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vector-toc-level-2"> <a class="vector-toc-link" href="#Special_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Special relativity</span> </div> </a> <ul id="toc-Special_relativity-sublist" class="vector-toc-list"> <li id="toc-Hyperbolic_orthogonality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hyperbolic_orthogonality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Hyperbolic orthogonality</span> </div> </a> <ul id="toc-Hyperbolic_orthogonality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Quantum mechanics</span> </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Art" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Art"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Art</span> </div> </a> <ul id="toc-Art-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Computer science</span> </div> </a> <button aria-controls="toc-Computer_science-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computer science subsection</span> </button> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> <li id="toc-Orthogonal_instruction_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthogonal_instruction_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Orthogonal instruction set</span> </div> </a> <ul id="toc-Orthogonal_instruction_set-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Telecommunications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Telecommunications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Telecommunications</span> </div> </a> <button aria-controls="toc-Telecommunications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Telecommunications subsection</span> </button> <ul id="toc-Telecommunications-sublist" class="vector-toc-list"> <li id="toc-Orthogonal_frequency-division_multiplexing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthogonal_frequency-division_multiplexing"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Orthogonal frequency-division multiplexing</span> </div> </a> <ul id="toc-Orthogonal_frequency-division_multiplexing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Statistics,_econometrics,_and_economics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statistics,_econometrics,_and_economics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Statistics, econometrics, and economics</span> </div> </a> <ul id="toc-Statistics,_econometrics,_and_economics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Taxonomy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Taxonomy"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Taxonomy</span> </div> </a> <ul id="toc-Taxonomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chemistry_and_biochemistry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Chemistry_and_biochemistry"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Chemistry and biochemistry</span> </div> </a> <button aria-controls="toc-Chemistry_and_biochemistry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Chemistry and biochemistry subsection</span> </button> <ul id="toc-Chemistry_and_biochemistry-sublist" class="vector-toc-list"> <li id="toc-Organic_synthesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Organic_synthesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Organic synthesis</span> </div> </a> <ul id="toc-Organic_synthesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bioorthogonal_chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bioorthogonal_chemistry"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Bioorthogonal chemistry</span> </div> </a> <ul id="toc-Bioorthogonal_chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Supramolecular_chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Supramolecular_chemistry"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Supramolecular chemistry</span> </div> </a> <ul id="toc-Supramolecular_chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analytical_chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytical_chemistry"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Analytical chemistry</span> </div> </a> <ul id="toc-Analytical_chemistry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-System_reliability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#System_reliability"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>System reliability</span> </div> </a> <ul id="toc-System_reliability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Neuroscience" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Neuroscience"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Neuroscience</span> </div> </a> <ul id="toc-Neuroscience-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Philosophy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Philosophy"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Philosophy</span> </div> </a> <ul id="toc-Philosophy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaming" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gaming"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Gaming</span> </div> </a> <ul id="toc-Gaming-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Other examples</span> </div> </a> <ul id="toc-Other_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</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">16</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 vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Orthogonality</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an 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Available in 38 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-38" 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">38 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B9%D8%A7%D9%85%D8%AF_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="تعامد (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="تعامد (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ortoqonal" title="Ortoqonal – Azerbaijani" lang="az" hreflang="az" data-title="Ortoqonal" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Ортогоналност – Bulgarian" lang="bg" hreflang="bg" data-title="Ортогоналност" 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/Ortogonal" title="Ortogonal – Catalan" lang="ca" hreflang="ca" data-title="Ortogonal" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BB%C4%95%D1%85" title="Ортогоналлĕх – Chuvash" lang="cv" hreflang="cv" data-title="Ортогоналлĕх" 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/Ortogonalita" title="Ortogonalita – Czech" lang="cs" hreflang="cs" data-title="Ortogonalita" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ortogonalitet" title="Ortogonalitet – Danish" lang="da" hreflang="da" data-title="Ortogonalitet" 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/Orthogonalit%C3%A4t" title="Orthogonalität – German" lang="de" hreflang="de" data-title="Orthogonalität" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ortogonalidad_(matem%C3%A1tica)" title="Ortogonalidad (matemática) – Spanish" lang="es" hreflang="es" data-title="Ortogonalidad (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Orteco" title="Orteco – Esperanto" lang="eo" hreflang="eo" data-title="Orteco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ortogonal" title="Ortogonal – Basque" lang="eu" hreflang="eu" data-title="Ortogonal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B9%D8%A7%D9%85%D8%AF_(%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%DB%8C)" title="تعامد (جبر خطی) – Persian" lang="fa" hreflang="fa" data-title="تعامد (جبر خطی)" 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/Orthogonalit%C3%A9" title="Orthogonalité – French" lang="fr" hreflang="fr" data-title="Orthogonalité" data-language-autonym="Français" data-language-local-name="French" 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/%EC%A7%81%EA%B5%90" title="직교 – Korean" lang="ko" hreflang="ko" data-title="직교" 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/%D5%88%D6%82%D5%B2%D5%B2%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ուղղանկյունություն – Armenian" lang="hy" hreflang="hy" data-title="Ուղղանկյունություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A8%D7%AA%D7%95%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%95%D7%AA" title="אורתוגונליות – Hebrew" lang="he" hreflang="he" data-title="אורתוגונליות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%B4%D1%83%D1%83%D0%BB%D1%83%D0%BA" title="Ортогоналдуулук – Kyrgyz" lang="ky" hreflang="ky" data-title="Ортогоналдуулук" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mer%C5%91legess%C3%A9g" title="Merőlegesség – Hungarian" lang="hu" hreflang="hu" data-title="Merőlegesség" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Ортогоналност – Macedonian" lang="mk" hreflang="mk" data-title="Ортогоналност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Orthogonaal" title="Orthogonaal – Dutch" lang="nl" hreflang="nl" data-title="Orthogonaal" 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/%E7%9B%B4%E4%BA%A4" title="直交 – Japanese" lang="ja" hreflang="ja" data-title="直交" 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/Ortogonalitet" title="Ortogonalitet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ortogonalitet" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ortogonalitet" title="Ortogonalitet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ortogonalitet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ortogonalno%C5%9B%C4%87" title="Ortogonalność – Polish" lang="pl" hreflang="pl" data-title="Ortogonalność" 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/Ortogonalidade" title="Ortogonalidade – Portuguese" lang="pt" hreflang="pt" data-title="Ortogonalidade" 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/Ortogonalitate" title="Ortogonalitate – Romanian" lang="ro" hreflang="ro" data-title="Ortogonalitate" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Ортогональность – Russian" lang="ru" hreflang="ru" data-title="Ортогональность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ortogonalnost" title="Ortogonalnost – Slovenian" lang="sl" hreflang="sl" data-title="Ortogonalnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Ортогоналност – Serbian" lang="sr" hreflang="sr" data-title="Ортогоналност" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ortogonalnost" title="Ortogonalnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Ortogonalnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kohtisuoruus" title="Kohtisuoruus – Finnish" lang="fi" hreflang="fi" data-title="Kohtisuoruus" 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/Ortogonalitet" title="Ortogonalitet – Swedish" lang="sv" hreflang="sv" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Orthogonal&redirect=no" class="mw-redirect" title="Orthogonal">Orthogonal</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Various meanings of the terms</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Orthogonal" redirects here. Not to be confused with <a href="/wiki/Orthogonal_(book_series)" class="mw-redirect" title="Orthogonal (book series)">Orthogonal (book series)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Perpendicular-coloured.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Perpendicular-coloured.svg/220px-Perpendicular-coloured.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Perpendicular-coloured.svg/330px-Perpendicular-coloured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Perpendicular-coloured.svg/440px-Perpendicular-coloured.svg.png 2x" data-file-width="723" data-file-height="501" /></a><figcaption>The line segments AB and CD are orthogonal to each other.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b><a href="/wiki/Orthogonality_(mathematics)" title="Orthogonality (mathematics)">orthogonality</a></b> is the generalization of the geometric notion of <i><a href="/wiki/Perpendicularity" class="mw-redirect" title="Perpendicularity">perpendicularity</a></i>. Whereas <i>perpendicular</i> is typically followed by <i>to</i> when relating two lines to one another (e.g., "line A is perpendicular to line B"),<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <i>orthogonal</i> is commonly used without <i>to</i> (e.g., "orthogonal lines A and B").<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><p><i>Orthogonality</i> is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word comes from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a> <span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BD%80%CF%81%CE%B8%CF%8C%CF%82#Ancient_Greek" class="extiw" title="wikt:ὀρθός">ὀρθός</a></span> (<i><span title="Ancient Greek transliteration" lang="grc-Latn"><i>orthós</i></span></i>), meaning "upright",<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> and <span lang="grc"><a href="https://en.wiktionary.org/wiki/%CE%B3%CF%89%CE%BD%CE%AF%CE%B1#Ancient_Greek" class="extiw" title="wikt:γωνία">γωνία</a></span> (<i><span title="Ancient Greek transliteration" lang="grc-Latn"><i>gōnía</i></span></i>), meaning "angle".<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> </p><p>The Ancient Greek <span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BD%80%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF%CE%BD#Ancient_Greek" class="extiw" title="wikt:ὀρθογώνιον">ὀρθογώνιον</a></span> (<i><span title="Ancient Greek transliteration" lang="grc-Latn"><i>orthogṓnion</i></span></i>) and <a href="/wiki/Classical_Latin" title="Classical Latin">Classical Latin</a> <i><i lang="la"><a href="https://en.wiktionary.org/wiki/orthogonium#Latin" class="extiw" title="wikt:orthogonium">orthogonium</a></i></i> originally denoted a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Later, they came to mean a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a>. In the 12th century, the post-classical Latin word <i>orthogonalis</i> came to mean a right angle or something related to a right angle.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=2" title="Edit section: Mathematics"><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/Orthogonality_(mathematics)" title="Orthogonality (mathematics)">Orthogonality (mathematics)</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=Orthogonality_(mathematics)&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Orthogonality_(mathematics)" title="Orthogonality (mathematics)">orthogonality</a> is the generalization of the geometric notion of <i><a href="/wiki/Perpendicularity" class="mw-redirect" title="Perpendicularity">perpendicularity</a></i> to the <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> of <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear forms</a>. </p><p>Two elements <span class="texhtml">u</span> and <span class="texhtml">v</span> of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> with bilinear form <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> are orthogonal when <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 B(\mathbf {u} ,\mathbf {v} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1c2cb00f26ace98d1bdfb600221a2a206e0b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.764ex; height:2.843ex;" alt="{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0}"></span>. Depending on the bilinear form, the vector space may contain <a href="/wiki/Null_vector" title="Null vector">null vectors</a>, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with <a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">hyperbolic orthogonality</a>. </p><p>In the case of <a href="/wiki/Function_space" title="Function space">function spaces</a>, families of functions are used to form an orthogonal <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>, such as in the contexts of <a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a>, <a href="/wiki/Orthogonal_functions" title="Orthogonal functions">orthogonal functions</a>, and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orthogonality_and_rotation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/350px-Orthogonality_and_rotation.svg.png" decoding="async" width="350" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/525px-Orthogonality_and_rotation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/700px-Orthogonality_and_rotation.svg.png 2x" data-file-width="1154" data-file-height="601" /></a><figcaption>Orthogonality and rotation of coordinate systems compared between left: <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> through circular <a href="/wiki/Angle" title="Angle">angle</a> <i>ϕ</i>, right: in <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a> through <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> <i>ϕ</i> (red lines labelled <i>c</i> denote the <a href="/wiki/Worldline" class="mw-redirect" title="Worldline">worldlines</a> of a light signal, a vector is orthogonal to itself if it lies on this line).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></figcaption></figure></div></div> <div class="mw-heading mw-heading2"><h2 id="Physics">Physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=3" title="Edit section: Physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Orthogonality&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">September 2022</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Optics">Optics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=4" title="Edit section: Optics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Optics" title="Optics">optics</a>, <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal <a href="/wiki/Linear_polarization" title="Linear polarization">linear polarization</a> or right- and left-handed <a href="/wiki/Circular_polarization" title="Circular polarization">circular polarization</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Special_relativity">Special relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=5" title="Edit section: Special relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, a time axis determined by a <a href="/wiki/Rapidity" title="Rapidity">rapidity</a> of motion is <a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">hyperbolic-orthogonal</a> to a space axis of simultaneous events, also determined by the rapidity. The theory features <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">relativity of simultaneity</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Hyperbolic_orthogonality">Hyperbolic orthogonality</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=6" title="Edit section: Hyperbolic orthogonality"><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/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</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=Hyperbolic_orthogonality&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orthogonality_and_rotation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/350px-Orthogonality_and_rotation.svg.png" decoding="async" width="350" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/525px-Orthogonality_and_rotation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/700px-Orthogonality_and_rotation.svg.png 2x" data-file-width="1154" data-file-height="601" /></a><figcaption>Euclidean orthogonality is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by <a href="/wiki/Hyperbolic_rotation" class="mw-redirect" title="Hyperbolic rotation">hyperbolic rotation</a> in the right diagram.</figcaption></figure> In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the relation of <a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">hyperbolic orthogonality</a> between two lines separated by the asymptotes of a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> is a concept used in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular timeline. This dependence on a certain timeline is determined by velocity, and is the basis for the <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">relativity of simultaneity</a>.</div></div> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=7" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, a sufficient (but not necessary) condition that two <a href="/wiki/Eigenstates" class="mw-redirect" title="Eigenstates">eigenstates</a> of a <a href="/wiki/Hermitian_operator" class="mw-redirect" title="Hermitian operator">Hermitian operator</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 \psi _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed229b721adfd0e48daa75f714788c1c7e37903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.188ex; height:2.509ex;" alt="{\displaystyle \psi _{m}}"></span> and <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 \psi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5ff71e5c69dbfebe7cd4530de07406eff55c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \psi _{n}}"></span>, are orthogonal is that they correspond to different eigenvalues. This means, in <a href="/wiki/Dirac_notation" class="mw-redirect" title="Dirac notation">Dirac notation</a>, that <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 \langle \psi _{m}|\psi _{n}\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebadcd7c0f3d94a601fd13fc013a1129e3bc145c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.637ex; height:2.843ex;" alt="{\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0}"></span> if <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 \psi _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed229b721adfd0e48daa75f714788c1c7e37903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.188ex; height:2.509ex;" alt="{\displaystyle \psi _{m}}"></span> and <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 \psi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5ff71e5c69dbfebe7cd4530de07406eff55c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \psi _{n}}"></span> correspond to different eigenvalues. This follows from the fact that <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger's equation</a> is a <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville</a> equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2012)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Art">Art</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=8" title="Edit section: Art"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In art, the <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a> (imaginary) lines pointing to the <a href="/wiki/Vanishing_point" title="Vanishing point">vanishing point</a> are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as <a href="/wiki/Piet_Mondrian" title="Piet Mondrian">Piet Mondrian</a> and <a href="/wiki/Burgoyne_Diller" title="Burgoyne Diller">Burgoyne Diller</a> are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the <a href="/wiki/Website" title="Website">web site</a> of the <a href="/wiki/Thyssen-Bornemisza_Museum" title="Thyssen-Bornemisza Museum">Thyssen-Bornemisza Museum</a> states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090131164818/https://www.museothyssen.org/thyssen_ing/coleccion/obras_ficha_texto_print497.html">Archived</a> 2009-01-31 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </p> <div class="mw-heading mw-heading2"><h2 id="Computer_science">Computer science</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=9" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Orthogonality_(programming)" title="Orthogonality (programming)">Orthogonality (programming)</a> and <a href="/wiki/Orthogonal_instruction_set" title="Orthogonal instruction set">Orthogonal instruction set</a></div> <p>Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This usage was introduced by <a href="/wiki/Adriaan_van_Wijngaarden" title="Adriaan van Wijngaarden">Van Wijngaarden</a> in the design of <a href="/wiki/Algol_68" class="mw-redirect" title="Algol 68">Algol 68</a>: </p> <blockquote> <p>The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> </blockquote> <p>Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the <a href="/wiki/Separation_of_concerns" title="Separation of concerns">separation of concerns</a> and <a href="/wiki/Information_Hiding#Encapsulation" class="mw-redirect" title="Information Hiding">encapsulation</a>, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them. </p> <div class="mw-heading mw-heading3"><h3 id="Orthogonal_instruction_set">Orthogonal instruction set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=10" title="Edit section: Orthogonal instruction set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orthogonal_instruction_set" title="Orthogonal instruction set">Orthogonal instruction set</a></div> <p>An <a href="/wiki/Instruction_set" class="mw-redirect" title="Instruction set">instruction set</a> is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task)<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and is designed such that instructions can use any <a href="/wiki/Processor_register" title="Processor register">register</a> in any <a href="/wiki/Addressing_mode" title="Addressing mode">addressing mode</a>. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An <a href="/wiki/Orthogonal_instruction_set" title="Orthogonal instruction set">orthogonal instruction set</a> uniquely encodes all combinations of registers and addressing modes.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Telecommunications">Telecommunications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=11" title="Edit section: Telecommunications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Orthogonality" title="Special:EditPage/Orthogonality">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Telecommunications" title="Telecommunications">telecommunications</a>, <a href="/wiki/Multiple_access" class="mw-redirect" title="Multiple access">multiple access</a> schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different <a href="/wiki/Basis_function" title="Basis function">basis functions</a>. One such scheme is <a href="/wiki/Time-division_multiple_access" title="Time-division multiple access">time-division multiple access</a> (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots"). </p> <div class="mw-heading mw-heading3"><h3 id="Orthogonal_frequency-division_multiplexing">Orthogonal frequency-division multiplexing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=12" title="Edit section: Orthogonal frequency-division multiplexing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orthogonal_frequency-division_multiplexing" title="Orthogonal frequency-division multiplexing">Orthogonal frequency-division multiplexing</a></div> <p>Another scheme is <a href="/wiki/Orthogonal_frequency-division_multiplexing" title="Orthogonal frequency-division multiplexing">orthogonal frequency-division multiplexing</a> (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (<b>a</b>, <b>g</b>, and <b>n</b>) versions of <a href="/wiki/802.11" class="mw-redirect" title="802.11">802.11</a> <a href="/wiki/Wi-Fi" title="Wi-Fi">Wi-Fi</a>; <a href="/wiki/WiMAX" title="WiMAX">WiMAX</a>; <a href="/wiki/ITU-T" title="ITU-T">ITU-T</a> <a href="/wiki/G.hn" title="G.hn">G.hn</a>, <a href="/wiki/DVB-T" title="DVB-T">DVB-T</a>, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of <a href="/wiki/ADSL" title="ADSL">ADSL</a>. </p><p>In OFDM, the <a href="/wiki/Subcarrier" title="Subcarrier">subcarrier</a> frequencies are chosen<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Please clarify the preceding statement or statements with a good explanation from a reliable source. (May 2019)">how?</span></a></i>]</sup> so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required. </p> <div class="mw-heading mw-heading2"><h2 id="Statistics,_econometrics,_and_economics"><span id="Statistics.2C_econometrics.2C_and_economics"></span>Statistics, econometrics, and economics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=13" title="Edit section: Statistics, econometrics, and economics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When performing statistical analysis, <a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">independent variables</a> that affect a particular <a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">dependent variable</a> are said to be orthogonal if they are uncorrelated,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with <a href="/wiki/Simple_linear_regression" title="Simple linear regression">simple regression</a> or simultaneously with <a href="/wiki/Multiple_regression" class="mw-redirect" title="Multiple regression">multiple regression</a>. If <a href="/wiki/Correlation" title="Correlation">correlation</a> is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the <a href="/wiki/Expected_value" title="Expected value">expected value</a> (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One <a href="/wiki/Econometrics" title="Econometrics">econometric</a> formalism that is alternative to the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> framework, the <a href="/wiki/Generalized_Method_of_Moments" class="mw-redirect" title="Generalized Method of Moments">Generalized Method of Moments</a>, relies on orthogonality conditions. In particular, the <a href="/wiki/Ordinary_Least_Squares" class="mw-redirect" title="Ordinary Least Squares">Ordinary Least Squares</a> estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals. </p> <div class="mw-heading mw-heading2"><h2 id="Taxonomy">Taxonomy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=14" title="Edit section: Taxonomy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Taxonomy_(general)" class="mw-redirect" title="Taxonomy (general)">taxonomy</a>, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive. </p> <div class="mw-heading mw-heading2"><h2 id="Chemistry_and_biochemistry">Chemistry and biochemistry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=15" title="Edit section: Chemistry and biochemistry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example, <a href="/wiki/DNA" title="DNA">DNA</a> has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Organic_synthesis">Organic synthesis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=16" title="Edit section: Organic synthesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Organic_synthesis" title="Organic synthesis">Organic synthesis</a> and <a href="/wiki/Orthogonal_protection" class="mw-redirect" title="Orthogonal protection">Orthogonal protection</a></div> <p>In <a href="/wiki/Organic_synthesis" title="Organic synthesis">organic synthesis</a>, <a href="/wiki/Orthogonal_protection" class="mw-redirect" title="Orthogonal protection">orthogonal protection</a> is a strategy allowing the deprotection of <a href="/wiki/Functional_group" title="Functional group">functional groups</a> independently of each other. </p> <div class="mw-heading mw-heading3"><h3 id="Bioorthogonal_chemistry">Bioorthogonal chemistry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=17" title="Edit section: Bioorthogonal chemistry"><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 paragraph is an excerpt from <a href="/wiki/Bioorthogonal_chemistry" title="Bioorthogonal chemistry">Bioorthogonal chemistry</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=Bioorthogonal_chemistry&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> The term <a href="/wiki/Bioorthogonal_chemistry" title="Bioorthogonal chemistry">bioorthogonal chemistry</a> refers to any <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reaction</a> that can occur inside of <a href="/wiki/Living_systems" title="Living systems">living systems</a> without interfering with native biochemical processes.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The term was coined by <a href="/wiki/Carolyn_R._Bertozzi" class="mw-redirect" title="Carolyn R. Bertozzi">Carolyn R. Bertozzi</a> in 2003.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bioorthogonal_chemistry_Doiarz_18-0" class="reference"><a href="#cite_note-Bioorthogonal_chemistry_Doiarz-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Since its introduction, the concept of the bioorthogonal reaction has enabled the study of biomolecules such as <a href="/wiki/Glycan" title="Glycan">glycans</a>, <a href="/wiki/Protein" title="Protein">proteins</a>,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Lipid" title="Lipid">lipids</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> in real time in living systems without cellular toxicity. A number of <a href="/wiki/Chemical_ligation" title="Chemical ligation">chemical ligation</a> strategies have been developed that fulfill the requirements of bioorthogonality, including the <a href="/wiki/1,3-dipolar_cycloaddition" class="mw-redirect" title="1,3-dipolar cycloaddition">1,3-dipolar cycloaddition</a> between <a href="/wiki/Azide" title="Azide">azides</a> and <a href="/wiki/Cyclooctyne" title="Cyclooctyne">cyclooctynes</a> (also termed <a href="/wiki/Copper-free_click_chemistry" title="Copper-free click chemistry">copper-free click chemistry</a>),<sup id="cite_ref-Bioorthogonal_chemistry_Doipnas_21-0" class="reference"><a href="#cite_note-Bioorthogonal_chemistry_Doipnas-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> between <a href="/wiki/Nitrone" title="Nitrone">nitrones</a> and cyclooctynes,<sup id="cite_ref-Bioorthogonal_chemistry_Doianie_22-0" class="reference"><a href="#cite_note-Bioorthogonal_chemistry_Doianie-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Oxime" title="Oxime">oxime</a>/<a href="/wiki/Hydrazone" title="Hydrazone">hydrazone</a> formation from <a href="/wiki/Aldehyde" title="Aldehyde">aldehydes</a> and <a href="/wiki/Ketone" title="Ketone">ketones</a>,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Tetrazine" title="Tetrazine">tetrazine</a> ligation,<sup id="cite_ref-Bioorthogonal_chemistry_Doija_a_24-0" class="reference"><a href="#cite_note-Bioorthogonal_chemistry_Doija_a-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Isocyanide" title="Isocyanide">isocyanide</a>-based click reaction,<sup id="cite_ref-Bioorthogonal_chemistry_DoiCOBJ_25-0" class="reference"><a href="#cite_note-Bioorthogonal_chemistry_DoiCOBJ-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and most recently, the <a href="/wiki/Quadricyclane" title="Quadricyclane">quadricyclane</a> ligation.<sup id="cite_ref-Bioorthogonal_chemistry_Doija_b_26-0" class="reference"><a href="#cite_note-Bioorthogonal_chemistry_Doija_b-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading3"><h3 id="Supramolecular_chemistry">Supramolecular chemistry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=18" title="Edit section: Supramolecular chemistry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Supramolecular_chemistry" title="Supramolecular chemistry">Supramolecular chemistry</a></div> <p>In <a href="/wiki/Supramolecular_chemistry" title="Supramolecular chemistry">supramolecular chemistry</a> the notion of orthogonality refers to the possibility of two or more supramolecular, often <a href="/wiki/Non-covalent" class="mw-redirect" title="Non-covalent">non-covalent</a>, interactions being compatible; reversibly forming without interference from the other. </p> <div class="mw-heading mw-heading3"><h3 id="Analytical_chemistry">Analytical chemistry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=19" title="Edit section: Analytical chemistry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Analytical_chemistry" title="Analytical chemistry">Analytical chemistry</a></div> <p>In <a href="/wiki/Analytical_chemistry" title="Analytical chemistry">analytical chemistry</a>, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. Orthogonal testing thus can be viewed as "cross-checking" of results, and the "cross" notion corresponds to the <a href="#Etymology">etymologic origin of <i>orthogonality</i></a>. Orthogonal testing is often required as a part of a <a href="/wiki/New_drug_application" class="mw-redirect" title="New drug application">new drug application</a>. </p> <div class="mw-heading mw-heading2"><h2 id="System_reliability">System reliability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=20" title="Edit section: System reliability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure. </p> <div class="mw-heading mw-heading2"><h2 id="Neuroscience">Neuroscience</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=21" title="Edit section: Neuroscience"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map. </p> <div class="mw-heading mw-heading2"><h2 id="Philosophy">Philosophy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=22" title="Edit section: Philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Philosophy" title="Philosophy">philosophy</a>, two topics, authors, or pieces of writing are said to be "orthogonal" to each other when they do not substantively cover what could be considered potentially overlapping or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they can be orthogonal to each other in cases where the scope, content, and purpose of the pieces of writing are entirely unrelated. </p> <div class="mw-heading mw-heading2"><h2 id="Gaming">Gaming</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=23" title="Edit section: Gaming"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Von_Neumann_neighborhood" title="Von Neumann neighborhood">von Neumann neighborhood</a></div> <p>In board games such as <a href="/wiki/Chess" title="Chess">chess</a> which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent".<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> In the ancient Chinese board game <a href="/wiki/Go_(game)" title="Go (game)">Go</a> a player can capture the stones of an opponent by occupying all orthogonally adjacent points. </p> <div class="mw-heading mw-heading2"><h2 id="Other_examples">Other examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonality&action=edit&section=24" title="Edit section: Other examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals. </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=Orthogonality&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media 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//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/orthogonal" class="extiw" title="wiktionary:orthogonal">orthogonal</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Orthogonal_ligand-protein_pair" title="Orthogonal ligand-protein pair">Orthogonal ligand-protein pair</a></li> <li><a href="/wiki/Up_tack" title="Up tack">Up tack</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=Orthogonality&action=edit&section=26" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/perpendicular">"perpendicular"</a>. <i><a href="/wiki/Merriam-Webster" title="Merriam-Webster">Merriam-Webster.com Dictionary</a></i>. Merriam-Webster.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Merriam-Webster.com+Dictionary&rft.atitle=perpendicular&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fperpendicular&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/orthogonal">"orthogonal"</a>. <i><a href="/wiki/Merriam-Webster" title="Merriam-Webster">Merriam-Webster.com Dictionary</a></i>. Merriam-Webster.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Merriam-Webster.com+Dictionary&rft.atitle=orthogonal&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Forthogonal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Liddell and Scott, <i><a href="/wiki/A_Greek%E2%80%93English_Lexicon" title="A Greek–English Lexicon">A Greek–English Lexicon</a></i> <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/morph?l=o%29rqos&la=greek#lexicon"><i>s.v.</i> ὀρθός</a></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">Liddell and Scott, <i><a href="/wiki/A_Greek%E2%80%93English_Lexicon" title="A Greek–English Lexicon">A Greek–English Lexicon</a></i> <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/morph?l=gwni%2Fa&la=greek#lexicon"><i>s.v.</i> γωνία</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Liddell and Scott, <i><a href="/wiki/A_Greek%E2%80%93English_Lexicon" title="A Greek–English Lexicon">A Greek–English Lexicon</a></i> <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/morph?l=o%29rqog%2Fwnion&la=greek#lexicon"><i>s.v.</i> ὀρθογώνιον</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"orthogonal". <i><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i> (3rd ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. September 2004.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=orthogonal&rft.btitle=Oxford+English+Dictionary&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=2004-09&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ.A._WheelerC._MisnerK.S._Thorne1973" class="citation book cs1">J.A. Wheeler; C. Misner; K.S. Thorne (1973). <i>Gravitation</i>. W.H. Freeman & Co. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0344-0" title="Special:BookSources/0-7167-0344-0"><bdi>0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pages=58&rft.pub=W.H.+Freeman+%26+Co&rft.date=1973&rft.isbn=0-7167-0344-0&rft.au=J.A.+Wheeler&rft.au=C.+Misner&rft.au=K.S.+Thorne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Michael L. Scott, <i>Programming Language Pragmatics</i>, p. 228.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">1968, Adriaan van Wijngaarden et al., Revised Report on the Algorithmic Language ALGOL 68, section 0.1.2, Orthogonal design</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNull,_LindaLobur,_Julia2006" class="citation book cs1">Null, Linda & Lobur, Julia (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QGPHAl9GE-IC&pg=PA257"><i>The essentials of computer organization and architecture</i></a> (2nd ed.). Jones & Bartlett Learning. p. 257. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-3769-6" title="Special:BookSources/978-0-7637-3769-6"><bdi>978-0-7637-3769-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+essentials+of+computer+organization+and+architecture&rft.pages=257&rft.edition=2nd&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2006&rft.isbn=978-0-7637-3769-6&rft.au=Null%2C+Linda&rft.au=Lobur%2C+Julia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQGPHAl9GE-IC%26pg%3DPA257&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLinda_Null2010" class="citation book cs1">Linda Null (2010). <a rel="nofollow" class="external text" href="https://samples.jbpub.com/9781449600068/00068_ch05_null3e.pdf"><i>The Essentials of Computer Organization and Architecture</i></a> <span class="cs1-format">(PDF)</span>. Jones & Bartlett Publishers. pp. 287–288. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1449600068" title="Special:BookSources/978-1449600068"><bdi>978-1449600068</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151010020907/http://samples.jbpub.com/9781449600068/00068_CH05_Null3e.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2015-10-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Essentials+of+Computer+Organization+and+Architecture&rft.pages=287-288&rft.pub=Jones+%26+Bartlett+Publishers&rft.date=2010&rft.isbn=978-1449600068&rft.au=Linda+Null&rft_id=https%3A%2F%2Fsamples.jbpub.com%2F9781449600068%2F00068_ch05_null3e.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAthanasios_PapoulisS._Unnikrishna_Pillai2002" class="citation book cs1">Athanasios Papoulis; S. 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Application to Cell Surface Glycoform Engineering"</a>. <i>Journal of Biological Chemistry</i>. <b>273</b> (47): 31168–79. <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/10.1074%2Fjbc.273.47.31168">10.1074/jbc.273.47.31168</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9813021">9813021</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Biological+Chemistry&rft.atitle=Metabolic+Delivery+of+Ketone+Groups+to+Sialic+Acid+Residues.+Application+to+Cell+Surface+Glycoform+Engineering&rft.volume=273&rft.issue=47&rft.pages=31168-79&rft.date=1998&rft_id=info%3Adoi%2F10.1074%2Fjbc.273.47.31168&rft_id=info%3Apmid%2F9813021&rft.aulast=Yarema&rft.aufirst=K.+J.&rft.au=Mahal%2C+LK&rft.au=Bruehl%2C+RE&rft.au=Rodriguez%2C+EC&rft.au=Bertozzi%2C+CR&rft_id=https%3A%2F%2Fdoi.org%2F10.1074%252Fjbc.273.47.31168&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-Bioorthogonal_chemistry_Doija_a-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bioorthogonal_chemistry_Doija_a_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlackmanRoyzenFox2008" class="citation journal cs1">Blackman, Melissa L.; Royzen, Maksim; Fox, Joseph M. (2008). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2653060">"The Tetrazine Ligation: Fast Bioconjugation based on Inverse-electron-demand Diels-Alder Reactivity"</a>. <i>Journal of the American Chemical Society</i>. <b>130</b> (41): 13518–9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fja8053805">10.1021/ja8053805</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2653060">2653060</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18798613">18798613</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Chemical+Society&rft.atitle=The+Tetrazine+Ligation%3A+Fast+Bioconjugation+based+on+Inverse-electron-demand+Diels-Alder+Reactivity&rft.volume=130&rft.issue=41&rft.pages=13518-9&rft.date=2008&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2653060%23id-name%3DPMC&rft_id=info%3Apmid%2F18798613&rft_id=info%3Adoi%2F10.1021%2Fja8053805&rft.aulast=Blackman&rft.aufirst=Melissa+L.&rft.au=Royzen%2C+Maksim&rft.au=Fox%2C+Joseph+M.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2653060&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-Bioorthogonal_chemistry_DoiCOBJ-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bioorthogonal_chemistry_DoiCOBJ_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStöckmannNevesStairsBrindle2011" class="citation journal cs1">Stöckmann, Henning; Neves, André A.; Stairs, Shaun; Brindle, Kevin M.; Leeper, Finian J. (2011). "Exploring isonitrile-based click chemistry for ligation with biomolecules". <i>Organic & Biomolecular Chemistry</i>. <b>9</b> (21): 7303–5. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FC1OB06424J">10.1039/C1OB06424J</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21915395">21915395</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Organic+%26+Biomolecular+Chemistry&rft.atitle=Exploring+isonitrile-based+click+chemistry+for+ligation+with+biomolecules&rft.volume=9&rft.issue=21&rft.pages=7303-5&rft.date=2011&rft_id=info%3Adoi%2F10.1039%2FC1OB06424J&rft_id=info%3Apmid%2F21915395&rft.aulast=St%C3%B6ckmann&rft.aufirst=Henning&rft.au=Neves%2C+Andr%C3%A9+A.&rft.au=Stairs%2C+Shaun&rft.au=Brindle%2C+Kevin+M.&rft.au=Leeper%2C+Finian+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-Bioorthogonal_chemistry_Doija_b-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bioorthogonal_chemistry_Doija_b_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlettenBertozzi2011" class="citation journal cs1">Sletten, Ellen M.; Bertozzi, Carolyn R. (2011). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3206493">"A Bioorthogonal Quadricyclane Ligation"</a>. <i>Journal of the American Chemical Society</i>. <b>133</b> (44): 17570–3. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fja2072934">10.1021/ja2072934</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3206493">3206493</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21962173">21962173</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Chemical+Society&rft.atitle=A+Bioorthogonal+Quadricyclane+Ligation&rft.volume=133&rft.issue=44&rft.pages=17570-3&rft.date=2011&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3206493%23id-name%3DPMC&rft_id=info%3Apmid%2F21962173&rft_id=info%3Adoi%2F10.1021%2Fja2072934&rft.aulast=Sletten&rft.aufirst=Ellen+M.&rft.au=Bertozzi%2C+Carolyn+R.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3206493&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.chessvariants.org/misc.dir/coreglossary.html#orthogonal_direction">"chessvariants.org chess glossary"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=chessvariants.org+chess+glossary&rft_id=http%3A%2F%2Fwww.chessvariants.org%2Fmisc.dir%2Fcoreglossary.html%23orthogonal_direction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonality" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">For an illustration, see <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=umu37m0qUiE">YouTube</a>.</span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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