"><span>Sideinformasjon</span></a></li><li id="t-cite" class="mw-list-item"><a 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</div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Sideverktøy"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Utseende"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Fra Wikipedia, den frie encyklopedi</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="nb" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Fano_mnemonic3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Fano_mnemonic3.png/250px-Fano_mnemonic3.png" decoding="async" width="250" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Fano_mnemonic3.png/375px-Fano_mnemonic3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d9/Fano_mnemonic3.png 2x" data-file-width="500" data-file-height="445" /></a><figcaption>De syv punktene i <a href="/wiki/Projektivt_plan#Eksempel:_Syvpunktsplanet" title="Projektivt plan">Fano-planet</a> kan direkte knyttes til oktonionene.</figcaption></figure> <p><b>Oktonion</b> (fra <a href="/wiki/Latin" title="Latin">latin</a> <i>octo</i> - åtte) eller <b>Cayley-tall</b> er et element i en åttedimensjonal utvidelse av de <a href="/wiki/Reelt_tall" title="Reelt tall">reelle tallene</a> på samme måte som <a href="/wiki/Kvaternion" title="Kvaternion">kvaternionene</a> er en firedimensjonal utvidelse. Den tilsvarende <a href="/wiki/Tallmengde" class="mw-redirect" title="Tallmengde">tallmengden</a> betegnes vanligvis med <b>O</b> eller <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 {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>. Tallene <a href="/wiki/Kommutativ_lov" title="Kommutativ lov">kommuterer</a> ikke ved <a href="/wiki/Multiplikasjon" title="Multiplikasjon">multiplikasjon</a> og denne operasjonen er heller ikke <a href="/wiki/Assosiativ_lov" title="Assosiativ lov">assosiativ</a>. I motsetning til kvaternionene kan de derfor ikke representeres ved <a href="/wiki/Matrise" title="Matrise">matriser</a> og har ingen kjente, praktiske anvendelser. Innen ren <a href="/wiki/Matematikk" title="Matematikk">matematikk</a> knyttes de til spesielle <a href="/wiki/Lie-gruppe" title="Lie-gruppe">Lie-grupper</a> og deres egenskaper. </p><p><a href="/wiki/Tallmengde" class="mw-redirect" title="Tallmengde">Tallmengdene</a> <b>R</b> av <a href="/wiki/Reelt_tall" title="Reelt tall">reelle tall</a>, <b>C</b> av <a href="/wiki/Komplekst_tall" title="Komplekst tall">komplekse tall</a>, <b>H</b> av <a href="/wiki/Kvaternion" title="Kvaternion">kvaternioner</a> og <b>O</b> av oktonioner er de eneste <a href="/wiki/Algebra" title="Algebra">divisjonsalgebraene</a> over de reelle tallene som er <a href="/wiki/Norm_(matematikk)" title="Norm (matematikk)">normerte</a>, det vil si at hvert element kan tilordnes en entydig, numerisk verdi. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historie">Historie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=1" title="Rediger avsnitt: Historie" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=1" title="Rediger kildekoden til seksjonen Historie"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dagen etter at <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> i oktober 1843 oppdaget <a href="/wiki/Kvaternion" title="Kvaternion">kvaternionene</a>, skrev han om disse i et brev til sin venn <a href="/w/index.php?title=John_Thomas_Graves&action=edit&redlink=1" class="new" title="John Thomas Graves (ikke skrevet ennå)">John Thomas Graves</a>. I desember samme år kunne Graves fortelle Hamilton at han hadde kommet frem til en lignende, åttedimensjonal utvidelse som han kalte for «oktaver». Komponentene oppfylte en «åtte-kvadraters identitet» av samme form som <a href="/wiki/Kvaternion#Konjugasjon_og_norm" title="Kvaternion">fire-kvadrats identiteten</a> for komponentene til to kvarterioner. </p><p>På samme tid hadde også den unge <a href="/w/index.php?title=Arthur_Cayley&action=edit&redlink=1" class="new" title="Arthur Cayley (ikke skrevet ennå)">Arthur Cayley</a> fattet interesse for Hamiltons kvaternioner. I et publisert arbeid om <a href="/wiki/Elliptisk_funksjon" title="Elliptisk funksjon">elliptiske funksjoner</a> i 1845 tok han med et appendiks hvor han presenterte egenskapene til dagens oktonioner som han hadde kommet frem til på egen hånd. Selv om Hamilton kunne bekrefte at Graves hadde gjort dette tidligere, ble det likevel arbeidet til Cayley som ble lagt merke til. Disse nye tallene fikk dermed Cayleys navn knyttet til seg.<sup id="cite_ref-Conway_1-0" class="reference"><a href="#cite_note-Conway-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>I tillegg viste det seg at åtte-kvadraters identiteten til Graves var tidligere funnet i en annen sammenheng av <a href="/wiki/Carl_Ferdinand_Degen#Åtte-kvadrats_likhet" title="Carl Ferdinand Degen">Carl F. Degen</a> som hadde hjulpet <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels H. Abel</a> da han ennå var student.<sup id="cite_ref-Baez_2-0" class="reference"><a href="#cite_note-Baez-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definisjon">Definisjon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=2" title="Rediger avsnitt: Definisjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=2" title="Rediger kildekoden til seksjonen Definisjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Oktonioner er element i et åttedimensjonalt <a href="/wiki/Vektorrom" title="Vektorrom">vektorrom</a> med basisvektorer 1, <b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, <b>e</b><sub>3</sub>, <b>e</b><sub>4</sub>, <b>e</b><sub>5</sub>, <b>e</b><sub>6</sub> og <b>e</b><sub>7</sub> slik at de kan skrives som </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=x_{0}+x_{1}\mathbf {e} _{1}+x_{2}\mathbf {e} _{2}+x_{3}\mathbf {e} _{3}+x_{4}\mathbf {e} _{4}+x_{5}\mathbf {e} _{5}+x_{6}\mathbf {e} _{6}+x_{7}\mathbf {e} _{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}+x_{1}\mathbf {e} _{1}+x_{2}\mathbf {e} _{2}+x_{3}\mathbf {e} _{3}+x_{4}\mathbf {e} _{4}+x_{5}\mathbf {e} _{5}+x_{6}\mathbf {e} _{6}+x_{7}\mathbf {e} _{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c59c23298094b186147fc8b172f461b99253b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:59.338ex; height:2.343ex;" alt="{\displaystyle x=x_{0}+x_{1}\mathbf {e} _{1}+x_{2}\mathbf {e} _{2}+x_{3}\mathbf {e} _{3}+x_{4}\mathbf {e} _{4}+x_{5}\mathbf {e} _{5}+x_{6}\mathbf {e} _{6}+x_{7}\mathbf {e} _{7}}"></span></dd></dl> <p>hvor de åtte komponentene (<i>x</i><sub>0</sub>,<i>x</i><sub>1</sub>,<i>x</i><sub>2</sub>,<i>x</i><sub>3</sub>,<i>x</i><sub>4</sub>,<i>x</i><sub>5</sub>,<i>x</i><sub>6</sub>,<i>x</i><sub>7</sub>) er <a href="/wiki/Reelt_tall" title="Reelt tall">reelle tall</a>. De første komponenten <i>x</i><sub>0</sub>  sies å tilhøre den <a href="/wiki/Skalar" title="Skalar">skalare</a> delen av oktonionen, mens de syv andre tilhører den vektorielle eller «imaginære» delen. </p><p>Basiselementene danner en <a href="/wiki/Algebra" title="Algebra">divisjonsalgebra</a> slik at produktet av to element alltid kan uttrykkes som en lineærkombinasjon av dem selv. To oktononioner <i>x</i> og <i>y </i> kan derfor multipliseres sammen med resultat <i>xy</i> eller <i>yx</i> som også er oktonioner. I allminnelighet er <i>xy</i> ≠ <i>yx</i> slik at de i allminnelighet ikke <a href="/wiki/Kommutativ_lov" title="Kommutativ lov">kommuterer</a> med hverandre. </p><p>Hvert element <i>x</i> skal også ha en bestemt <a href="/wiki/Norm_(matematikk)" title="Norm (matematikk)">norm</a> |<i>x</i> | slik at |<i>xy</i> | = |<i>x</i> | |<i>y</i> |. Den kan beregnes fra den konjugerte <i>x</i>*  av hver oktonion, definert </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^{*}=x_{0}-x_{1}\mathbf {e} _{1}-x_{2}\mathbf {e} _{2}-x_{3}\mathbf {e} _{3}-x_{4}\mathbf {e} _{4}-x_{5}\mathbf {e} _{5}-x_{6}\mathbf {e} _{6}-x_{7}\mathbf {e} _{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{*}=x_{0}-x_{1}\mathbf {e} _{1}-x_{2}\mathbf {e} _{2}-x_{3}\mathbf {e} _{3}-x_{4}\mathbf {e} _{4}-x_{5}\mathbf {e} _{5}-x_{6}\mathbf {e} _{6}-x_{7}\mathbf {e} _{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd02209514f8abe956a845ff0cac7cf8d84afdbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:60.392ex; height:2.676ex;" alt="{\displaystyle x^{*}=x_{0}-x_{1}\mathbf {e} _{1}-x_{2}\mathbf {e} _{2}-x_{3}\mathbf {e} _{3}-x_{4}\mathbf {e} _{4}-x_{5}\mathbf {e} _{5}-x_{6}\mathbf {e} _{6}-x_{7}\mathbf {e} _{7}}"></span></dd></dl> <p>hvor produktet må være slik at </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^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>x</mi> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af0aaad3c74372210a30375952857ec6a1dad8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.766ex; height:3.176ex;" alt="{\displaystyle x^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}}"></span></dd></dl> <p>Da kan den inverse <i>x</i><sup> -1</sup> til oktonion <i>x </i> finnes som </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^{-1}={x^{*} \over |x|^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}={x^{*} \over |x|^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576a9966af3963d50bc2b6a353604f0d78fb3e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.275ex; height:6.509ex;" alt="{\displaystyle x^{-1}={x^{*} \over |x|^{2}}}"></span></dd></dl> <p>slik at den oppfyller betingelsen <span class="nowrap"><i>x</i> <i>x</i><sup> -1</sup> = <i>x</i><sup> -1</sup><i>x</i></span> = 1. Graves og Cayley viste at produkter mellom de forskjellige basiselementene kan finnes slik at disse egenskapene ved oktonionene er oppfylt.<sup id="cite_ref-Conway_1-1" class="reference"><a href="#cite_note-Conway-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Cayley-Dicksons_konstruksjon">Cayley-Dicksons konstruksjon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=3" title="Rediger avsnitt: Cayley-Dicksons konstruksjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=3" title="Rediger kildekoden til seksjonen Cayley-Dicksons konstruksjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <table border="" cellspacing="0" cellpadding="5" bgcolor="#DDEEFF" align="right"> <tbody><tr> <td align="center" bgcolor="#FFFFFF"> </td> <td align="center" bgcolor="#EEEEEE">1 </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>1</sub> </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>2</sub> </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>3</sub> </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>4</sub> </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>5</sub> </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>6</sub> </td> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>7</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE">1 </td> <td align="center">1</td> <td align="center"><b>e</b><sub>1</sub> </td> <td align="center"><b>e</b><sub>2</sub> </td> <td align="center"><b>e</b><sub>3</sub></td> <td align="center"><b>e</b><sub>4</sub> </td> <td align="center"><b>e</b><sub>5</sub> </td> <td align="center"><b>e</b><sub>6</sub></td> <td align="center"><b>e</b><sub>7</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>1</sub> </td> <td align="center"><b>e</b><sub>1</sub></td> <td align="center">−1 </td> <td align="center"><b>e</b><sub>3</sub></td> <td align="center">-<b>e</b><sub>2</sub> </td> <td align="center"><b>e</b><sub>5</sub> </td> <td align="center">-<b>e</b><sub>4</sub> </td> <td align="center">-<b>e</b><sub>7</sub> </td> <td align="center"><b>e</b><sub>6</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>2</sub> </td> <td align="center"><b>e</b><sub>2</sub> </td> <td align="center">−<b>e</b><sub>3</sub> </td> <td align="center">−1</td> <td align="center"><b>e</b><sub>1</sub> </td> <td align="center"><b>e</b><sub>6</sub> </td> <td align="center"><b>e</b><sub>7</sub> </td> <td align="center">-<b>e</b><sub>4</sub> </td> <td align="center">-<b>e</b><sub>5</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>3</sub> </td> <td align="center"><b>e</b><sub>3</sub> </td> <td align="center"><b>e</b><sub>2</sub> </td> <td align="center">−<b>e</b><sub>1</sub> </td> <td align="center">−1</td> <td align="center"><b>e</b><sub>7</sub> </td> <td align="center">-<b>e</b><sub>6</sub> </td> <td align="center"><b>e</b><sub>5</sub> </td> <td align="center">-<b>e</b><sub>4</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>4</sub> </td> <td align="center"><b>e</b><sub>4</sub></td> <td align="center">-<b>e</b><sub>5</sub> </td> <td align="center">−<b>e</b><sub>6</sub> </td> <td align="center">−<b>e</b><sub>7</sub> </td> <td align="center">−1</td> <td align="center"><b>e</b><sub>1</sub> </td> <td align="center"><b>e</b><sub>2</sub> </td> <td align="center"><b>e</b><sub>3</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>5</sub> </td> <td align="center"><b>e</b><sub>5</sub> </td> <td align="center"><b>e</b><sub>4</sub> </td> <td align="center">-<b>e</b><sub>7</sub> </td> <td align="center"><b>e</b><sub>6</sub> </td> <td align="center">−<b>e</b><sub>1</sub> </td> <td align="center">−1</td> <td align="center">-<b>e</b><sub>3</sub> </td> <td align="center"><b>e</b><sub>2</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>6</sub> </td> <td align="center"><b>e</b><sub>6</sub></td> <td align="center"><b>e</b><sub>7</sub> </td> <td align="center"><b>e</b><sub>4</sub> </td> <td align="center">-<b>e</b><sub>5</sub> </td> <td align="center">−<b>e</b><sub>2</sub> </td> <td align="center"><b>e</b><sub>3</sub> </td> <td align="center">−1</td> <td align="center">-<b>e</b><sub>1</sub> </td></tr> <tr> <td align="center" bgcolor="#EEEEEE"><b>e</b><sub>7</sub> </td> <td align="center"><b>e</b><sub>7</sub></td> <td align="center">-<b>e</b><sub>6</sub> </td> <td align="center"><b>e</b><sub>5</sub> </td> <td align="center"><b>e</b><sub>4</sub> </td> <td align="center">-<b>e</b><sub>3</sub> </td> <td align="center">−<b>e</b><sub>2</sub> </td> <td align="center"><b>e</b><sub>1</sub> </td> <td align="center">−1 </td></tr></tbody></table> <p>Den enkleste og meste brukte fremgangsmåte for å finne multiplikasjonsregler mellom basiselementene som tilfredsstiller disse betingelsene, går tilbake til det første arbeidet til Cayley<sup id="cite_ref-Cayley_3-0" class="reference"><a href="#cite_note-Cayley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> som senere ble videreført av Dickson.<sup id="cite_ref-Dickson_4-0" class="reference"><a href="#cite_note-Dickson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>På samme måte som <a href="/wiki/Kvaternion" title="Kvaternion">kvaternioner</a> kan defineres som et par av <a href="/wiki/Komplekst_tall" title="Komplekst tall">komplekse tall</a>, kan man tenke seg oktonioner som bestående av et tallpar (<i>P,Q</i> ) der <i>P</i> og <i>Q</i>  er kvaternioner. Det tilsvarer å utvide <a href="/wiki/Vektorrom" title="Vektorrom">vektorrommet</a> for disse som er gitt ved basiselementene 1, <b>i</b>, <b>j</b> og <b>k</b> med et nytt, imaginært basiselement <b>e</b>. En oktonion kan dermed skrives som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=(P,Q)=P+Q\mathbf {e} \\&=p_{0}+p_{1}\mathbf {i} +p_{2}\mathbf {j} +p_{3}\mathbf {k} +(q_{0}+q_{1}\mathbf {i} +q_{2}\mathbf {j} +q_{3}\mathbf {k} )\mathbf {e} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=(P,Q)=P+Q\mathbf {e} \\&=p_{0}+p_{1}\mathbf {i} +p_{2}\mathbf {j} +p_{3}\mathbf {k} +(q_{0}+q_{1}\mathbf {i} +q_{2}\mathbf {j} +q_{3}\mathbf {k} )\mathbf {e} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4011f895e9550da9fab2e045da94b1bb03441137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.295ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}x&=(P,Q)=P+Q\mathbf {e} \\&=p_{0}+p_{1}\mathbf {i} +p_{2}\mathbf {j} +p_{3}\mathbf {k} +(q_{0}+q_{1}\mathbf {i} +q_{2}\mathbf {j} +q_{3}\mathbf {k} )\mathbf {e} \end{aligned}}}"></span></dd></dl> <p>der <b>e</b> <b>e</b> = <b>e</b><sup> 2</sup> = -1. De syv vektorielle basiselementene for oktonionene er dermed <span class="nowrap"><b>e</b><sub>1</sub> = <b>i</b></span>, <span class="nowrap"><b>e</b><sub>2</sub> = <b>j</b></span>, <span class="nowrap"><b>e</b><sub>3</sub> = <b>k</b></span>, <span class="nowrap"><b>e</b><sub>4</sub> = <b>e</b></span>, <span class="nowrap"><b>e</b><sub>5</sub> = <b>i</b> <b>e</b></span>, <span class="nowrap"><b>e</b><sub>6</sub> = <b>j</b> <b>e</b> </span> og <span class="nowrap"><b>e</b><sub>7</sub> = <b>k</b> <b>e</b></span>. Disse kan multipliseres sammen basert på en generalisering av produktregelen for kvaternioner. Den gir </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(P_{1},Q_{1})(P_{2},Q_{2})&=(P_{1}+Q_{1}\mathbf {e} )(P_{2}+Q_{2}\mathbf {e} )\\&=P_{1}P_{2}-Q_{2}^{*}Q_{1}+(Q_{2}P_{1}+Q_{1}P_{2}^{*})\mathbf {e} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(P_{1},Q_{1})(P_{2},Q_{2})&=(P_{1}+Q_{1}\mathbf {e} )(P_{2}+Q_{2}\mathbf {e} )\\&=P_{1}P_{2}-Q_{2}^{*}Q_{1}+(Q_{2}P_{1}+Q_{1}P_{2}^{*})\mathbf {e} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794aae7ee1ce79140b39ad0cfbb84a429bb6c99c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.275ex; margin-bottom: -0.23ex; width:54.056ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(P_{1},Q_{1})(P_{2},Q_{2})&=(P_{1}+Q_{1}\mathbf {e} )(P_{2}+Q_{2}\mathbf {e} )\\&=P_{1}P_{2}-Q_{2}^{*}Q_{1}+(Q_{2}P_{1}+Q_{1}P_{2}^{*})\mathbf {e} \end{aligned}}}"></span></dd></dl> <p>hvor rekkefølgen av faktorene i produktene er viktig da kvaternionene ikke kommuterer med hverandre. Med denne notasjonen er den konjugerte av oktonion <span class="nowrap"><i>x</i> = <i>P</i> + <i>Q</i> <b>e</b> </span> gitt som </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^{*}=P^{*}-Q\mathbf {e} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{*}=P^{*}-Q\mathbf {e} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777e79956780170626e26ae4633429f278c9bbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.262ex; height:2.676ex;" alt="{\displaystyle x^{*}=P^{*}-Q\mathbf {e} }"></span></dd></dl> <p>slik at produktregelen gir den ønskede normen <span class="nowrap"><i>x</i>*<i>x</i> = <i>P</i> *<i>P</i> + <i>Q</i> *<i>Q</i></span>. Samme regel gir også som forventet at <span class="nowrap"><b>e</b><sub>5</sub> <b>e</b><sub>5</sub> = (<b>i</b> <b>e</b>)(<b>i</b> <b>e</b>)</span> = <span class="nowrap">-<b>i</b>*<b>i</b> = <b>i</b><sup> 2</sup></span> = - 1 og likedan <span class="nowrap"><b>e</b><sub>6</sub> <b>e</b><sub>6</sub> = <b>e</b><sub>7</sub> <b>e</b><sub>7</sub></span> = -1. Alle produktene kan samles i en kvadratisk tabell hvor elementene er plassert antisymmetrisk om diagonalen med negative enere.<sup id="cite_ref-Baez_2-1" class="reference"><a href="#cite_note-Baez-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>En konsekvens av disse regnereglene er at multiplikasjon ikke er <a href="/wiki/Assosiativ_lov" title="Assosiativ lov">assosiativ</a>. For eksempel kan man fra tabellen se at <span class="nowrap"><b>e</b><sub>4</sub>(<b>e</b><sub>5</sub> <b>e</b><sub>6</sub>) = <b>e</b><sub>7</sub>,</span> mens <span class="nowrap">(<b>e</b><sub>4</sub> <b>e</b><sub>5</sub>)<b>e</b><sub>6</sub> = -<b>e</b><sub>7</sub>.</span> For tre vilkårlige oktonioner <i>x</i>, <i>y</i> og <i>z</i> gjelder derfor ikke lenger at <i>x</i>(<i>y</i> <i>z</i>) kan settes lik med (<i>x</i> <i>y</i>)<i>z</i>. Derimot har man generelt de mer spesielle egenskapene </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(xy)=(xx)y,\;\;\;x(yy)=(xy)y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(xy)=(xx)y,\;\;\;x(yy)=(xy)y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4775671fc1d55590ea24dbcc3d5683cb9a9b7917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.961ex; height:2.843ex;" alt="{\displaystyle x(xy)=(xx)y,\;\;\;x(yy)=(xy)y,}"></span></dd></dl> <p>som uttrykker en form for «alternativitet».<sup id="cite_ref-Baez_2-2" class="reference"><a href="#cite_note-Baez-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fano-planet">Fano-planet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=4" title="Rediger avsnitt: Fano-planet" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=4" title="Rediger kildekoden til seksjonen Fano-planet"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:FanoMnemonic.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/FanoMnemonic.PNG/250px-FanoMnemonic.PNG" decoding="async" width="250" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/FanoMnemonic.PNG/375px-FanoMnemonic.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/FanoMnemonic.PNG/500px-FanoMnemonic.PNG 2x" data-file-width="565" data-file-height="559" /></a><figcaption>De vektorielle basiselementene utgjør de syv punktene i <a href="/wiki/Projektivt_plan#Eksempel:_Syvpunktsplanet" title="Projektivt plan">Fano-planet</a>. Multiplikasjon av oktonioner følger fra de tre punktene på hver av de syv linjene i planet.</figcaption></figure> <p>Produktene av de vektorielle basiselementene som følger fra konstruksjonen til Cayley og Dickson, kan også elegant gjengis i <a href="/wiki/Projektivt_plan#Eksempel:_Syvpunktsplanet" title="Projektivt plan">Fano-planet</a>. Det består av syv punkter knyttet sammen ved syv linjer bestående av tre punkt <i>abc</i>. Hvis punktene angis som vist i figuren, er de tilsvarende linjene 123, 145, 176, 246, 257, 347, 365. Hver linje har en syklisk retning slik at produktet av to basisvektorer </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 \mathbf {e} _{a}\mathbf {e} _{b}=\mathbf {e} _{c}=-\mathbf {e} _{b}\mathbf {e} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{a}\mathbf {e} _{b}=\mathbf {e} _{c}=-\mathbf {e} _{b}\mathbf {e} _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05917feacd5022b7eefa7cdc96f4bfb369da329d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.154ex; height:2.343ex;" alt="{\displaystyle \mathbf {e} _{a}\mathbf {e} _{b}=\mathbf {e} _{c}=-\mathbf {e} _{b}\mathbf {e} _{a}}"></span></dd></dl> <p>når indeksene <i>abc</i> følger denne retningen. For eksempel inkluderer dette den fundamentale sammenhengen <span class="nowrap"><b>e</b><sub>1</sub> <b>e</b><sub>2</sub> = <b>e</b><sub>3</sub>,</span> men også <span class="nowrap"><b>e</b><sub>2</sub> <b>e</b><sub>4</sub> = <b>e</b><sub>6</sub></span> og <span class="nowrap"><b>e</b><sub>3</sub> <b>e</b><sub>4</sub> = <b>e</b><sub>7</sub>.</span> </p><p>Det er forbindelsen til Fano-planet som i stor grad forbinder oktonionene med egenskaper til noen av de mest spesielle <a href="/wiki/Lie-gruppe" title="Lie-gruppe">Lie-gruppene</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Se_også"><span id="Se_ogs.C3.A5"></span>Se også</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=5" title="Rediger avsnitt: Se også" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=5" title="Rediger kildekoden til seksjonen Se også"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Komplekst_tall" title="Komplekst tall">Komplekse tall</a></li> <li><a href="/wiki/Kvaternion" title="Kvaternion">Kvaternioner</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=6" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=6" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Conway-1"><b>^</b> <a href="#cite_ref-Conway_1-0"><sup>a</sup></a> <a href="#cite_ref-Conway_1-1"><sup>b</sup></a> <span class="reference-text"> J.H. Conway and D.A. Smith, <i>On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry</i>, CRC Press, Boca Raton (2003). <a href="/wiki/Spesial:Bokkilder/9781568811345" class="internal mw-magiclink-isbn">ISBN 978-1-56881-134-5</a>.</span> </li> <li id="cite_note-Baez-2"><b>^</b> <a href="#cite_ref-Baez_2-0"><sup>a</sup></a> <a href="#cite_ref-Baez_2-1"><sup>b</sup></a> <a href="#cite_ref-Baez_2-2"><sup>c</sup></a> <span class="reference-text"> J.C. Baez, <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/math/0105155.pdf"><i>The Octonions</i></a>, Bull. Amer. Math. Soc. <b>39</b>, 145-205 (2002). <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/math/0105155.pdf">arXiv:math/0105155</a></span> </li> <li id="cite_note-Cayley-3"><b><a href="#cite_ref-Cayley_3-0">^</a></b> <span class="reference-text"> A. Cayley, <a rel="nofollow" class="external text" href="https://zs.thulb.uni-jena.de/receive/jportal_jparticle_00207172"><i>On certain Results relating to Quaternions</i></a>, Philosophical Magazine <b>26</b>(3), 141-145 (1845). </span> </li> <li id="cite_note-Dickson-4"><b><a href="#cite_ref-Dickson_4-0">^</a></b> <span class="reference-text"> L.E. Dickson, <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/pdf/1967865.pdf?refreqid=excelsior%3A1704d6c90b6d007e1160caaa8354163d"><i>On Quaternions and Their Generalization and the History of the Eight Square Theorem</i></a>, Annals of Mathematics, <b>20</b>(3), 155–171 (1919).</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Eksterne_lenker">Eksterne lenker</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Oktonion&veaction=edit&section=7" title="Rediger avsnitt: Eksterne lenker" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Oktonion&action=edit&section=7" title="Rediger kildekoden til seksjonen Eksterne lenker"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>J.C. Baez, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/octonions/octonions.html"><i>Octonions</i></a>, informative websider</li> <li>Quanta Magazine, <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/"><i>Octonions in particle physics</i>?</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23230704">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output 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