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class="vector-toc-link" href="#Points"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Points</span> </div> </a> <ul id="toc-Points-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lines" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lines"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Lines</span> </div> </a> <ul id="toc-Lines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Planes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Planes</span> </div> </a> <ul id="toc-Planes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curves" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.4</span> <span>Curves</span> </div> </a> <ul id="toc-Curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surfaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.5</span> <span>Surfaces</span> </div> </a> <ul id="toc-Surfaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solids" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.6</span> <span>Solids</span> </div> </a> <ul id="toc-Solids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Manifolds" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.7</span> <span>Manifolds</span> </div> </a> <ul id="toc-Manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Angles</span> </div> </a> <ul id="toc-Angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measures:_length,_area,_and_volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measures:_length,_area,_and_volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Measures: length, area, and volume</span> </div> </a> <ul id="toc-Measures:_length,_area,_and_volume-sublist" class="vector-toc-list"> <li id="toc-Metrics_and_measures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Metrics_and_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.1</span> <span>Metrics and measures</span> </div> </a> <ul id="toc-Metrics_and_measures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Congruence_and_similarity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congruence_and_similarity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Congruence and similarity</span> </div> </a> <ul id="toc-Congruence_and_similarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compass_and_straightedge_constructions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compass_and_straightedge_constructions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Compass and straightedge constructions</span> </div> </a> <ul id="toc-Compass_and_straightedge_constructions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotation_and_orientation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotation_and_orientation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Rotation and orientation</span> </div> </a> <ul id="toc-Rotation_and_orientation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimension"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Dimension</span> </div> </a> <ul id="toc-Dimension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>Symmetry</span> </div> </a> <ul id="toc-Symmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Contemporary_geometry" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Contemporary_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Contemporary geometry</span> </div> </a> <button aria-controls="toc-Contemporary_geometry-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 Contemporary geometry subsection</span> </button> <ul id="toc-Contemporary_geometry-sublist" class="vector-toc-list"> <li id="toc-Euclidean_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Euclidean geometry</span> </div> </a> <ul id="toc-Euclidean_geometry-sublist" class="vector-toc-list"> <li id="toc-Euclidean_vectors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Euclidean_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Euclidean vectors</span> </div> </a> <ul id="toc-Euclidean_vectors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Differential_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Differential geometry</span> </div> </a> <ul id="toc-Differential_geometry-sublist" class="vector-toc-list"> <li id="toc-Non-Euclidean_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Non-Euclidean_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Non-Euclidean geometry</span> </div> </a> <ul id="toc-Non-Euclidean_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Topology</span> </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Algebraic geometry</span> </div> </a> <ul id="toc-Algebraic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Complex geometry</span> </div> </a> <ul id="toc-Complex_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Discrete geometry</span> </div> </a> <ul id="toc-Discrete_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Computational geometry</span> </div> </a> <ul id="toc-Computational_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_group_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Geometric group theory</span> </div> </a> <ul id="toc-Geometric_group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convex_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convex_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Convex geometry</span> </div> </a> <ul id="toc-Convex_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Art" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Art"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Art</span> </div> </a> <ul id="toc-Art-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Architecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Architecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Architecture</span> </div> </a> <ul id="toc-Architecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Physics</span> </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_fields_of_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_fields_of_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Other fields of mathematics</span> </div> </a> <ul id="toc-Other_fields_of_mathematics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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">Geometry</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 article in another language. Available in 179 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-179" 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">179 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B5" title="Геометрие – Kabardian" lang="kbd" hreflang="kbd" data-title="Геометрие" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target"><span>Адыгэбзэ</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Meetkunde" title="Meetkunde – Afrikaans" lang="af" hreflang="af" data-title="Meetkunde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Geometrie" title="Geometrie – Alemannic" lang="gsw" hreflang="gsw" data-title="Geometrie" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8C%82%E1%8B%8E%E1%88%9C%E1%89%B5%E1%88%AA" title="ጂዎሜትሪ – Amharic" lang="am" hreflang="am" data-title="ጂዎሜትሪ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Geometria" title="Geometria – Inari Sami" lang="smn" hreflang="smn" data-title="Geometria" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF" title="ज्यामिति – Angika" lang="anp" hreflang="anp" data-title="ज्यामिति" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9" title="هندسة رياضية – Arabic" lang="ar" hreflang="ar" data-title="هندسة رياضية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Cheometr%C3%ADa" title="Cheometría – Aragonese" lang="an" hreflang="an" data-title="Cheometría" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF" title="জ্যামিতি – Assamese" lang="as" hreflang="as" data-title="জ্যামিতি" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Xeometr%C3%ADa" title="Xeometría – Asturian" lang="ast" hreflang="ast" data-title="Xeometría" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Ysajarekokuaa" title="Ysajarekokuaa – Guarani" lang="gn" hreflang="gn" data-title="Ysajarekokuaa" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99nd%C9%99s%C9%99" title="Həndəsə – Azerbaijani" lang="az" hreflang="az" data-title="Həndəsə" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D9%87" title="هندسه – South Azerbaijani" lang="azb" hreflang="azb" data-title="هندسه" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF" title="জ্যামিতি – Bangla" lang="bn" hreflang="bn" data-title="জ্যামিতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C3%AD-h%C3%B4-ha%CC%8Dk" title="Kí-hô-ha̍k – Minnan" lang="nan" hreflang="nan" data-title="Kí-hô-ha̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Bashkir" lang="ba" hreflang="ba" data-title="Геометрия" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F" title="Геаметрыя – Belarusian" lang="be" hreflang="be" data-title="Геаметрыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%93%D0%B5%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F" title="Геамэтрыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Геамэтрыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A5%89%E0%A4%AE%E0%A5%87%E0%A4%9F%E0%A5%8D%E0%A4%B0%E0%A5%80" title="ज्यॉमेट्री – Bhojpuri" lang="bh" hreflang="bh" data-title="ज्यॉमेट्री" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Heometriya" title="Heometriya – Central Bikol" lang="bcl" hreflang="bcl" data-title="Heometriya" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bi mw-list-item"><a href="https://bi.wikipedia.org/wiki/Jiometri" title="Jiometri – Bislama" lang="bi" hreflang="bi" data-title="Jiometri" data-language-autonym="Bislama" data-language-local-name="Bislama" class="interlanguage-link-target"><span>Bislama</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Bulgarian" lang="bg" hreflang="bg" data-title="Геометрия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%91%E0%BD%96%E0%BE%B1%E0%BD%B2%E0%BD%96%E0%BD%A6%E0%BC%8B%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8B%E0%BD%A2%E0%BD%B2%E0%BD%82%E0%BC%8B%E0%BD%94%E0%BC%8D" title="དབྱིབས་རྩིས་རིག་པ། – Tibetan" lang="bo" hreflang="bo" data-title="དབྱིབས་རྩིས་རིག་པ།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Geometrija" title="Geometrija – Bosnian" lang="bs" hreflang="bs" data-title="Geometrija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Mentoniezh" title="Mentoniezh – Breton" lang="br" hreflang="br" data-title="Mentoniezh" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8" title="Геометри – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Геометри" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geometria" title="Geometria – Catalan" lang="ca" hreflang="ca" data-title="Geometria" 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%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8" title="Геометри – Chuvash" lang="cv" hreflang="cv" data-title="Геометри" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://cs.wikipedia.org/wiki/Geometrie" title="Geometrie – Czech" lang="cs" hreflang="cs" data-title="Geometrie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Pimachisi" title="Pimachisi – Shona" lang="sn" hreflang="sn" data-title="Pimachisi" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Geumitria" title="Geumitria – Corsican" lang="co" hreflang="co" data-title="Geumitria" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Geometreg" title="Geometreg – Welsh" lang="cy" hreflang="cy" data-title="Geometreg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Geometri" title="Geometri – Danish" lang="da" hreflang="da" data-title="Geometri" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AA%D8%B3%D8%B7%D8%A7%D8%B1" title="تسطار – Moroccan Arabic" lang="ary" hreflang="ary" data-title="تسطار" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geometrie" title="Geometrie – German" lang="de" hreflang="de" data-title="Geometrie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geomeetria" title="Geomeetria – Estonian" lang="et" hreflang="et" data-title="Geomeetria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Γεωμετρία – Greek" lang="el" hreflang="el" data-title="Γεωμετρία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Geometr%C3%AE" title="Geometrî – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Geometrî" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Erzya" lang="myv" hreflang="myv" data-title="Геометрия" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Geometr%C3%ADa" title="Geometría – Spanish" lang="es" hreflang="es" data-title="Geometría" 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/Geometrio" title="Geometrio – Esperanto" lang="eo" hreflang="eo" data-title="Geometrio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Geometria" title="Geometria – Extremaduran" lang="ext" hreflang="ext" data-title="Geometria" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Geometria" title="Geometria – Basque" lang="eu" hreflang="eu" data-title="Geometria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D9%87" title="هندسه – Persian" lang="fa" hreflang="fa" data-title="هندسه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Geometry" title="Geometry – Fiji Hindi" lang="hif" hreflang="hif" data-title="Geometry" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Geometri" title="Geometri – Faroese" lang="fo" hreflang="fo" data-title="Geometri" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie" title="Géométrie – French" lang="fr" hreflang="fr" data-title="Géométrie" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Geoim%C3%A9adracht" title="Geoiméadracht – Irish" lang="ga" hreflang="ga" data-title="Geoiméadracht" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Towse-oaylleeaght" title="Towse-oaylleeaght – Manx" lang="gv" hreflang="gv" data-title="Towse-oaylleeaght" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Geoimeatras" title="Geoimeatras – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Geoimeatras" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Xeometr%C3%ADa" title="Xeometría – Galician" lang="gl" hreflang="gl" data-title="Xeometría" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E5%AD%B8" title="幾何學 – Gan" lang="gan" hreflang="gan" data-title="幾何學" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/M%C5%A9thun%C5%A9r%C5%A9rio_(geometry)" title="Mũthunũrũrio (geometry) – Kikuyu" lang="ki" hreflang="ki" data-title="Mũthunũrũrio (geometry)" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AD%E0%AB%82%E0%AA%AE%E0%AA%BF%E0%AA%A4%E0%AA%BF" title="ભૂમિતિ – Gujarati" lang="gu" hreflang="gu" data-title="ભૂમિતિ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/K%C3%AD-h%C3%B2-ho%CC%8Dk" title="Kí-hò-ho̍k – Hakka Chinese" lang="hak" hreflang="hak" data-title="Kí-hò-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%B0%ED%95%98%ED%95%99" title="기하학 – Korean" lang="ko" hreflang="ko" data-title="기하학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Երկրաչափություն – Armenian" lang="hy" hreflang="hy" data-title="Երկրաչափություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF" title="ज्यामिति – Hindi" lang="hi" hreflang="hi" data-title="ज्यामिति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Geometrija" title="Geometrija – Croatian" lang="hr" hreflang="hr" data-title="Geometrija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Geometrio" title="Geometrio – Ido" lang="io" hreflang="io" data-title="Geometrio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Heometria" title="Heometria – Iloko" lang="ilo" hreflang="ilo" data-title="Heometria" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geometri" title="Geometri – Indonesian" lang="id" hreflang="id" data-title="Geometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ia.wikipedia.org/wiki/Geometria" title="Geometria – Interlingua" lang="ia" hreflang="ia" data-title="Geometria" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Geometrie" title="Geometrie – Interlingue" lang="ie" hreflang="ie" data-title="Geometrie" data-language-autonym="Interlingue" data-language-local-name="Interlingue" class="interlanguage-link-target"><span>Interlingue</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Umchazabukhulu" title="Umchazabukhulu – Zulu" lang="zu" hreflang="zu" data-title="Umchazabukhulu" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%BAmfr%C3%A6%C3%B0i" title="Rúmfræði – Icelandic" lang="is" hreflang="is" data-title="Rúmfræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geometria" title="Geometria – Italian" lang="it" hreflang="it" data-title="Geometria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94" title="גאומטריה – Hebrew" lang="he" hreflang="he" data-title="גאומטריה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/G%C3%A9om%C3%A8tri" title="Géomètri – Javanese" lang="jv" hreflang="jv" data-title="Géomètri" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Sii%C5%8B_l%C9%A9z%CA%8A%CA%8A" title="Siiŋ lɩzʊʊ – Kabiye" lang="kbp" hreflang="kbp" data-title="Siiŋ lɩzʊʊ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B0%E0%B3%87%E0%B2%96%E0%B2%BE%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ರೇಖಾಗಣಿತ – Kannada" lang="kn" hreflang="kn" data-title="ರೇಖಾಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%94%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90" title="გეომეტრია – Georgian" lang="ka" hreflang="ka" data-title="გეომეტრია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Kazakh" lang="kk" hreflang="kk" data-title="Геометрия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Mynsonieth" title="Mynsonieth – Cornish" lang="kw" hreflang="kw" data-title="Mynsonieth" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Jiometri" title="Jiometri – Swahili" lang="sw" hreflang="sw" data-title="Jiometri" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Jewometri" title="Jewometri – Haitian Creole" lang="ht" hreflang="ht" data-title="Jewometri" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/J%C3%A9om%C3%A9tri" title="Jéométri – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Jéométri" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Geometr%C3%AE" title="Geometrî – Kurdish" lang="ku" hreflang="ku" data-title="Geometrî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Kyrgyz" lang="ky" hreflang="ky" data-title="Геометрия" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%A5%E0%BA%82%E0%BA%B2%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94" title="ເລຂາຄະນິດ – Lao" lang="lo" hreflang="lo" data-title="ເລຂາຄະນິດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Geometria" title="Geometria – Latin" lang="la" hreflang="la" data-title="Geometria" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%A2eometrija" title="Ģeometrija – Latvian" lang="lv" hreflang="lv" data-title="Ģeometrija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Geometrie" title="Geometrie – Luxembourgish" lang="lb" hreflang="lb" data-title="Geometrie" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Geometrija" title="Geometrija – Lithuanian" lang="lt" hreflang="lt" data-title="Geometrija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nia mw-list-item"><a href="https://nia.wikipedia.org/wiki/Geometris" title="Geometris – Nias" lang="nia" hreflang="nia" data-title="Geometris" data-language-autonym="Li Niha" data-language-local-name="Nias" class="interlanguage-link-target"><span>Li Niha</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Geometria" title="Geometria – Ligurian" lang="lij" hreflang="lij" data-title="Geometria" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Maetk%C3%B3nde" title="Maetkónde – Limburgish" lang="li" hreflang="li" data-title="Maetkónde" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Zom%C9%9Bt%C9%9Bl%C3%AD" title="Zomɛtɛlí – Lingala" lang="ln" hreflang="ln" data-title="Zomɛtɛlí" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Jeometria" title="Jeometria – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Jeometria" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Essomampimo_(Geometry)" title="Essomampimo (Geometry) – Ganda" lang="lg" hreflang="lg" data-title="Essomampimo (Geometry)" data-language-autonym="Luganda" data-language-local-name="Ganda" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Geometr%C3%ACa" title="Geometrìa – Lombard" lang="lmo" hreflang="lmo" data-title="Geometrìa" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Geometria" title="Geometria – Hungarian" lang="hu" hreflang="hu" data-title="Geometria" 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%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Геометрија – Macedonian" lang="mk" hreflang="mk" data-title="Геометрија" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Je%C3%B4metria" title="Jeômetria – Malagasy" lang="mg" hreflang="mg" data-title="Jeômetria" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF" title="ജ്യാമിതി – Malayalam" lang="ml" hreflang="ml" data-title="ജ്യാമിതി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/%C4%A0eometrija" title="Ġeometrija – Maltese" lang="mt" hreflang="mt" data-title="Ġeometrija" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AD%E0%A5%82%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80" title="भूमिती – Marathi" lang="mr" hreflang="mr" data-title="भूमिती" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%92%E1%83%94%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90" title="გეომეტრია – Mingrelian" lang="xmf" hreflang="xmf" data-title="გეომეტრია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-mnw mw-list-item"><a href="https://mnw.wikipedia.org/wiki/%E1%80%82%E1%80%B1%E1%80%9E%E1%80%BC%E1%80%99%E1%80%B1%E1%80%90%E1%80%BC%E1%80%B3" title="ဂေသြမေတြဳ – Mon" lang="mnw" hreflang="mnw" data-title="ဂေသြမေတြဳ" data-language-autonym="ဘာသာမန်" data-language-local-name="Mon" class="interlanguage-link-target"><span>ဘာသာမန်</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Geometri" title="Geometri – Malay" lang="ms" hreflang="ms" data-title="Geometri" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Ilmu_ukua" title="Ilmu ukua – Minangkabau" lang="min" hreflang="min" data-title="Ilmu ukua" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Geometrie" title="Geometrie – Mirandese" lang="mwl" hreflang="mwl" data-title="Geometrie" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D1%81%D1%8C" title="Геометриясь – Moksha" lang="mdf" hreflang="mdf" data-title="Геометриясь" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80" title="Геометр – Mongolian" lang="mn" hreflang="mn" data-title="Геометр" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%82%E1%80%BB%E1%80%AE%E1%80%A9%E1%80%99%E1%80%B1%E1%80%90%E1%80%BC%E1%80%AE" title="ဂျီဩမေတြီ – Burmese" lang="my" hreflang="my" data-title="ဂျီဩမေတြီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Geometry" title="Geometry – Fijian" lang="fj" hreflang="fj" data-title="Geometry" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Meetkunde" title="Meetkunde – Dutch" lang="nl" hreflang="nl" data-title="Meetkunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF" title="ज्यामिति – Nepali" lang="ne" hreflang="ne" data-title="ज्यामिति" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="रेखागणित – Newari" lang="new" hreflang="new" data-title="रेखागणित" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E5%AD%A6" title="幾何学 – Japanese" lang="ja" hreflang="ja" data-title="幾何学" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Geometrii" title="Geometrii – Northern Frisian" lang="frr" hreflang="frr" data-title="Geometrii" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Geometri" title="Geometri – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Geometri" 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/Geometri" title="Geometri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Geometri" 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-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Geometria" title="Geometria – Novial" lang="nov" hreflang="nov" data-title="Geometria" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Geometria" title="Geometria – Occitan" lang="oc" hreflang="oc" data-title="Geometria" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B9" title="Геометрий – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Геометрий" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%9C%E0%AD%8D%E0%AD%9F%E0%AC%BE%E0%AC%AE%E0%AC%BF%E0%AC%A4%E0%AC%BF" title="ଜ୍ୟାମିତି – Odia" lang="or" hreflang="or" data-title="ଜ୍ୟାମିତି" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Ji%27oomeetirii" title="Ji'oomeetirii – Oromo" lang="om" hreflang="om" data-title="Ji'oomeetirii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geometriya" title="Geometriya – Uzbek" lang="uz" hreflang="uz" data-title="Geometriya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A8%BE_%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਰੇਖਾ ਗਣਿਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਰੇਖਾ ਗਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AC%DB%8C%D9%88%D9%85%DB%8C%D9%B9%D8%B1%DB%8C" title="جیومیٹری – Western Punjabi" lang="pnb" hreflang="pnb" data-title="جیومیٹری" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%85%DB%90%DA%86%D9%BE%D9%88%D9%87%D9%86%D9%87" title="مېچپوهنه – Pashto" lang="ps" hreflang="ps" data-title="مېچپوهنه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Jaamichri" title="Jaamichri – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Jaamichri" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%92%E1%9E%9A%E1%9E%8E%E1%9E%B8%E1%9E%98%E1%9E%B6%E1%9E%8F%E1%9F%92%E1%9E%9A" title="ធរណីមាត្រ – Khmer" lang="km" hreflang="km" data-title="ធរណីមាត្រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Geometr%C3%ACa" title="Geometrìa – Piedmontese" lang="pms" hreflang="pms" data-title="Geometrìa" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Geometrie" title="Geometrie – Low German" lang="nds" hreflang="nds" data-title="Geometrie" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Geometria" title="Geometria – Polish" lang="pl" hreflang="pl" data-title="Geometria" 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/Geometria" title="Geometria – Portuguese" lang="pt" hreflang="pt" data-title="Geometria" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Geometriya" title="Geometriya – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Geometriya" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geometrie" title="Geometrie – Romanian" lang="ro" hreflang="ro" data-title="Geometrie" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Pacha_tupuy" title="Pacha tupuy – Quechua" lang="qu" hreflang="qu" data-title="Pacha tupuy" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D2%90%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Ґеометрія – Rusyn" lang="rue" hreflang="rue" data-title="Ґеометрія" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Russian" lang="ru" hreflang="ru" data-title="Геометрия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Yakut" lang="sah" hreflang="sah" data-title="Геометрия" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Geometry" title="Geometry – Scots" lang="sco" hreflang="sco" data-title="Geometry" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Geometrie" title="Geometrie – Saterland Frisian" lang="stq" hreflang="stq" data-title="Geometrie" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target"><span>Seeltersk</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gjeometria" title="Gjeometria – Albanian" lang="sq" hreflang="sq" data-title="Gjeometria" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Giometr%C3%ACa" title="Giometrìa – Sicilian" lang="scn" hreflang="scn" data-title="Giometrìa" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%A2%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B6%B8%E0%B7%92%E0%B6%AD%E0%B7%92%E0%B6%BA" title="ජ්‍යාමිතිය – Sinhala" lang="si" hreflang="si" data-title="ජ්‍යාමිතිය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Geometry" title="Geometry – Simple English" lang="en-simple" hreflang="en-simple" data-title="Geometry" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%AC%D8%A7%D9%85%D9%8A%D9%BD%D8%B1%D9%8A" title="جاميٽري – Sindhi" lang="sd" hreflang="sd" data-title="جاميٽري" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Geometria" title="Geometria – Slovak" lang="sk" hreflang="sk" data-title="Geometria" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Geometrija" title="Geometrija – Slovenian" lang="sl" hreflang="sl" data-title="Geometrija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Geometryjo" title="Geometryjo – Silesian" lang="szl" hreflang="szl" data-title="Geometryjo" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95" title="ئەندازە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئەندازە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Геометрија – Serbian" lang="sr" hreflang="sr" data-title="Геометрија" 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/Geometrija" title="Geometrija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Geometrija" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/%C3%89lmu_ukur" title="Élmu ukur – Sundanese" lang="su" hreflang="su" data-title="Élmu ukur" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Geometria" title="Geometria – Finnish" lang="fi" hreflang="fi" data-title="Geometria" 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/Geometri" title="Geometri – Swedish" lang="sv" hreflang="sv" data-title="Geometri" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Heometriya" title="Heometriya – Tagalog" lang="tl" hreflang="tl" data-title="Heometriya" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="வடிவவியல் – Tamil" lang="ta" hreflang="ta" data-title="வடிவவியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/As%C9%A3kl" title="Asɣkl – Tachelhit" lang="shi" hreflang="shi" data-title="Asɣkl" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tanzeggit" title="Tanzeggit – Kabyle" lang="kab" hreflang="kab" data-title="Tanzeggit" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Tatar" lang="tt" hreflang="tt" data-title="Геометрия" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B0%E0%B1%87%E0%B0%96%E0%B0%BE%E0%B0%97%E0%B0%A3%E0%B0%BF%E0%B0%A4%E0%B0%82" title="రేఖాగణితం – Telugu" lang="te" hreflang="te" data-title="రేఖాగణితం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="เรขาคณิต – Thai" lang="th" hreflang="th" data-title="เรขาคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D2%B2%D0%B0%D0%BD%D0%B4%D0%B0%D1%81%D0%B0" title="Ҳандаса – Tajik" lang="tg" hreflang="tg" data-title="Ҳандаса" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8F%97%E1%8F%8E%E1%8F%8D%E1%8F%97_%E1%8F%93%E1%8F%8D%E1%8F%93%E1%8F%85%E1%8F%85" title="ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ – Cherokee" lang="chr" hreflang="chr" data-title="ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Geometri" title="Geometri – Turkish" lang="tr" hreflang="tr" data-title="Geometri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Geometri%C3%BDa" title="Geometriýa – Turkmen" lang="tk" hreflang="tk" data-title="Geometriýa" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tyv mw-list-item"><a href="https://tyv.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия – Tuvinian" lang="tyv" hreflang="tyv" data-title="Геометрия" data-language-autonym="Тыва дыл" data-language-local-name="Tuvinian" class="interlanguage-link-target"><span>Тыва дыл</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Геометрія – Ukrainian" lang="uk" hreflang="uk" data-title="Геометрія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DB%81%D9%86%D8%AF%D8%B3%DB%81" title="ہندسہ – Urdu" lang="ur" hreflang="ur" data-title="ہندسہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Gijhozyoz" title="Gijhozyoz – Zhuang" lang="za" hreflang="za" data-title="Gijhozyoz" data-language-autonym="Vahcuengh" data-language-local-name="Zhuang" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Zeometria" title="Zeometria – Venetian" lang="vec" hreflang="vec" data-title="Zeometria" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Geometrii" title="Geometrii – Veps" lang="vep" hreflang="vep" data-title="Geometrii" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_h%E1%BB%8Dc" title="Hình học – Vietnamese" lang="vi" hreflang="vi" data-title="Hình học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Geomeetri%C3%A4" title="Geomeetriä – Võro" lang="vro" hreflang="vro" data-title="Geomeetriä" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95" title="幾何 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="幾何" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Heyometriya" title="Heyometriya – Waray" lang="war" hreflang="war" data-title="Heyometriya" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a 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href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Geometry</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size notpageimage" typeof="mw:File/Frameless"><a href="/wiki/File:Stereographic_projection_in_3D.svg" class="mw-file-description"><img alt="Stereographic projection from the top of a sphere onto a plane beneath it" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/220px-Stereographic_projection_in_3D.svg.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/330px-Stereographic_projection_in_3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/440px-Stereographic_projection_in_3D.svg.png 2x" data-file-width="870" data-file-height="639" /></a></span><div class="sidebar-caption"><a href="/wiki/Projective_geometry" title="Projective geometry">Projecting</a> a <a href="/wiki/Sphere" title="Sphere">sphere</a> to a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a> <ul><li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li> <li><a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a> <ul><li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a> <ul><li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a> <ul><li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital</a></li></ul></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence </a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a> <ul><li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="/wiki/Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a> <ul><li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a></li> <li><a href="/wiki/Curve" title="Curve">Curve</a></li> <li><a href="/wiki/Diagonal" title="Diagonal">Diagonal</a></li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a>)</li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Line_(geometry)" title="Line (geometry)">Line</a> <ul><li><a href="/wiki/Line_segment" title="Line segment">segment</a></li> <li><a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li> <li><a href="/wiki/Length" title="Length">Length</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li> <li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li> <li><a href="/wiki/Hypotenuse" title="Hypotenuse">Hypotenuse</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Diameter" title="Diameter">Diameter</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Volume" title="Volume">Volume</a></li></ul> <ul><li><a href="/wiki/Cube" title="Cube">Cube</a> <ul><li><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></li></ul></li> <li><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li> <li><a href="/wiki/Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a>- / other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li> <li><a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;"> <a href="/wiki/List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1–1400s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1400s–1700s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1700s–1900s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Present day</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output 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hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th></tr><tr><td class="sidebar-above" style="padding-bottom:0.35em;"> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup"> <table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="/wiki/Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> 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class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Physics</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Geomathematics" title="Geomathematics">Geosciences</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computation</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Computational_linguistics" title="Computational linguistics">Linguistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Economics</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Education</a></li></ul></div></div></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <span typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/20px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/30px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics Portal</a></th></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Math_topics_sidebar" title="Template:Math topics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Geometry</b> (from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1#Ancient_Greek" class="extiw" title="wikt:γεωμετρία">γεωμετρία</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>geōmetría</i></span>)</i> 'land measurement'; from <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%CE%B3%E1%BF%86#Ancient_Greek" class="extiw" title="wikt:γῆ">γῆ</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>gê</i></span>)</i> 'earth, land' and <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%84%CF%81%CE%BF%CE%BD#Ancient_Greek" class="extiw" title="wikt:μέτρον">μέτρον</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>métron</i></span>)</i> 'a measure')<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> is a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> concerned with properties of space such as the distance, shape, size, and relative position of figures.<sup id="cite_ref-Risi2015_2-0" class="reference"><a href="#cite_note-Risi2015-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Geometry is, along with <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a <i><a href="/wiki/List_of_geometers" title="List of geometers">geometer</a></i>. Until the 19th century, geometry was almost exclusively devoted to <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> which includes the notions of <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a>, <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a>, <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, <a href="/wiki/Distance" title="Distance">distance</a>, <a href="/wiki/Angle" title="Angle">angle</a>, <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a>, and <a href="/wiki/Curve" title="Curve">curve</a>, as fundamental concepts.<sup id="cite_ref-Tabak_2014_xiv_4-0" class="reference"><a href="#cite_note-Tabak_2014_xiv-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, <a href="/wiki/Architecture" title="Architecture">architecture</a>, and other activities that are related to graphics.<sup id="cite_ref-Meyer2006_5-0" class="reference"><a href="#cite_note-Meyer2006-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof</a> of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>, a problem that was stated in terms of <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a>, and remained unsolved for several centuries. </p><p>During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>'s <span title="Latin-language text"><span lang="la" style="font-style: normal;"><a href="/wiki/Theorema_Egregium" title="Theorema Egregium">Theorema Egregium</a></span></span> ("remarkable theorem") that asserts roughly that the <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> of a surface is independent from any specific <a href="/wiki/Embedding" title="Embedding">embedding</a> in a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. This implies that surfaces can be studied <i>intrinsically</i>, that is, as stand-alone spaces, and has been expanded into the theory of <a href="/wiki/Manifold" title="Manifold">manifolds</a> and <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>. Later in the 19th century, it appeared that geometries without the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> (<a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a>) can be developed without introducing any contradiction. The geometry that underlies <a href="/wiki/General_relativity" title="General relativity">general relativity</a> is a famous application of non-Euclidean geometry. </p><p>Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—<a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>, <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, <a href="/wiki/Discrete_geometry" title="Discrete geometry">discrete geometry</a> (also known as <i>combinatorial geometry</i>), etc.—or on the properties of Euclidean spaces that are disregarded—<a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> that consider only alignment of points but not distance and parallelism, <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a> that omits the concept of angle and distance, <a href="/wiki/Finite_geometry" title="Finite geometry">finite geometry</a> that omits <a href="/wiki/Continuity_(mathematics)" class="mw-redirect" title="Continuity (mathematics)">continuity</a>, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional <a href="/wiki/Space" title="Space">space</a> of the physical world and its <a href="/wiki/Model" title="Model">model</a> provided by Euclidean geometry; presently a <b>geometric space</b>, or simply a <i>space</i> is a <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a> on which some geometry is defined. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></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/History_of_geometry" title="History of geometry">History of geometry</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg/220px-Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg/330px-Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg/440px-Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpg 2x" data-file-width="2130" data-file-height="2083" /></a><figcaption>A <a href="/wiki/Ethnic_groups_in_Europe" title="Ethnic groups in Europe">European</a> and an <a href="/wiki/Arab" class="mw-redirect" title="Arab">Arab</a> practicing geometry in the 15th century</figcaption></figure> <p>The earliest recorded beginnings of geometry can be traced to ancient <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a> and <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a> in the 2nd millennium BC.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in <a href="/wiki/Surveying" title="Surveying">surveying</a>, <a href="/wiki/Construction" title="Construction">construction</a>, <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, and various crafts. The earliest known texts on geometry are the <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian</a> <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Papyrus</a> (2000–1800 BC) and <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Papyrus</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1890 BC</span>), and the <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian clay tablets</a>, such as <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a> (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or <a href="/wiki/Frustum" title="Frustum">frustum</a>.<sup id="cite_ref-Boyer_1991_loc=Egypt_p._19_8-0" class="reference"><a href="#cite_note-Boyer_1991_loc=Egypt_p._19-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> procedures for computing Jupiter's position and <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">motion</a> within time-velocity space. These geometric procedures anticipated the <a href="/wiki/Oxford_Calculators" title="Oxford Calculators">Oxford Calculators</a>, including the <a href="/wiki/Mean_speed_theorem" title="Mean speed theorem">mean speed theorem</a>, by 14 centuries.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> South of Egypt the <a href="/wiki/Nubia" title="Nubia">ancient Nubians</a> established a system of geometry including early versions of sun clocks.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In the 7th century BC, the <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek</a> mathematician <a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a> used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to <a href="/wiki/Thales%27s_theorem" title="Thales's theorem">Thales's theorem</a>.<sup id="cite_ref-Boyer_1991_loc=Ionia_and_the_Pythagoreans_p._43_12-0" class="reference"><a href="#cite_note-Boyer_1991_loc=Ionia_and_the_Pythagoreans_p._43-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> established the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagorean School</a>, which is credited with the first proof of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> though the statement of the theorem has a long history.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a> (408–<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 355 BC</span>) developed the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>, which allowed the calculation of areas and volumes of curvilinear figures,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> as well as a theory of ratios that avoided the problem of <a href="/wiki/Incommensurable_magnitudes" class="mw-redirect" title="Incommensurable magnitudes">incommensurable magnitudes</a>, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i>, widely considered the most successful and influential textbook of all time,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> introduced <a href="/wiki/Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">mathematical rigor</a> through the <a href="/wiki/Axiomatic_method" class="mw-redirect" title="Axiomatic method">axiomatic method</a> and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the <i>Elements</i> were already known, Euclid arranged them into a single, coherent logical framework.<sup id="cite_ref-Boyer_1991_loc=Euclid_of_Alexandria_p._104_18-0" class="reference"><a href="#cite_note-Boyer_1991_loc=Euclid_of_Alexandria_p._104-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The <i>Elements</i> was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 287–212 BC</span>) of <a href="/wiki/Syracuse,_Italy" class="mw-redirect" title="Syracuse, Italy">Syracuse, Italy</a> used the method of exhaustion to calculate the <a href="/wiki/Area" title="Area">area</a> under the arc of a <a href="/wiki/Parabola" title="Parabola">parabola</a> with the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">summation of an infinite series</a>, and gave remarkably accurate approximations of <a href="/wiki/Pi" title="Pi">pi</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> He also studied the <a href="/wiki/Archimedes_spiral" class="mw-redirect" title="Archimedes spiral">spiral</a> bearing his name and obtained formulas for the <a href="/wiki/Volume" title="Volume">volumes</a> of <a href="/wiki/Surface_of_revolution" title="Surface of revolution">surfaces of revolution</a>. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Woman_teaching_geometry.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Woman_teaching_geometry.jpg/190px-Woman_teaching_geometry.jpg" decoding="async" width="190" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Woman_teaching_geometry.jpg/285px-Woman_teaching_geometry.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Woman_teaching_geometry.jpg/380px-Woman_teaching_geometry.jpg 2x" data-file-width="1039" data-file-height="1148" /></a><figcaption><i>Woman teaching geometry</i>. Illustration at the beginning of a medieval translation of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1310</span>).</figcaption></figure> <p><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian</a> mathematicians also made many important contributions in geometry. The <i><a href="/wiki/Shatapatha_Brahmana" title="Shatapatha Brahmana">Shatapatha Brahmana</a></i> (3rd century BC) contains rules for ritual geometric constructions that are similar to the <i><a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Sulba Sutras</a></i>.<sup id="cite_ref-Staal_1999_21-0" class="reference"><a href="#cite_note-Staal_1999-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> According to (<a href="#CITEREFHayashi2005">Hayashi 2005</a>, p. 363), the <i>Śulba Sūtras</i> contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of <a href="/wiki/Pythagorean_triples" class="mw-redirect" title="Pythagorean triples">Pythagorean triples</a>,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> which are particular cases of <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a>.<sup id="cite_ref-cooke198_23-0" class="reference"><a href="#cite_note-cooke198-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In the <a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a>, there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."<sup id="cite_ref-hayashi2005-371_24-0" class="reference"><a href="#cite_note-hayashi2005-371-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>'s <i><a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></i> (499) includes the computation of areas and volumes. <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> wrote his astronomical work <i><a href="/wiki/Brahmasphutasiddhanta" class="mw-redirect" title="Brahmasphutasiddhanta"><span title="International Alphabet of Sanskrit transliteration"><i lang="sa-Latn">Brāhmasphuṭasiddhānta</i></span></a></i> in 628. Chapter 12, containing 66 <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).<sup id="cite_ref-hayashi2003-p121-122_25-0" class="reference"><a href="#cite_note-hayashi2003-p121-122-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In the latter section, he stated his famous theorem on the diagonals of a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a>), as well as a complete description of <a href="/wiki/Rational_triangle" class="mw-redirect" title="Rational triangle">rational triangles</a> (<i>i.e.</i> triangles with rational sides and rational areas).<sup id="cite_ref-hayashi2003-p121-122_25-1" class="reference"><a href="#cite_note-hayashi2003-p121-122-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="/wiki/Middle_Ages" title="Middle Ages">Middle Ages</a>, <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">mathematics in medieval Islam</a> contributed to the development of geometry, especially <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Al-Mahani" title="Al-Mahani">Al-Mahani</a> (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Th%C4%81bit_ibn_Qurra" title="Thābit ibn Qurra">Thābit ibn Qurra</a> (known as Thebit in <a href="/wiki/Latin" title="Latin">Latin</a>) (836–901) dealt with <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> operations applied to <a href="/wiki/Ratio" title="Ratio">ratios</a> of geometrical quantities, and contributed to the development of <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>.<sup id="cite_ref-ReferenceA_29-0" class="reference"><a href="#cite_note-ReferenceA-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> (1048–1131) found geometric solutions to <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equations</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The theorems of <a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a> (Alhazen), Omar Khayyam and <a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">Nasir al-Din al-Tusi</a> on <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilaterals</a>, including the <a href="/wiki/Lambert_quadrilateral" title="Lambert quadrilateral">Lambert quadrilateral</a> and <a href="/wiki/Saccheri_quadrilateral" title="Saccheri quadrilateral">Saccheri quadrilateral</a>, were part of a line of research on the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> continued by later European geometers, including <a href="/wiki/Vitello" title="Vitello">Vitello</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1230</span> – c.<span style="white-space:nowrap;"> 1314</span>), <a href="/wiki/Gersonides" title="Gersonides">Gersonides</a> (1288–1344), Alfonso, <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a>, and <a href="/wiki/Giovanni_Girolamo_Saccheri" title="Giovanni Girolamo Saccheri">Giovanni Girolamo Saccheri</a>, that by the 19th century led to the discovery of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with <a href="/wiki/Coordinate_system" title="Coordinate system">coordinates</a> and <a href="/wiki/Equation" title="Equation">equations</a>, by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (1596–1650) and <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> (1601–1665).<sup id="cite_ref-Boyer2012_32-0" class="reference"><a href="#cite_note-Boyer2012-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This was a necessary precursor to the development of <a href="/wiki/Calculus" title="Calculus">calculus</a> and a precise quantitative science of <a href="/wiki/Physics" title="Physics">physics</a>.<sup id="cite_ref-Edwards2012_33-0" class="reference"><a href="#cite_note-Edwards2012-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The second geometric development of this period was the systematic study of <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> by <a href="/wiki/Girard_Desargues" title="Girard Desargues">Girard Desargues</a> (1591–1661).<sup id="cite_ref-FieldGray2012_34-0" class="reference"><a href="#cite_note-FieldGray2012-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Projective geometry studies properties of shapes which are unchanged under <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projections</a> and <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a>, especially as they relate to <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">artistic perspective</a>.<sup id="cite_ref-Wylie2011_35-0" class="reference"><a href="#cite_note-Wylie2011-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>Two developments in geometry in the 19th century changed the way it had been studied previously.<sup id="cite_ref-Gray2011_36-0" class="reference"><a href="#cite_note-Gray2011-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> These were the discovery of <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometries</a> by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> as the central consideration in the <a href="/wiki/Erlangen_programme" class="mw-redirect" title="Erlangen programme">Erlangen programme</a> of <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> (1826–1866), working primarily with tools from <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, and introducing the <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a>, and <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>, the founder of <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> and the geometric theory of <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. As a consequence of these major changes in the conception of geometry, the concept of "<a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a>" became something rich and varied, and the natural background for theories as different as <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> and <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>.<sup id="cite_ref-Bayro-Corrochano2018_37-0" class="reference"><a href="#cite_note-Bayro-Corrochano2018-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Main_concepts">Main concepts</h2></div> <p>The following are some of the most important concepts in geometry.<sup id="cite_ref-Tabak_2014_xiv_4-1" class="reference"><a href="#cite_note-Tabak_2014_xiv-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kline1990_38-0" class="reference"><a href="#cite_note-Kline1990-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Axioms">Axioms</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Parallel_postulate_en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parallel_postulate_en.svg/220px-Parallel_postulate_en.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parallel_postulate_en.svg/330px-Parallel_postulate_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parallel_postulate_en.svg/440px-Parallel_postulate_en.svg.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>An illustration of Euclid's <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> and <a href="/wiki/Axiom" title="Axiom">Axiom</a></div> <p><a href="/wiki/Euclid" title="Euclid">Euclid</a> took an abstract approach to geometry in his <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>,<sup id="cite_ref-Katz2000_39-0" class="reference"><a href="#cite_note-Katz2000-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> one of the most influential books ever written.<sup id="cite_ref-Berlinski2014_40-0" class="reference"><a href="#cite_note-Berlinski2014-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Euclid introduced certain <a href="/wiki/Axiom" title="Axiom">axioms</a>, or <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a>, expressing primary or self-evident properties of points, lines, and planes.<sup id="cite_ref-Hartshorne2013_41-0" class="reference"><a href="#cite_note-Hartshorne2013-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as <i>axiomatic</i> or <i><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic</a></i> geometry.<sup id="cite_ref-HerbstFujita2017_42-0" class="reference"><a href="#cite_note-HerbstFujita2017-42"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> At the start of the 19th century, the discovery of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a> by <a href="/wiki/Nikolai_Ivanovich_Lobachevsky" class="mw-redirect" title="Nikolai Ivanovich Lobachevsky">Nikolai Ivanovich Lobachevsky</a> (1792–1856), <a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">János Bolyai</a> (1802–1860), <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1777–1855) and others<sup id="cite_ref-Yaglom2012_43-0" class="reference"><a href="#cite_note-Yaglom2012-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> led to a revival of interest in this discipline, and in the 20th century, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.<sup id="cite_ref-Holme2010_44-0" class="reference"><a href="#cite_note-Holme2010-44"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Spaces_and_subspaces">Spaces and subspaces</h3></div> <div class="mw-heading mw-heading4"><h4 id="Points">Points</h4></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/Point_(geometry)" title="Point (geometry)">Point (geometry)</a></div> <p>Points are generally considered fundamental objects for building geometry. They may be defined by the properties that they must have, as in Euclid's definition as "that which has no part",<sup id="cite_ref-EuclidAll_45-0" class="reference"><a href="#cite_note-EuclidAll-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> or in <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>. In modern mathematics, they are generally defined as <a href="/wiki/Element_(set_theory)" class="mw-redirect" title="Element (set theory)">elements</a> of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> called <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a>, which is itself <a href="/wiki/Axiomatically" class="mw-redirect" title="Axiomatically">axiomatically</a> defined. </p><p>With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes. </p><p>However, there are modern geometries in which points are not primitive objects, or even without points.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> One of the oldest such geometries is <a href="/wiki/Whitehead%27s_point-free_geometry" title="Whitehead's point-free geometry">Whitehead's point-free geometry</a>, formulated by <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> in 1919–1920. </p> <div class="mw-heading mw-heading4"><h4 id="Lines">Lines</h4></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/Line_(geometry)" title="Line (geometry)">Line (geometry)</a></div> <p><a href="/wiki/Euclid" title="Euclid">Euclid</a> described a line as "breadthless length" which "lies equally with respect to the points on itself".<sup id="cite_ref-EuclidAll_45-1" class="reference"><a href="#cite_note-EuclidAll-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, a line in the plane is often defined as the set of points whose coordinates satisfy a given <a href="/wiki/Linear_equation" title="Linear equation">linear equation</a>,<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> but in a more abstract setting, such as <a href="/wiki/Incidence_geometry" title="Incidence geometry">incidence geometry</a>, a line may be an independent object, distinct from the set of points which lie on it.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> In differential geometry, a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> is a generalization of the notion of a line to <a href="/wiki/Manifold" title="Manifold">curved spaces</a>.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Planes">Planes</h4></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/Euclidean_plane" title="Euclidean plane">Euclidean plane</a></div> <p>In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely;<sup id="cite_ref-EuclidAll_45-2" class="reference"><a href="#cite_note-EuclidAll-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a <a href="/wiki/Surface_(topology)" title="Surface (topology)">topological surface</a> without reference to distances or angles;<sup id="cite_ref-Munkres_51-0" class="reference"><a href="#cite_note-Munkres-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> it can be studied as an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, where collinearity and ratios can be studied but not distances;<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> it can be studied as the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> using techniques of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>;<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> and so on. </p> <div class="mw-heading mw-heading4"><h4 id="Curves">Curves</h4></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/Curve_(geometry)" class="mw-redirect" title="Curve (geometry)">Curve (geometry)</a></div> <p>A <a href="/wiki/Curve_(geometry)" class="mw-redirect" title="Curve (geometry)">curve</a> is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called <a href="/wiki/Plane_curve" title="Plane curve">plane curves</a> and those in 3-dimensional space are called <a href="/wiki/Space_curve" class="mw-redirect" title="Space curve">space curves</a>.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>In topology, a curve is defined by a function from an interval of the real numbers to another space.<sup id="cite_ref-Munkres_51-1" class="reference"><a href="#cite_note-Munkres-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> In differential geometry, the same definition is used, but the defining function is required to be differentiable.<sup id="cite_ref-Carmo_55-0" class="reference"><a href="#cite_note-Carmo-55"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> Algebraic geometry studies <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a>, which are defined as <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a> of <a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">dimension</a> one.<sup id="cite_ref-mumford_56-0" class="reference"><a href="#cite_note-mumford-56"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Surfaces">Surfaces</h4></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/Surface_(mathematics)" title="Surface (mathematics)">Surface (mathematics)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_wireframe.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Sphere_wireframe.svg/190px-Sphere_wireframe.svg.png" decoding="async" width="190" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Sphere_wireframe.svg/285px-Sphere_wireframe.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Sphere_wireframe.svg/380px-Sphere_wireframe.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>A sphere is a surface that can be defined parametrically (by <span class="nowrap"><i>x</i> = <i>r</i> sin <i>θ</i> cos <i>φ</i>,</span> <span class="nowrap"><i>y</i> = <i>r</i> sin <i>θ</i> sin <i>φ</i>,</span> <span class="nowrap"><i>z</i> = <i>r</i> cos <i>θ</i>)</span> or implicitly (by <span class="nowrap"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> − <i>r</i><sup>2</sup> = 0</span>).</figcaption></figure> <p>A <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> is a two-dimensional object, such as a sphere or paraboloid.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a><sup id="cite_ref-Carmo_55-1" class="reference"><a href="#cite_note-Carmo-55"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Topology" title="Topology">topology</a>,<sup id="cite_ref-Munkres_51-2" class="reference"><a href="#cite_note-Munkres-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> surfaces are described by two-dimensional 'patches' (or <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhoods</a>) that are assembled by <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> or <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>, respectively. In algebraic geometry, surfaces are described by <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a>.<sup id="cite_ref-mumford_56-1" class="reference"><a href="#cite_note-mumford-56"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Solids">Solids</h4></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/Solid_geometry" title="Solid geometry">Solid geometry</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Blue_ball.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/cf/Blue_ball.png" decoding="async" width="95" height="94" class="mw-file-element" data-file-width="95" data-file-height="94" /></a><figcaption>In <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, a ball is the volume bounded by a sphere.</figcaption></figure> <p>A <a href="/wiki/Solid_(mathematics)" class="mw-redirect" title="Solid (mathematics)">solid</a> is a three-dimensional object bounded by a closed surface; for example, a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a> is the volume bounded by a sphere. </p> <div class="mw-heading mw-heading4"><h4 id="Manifolds">Manifolds</h4></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/Manifold" title="Manifold">Manifold</a></div> <p>A <a href="/wiki/Manifold" title="Manifold">manifold</a> is a generalization of the concepts of curve and surface. In <a href="/wiki/Topology" title="Topology">topology</a>, a manifold is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> where every point has a <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> that is <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> to Euclidean space.<sup id="cite_ref-Munkres_51-3" class="reference"><a href="#cite_note-Munkres-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> is a space where each neighborhood is <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> to Euclidean space.<sup id="cite_ref-Carmo_55-2" class="reference"><a href="#cite_note-Carmo-55"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p><p>Manifolds are used extensively in physics, including in <a href="/wiki/General_relativity" title="General relativity">general relativity</a> and <a href="/wiki/String_theory" title="String theory">string theory</a>.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Angles">Angles</h3></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/Angle" title="Angle">Angle</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angle_obtuse_acute_straight.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/220px-Angle_obtuse_acute_straight.svg.png" decoding="async" width="220" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/330px-Angle_obtuse_acute_straight.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/440px-Angle_obtuse_acute_straight.svg.png 2x" data-file-width="800" data-file-height="445" /></a><figcaption>Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.</figcaption></figure> <p><a href="/wiki/Euclid" title="Euclid">Euclid</a> defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other.<sup id="cite_ref-EuclidAll_45-3" class="reference"><a href="#cite_note-EuclidAll-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> In modern terms, an angle is the figure formed by two <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">rays</a>, called the <i>sides</i> of the angle, sharing a common endpoint, called the <i><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a></i> of the angle.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> The size of an angle is formalized as an <a href="/wiki/Angular_measure" class="mw-redirect" title="Angular measure">angular measure</a>. </p><p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, angles are used to study <a href="/wiki/Polygon" title="Polygon">polygons</a> and <a href="/wiki/Triangle" title="Triangle">triangles</a>, as well as forming an object of study in their own right.<sup id="cite_ref-EuclidAll_45-4" class="reference"><a href="#cite_note-EuclidAll-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The study of the angles of a triangle or of angles in a <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> forms the basis of <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="/wiki/Calculus" title="Calculus">calculus</a>, the angles between <a href="/wiki/Plane_curve" title="Plane curve">plane curves</a> or <a href="/wiki/Space_curve" class="mw-redirect" title="Space curve">space curves</a> or <a href="/wiki/Surface_(geometry)" class="mw-redirect" title="Surface (geometry)">surfaces</a> can be calculated using the <a href="/wiki/Derivative_(calculus)" class="mw-redirect" title="Derivative (calculus)">derivative</a>.<sup id="cite_ref-Stewart_61-0" class="reference"><a href="#cite_note-Stewart-61"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Measures:_length,_area,_and_volume"><span id="Measures:_length.2C_area.2C_and_volume"></span>Measures: length, area, and volume<span class="anchor" id="Measures"></span></h3></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/Length" title="Length">Length</a>, <a href="/wiki/Area" title="Area">Area</a>, and <a href="/wiki/Volume" title="Volume">Volume</a></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/Area#List_of_formulas" title="Area">Area § List of formulas</a>, and <a href="/wiki/Volume#Volume_formulas" title="Volume">Volume § Volume formulas</a></div> <p><a href="/wiki/Length" title="Length">Length</a>, <a href="/wiki/Area" title="Area">area</a>, and <a href="/wiki/Volume" title="Volume">volume</a> describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.<sup id="cite_ref-Treese2018_63-0" class="reference"><a href="#cite_note-Treese2018-63"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> and <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, the length of a line segment can often be calculated by the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<sup id="cite_ref-Cannon2017_64-0" class="reference"><a href="#cite_note-Cannon2017-64"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> </p><p>Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space.<sup id="cite_ref-Treese2018_63-1" class="reference"><a href="#cite_note-Treese2018-63"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Mathematicians have found many explicit <a href="/wiki/Area#List_of_formulas" title="Area">formulas for area</a> and <a href="/wiki/Volume#Formulas" title="Volume">formulas for volume</a> of various geometric objects. In <a href="/wiki/Calculus" title="Calculus">calculus</a>, area and volume can be defined in terms of <a href="/wiki/Integral" title="Integral">integrals</a>, such as the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a><sup id="cite_ref-Strang1991_65-0" class="reference"><a href="#cite_note-Strang1991-65"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> or the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>.<sup id="cite_ref-Bear2002_66-0" class="reference"><a href="#cite_note-Bear2002-66"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p><p>Other geometrical measures include the <a href="/wiki/Curvature" title="Curvature">curvature</a> and <a href="/wiki/Compactness_measure" title="Compactness measure">compactness</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Metrics_and_measures">Metrics and measures</h4></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/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">Metric (mathematics)</a> and <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure (mathematics)</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Chinese_pythagoras.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Chinese_pythagoras.jpg/220px-Chinese_pythagoras.jpg" decoding="async" width="220" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Chinese_pythagoras.jpg/330px-Chinese_pythagoras.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Chinese_pythagoras.jpg/440px-Chinese_pythagoras.jpg 2x" data-file-width="871" data-file-height="475" /></a><figcaption>Visual checking of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> for the (3, 4, 5) <a href="/wiki/Triangle" title="Triangle">triangle</a> as in the <a href="/wiki/Zhoubi_Suanjing" title="Zhoubi Suanjing">Zhoubi Suanjing</a> 500–200 BC. The Pythagorean theorem is a consequence of the <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>.</figcaption></figure> <p>The concept of length or distance can be generalized, leading to the idea of <a href="/wiki/Metric_space" title="Metric space">metrics</a>.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> For instance, the <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a> measures the distance between points in the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, while the <a href="/wiki/Hyperbolic_metric" class="mw-redirect" title="Hyperbolic metric">hyperbolic metric</a> measures the distance in the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a>. Other important examples of metrics include the <a href="/wiki/Lorentz_metric" class="mw-redirect" title="Lorentz metric">Lorentz metric</a> of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> and the semi-<a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metrics</a> of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p><p>In a different direction, the concepts of length, area and volume are extended by <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, which studies methods of assigning a size or <i>measure</i> to <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, where the measures follow rules similar to those of classical area and volume.<sup id="cite_ref-Tao2011_69-0" class="reference"><a href="#cite_note-Tao2011-69"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Congruence_and_similarity">Congruence and similarity</h3></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/Congruence_(geometry)" title="Congruence (geometry)">Congruence (geometry)</a> and <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity (geometry)</a></div> <p><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a> and <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a> are concepts that describe when two shapes have similar characteristics.<sup id="cite_ref-Libeskind2008_70-0" class="reference"><a href="#cite_note-Libeskind2008-70"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape.<sup id="cite_ref-Freitag2013_71-0" class="reference"><a href="#cite_note-Freitag2013-71"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Hilbert" class="mw-redirect" title="Hilbert">Hilbert</a>, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by <a href="/wiki/Axiom" title="Axiom">axioms</a>. </p><p>Congruence and similarity are generalized in <a href="/wiki/Transformation_geometry" title="Transformation geometry">transformation geometry</a>, which studies the properties of geometric objects that are preserved by different kinds of transformations.<sup id="cite_ref-Martin2012_72-0" class="reference"><a href="#cite_note-Martin2012-72"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Compass_and_straightedge_constructions">Compass and straightedge constructions</h3></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/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">Compass and straightedge constructions</a></div> <p>Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the <a href="/wiki/Compass_(drafting)" class="mw-redirect" title="Compass (drafting)">compass</a> and <a href="/wiki/Ruler" title="Ruler">straightedge</a>.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using <a href="/wiki/Neusis_construction" title="Neusis construction">neusis</a>, parabolas and other curves, or mechanical devices, were found. </p> <div class="mw-heading mw-heading3"><h3 id="Rotation_and_orientation">Rotation and orientation</h3></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/Rotation_(geometry)" class="mw-redirect" title="Rotation (geometry)">Rotation (geometry)</a> and <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">Orientation (geometry)</a></div> <p>The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space. </p> <div class="mw-heading mw-heading3"><h3 id="Dimension">Dimension</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">Dimension (mathematics)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Von_Koch_curve.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Von_Koch_curve.gif/220px-Von_Koch_curve.gif" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fd/Von_Koch_curve.gif 1.5x" data-file-width="300" data-file-height="312" /></a><figcaption>The <a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a>, with <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a>=log4/log3 and <a href="/wiki/Topological_dimension" class="mw-redirect" title="Topological dimension">topological dimension</a>=1</figcaption></figure> <p>Traditional geometry allowed dimensions 1 (a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> or curve), 2 (a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> or surface), and 3 (our ambient world conceived of as <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>). Furthermore, mathematicians and physicists have used <a href="/wiki/Higher_dimension" class="mw-redirect" title="Higher dimension">higher dimensions</a> for nearly two centuries.<sup id="cite_ref-Blacklock2018_74-0" class="reference"><a href="#cite_note-Blacklock2018-74"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> One example of a mathematical use for higher dimensions is the <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> of a physical system, which has a dimension equal to the system's <a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a>. For instance, the configuration of a screw can be described by five coordinates.<sup id="cite_ref-Joly1895_75-0" class="reference"><a href="#cite_note-Joly1895-75"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/General_topology" title="General topology">general topology</a>, the concept of dimension has been extended from <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, to infinite dimension (<a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, for example) and positive <a href="/wiki/Real_number" title="Real number">real numbers</a> (in <a href="/wiki/Fractal_geometry" class="mw-redirect" title="Fractal geometry">fractal geometry</a>).<sup id="cite_ref-Temam2013_76-0" class="reference"><a href="#cite_note-Temam2013-76"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the <a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">dimension of an algebraic variety</a> has received a number of apparently different definitions, which are all equivalent in the most common cases.<sup id="cite_ref-JacobLam1994_77-0" class="reference"><a href="#cite_note-JacobLam1994-77"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3></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/Symmetry" title="Symmetry">Symmetry</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Order-3_heptakis_heptagonal_tiling.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Order-3_heptakis_heptagonal_tiling.png/220px-Order-3_heptakis_heptagonal_tiling.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Order-3_heptakis_heptagonal_tiling.png/330px-Order-3_heptakis_heptagonal_tiling.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Order-3_heptakis_heptagonal_tiling.png/440px-Order-3_heptakis_heptagonal_tiling.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>A <a href="/wiki/Order-3_bisected_heptagonal_tiling" class="mw-redirect" title="Order-3 bisected heptagonal tiling">tiling</a> of the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic plane</a></figcaption></figure> <p>The theme of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> in geometry is nearly as old as the science of geometry itself.<sup id="cite_ref-Stewart2008_78-0" class="reference"><a href="#cite_note-Stewart2008-78"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> Symmetric shapes such as the <a href="/wiki/Circle" title="Circle">circle</a>, <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> and <a href="/wiki/Platonic_solid" title="Platonic solid">platonic solids</a> held deep significance for many ancient philosophers<sup id="cite_ref-Alexey2009_79-0" class="reference"><a href="#cite_note-Alexey2009-79"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> and were investigated in detail before the time of Euclid.<sup id="cite_ref-Hartshorne2013_41-1" class="reference"><a href="#cite_note-Hartshorne2013-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a>, <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>, and others.<sup id="cite_ref-Hahn1998_80-0" class="reference"><a href="#cite_note-Hahn1998-80"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>'s <a href="/wiki/Erlangen_program" title="Erlangen program">Erlangen program</a> proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, determines what geometry <i>is</i>.<sup id="cite_ref-Cantwell2002_81-0" class="reference"><a href="#cite_note-Cantwell2002-81"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> Symmetry in classical <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is represented by <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruences</a> and rigid motions, whereas in <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> an analogous role is played by <a href="/wiki/Collineation" title="Collineation">collineations</a>, <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformations</a> that take straight lines into straight lines.<sup id="cite_ref-RosenfeldWiebe2013_82-0" class="reference"><a href="#cite_note-RosenfeldWiebe2013-82"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> However it was in the new geometries of Bolyai and Lobachevsky, Riemann, <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">Clifford</a> and Klein, and <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> that Klein's idea to 'define a geometry via its <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a>' found its inspiration.<sup id="cite_ref-Pesic2007_83-0" class="reference"><a href="#cite_note-Pesic2007-83"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> Both discrete and continuous symmetries play prominent roles in geometry, the former in <a href="/wiki/Topology" title="Topology">topology</a> and <a href="/wiki/Geometric_group_theory" title="Geometric group theory">geometric group theory</a>,<sup id="cite_ref-Kaku2012_84-0" class="reference"><a href="#cite_note-Kaku2012-84"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BestvinaSageev2014_85-0" class="reference"><a href="#cite_note-BestvinaSageev2014-85"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> the latter in <a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a> and <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>.<sup id="cite_ref-Steeb1996_86-0" class="reference"><a href="#cite_note-Steeb1996-86"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Misner2005_87-0" class="reference"><a href="#cite_note-Misner2005-87"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> </p><p>A different type of symmetry is the principle of <a href="/wiki/Duality_(projective_geometry)" title="Duality (projective geometry)">duality</a> in <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, among other fields. This meta-phenomenon can roughly be described as follows: in any <a href="/wiki/Theorem" title="Theorem">theorem</a>, exchange <i>point</i> with <i>plane</i>, <i>join</i> with <i>meet</i>, <i>lies in</i> with <i>contains</i>, and the result is an equally true theorem.<sup id="cite_ref-Dowling1917_88-0" class="reference"><a href="#cite_note-Dowling1917-88"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> A similar and closely related form of duality exists between a <a href="/wiki/Vector_space" title="Vector space">vector space</a> and its <a href="/wiki/Dual_space" title="Dual space">dual space</a>.<sup id="cite_ref-Gierz2006_89-0" class="reference"><a href="#cite_note-Gierz2006-89"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Contemporary_geometry">Contemporary geometry</h2></div> <div class="mw-heading mw-heading3"><h3 id="Euclidean_geometry">Euclidean geometry</h3></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/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></div> <p><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is geometry in its classical sense.<sup id="cite_ref-ButtsBrown2012_90-0" class="reference"><a href="#cite_note-ButtsBrown2012-90"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> As it models the space of the physical world, it is used in many scientific areas, such as <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>,<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> and many technical fields, such as <a href="/wiki/Engineering" title="Engineering">engineering</a>,<sup id="cite_ref-Abbot2013_92-0" class="reference"><a href="#cite_note-Abbot2013-92"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Architecture" title="Architecture">architecture</a>,<sup id="cite_ref-HerseyHersey2001_93-0" class="reference"><a href="#cite_note-HerseyHersey2001-93"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Geodesy" title="Geodesy">geodesy</a>,<sup id="cite_ref-VanícekKrakiwsky2015_94-0" class="reference"><a href="#cite_note-VanícekKrakiwsky2015-94"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Aerodynamics" title="Aerodynamics">aerodynamics</a>,<sup id="cite_ref-CummingsMorton2015_95-0" class="reference"><a href="#cite_note-CummingsMorton2015-95"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Navigation" title="Navigation">navigation</a>.<sup id="cite_ref-Williams1998_96-0" class="reference"><a href="#cite_note-Williams1998-96"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planes</a>, <a href="/wiki/Angle" title="Angle">angles</a>, <a href="/wiki/Triangle" title="Triangle">triangles</a>, <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a>, <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a>, <a href="/wiki/Solid_figure" class="mw-redirect" title="Solid figure">solid figures</a>, <a href="/wiki/Circle" title="Circle">circles</a>, and <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>.<sup id="cite_ref-Schmidt,_W._2002_97-0" class="reference"><a href="#cite_note-Schmidt,_W._2002-97"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Euclidean_vectors">Euclidean vectors</h4></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/Euclidean_vector" title="Euclidean vector">Euclidean vector</a></div> <p>Euclidean vectors are used for a myriad of applications in physics and engineering, such as <a href="/wiki/Position_(geometry)" title="Position (geometry)">position</a>, <a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">displacement</a>, <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">deformation</a>, <a href="/wiki/Velocity" title="Velocity">velocity</a>, <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>, <a href="/wiki/Force" title="Force">force</a>, etc. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_geometry">Differential geometry</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hyperbolic_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/220px-Hyperbolic_triangle.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/330px-Hyperbolic_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/440px-Hyperbolic_triangle.svg.png 2x" data-file-width="809" data-file-height="559" /></a><figcaption><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> uses tools from <a href="/wiki/Calculus" title="Calculus">calculus</a> to study problems involving curvature.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></div> <p><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> uses techniques of <a href="/wiki/Calculus" title="Calculus">calculus</a> and <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> to study problems in geometry.<sup id="cite_ref-Walschap2015_98-0" class="reference"><a href="#cite_note-Walschap2015-98"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> It has applications in <a href="/wiki/Physics" title="Physics">physics</a>,<sup id="cite_ref-Flanders2012_99-0" class="reference"><a href="#cite_note-Flanders2012-99"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>,<sup id="cite_ref-MarriottSalmon2000_100-0" class="reference"><a href="#cite_note-MarriottSalmon2000-100"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>,<sup id="cite_ref-HePetoukhov2011_101-0" class="reference"><a href="#cite_note-HePetoukhov2011-101"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> among others. </p><p>In particular, differential geometry is of importance to <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> due to <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="/wiki/General_relativity" title="General relativity">general relativity</a> postulation that the <a href="/wiki/Universe" title="Universe">universe</a> is <a href="/wiki/Curvature" title="Curvature">curved</a>.<sup id="cite_ref-Dirac2016_102-0" class="reference"><a href="#cite_note-Dirac2016-102"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> Differential geometry can either be <i>intrinsic</i> (meaning that the spaces it considers are <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a> whose geometric structure is governed by a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a>, which determines how distances are measured near each point) or <i>extrinsic</i> (where the object under study is a part of some ambient flat Euclidean space).<sup id="cite_ref-AyJost2017_103-0" class="reference"><a href="#cite_note-AyJost2017-103"><span class="cite-bracket">[</span>100<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Non-Euclidean_geometry">Non-Euclidean geometry</h4></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/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</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=Non-Euclidean_geometry&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Noneuclid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/400px-Noneuclid.svg.png" decoding="async" width="400" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/600px-Noneuclid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/800px-Noneuclid.svg.png 2x" data-file-width="663" data-file-height="167" /></a><figcaption><div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Behavior of lines with a common perpendicular in each of the three types of geometry</div></figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> consists of two geometries based on <a href="/wiki/Axiom" title="Axiom">axioms</a> closely related to those that specify <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. As Euclidean geometry lies at the intersection of <a href="/wiki/Metric_geometry" class="mw-redirect" title="Metric geometry">metric geometry</a> and <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>, non-Euclidean geometry arises by either replacing the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> with an alternative, or relaxing the metric requirement. In the former case, one obtains <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> and <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the <a href="#Planar_algebras">planar algebras</a>, which give rise to <a href="#Kinematic_geometries">kinematic geometries</a> that have also been called non-Euclidean geometry.</div></div> <div class="mw-heading mw-heading3"><h3 id="Topology">Topology</h3></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/Topology" title="Topology">Topology</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Trefoil_knot_arb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Trefoil_knot_arb.png/220px-Trefoil_knot_arb.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Trefoil_knot_arb.png/330px-Trefoil_knot_arb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Trefoil_knot_arb.png/440px-Trefoil_knot_arb.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>A thickening of the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a></figcaption></figure> <p>Topology is the field concerned with the properties of <a href="/wiki/Continuous_mapping" class="mw-redirect" title="Continuous mapping">continuous mappings</a>,<sup id="cite_ref-Crossley2011_104-0" class="reference"><a href="#cite_note-Crossley2011-104"><span class="cite-bracket">[</span>101<span class="cite-bracket">]</span></a></sup> and can be considered a generalization of Euclidean geometry.<sup id="cite_ref-NashSen1988_105-0" class="reference"><a href="#cite_note-NashSen1988-105"><span class="cite-bracket">[</span>102<span class="cite-bracket">]</span></a></sup> In practice, topology often means dealing with large-scale properties of spaces, such as <a href="/wiki/Connectedness" title="Connectedness">connectedness</a> and <a href="/wiki/Compact_(topology)" class="mw-redirect" title="Compact (topology)">compactness</a>.<sup id="cite_ref-Munkres_51-4" class="reference"><a href="#cite_note-Munkres-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>The field of topology, which saw massive development in the 20th century, is in a technical sense a type of <a href="/wiki/Transformation_geometry" title="Transformation geometry">transformation geometry</a>, in which transformations are <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>.<sup id="cite_ref-Martin1996_106-0" class="reference"><a href="#cite_note-Martin1996-106"><span class="cite-bracket">[</span>103<span class="cite-bracket">]</span></a></sup> This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a>, <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> and <a href="/wiki/General_topology" title="General topology">general topology</a>.<sup id="cite_ref-May1999_107-0" class="reference"><a href="#cite_note-May1999-107"><span class="cite-bracket">[</span>104<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_geometry">Algebraic geometry</h3></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/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Calabi_yau.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/220px-Calabi_yau.jpg" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/330px-Calabi_yau.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/440px-Calabi_yau.jpg 2x" data-file-width="1432" data-file-height="1493" /></a><figcaption>Quintic <a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau threefold</a></figcaption></figure> <p>Algebraic geometry is fundamentally the study by means of <a href="/wiki/Algebra" title="Algebra">algebraic</a> methods of some geometrical shapes, called <a href="/wiki/Algebraic_set" class="mw-redirect" title="Algebraic set">algebraic sets</a>, and defined as common <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of <a href="/wiki/Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">multivariate polynomials</a>.<sup id="cite_ref-AHartshorne2013_108-0" class="reference"><a href="#cite_note-AHartshorne2013-108"><span class="cite-bracket">[</span>105<span class="cite-bracket">]</span></a></sup> Algebraic geometry became an autonomous subfield of geometry <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1900</span>, with a theorem called <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a> that establishes a strong correspondence between algebraic sets and <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> of <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a>. This led to a parallel development of algebraic geometry, and its algebraic counterpart, called <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>.<sup id="cite_ref-Dieudonne1985_109-0" class="reference"><a href="#cite_note-Dieudonne1985-109"><span class="cite-bracket">[</span>106<span class="cite-bracket">]</span></a></sup> From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> of <a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a>, which allows using <a href="/wiki/Algebraic_topology" title="Algebraic topology">topological methods</a>, including <a href="/wiki/Cohomology_theory" class="mw-redirect" title="Cohomology theory">cohomology theories</a> in a purely algebraic context.<sup id="cite_ref-Dieudonne1985_109-1" class="reference"><a href="#cite_note-Dieudonne1985-109"><span class="cite-bracket">[</span>106<span class="cite-bracket">]</span></a></sup> Scheme theory allowed to solve many difficult problems not only in geometry, but also in <a href="/wiki/Number_theory" title="Number theory">number theory</a>. <a href="/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem" class="mw-redirect" title="Wiles' proof of Fermat's Last Theorem">Wiles' proof of Fermat's Last Theorem</a> is a famous example of a long-standing problem of <a href="/wiki/Number_theory" title="Number theory">number theory</a> whose solution uses scheme theory and its extensions such as <a href="/wiki/Stack_(mathematics)" title="Stack (mathematics)">stack theory</a>. One of seven <a href="/wiki/Millennium_Prize_problems" class="mw-redirect" title="Millennium Prize problems">Millennium Prize problems</a>, the <a href="/wiki/Hodge_conjecture" title="Hodge conjecture">Hodge conjecture</a>, is a question in algebraic geometry.<sup id="cite_ref-CarlsonCarlson2006_110-0" class="reference"><a href="#cite_note-CarlsonCarlson2006-110"><span class="cite-bracket">[</span>107<span class="cite-bracket">]</span></a></sup> </p><p>Algebraic geometry has applications in many areas, including <a href="/wiki/Cryptography" title="Cryptography">cryptography</a><sup id="cite_ref-HoweLauter2017_111-0" class="reference"><a href="#cite_note-HoweLauter2017-111"><span class="cite-bracket">[</span>108<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/String_theory" title="String theory">string theory</a>.<sup id="cite_ref-MarinoThaddeus2008_112-0" class="reference"><a href="#cite_note-MarinoThaddeus2008-112"><span class="cite-bracket">[</span>109<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Complex_geometry">Complex geometry</h3></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/Complex_geometry" title="Complex geometry">Complex geometry</a></div> <p><a href="/wiki/Complex_geometry" title="Complex geometry">Complex geometry</a> studies the nature of geometric structures modelled on, or arising out of, the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>.<sup id="cite_ref-113" class="reference"><a href="#cite_note-113"><span class="cite-bracket">[</span>110<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-114" class="reference"><a href="#cite_note-114"><span class="cite-bracket">[</span>111<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-115" class="reference"><a href="#cite_note-115"><span class="cite-bracket">[</span>112<span class="cite-bracket">]</span></a></sup> Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of <a href="/wiki/Several_complex_variables" class="mw-redirect" title="Several complex variables">several complex variables</a>, and has found applications to <a href="/wiki/String_theory" title="String theory">string theory</a> and <a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">mirror symmetry</a>.<sup id="cite_ref-116" class="reference"><a href="#cite_note-116"><span class="cite-bracket">[</span>113<span class="cite-bracket">]</span></a></sup> </p><p>Complex geometry first appeared as a distinct area of study in the work of <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> in his study of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>.<sup id="cite_ref-117" class="reference"><a href="#cite_note-117"><span class="cite-bracket">[</span>114<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-118" class="reference"><a href="#cite_note-118"><span class="cite-bracket">[</span>115<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-119" class="reference"><a href="#cite_note-119"><span class="cite-bracket">[</span>116<span class="cite-bracket">]</span></a></sup> Work in the spirit of Riemann was carried out by the <a href="/wiki/Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry</a> in the early 1900s. Contemporary treatment of complex geometry began with the work of <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a>, who introduced the concept of <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> to the subject, and illuminated the relations between complex geometry and algebraic geometry.<sup id="cite_ref-120" class="reference"><a href="#cite_note-120"><span class="cite-bracket">[</span>117<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-121" class="reference"><a href="#cite_note-121"><span class="cite-bracket">[</span>118<span class="cite-bracket">]</span></a></sup> The primary objects of study in complex geometry are <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a>, <a href="/wiki/Complex_algebraic_varieties" class="mw-redirect" title="Complex algebraic varieties">complex algebraic varieties</a>, and <a href="/wiki/Complex_analytic_varieties" class="mw-redirect" title="Complex analytic varieties">complex analytic varieties</a>, and <a href="/wiki/Holomorphic_vector_bundles" class="mw-redirect" title="Holomorphic vector bundles">holomorphic vector bundles</a> and <a href="/wiki/Coherent_sheaves" class="mw-redirect" title="Coherent sheaves">coherent sheaves</a> over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and <a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifolds</a>, and these spaces find uses in string theory. In particular, <a href="/wiki/Worldsheet" title="Worldsheet">worldsheets</a> of strings are modelled by Riemann surfaces, and <a href="/wiki/Superstring_theory" title="Superstring theory">superstring theory</a> predicts that the extra 6 dimensions of 10 dimensional <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> may be modelled by Calabi–Yau manifolds. </p> <div class="mw-heading mw-heading3"><h3 id="Discrete_geometry">Discrete geometry</h3></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/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Closepacking.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Closepacking.svg/220px-Closepacking.svg.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Closepacking.svg/330px-Closepacking.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Closepacking.svg/440px-Closepacking.svg.png 2x" data-file-width="703" data-file-height="534" /></a><figcaption>Discrete geometry includes the study of various <a href="/wiki/Sphere_packing" title="Sphere packing">sphere packings</a>.</figcaption></figure> <p><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a> is a subject that has close connections with <a href="/wiki/Convex_geometry" title="Convex geometry">convex geometry</a>.<sup id="cite_ref-Matoušek2013_122-0" class="reference"><a href="#cite_note-Matoušek2013-122"><span class="cite-bracket">[</span>119<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Zong2006_123-0" class="reference"><a href="#cite_note-Zong2006-123"><span class="cite-bracket">[</span>120<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gruber2007_124-0" class="reference"><a href="#cite_note-Gruber2007-124"><span class="cite-bracket">[</span>121<span class="cite-bracket">]</span></a></sup> It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of <a href="/wiki/Sphere_packing" title="Sphere packing">sphere packings</a>, <a href="/wiki/Triangulation_(geometry)" title="Triangulation (geometry)">triangulations</a>, the Kneser-Poulsen conjecture, etc.<sup id="cite_ref-DevadossO'Rourke2011_125-0" class="reference"><a href="#cite_note-DevadossO'Rourke2011-125"><span class="cite-bracket">[</span>122<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bezdek2010_126-0" class="reference"><a href="#cite_note-Bezdek2010-126"><span class="cite-bracket">[</span>123<span class="cite-bracket">]</span></a></sup> It shares many methods and principles with <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Computational_geometry">Computational geometry</h3></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/Computational_geometry" title="Computational geometry">Computational geometry</a></div> <p><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a> deals with <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> and their <a href="/wiki/Implementation_(computer_science)" class="mw-redirect" title="Implementation (computer science)">implementations</a> for manipulating geometrical objects. Important problems historically have included the <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a>, <a href="/wiki/Minimum_spanning_tree" title="Minimum spanning tree">minimum spanning trees</a>, <a href="/wiki/Hidden-line_removal" title="Hidden-line removal">hidden-line removal</a>, and <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a>.<sup id="cite_ref-PreparataShamos2012_127-0" class="reference"><a href="#cite_note-PreparataShamos2012-127"><span class="cite-bracket">[</span>124<span class="cite-bracket">]</span></a></sup> </p><p>Although being a young area of geometry, it has many applications in <a href="/wiki/Computer_vision" title="Computer vision">computer vision</a>, <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, <a href="/wiki/Computer-aided_design" title="Computer-aided design">computer-aided design</a>, <a href="/wiki/Medical_imaging" title="Medical imaging">medical imaging</a>, etc.<sup id="cite_ref-GuYau2008_128-0" class="reference"><a href="#cite_note-GuYau2008-128"><span class="cite-bracket">[</span>125<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_group_theory">Geometric group theory</h3></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/Geometric_group_theory" title="Geometric group theory">Geometric group theory</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cayley_graph_of_F2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cayley_graph_of_F2.svg/220px-Cayley_graph_of_F2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cayley_graph_of_F2.svg/330px-Cayley_graph_of_F2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cayley_graph_of_F2.svg/440px-Cayley_graph_of_F2.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>The Cayley graph of the <a href="/wiki/Free_group" title="Free group">free group</a> on two generators <i>a</i> and <i>b</i></figcaption></figure> <p><a href="/wiki/Geometric_group_theory" title="Geometric group theory">Geometric group theory</a> uses large-scale geometric techniques to study <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated groups</a>.<sup id="cite_ref-Löh2017_129-0" class="reference"><a href="#cite_note-Löh2017-129"><span class="cite-bracket">[</span>126<span class="cite-bracket">]</span></a></sup> It is closely connected to <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional topology</a>, such as in <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>'s proof of the <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">Geometrization conjecture</a>, which included the proof of the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>, a <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problem</a>.<sup id="cite_ref-MorganTian2014_130-0" class="reference"><a href="#cite_note-MorganTian2014-130"><span class="cite-bracket">[</span>127<span class="cite-bracket">]</span></a></sup> </p><p>Geometric group theory often revolves around the <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a>, which is a geometric representation of a group. Other important topics include <a href="/wiki/Quasi-isometry" title="Quasi-isometry">quasi-isometries</a>, <a href="/wiki/Gromov-hyperbolic_group" class="mw-redirect" title="Gromov-hyperbolic group">Gromov-hyperbolic groups</a>, and <a href="/wiki/Right_angled_Artin_group" class="mw-redirect" title="Right angled Artin group">right angled Artin groups</a>.<sup id="cite_ref-Löh2017_129-1" class="reference"><a href="#cite_note-Löh2017-129"><span class="cite-bracket">[</span>126<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wise2012_131-0" class="reference"><a href="#cite_note-Wise2012-131"><span class="cite-bracket">[</span>128<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Convex_geometry">Convex geometry</h3></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/Convex_geometry" title="Convex geometry">Convex geometry</a></div> <p><a href="/wiki/Convex_geometry" title="Convex geometry">Convex geometry</a> investigates <a href="/wiki/Convex_set" title="Convex set">convex</a> shapes in the Euclidean space and its more abstract analogues, often using techniques of <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> and <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a>.<sup id="cite_ref-Meurant2014_132-0" class="reference"><a href="#cite_note-Meurant2014-132"><span class="cite-bracket">[</span>129<span class="cite-bracket">]</span></a></sup> It has close connections to <a href="/wiki/Convex_analysis" title="Convex analysis">convex analysis</a>, <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimization</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> and important applications in <a href="/wiki/Number_theory" title="Number theory">number theory</a>. </p><p>Convex geometry dates back to antiquity.<sup id="cite_ref-Meurant2014_132-1" class="reference"><a href="#cite_note-Meurant2014-132"><span class="cite-bracket">[</span>129<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> gave the first known precise definition of convexity. The <a href="/wiki/Isoperimetric_problem" class="mw-redirect" title="Isoperimetric problem">isoperimetric problem</a>, a recurring concept in convex geometry, was studied by the Greeks as well, including <a href="/wiki/Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a>. Archimedes, <a href="/wiki/Plato" title="Plato">Plato</a>, <a href="/wiki/Euclid" title="Euclid">Euclid</a>, and later <a href="/wiki/Kepler" class="mw-redirect" title="Kepler">Kepler</a> and <a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a> all studied <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytopes</a> and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>, <a href="/wiki/Algorithms" class="mw-redirect" title="Algorithms">algorithms</a>, <a href="/wiki/Tiling_(geometry)" class="mw-redirect" title="Tiling (geometry)">tilings</a> and <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattices</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div> <p>Geometry has found applications in many fields, some of which are described below. </p> <div class="mw-heading mw-heading3"><h3 id="Art">Art</h3></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/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fes_Medersa_Bou_Inania_Mosaique2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Fes_Medersa_Bou_Inania_Mosaique2.jpg/220px-Fes_Medersa_Bou_Inania_Mosaique2.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Fes_Medersa_Bou_Inania_Mosaique2.jpg/330px-Fes_Medersa_Bou_Inania_Mosaique2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Fes_Medersa_Bou_Inania_Mosaique2.jpg/440px-Fes_Medersa_Bou_Inania_Mosaique2.jpg 2x" data-file-width="1536" data-file-height="2048" /></a><figcaption>Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations</figcaption></figure> <p>Mathematics and art are related in a variety of ways. For instance, the theory of <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a> showed that there is more to geometry than just the metric properties of figures: perspective is the origin of <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>.<sup id="cite_ref-Richter-Gebert2011_133-0" class="reference"><a href="#cite_note-Richter-Gebert2011-133"><span class="cite-bracket">[</span>130<span class="cite-bracket">]</span></a></sup> </p><p>Artists have long used concepts of <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportion</a> in design. <a href="/wiki/Vitruvius" title="Vitruvius">Vitruvius</a> developed a complicated theory of <i>ideal proportions</i> for the human figure.<sup id="cite_ref-Elam2001_134-0" class="reference"><a href="#cite_note-Elam2001-134"><span class="cite-bracket">[</span>131<span class="cite-bracket">]</span></a></sup> These concepts have been used and adapted by artists from <a href="/wiki/Michelangelo" title="Michelangelo">Michelangelo</a> to modern comic book artists.<sup id="cite_ref-Guigar2004_135-0" class="reference"><a href="#cite_note-Guigar2004-135"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.<sup id="cite_ref-Livio2008_136-0" class="reference"><a href="#cite_note-Livio2008-136"><span class="cite-bracket">[</span>133<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Tiling_(geometry)" class="mw-redirect" title="Tiling (geometry)">Tilings</a>, or tessellations, have been used in art throughout history. <a href="/wiki/Islamic_art" title="Islamic art">Islamic art</a> makes frequent use of tessellations, as did the art of <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>.<sup id="cite_ref-EmmerSchattschneider2007_137-0" class="reference"><a href="#cite_note-EmmerSchattschneider2007-137"><span class="cite-bracket">[</span>134<span class="cite-bracket">]</span></a></sup> Escher's work also made use of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>. </p><p><a href="/wiki/C%C3%A9zanne" class="mw-redirect" title="Cézanne">Cézanne</a> advanced the theory that all images can be built up from the <a href="/wiki/Sphere" title="Sphere">sphere</a>, the <a href="/wiki/Cone" title="Cone">cone</a>, and the <a href="/wiki/Cylinder" title="Cylinder">cylinder</a>. This is still used in art theory today, although the exact list of shapes varies from author to author.<sup id="cite_ref-CapitoloSchwab2004_138-0" class="reference"><a href="#cite_note-CapitoloSchwab2004-138"><span class="cite-bracket">[</span>135<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gelineau2011_139-0" class="reference"><a href="#cite_note-Gelineau2011-139"><span class="cite-bracket">[</span>136<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Architecture">Architecture</h3></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/Mathematics_and_architecture" title="Mathematics and architecture">Mathematics and architecture</a> and <a href="/wiki/Architectural_geometry" title="Architectural geometry">Architectural geometry</a></div> <p>Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.<sup id="cite_ref-CeccatoHesselgren2016_140-0" class="reference"><a href="#cite_note-CeccatoHesselgren2016-140"><span class="cite-bracket">[</span>137<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pottmann2007_141-0" class="reference"><a href="#cite_note-Pottmann2007-141"><span class="cite-bracket">[</span>138<span class="cite-bracket">]</span></a></sup> Applications of geometry to architecture include the use of <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> to create <a href="/wiki/Forced_perspective" title="Forced perspective">forced perspective</a>,<sup id="cite_ref-MoffettFazio2003_142-0" class="reference"><a href="#cite_note-MoffettFazio2003-142"><span class="cite-bracket">[</span>139<span class="cite-bracket">]</span></a></sup> the use of <a href="/wiki/Conic_section" title="Conic section">conic sections</a> in constructing domes and similar objects,<sup id="cite_ref-HerseyHersey2001_93-1" class="reference"><a href="#cite_note-HerseyHersey2001-93"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> the use of <a href="/wiki/Tessellations" class="mw-redirect" title="Tessellations">tessellations</a>,<sup id="cite_ref-HerseyHersey2001_93-2" class="reference"><a href="#cite_note-HerseyHersey2001-93"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> and the use of symmetry.<sup id="cite_ref-HerseyHersey2001_93-3" class="reference"><a href="#cite_note-HerseyHersey2001-93"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3></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/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></div> <p>The field of <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, especially as it relates to mapping the positions of <a href="/wiki/Star" title="Star">stars</a> and <a href="/wiki/Planet" title="Planet">planets</a> on the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.<sup id="cite_ref-GreenGreen1985_143-0" class="reference"><a href="#cite_note-GreenGreen1985-143"><span class="cite-bracket">[</span>140<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a> and <a href="/wiki/Pseudo-Riemannian" class="mw-redirect" title="Pseudo-Riemannian">pseudo-Riemannian</a> geometry are used in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>.<sup id="cite_ref-Alekseevskiĭ2008_144-0" class="reference"><a href="#cite_note-Alekseevskiĭ2008-144"><span class="cite-bracket">[</span>141<span class="cite-bracket">]</span></a></sup> <a href="/wiki/String_theory" title="String theory">String theory</a> makes use of several variants of geometry,<sup id="cite_ref-YauNadis2010_145-0" class="reference"><a href="#cite_note-YauNadis2010-145"><span class="cite-bracket">[</span>142<span class="cite-bracket">]</span></a></sup> as does <a href="/wiki/Quantum_information_theory" class="mw-redirect" title="Quantum information theory">quantum information theory</a>.<sup id="cite_ref-146" class="reference"><a href="#cite_note-146"><span class="cite-bracket">[</span>143<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_fields_of_mathematics">Other fields of mathematics</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Square_root_of_2_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/220px-Square_root_of_2_triangle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/330px-Square_root_of_2_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/440px-Square_root_of_2_triangle.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>The Pythagoreans discovered that the sides of a triangle could have <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurable</a> lengths.</figcaption></figure> <p><a href="/wiki/Calculus" title="Calculus">Calculus</a> was strongly influenced by geometry.<sup id="cite_ref-Boyer2012_32-1" class="reference"><a href="#cite_note-Boyer2012-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> For instance, the introduction of <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> and the concurrent developments of <a href="/wiki/Algebra" title="Algebra">algebra</a> marked a new stage for geometry, since geometric figures such as <a href="/wiki/Plane_curve" title="Plane curve">plane curves</a> could now be represented <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytically</a> in the form of functions and equations. This played a key role in the emergence of <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a> in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.<sup id="cite_ref-FlandersPrice2014_147-0" class="reference"><a href="#cite_note-FlandersPrice2014-147"><span class="cite-bracket">[</span>144<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-RogawskiAdams2015_148-0" class="reference"><a href="#cite_note-RogawskiAdams2015-148"><span class="cite-bracket">[</span>145<span class="cite-bracket">]</span></a></sup> </p><p>Another important area of application is <a href="/wiki/Number_theory" title="Number theory">number theory</a>.<sup id="cite_ref-Lozano-Robledo2019_149-0" class="reference"><a href="#cite_note-Lozano-Robledo2019-149"><span class="cite-bracket">[</span>146<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greece</a> the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a> considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.<sup id="cite_ref-Sangalli2009_150-0" class="reference"><a href="#cite_note-Sangalli2009-150"><span class="cite-bracket">[</span>147<span class="cite-bracket">]</span></a></sup> Since the 19th century, geometry has been used for solving problems in number theory, for example through the <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">geometry of numbers</a> or, more recently, <a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a>, which is used in <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof of Fermat's Last Theorem</a>.<sup id="cite_ref-CornellSilverman2013_151-0" class="reference"><a href="#cite_note-CornellSilverman2013-151"><span class="cite-bracket">[</span>148<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main category: <a href="/wiki/Category:Geometry" title="Category:Geometry">Geometry</a></div> <dl><dt>Lists</dt></dl> <ul><li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a> <ul><li><a href="/wiki/Category:Algebraic_geometers" title="Category:Algebraic geometers">Category:Algebraic geometers</a></li> <li><a href="/wiki/Category:Differential_geometers" title="Category:Differential geometers">Category:Differential geometers</a></li> <li><a href="/wiki/Category:Geometers" title="Category:Geometers">Category:Geometers</a></li> <li><a href="/wiki/Category:Topologists" title="Category:Topologists">Category:Topologists</a></li></ul></li> <li><a href="/wiki/List_of_formulas_in_elementary_geometry" title="List of formulas in elementary geometry">List of formulas in elementary geometry</a></li> <li><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">List of geometry topics</a></li> <li><a href="/wiki/List_of_important_publications_in_mathematics#Geometry" title="List of important publications in mathematics">List of important publications in geometry</a></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists of mathematics topics</a></li></ul> <dl><dt>Related topics</dt></dl> <ul><li><a href="/wiki/Descriptive_geometry" title="Descriptive geometry">Descriptive geometry</a></li> <li><i><a href="/wiki/Flatland" title="Flatland">Flatland</a></i>, a book written by <a href="/wiki/Edwin_Abbott_Abbott" title="Edwin Abbott Abbott">Edwin Abbott Abbott</a> about two- and <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>, to understand the concept of four dimensions</li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">List of interactive geometry software</a></li></ul> <dl><dt>Other applications</dt></dl> <ul><li><a href="/wiki/Molecular_geometry" title="Molecular geometry">Molecular geometry</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. In the 19th century and later, this was challenged by the development of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> by <a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a> and other <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a> by <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a> and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of <a href="/wiki/Desargues" class="mw-redirect" title="Desargues">Desargues</a> in the 17th century, all the way back to the implicit use of <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a> to understand the <a href="/wiki/Geodesy" title="Geodesy">Earth geodesy</a> and to navigate the oceans since antiquity.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Pythagorean triples are triples of integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}"></span> with the property: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span>. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}+4^{2}=5^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}+4^{2}=5^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53caa1a580c8f5b70212931e47fc229315e7b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.589ex; height:2.843ex;" alt="{\displaystyle 3^{2}+4^{2}=5^{2}}"></span>, <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 8^{2}+15^{2}=17^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>17</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{2}+15^{2}=17^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfbfc4b1eb1d33551df39e0c92c1123ccc026b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.914ex; height:2.843ex;" alt="{\displaystyle 8^{2}+15^{2}=17^{2}}"></span>, <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 12^{2}+35^{2}=37^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>35</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>37</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2}+35^{2}=37^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93495f13efb876cfbc2cbaaf7548e9a701f05c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.076ex; height:2.843ex;" alt="{\displaystyle 12^{2}+35^{2}=37^{2}}"></span> etc.</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text">The ancient Greeks had some constructions using other instruments.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><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://cuemath.com/geometry">"Geometry - Formulas, Examples | Plane and Solid Geometry"</a>. <i>Cuemath</i><span class="reference-accessdate">. Retrieved <span class="nowrap">31 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Cuemath&rft.atitle=Geometry+-+Formulas%2C+Examples+%7C+Plane+and+Solid+Geometry&rft_id=https%3A%2F%2Fcuemath.com%2Fgeometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Risi2015-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Risi2015_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVincenzo_De_Risi2015" class="citation book cs1">Vincenzo De Risi (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1m11BgAAQBAJ&pg=PA1"><i>Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age</i></a>. Birkhäuser. pp. 1–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-12102-4" title="Special:BookSources/978-3-319-12102-4"><bdi>978-3-319-12102-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210220094741/https://books.google.com/books?id=1m11BgAAQBAJ&pg=PA1">Archived</a> from the original on 20 February 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematizing+Space%3A+The+Objects+of+Geometry+from+Antiquity+to+the+Early+Modern+Age&rft.pages=1-&rft.pub=Birkh%C3%A4user&rft.date=2015&rft.isbn=978-3-319-12102-4&rft.au=Vincenzo+De+Risi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1m11BgAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Tabak_2014_xiv-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tabak_2014_xiv_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Tabak_2014_xiv_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTabak2014" class="citation book cs1">Tabak, John (2014). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometrylanguage0000taba"><i>Geometry: the language of space and form</i></a></span>. Infobase Publishing. p. xiv. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8160-4953-0" title="Special:BookSources/978-0-8160-4953-0"><bdi>978-0-8160-4953-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+the+language+of+space+and+form&rft.pages=xiv&rft.pub=Infobase+Publishing&rft.date=2014&rft.isbn=978-0-8160-4953-0&rft.aulast=Tabak&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometrylanguage0000taba&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Meyer2006-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Meyer2006_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalter_A._Meyer2006" class="citation book cs1">Walter A. Meyer (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ez6H5Ho6E3cC"><i>Geometry and Its Applications</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-047803-6" title="Special:BookSources/978-0-08-047803-6"><bdi>978-0-08-047803-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183207/https://books.google.com/books?id=ez6H5Ho6E3cC">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+and+Its+Applications&rft.pub=Elsevier&rft.date=2006&rft.isbn=978-0-08-047803-6&rft.au=Walter+A.+Meyer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dez6H5Ho6E3cC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></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 id="CITEREFFriberg1981" class="citation journal cs1">Friberg, Jöran (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2881%2990069-0">"Methods and traditions of Babylonian mathematics"</a>. <i>Historia Mathematica</i>. <b>8</b> (3): 277–318. <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/10.1016%2F0315-0860%2881%2990069-0">10.1016/0315-0860(81)90069-0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Methods+and+traditions+of+Babylonian+mathematics&rft.volume=8&rft.issue=3&rft.pages=277-318&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2881%2990069-0&rft.aulast=Friberg&rft.aufirst=J%C3%B6ran&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252881%252990069-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" 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="CITEREFNeugebauer1969" class="citation book cs1"><a href="/wiki/Otto_E._Neugebauer" title="Otto E. Neugebauer">Neugebauer, Otto</a> (1969) [1957]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA71">"Chap. IV Egyptian Mathematics and Astronomy"</a>. <i>The Exact Sciences in Antiquity</i> (2 ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. pp. 71–96. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-22332-2" title="Special:BookSources/978-0-486-22332-2"><bdi>978-0-486-22332-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200814151056/https://books.google.com/books?id=JVhTtVA2zr8C">Archived</a> from the original on 14 August 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">27 February</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chap.+IV+Egyptian+Mathematics+and+Astronomy&rft.btitle=The+Exact+Sciences+in+Antiquity&rft.pages=71-96&rft.edition=2&rft.pub=Dover+Publications&rft.date=1969&rft.isbn=978-0-486-22332-2&rft.aulast=Neugebauer&rft.aufirst=Otto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJVhTtVA2zr8C%26pg%3DPA71&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span>.</span> </li> <li id="cite_note-Boyer_1991_loc=Egypt_p._19-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer_1991_loc=Egypt_p._19_8-0">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Egypt" p. 19)</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOssendrijver2016" class="citation journal cs1">Ossendrijver, Mathieu (29 January 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph". <i>Science</i>. <b>351</b> (6272): 482–484. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016Sci...351..482O">2016Sci...351..482O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.aad8085">10.1126/science.aad8085</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/26823423">26823423</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206644971">206644971</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Ancient+Babylonian+astronomers+calculated+Jupiter%27s+position+from+the+area+under+a+time-velocity+graph&rft.volume=351&rft.issue=6272&rft.pages=482-484&rft.date=2016-01-29&rft_id=info%3Adoi%2F10.1126%2Fscience.aad8085&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A206644971%23id-name%3DS2CID&rft_id=info%3Apmid%2F26823423&rft_id=info%3Abibcode%2F2016Sci...351..482O&rft.aulast=Ossendrijver&rft.aufirst=Mathieu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></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="CITEREFDepuydt1998" class="citation journal cs1">Depuydt, Leo (1 January 1998). "Gnomons at Meroë and Early Trigonometry". <i>The Journal of Egyptian Archaeology</i>. <b>84</b>: 171–180. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3822211">10.2307/3822211</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3822211">3822211</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Egyptian+Archaeology&rft.atitle=Gnomons+at+Mero%C3%AB+and+Early+Trigonometry&rft.volume=84&rft.pages=171-180&rft.date=1998-01-01&rft_id=info%3Adoi%2F10.2307%2F3822211&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3822211%23id-name%3DJSTOR&rft.aulast=Depuydt&rft.aufirst=Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" 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="CITEREFSlayman1998" class="citation web cs1">Slayman, Andrew (27 May 1998). <a rel="nofollow" class="external text" href="http://www.archaeology.org/online/news/nubia.html">"Neolithic Skywatchers"</a>. <i>Archaeology Magazine Archive</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110605234044/http://www.archaeology.org/online/news/nubia.html">Archived</a> from the original on 5 June 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">17 April</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Archaeology+Magazine+Archive&rft.atitle=Neolithic+Skywatchers&rft.date=1998-05-27&rft.aulast=Slayman&rft.aufirst=Andrew&rft_id=http%3A%2F%2Fwww.archaeology.org%2Fonline%2Fnews%2Fnubia.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Boyer_1991_loc=Ionia_and_the_Pythagoreans_p._43-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer_1991_loc=Ionia_and_the_Pythagoreans_p._43_12-0">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Ionia and the Pythagoreans" p. 43)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Eves, Howard, <i><a href="//archive.org/details/introductiontohi0000eves" class="extiw" title="iarchive:introductiontohi0000eves">An Introduction to the History of Mathematics</a></i>, Saunders, 1990, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-029558-0" title="Special:BookSources/0-03-029558-0">0-03-029558-0</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurt_Von_Fritz1945" class="citation book cs1">Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". <i>Classics in the History of Greek Mathematics</i>. Annals of Mathematics; Boston Studies in the Philosophy of Science. Vol. 240. pp. 211–231. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4020-2640-9_11">10.1007/978-1-4020-2640-9_11</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-481-5850-8" title="Special:BookSources/978-90-481-5850-8"><bdi>978-90-481-5850-8</bdi></a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969021">1969021</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Discovery+of+Incommensurability+by+Hippasus+of+Metapontum&rft.btitle=Classics+in+the+History+of+Greek+Mathematics&rft.series=Annals+of+Mathematics%3B+Boston+Studies+in+the+Philosophy+of+Science&rft.pages=211-231&rft.date=1945&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969021%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2F978-1-4020-2640-9_11&rft.isbn=978-90-481-5850-8&rft.au=Kurt+Von+Fritz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_R._Choike1980" class="citation journal cs1">James R. Choike (1980). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00494925.1980.11972468">"The Pentagram and the Discovery of an Irrational Number"</a>. <i>The Two-Year College Mathematics Journal</i>. <b>11</b> (5): 312–316. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3026893">10.2307/3026893</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3026893">3026893</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220909203418/https://www.tandfonline.com/doi/abs/10.1080/00494925.1980.11972468">Archived</a> from the original on 9 September 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Two-Year+College+Mathematics+Journal&rft.atitle=The+Pentagram+and+the+Discovery+of+an+Irrational+Number&rft.volume=11&rft.issue=5&rft.pages=312-316&rft.date=1980&rft_id=info%3Adoi%2F10.2307%2F3026893&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3026893%23id-name%3DJSTOR&rft.au=James+R.+Choike&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00494925.1980.11972468&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBoyer1991">Boyer 1991</a>, "The Age of Plato and Aristotle" p. 92)</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Euclid of Alexandria" p. 119)</span> </li> <li id="cite_note-Boyer_1991_loc=Euclid_of_Alexandria_p._104-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer_1991_loc=Euclid_of_Alexandria_p._104_18-0">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Euclid of Alexandria" p. 104)</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="/wiki/Howard_Eves" title="Howard Eves">Howard Eves</a>, <i><a href="//archive.org/details/introductiontohi0000eves" class="extiw" title="iarchive:introductiontohi0000eves">An Introduction to the History of Mathematics</a></i>, Saunders, 1990, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-029558-0" title="Special:BookSources/0-03-029558-0">0-03-029558-0</a> p. 141: "No work, except <a href="/wiki/The_Bible" class="mw-redirect" title="The Bible">The Bible</a>, has been more widely used...."</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Connor,_J.J.Robertson,_E.F.1996" class="citation web cs1">O'Connor, J.J.; Robertson, E.F. (February 1996). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070715191704/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html">"A history of calculus"</a>. <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>. Archived from <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html">the original</a> on 15 July 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">7 August</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+history+of+calculus&rft.pub=University+of+St+Andrews&rft.date=1996-02&rft.au=O%27Connor%2C+J.J.&rft.au=Robertson%2C+E.F.&rft_id=http%3A%2F%2Fwww-groups.dcs.st-and.ac.uk%2F~history%2FHistTopics%2FThe_rise_of_calculus.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Staal_1999-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Staal_1999_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStaal1999" class="citation journal cs1"><a href="/wiki/Frits_Staal" title="Frits Staal">Staal, Frits</a> (1999). "Greek and Vedic Geometry". <i>Journal of Indian Philosophy</i>. <b>27</b> (1–2): 105–127. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1004364417713">10.1023/A:1004364417713</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:170894641">170894641</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+Indian+Philosophy&rft.atitle=Greek+and+Vedic+Geometry&rft.volume=27&rft.issue=1%E2%80%932&rft.pages=105-127&rft.date=1999&rft_id=info%3Adoi%2F10.1023%2FA%3A1004364417713&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A170894641%23id-name%3DS2CID&rft.aulast=Staal&rft.aufirst=Frits&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-cooke198-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-cooke198_23-0">^</a></b></span> <span class="reference-text">(<a href="#CITEREFCooke2005">Cooke 2005</a>, p. 198): "The arithmetic content of the <i>Śulva Sūtras</i> consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."</span> </li> <li id="cite_note-hayashi2005-371-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-hayashi2005-371_24-0">^</a></b></span> <span class="reference-text">(<a href="#CITEREFHayashi2005">Hayashi 2005</a>, p. 371)</span> </li> <li id="cite_note-hayashi2003-p121-122-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-hayashi2003-p121-122_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hayashi2003-p121-122_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(<a href="#CITEREFHayashi2003">Hayashi 2003</a>, pp. 121–122)</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRāshid1994" class="citation book cs1">Rāshid, Rushdī (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/page/n43/mode/2up"><i>The development of Arabic mathematics : between arithmetic and algebra</i></a>. Boston Studies in the Philosophy of Science. Vol. 156. p. 35. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-017-3274-1">10.1007/978-94-017-3274-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-2565-9" title="Special:BookSources/978-0-7923-2565-9"><bdi>978-0-7923-2565-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/29181926">29181926</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+development+of+Arabic+mathematics+%3A+between+arithmetic+and+algebra&rft.series=Boston+Studies+in+the+Philosophy+of+Science&rft.pages=35&rft.date=1994&rft_id=info%3Aoclcnum%2F29181926&rft_id=info%3Adoi%2F10.1007%2F978-94-017-3274-1&rft.isbn=978-0-7923-2565-9&rft.aulast=R%C4%81shid&rft.aufirst=Rushd%C4%AB&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FRoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994%2Fpage%2Fn43%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" 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">(<a href="#CITEREFBoyer1991">Boyer 1991</a>, "The Arabic Hegemony" pp. 241–242) "Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an <i>Algebra</i> that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."".</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs1">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Al-Mahani.html">"Al-Mahani"</a>. <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>. <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Al-Mahani&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FAl-Mahani.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-ReferenceA-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-ReferenceA_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs1">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Thabit.html">"Al-Sabi Thabit ibn Qurra al-Harrani"</a>. <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>. <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Al-Sabi+Thabit+ibn+Qurra+al-Harrani&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FThabit.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs1">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Khayyam.html">"Omar Khayyam"</a>. <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>. <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Omar+Khayyam&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FKhayyam.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., <i><a href="/wiki/Encyclopedia_of_the_History_of_Arabic_Science" title="Encyclopedia of the History of Arabic Science">Encyclopedia of the History of Arabic Science</a></i>, Vol. 2, pp. 447–494 [470], <a href="/wiki/Routledge" title="Routledge">Routledge</a>, London and New York: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence, their propositions concerning the properties of quadrangles which they considered, assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines—made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's <i><a href="/wiki/Book_of_Optics" title="Book of Optics">Book of Optics</a></i> (<i>Kitab al-Manazir</i>)—was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that <i>Pseudo-Tusi's Exposition of Euclid</i> had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."</p></blockquote></span> </li> <li id="cite_note-Boyer2012-32"><span class="mw-cite-backlink">^ <a href="#cite_ref-Boyer2012_32-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Boyer2012_32-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarl_B._Boyer2012" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Carl B. Boyer</a> (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2T4i5fXZbOYC"><i>History of Analytic Geometry</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-15451-0" title="Special:BookSources/978-0-486-15451-0"><bdi>978-0-486-15451-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191226215605/https://books.google.com/books?id=2T4i5fXZbOYC">Archived</a> from the original on 26 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Analytic+Geometry&rft.pub=Courier+Corporation&rft.date=2012&rft.isbn=978-0-486-15451-0&rft.au=Carl+B.+Boyer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2T4i5fXZbOYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Edwards2012-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Edwards2012_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._H._Edwards_Jr.2012" class="citation book cs1">C. H. Edwards Jr. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ilrlBwAAQBAJ&pg=PA95"><i>The Historical Development of the Calculus</i></a>. Springer Science & Business Media. p. 95. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-6230-5" title="Special:BookSources/978-1-4612-6230-5"><bdi>978-1-4612-6230-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191229201529/https://books.google.com/books?id=ilrlBwAAQBAJ&pg=PA95">Archived</a> from the original on 29 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Historical+Development+of+the+Calculus&rft.pages=95&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4612-6230-5&rft.au=C.+H.+Edwards+Jr.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DilrlBwAAQBAJ%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-FieldGray2012-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FieldGray2012_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJudith_V._FieldJeremy_Gray2012" class="citation book cs1"><a href="/wiki/Judith_V._Field" title="Judith V. Field">Judith V. Field</a>; Jeremy Gray (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zSvSBwAAQBAJ&pg=PA43"><i>The Geometrical Work of Girard Desargues</i></a>. Springer Science & Business Media. p. 43. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-8692-6" title="Special:BookSources/978-1-4613-8692-6"><bdi>978-1-4613-8692-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227054645/https://books.google.com/books?id=zSvSBwAAQBAJ&pg=PA43">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometrical+Work+of+Girard+Desargues&rft.pages=43&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4613-8692-6&rft.au=Judith+V.+Field&rft.au=Jeremy+Gray&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzSvSBwAAQBAJ%26pg%3DPA43&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Wylie2011-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wylie2011_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._R._Wylie2011" class="citation book cs1">C. R. Wylie (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VVvGc8kaajEC"><i>Introduction to Projective Geometry</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-14170-1" title="Special:BookSources/978-0-486-14170-1"><bdi>978-0-486-14170-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228051716/https://books.google.com/books?id=VVvGc8kaajEC">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Projective+Geometry&rft.pub=Courier+Corporation&rft.date=2011&rft.isbn=978-0-486-14170-1&rft.au=C.+R.+Wylie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVVvGc8kaajEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Gray2011-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gray2011_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJeremy_Gray2011" class="citation book cs1">Jeremy Gray (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3UeSCvazV0QC"><i>Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85729-060-1" title="Special:BookSources/978-0-85729-060-1"><bdi>978-0-85729-060-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191207041658/https://books.google.com/books?id=3UeSCvazV0QC">Archived</a> from the original on 7 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Worlds+Out+of+Nothing%3A+A+Course+in+the+History+of+Geometry+in+the+19th+Century&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-0-85729-060-1&rft.au=Jeremy+Gray&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3UeSCvazV0QC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Bayro-Corrochano2018-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bayro-Corrochano2018_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEduardo_Bayro-Corrochano2018" class="citation book cs1">Eduardo Bayro-Corrochano (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SSVhDwAAQBAJ&pg=PA4"><i>Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing</i></a>. Springer. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-74830-6" title="Special:BookSources/978-3-319-74830-6"><bdi>978-3-319-74830-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228052142/https://books.google.com/books?id=SSVhDwAAQBAJ&pg=PA4">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Algebra+Applications+Vol.+I%3A+Computer+Vision%2C+Graphics+and+Neurocomputing&rft.pages=4&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-319-74830-6&rft.au=Eduardo+Bayro-Corrochano&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSSVhDwAAQBAJ%26pg%3DPA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Kline1990-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kline1990_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorris_Kline1990" class="citation book cs1">Morris Kline (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010"><i>Mathematical Thought From Ancient to Modern Times: Volume 3</i></a>. US: Oxford University Press. pp. 1010–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-506137-6" title="Special:BookSources/978-0-19-506137-6"><bdi>978-0-19-506137-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183204/https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thought+From+Ancient+to+Modern+Times%3A+Volume+3&rft.place=US&rft.pages=1010-&rft.pub=Oxford+University+Press&rft.date=1990&rft.isbn=978-0-19-506137-6&rft.au=Morris+Kline&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8YaBuGcmLb0C%26pg%3DPA1010&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Katz2000-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-Katz2000_39-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVictor_J._Katz2000" class="citation book cs1">Victor J. Katz (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45"><i>Using History to Teach Mathematics: An International Perspective</i></a>. Cambridge University Press. pp. 45–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-163-0" title="Special:BookSources/978-0-88385-163-0"><bdi>978-0-88385-163-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183205/https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Using+History+to+Teach+Mathematics%3A+An+International+Perspective&rft.pages=45-&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-88385-163-0&rft.au=Victor+J.+Katz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCbZ_YsdCmP0C%26pg%3DPA45&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Berlinski2014-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-Berlinski2014_40-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Berlinski2014" class="citation book cs1"><a href="/wiki/David_Berlinski" title="David Berlinski">David Berlinski</a> (2014). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/kingofinfinitesp00davi"><i>The King of Infinite Space: Euclid and His Elements</i></a></span>. Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-03863-3" title="Special:BookSources/978-0-465-03863-3"><bdi>978-0-465-03863-3</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+King+of+Infinite+Space%3A+Euclid+and+His+Elements&rft.pub=Basic+Books&rft.date=2014&rft.isbn=978-0-465-03863-3&rft.au=David+Berlinski&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fkingofinfinitesp00davi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Hartshorne2013-41"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hartshorne2013_41-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hartshorne2013_41-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobin_Hartshorne2013" class="citation book cs1"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Robin Hartshorne</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29"><i>Geometry: Euclid and Beyond</i></a>. Springer Science & Business Media. pp. 29–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-22676-7" title="Special:BookSources/978-0-387-22676-7"><bdi>978-0-387-22676-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183205/https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+Euclid+and+Beyond&rft.pages=29-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-0-387-22676-7&rft.au=Robin+Hartshorne&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DC5fSBwAAQBAJ%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-HerbstFujita2017-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-HerbstFujita2017_42-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPat_HerbstTaro_FujitaStefan_HalverscheidMichael_Weiss2017" class="citation book cs1">Pat Herbst; Taro Fujita; Stefan Halverscheid; Michael Weiss (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20"><i>The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective</i></a>. Taylor & Francis. pp. 20–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-97353-3" title="Special:BookSources/978-1-351-97353-3"><bdi>978-1-351-97353-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183206/https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Learning+and+Teaching+of+Geometry+in+Secondary+Schools%3A+A+Modeling+Perspective&rft.pages=20-&rft.pub=Taylor+%26+Francis&rft.date=2017&rft.isbn=978-1-351-97353-3&rft.au=Pat+Herbst&rft.au=Taro+Fujita&rft.au=Stefan+Halverscheid&rft.au=Michael+Weiss&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6DAlDwAAQBAJ%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Yaglom2012-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yaglom2012_43-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFI._M._Yaglom2012" class="citation book cs1"><a href="/wiki/Isaak_Yaglom" title="Isaak Yaglom">I. M. Yaglom</a> (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6"><i>A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity</i></a>. Springer Science & Business Media. pp. 6–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-6135-3" title="Special:BookSources/978-1-4612-6135-3"><bdi>978-1-4612-6135-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183221/https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Simple+Non-Euclidean+Geometry+and+Its+Physical+Basis%3A+An+Elementary+Account+of+Galilean+Geometry+and+the+Galilean+Principle+of+Relativity&rft.pages=6-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4612-6135-3&rft.au=I.+M.+Yaglom&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFyToBwAAQBAJ%26pg%3DPR6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Holme2010-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-Holme2010_44-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAudun_Holme2010" class="citation book cs1">Audun Holme (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254"><i>Geometry: Our Cultural Heritage</i></a>. Springer Science & Business Media. pp. 254–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-14441-7" title="Special:BookSources/978-3-642-14441-7"><bdi>978-3-642-14441-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183209/https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+Our+Cultural+Heritage&rft.pages=254-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2010&rft.isbn=978-3-642-14441-7&rft.au=Audun+Holme&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzXwQGo8jyHUC%26pg%3DPA254&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-EuclidAll-45"><span class="mw-cite-backlink">^ <a href="#cite_ref-EuclidAll_45-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-EuclidAll_45-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-EuclidAll_45-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-EuclidAll_45-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-EuclidAll_45-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><i>Euclid's Elements – All thirteen books in one volume</i>, Based on Heath's translation, Green Lion Press <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-888009-18-7" title="Special:BookSources/1-888009-18-7">1-888009-18-7</a>.</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerla,_G.1995" class="citation book cs1">Gerla, G. (1995). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110717210751/http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf">"Pointless Geometries"</a> <span class="cs1-format">(PDF)</span>. In Buekenhout, F.; Kantor, W. (eds.). <i>Handbook of incidence geometry: buildings and foundations</i>. North-Holland. pp. 1015–1031. Archived from <a rel="nofollow" class="external text" href="http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 17 July 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Pointless+Geometries&rft.btitle=Handbook+of+incidence+geometry%3A+buildings+and+foundations&rft.pages=1015-1031&rft.pub=North-Holland&rft.date=1995&rft.au=Gerla%2C+G.&rft_id=http%3A%2F%2Fwww.dmi.unisa.it%2Fpeople%2Fgerla%2Fwww%2FDown%2Fpoint-free.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClark1985" class="citation journal cs1">Clark, Bowman L. (January 1985). <a rel="nofollow" class="external text" href="https://doi.org/10.1305%2Fndjfl%2F1093870761">"Individuals and Points"</a>. <i>Notre Dame Journal of Formal Logic</i>. <b>26</b> (1): 61–75. <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/10.1305%2Fndjfl%2F1093870761">10.1305/ndjfl/1093870761</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notre+Dame+Journal+of+Formal+Logic&rft.atitle=Individuals+and+Points&rft.volume=26&rft.issue=1&rft.pages=61-75&rft.date=1985-01&rft_id=info%3Adoi%2F10.1305%2Fndjfl%2F1093870761&rft.aulast=Clark&rft.aufirst=Bowman+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1305%252Fndjfl%252F1093870761&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Casey1885" class="citation book cs1"><a href="/wiki/John_Casey_(mathematician)" title="John Casey (mathematician)">John Casey</a> (1885). <a rel="nofollow" class="external text" href="https://archive.org/details/cu31924001520455"><i>Analytic Geometry of the Point, Line, Circle, and Conic Sections</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+Geometry+of+the+Point%2C+Line%2C+Circle%2C+and+Conic+Sections&rft.date=1885&rft.au=John+Casey&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcu31924001520455&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrancis_Buekenhout1995" class="citation book cs1">Francis Buekenhout, ed. (1995). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/162589397"><i>Handbook of incidence geometry : buildings and foundations</i></a>. Amsterdam: Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-88355-1" title="Special:BookSources/978-0-444-88355-1"><bdi>978-0-444-88355-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/162589397">162589397</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145150/https://www.worldcat.org/title/162589397">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+incidence+geometry+%3A+buildings+and+foundations&rft.place=Amsterdam&rft.pub=Elsevier&rft.date=1995&rft_id=info%3Aoclcnum%2F162589397&rft.isbn=978-0-444-88355-1&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F162589397&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</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://web.archive.org/web/20160715034047/http://www.oxforddictionaries.com/definition/english/geodesic">"geodesic – definition of geodesic in English from the Oxford dictionary"</a>. <a href="/wiki/OxfordDictionaries.com" class="mw-redirect" title="OxfordDictionaries.com">OxfordDictionaries.com</a>. Archived from <a rel="nofollow" class="external text" href="https://www.oxforddictionaries.com/definition/english/geodesic">the original</a> on 15 July 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">20 January</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=geodesic+%E2%80%93+definition+of+geodesic+in+English+from+the+Oxford+dictionary&rft.pub=OxfordDictionaries.com&rft_id=https%3A%2F%2Fwww.oxforddictionaries.com%2Fdefinition%2Fenglish%2Fgeodesic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Munkres-51"><span class="mw-cite-backlink">^ <a href="#cite_ref-Munkres_51-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Munkres_51-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Munkres_51-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Munkres_51-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Munkres_51-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/42683260"><i>Topology</i></a>. Vol. 2 (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-181629-2" title="Special:BookSources/0-13-181629-2"><bdi>0-13-181629-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42683260">42683260</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.place=Upper+Saddle+River%2C+NJ&rft.edition=2nd&rft.pub=Prentice+Hall%2C+Inc&rft.date=2000&rft_id=info%3Aoclcnum%2F42683260&rft.isbn=0-13-181629-2&rft.aulast=Munkres&rft.aufirst=James+R.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F42683260&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzmielew1983" class="citation book cs1"><a href="/wiki/Wanda_Szmielew" title="Wanda Szmielew">Szmielew, Wanda</a> (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xDJPAQAAIAAJ"><i>From Affine to Euclidean Geometry</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-277-1243-1" title="Special:BookSources/978-90-277-1243-1"><bdi>978-90-277-1243-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145204/https://books.google.com/books?id=xDJPAQAAIAAJ">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Affine+to+Euclidean+Geometry&rft.pub=Springer&rft.date=1983&rft.isbn=978-90-277-1243-1&rft.aulast=Szmielew&rft.aufirst=Wanda&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxDJPAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlfors1979" class="citation book cs1"><a href="/wiki/Lars_Ahlfors" title="Lars Ahlfors">Ahlfors, Lars V.</a> (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2MRuus-5GGoC"><i>Complex analysis : an introduction to the theory of analytic functions of one complex variable</i></a> (3rd ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780070006577" title="Special:BookSources/9780070006577"><bdi>9780070006577</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/4036464">4036464</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145208/https://books.google.com/books?id=2MRuus-5GGoC">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis+%3A+an+introduction+to+the+theory+of+analytic+functions+of+one+complex+variable&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1979&rft_id=info%3Aoclcnum%2F4036464&rft.isbn=9780070006577&rft.aulast=Ahlfors&rft.aufirst=Lars+V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2MRuus-5GGoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text">Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.</span> </li> <li id="cite_note-Carmo-55"><span class="mw-cite-backlink">^ <a href="#cite_ref-Carmo_55-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Carmo_55-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Carmo_55-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarmo1976" class="citation book cs1">Carmo, Manfredo Perdigão do (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1v0YAQAAIAAJ"><i>Differential geometry of curves and surfaces</i></a>. Vol. 2. Englewood Cliffs, N.J.: Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-212589-7" title="Special:BookSources/0-13-212589-7"><bdi>0-13-212589-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1529515">1529515</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145145/https://books.google.com/books?id=1v0YAQAAIAAJ">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+geometry+of+curves+and+surfaces&rft.place=Englewood+Cliffs%2C+N.J.&rft.pub=Prentice-Hall&rft.date=1976&rft_id=info%3Aoclcnum%2F1529515&rft.isbn=0-13-212589-7&rft.aulast=Carmo&rft.aufirst=Manfredo+Perdig%C3%A3o+do&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1v0YAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-mumford-56"><span class="mw-cite-backlink">^ <a href="#cite_ref-mumford_56-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mumford_56-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMumford1999" class="citation book cs1"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a> (1999). <i>The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians</i> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-63293-1" title="Special:BookSources/978-3-540-63293-1"><bdi>978-3-540-63293-1</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0945.14001">0945.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Red+Book+of+Varieties+and+Schemes+Includes+the+Michigan+Lectures+on+Curves+and+Their+Jacobians&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0945.14001%23id-name%3DZbl&rft.isbn=978-3-540-63293-1&rft.aulast=Mumford&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text">Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-57056-7" title="Special:BookSources/978-0-321-57056-7">978-0-321-57056-7</a>.</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau, Shing-Tung</a>; Nadis, Steve (2010). <i>The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions</i>. Basic Books. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-02023-2" title="Special:BookSources/978-0-465-02023-2">978-0-465-02023-2</a>.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSidorov2001" class="citation cs1">Sidorov, L.A. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Angle">"Angle"</a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Angle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Sidorov&rft.aufirst=L.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAngle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelʹfand2001" class="citation book cs1"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelʹfand, I. M.</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZCYtwHFVZHgC"><i>Trigonometry</i></a>. Mark E. Saul. Boston: Birkhäuser. pp. 1–20. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3914-4" title="Special:BookSources/0-8176-3914-4"><bdi>0-8176-3914-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/41355833">41355833</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145230/https://books.google.com/books?id=ZCYtwHFVZHgC">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">10 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometry&rft.place=Boston&rft.pages=1-20&rft.pub=Birkh%C3%A4user&rft.date=2001&rft_id=info%3Aoclcnum%2F41355833&rft.isbn=0-8176-3914-4&rft.aulast=Gel%CA%B9fand&rft.aufirst=I.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZCYtwHFVZHgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Stewart-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stewart_61-0">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2012). <i>Calculus: Early Transcendentals</i>, 7th ed., Brooks Cole Cengage Learning. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-538-49790-9" title="Special:BookSources/978-0-538-49790-9">978-0-538-49790-9</a></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJost2002" class="citation book cs1">Jost, Jürgen (2002). <i>Riemannian Geometry and Geometric Analysis</i>. Berlin: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42627-1" title="Special:BookSources/978-3-540-42627-1"><bdi>978-3-540-42627-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemannian+Geometry+and+Geometric+Analysis&rft.place=Berlin&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-3-540-42627-1&rft.aulast=Jost&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span>.</span> </li> <li id="cite_note-Treese2018-63"><span class="mw-cite-backlink">^ <a href="#cite_ref-Treese2018_63-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Treese2018_63-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteven_A._Treese2018" class="citation book cs1">Steven A. Treese (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bi1bDwAAQBAJ&pg=PA101"><i>History and Measurement of the Base and Derived Units</i></a>. Springer International Publishing. pp. 101–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-77577-7" title="Special:BookSources/978-3-319-77577-7"><bdi>978-3-319-77577-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191230065433/https://books.google.com/books?id=bi1bDwAAQBAJ&pg=PA101">Archived</a> from the original on 30 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+and+Measurement+of+the+Base+and+Derived+Units&rft.pages=101-&rft.pub=Springer+International+Publishing&rft.date=2018&rft.isbn=978-3-319-77577-7&rft.au=Steven+A.+Treese&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dbi1bDwAAQBAJ%26pg%3DPA101&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Cannon2017-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cannon2017_64-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_W._Cannon2017" class="citation book cs1"><a href="/wiki/James_W._Cannon" title="James W. Cannon">James W. Cannon</a> (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11"><i>Geometry of Lengths, Areas, and Volumes</i></a>. American Mathematical Soc. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-3714-5" title="Special:BookSources/978-1-4704-3714-5"><bdi>978-1-4704-3714-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191231135911/https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11">Archived</a> from the original on 31 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Lengths%2C+Areas%2C+and+Volumes&rft.pages=11&rft.pub=American+Mathematical+Soc.&rft.date=2017&rft.isbn=978-1-4704-3714-5&rft.au=James+W.+Cannon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsSI_DwAAQBAJ%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Strang1991-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-Strang1991_65-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilbert_Strang1991" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a> (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OisInC1zvEMC"><i>Calculus</i></a>. SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9614088-2-4" title="Special:BookSources/978-0-9614088-2-4"><bdi>978-0-9614088-2-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224134500/https://books.google.com/books?id=OisInC1zvEMC">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.pub=SIAM&rft.date=1991&rft.isbn=978-0-9614088-2-4&rft.au=Gilbert+Strang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOisInC1zvEMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Bear2002-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bear2002_66-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFH._S._Bear2002" class="citation book cs1">H. S. Bear (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=__AmiGnEEewC"><i>A Primer of Lebesgue Integration</i></a>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-083971-1" title="Special:BookSources/978-0-12-083971-1"><bdi>978-0-12-083971-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225032645/https://books.google.com/books?id=__AmiGnEEewC">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Primer+of+Lebesgue+Integration&rft.pub=Academic+Press&rft.date=2002&rft.isbn=978-0-12-083971-1&rft.au=H.+S.+Bear&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D__AmiGnEEewC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text">Dmitri Burago, <a href="/wiki/Yuri_Dmitrievich_Burago" class="mw-redirect" title="Yuri Dmitrievich Burago">Yu D Burago</a>, Sergei Ivanov, <i>A Course in Metric Geometry</i>, American Mathematical Society, 2001, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2129-6" title="Special:BookSources/0-8218-2129-6">0-8218-2129-6</a>.</span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWald1984" class="citation book cs1"><a href="/wiki/Robert_Wald" title="Robert Wald">Wald, Robert M.</a> (1984). <a href="/wiki/General_Relativity_(book)" title="General Relativity (book)"><i>General Relativity</i></a>. University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-87033-5" title="Special:BookSources/978-0-226-87033-5"><bdi>978-0-226-87033-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity&rft.pub=University+of+Chicago+Press&rft.date=1984&rft.isbn=978-0-226-87033-5&rft.aulast=Wald&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Tao2011-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tao2011_69-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTerence_Tao2011" class="citation book cs1"><a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a> (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HoGDAwAAQBAJ"><i>An Introduction to Measure Theory</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-6919-2" title="Special:BookSources/978-0-8218-6919-2"><bdi>978-0-8218-6919-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Measure+Theory&rft.pub=American+Mathematical+Soc.&rft.date=2011&rft.isbn=978-0-8218-6919-2&rft.au=Terence+Tao&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHoGDAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Libeskind2008-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-Libeskind2008_70-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShlomo_Libeskind2008" class="citation book cs1">Shlomo Libeskind (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255"><i>Euclidean and Transformational Geometry: A Deductive Inquiry</i></a>. Jones & Bartlett Learning. p. 255. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-4366-6" title="Special:BookSources/978-0-7637-4366-6"><bdi>978-0-7637-4366-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225090248/https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euclidean+and+Transformational+Geometry%3A+A+Deductive+Inquiry&rft.pages=255&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2008&rft.isbn=978-0-7637-4366-6&rft.au=Shlomo+Libeskind&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Det6WMlkQlFcC%26pg%3DPA255&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Freitag2013-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-Freitag2013_71-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_A._Freitag2013" class="citation book cs1">Mark A. Freitag (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614"><i>Mathematics for Elementary School Teachers: A Process Approach</i></a>. Cengage Learning. p. 614. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-618-61008-2" title="Special:BookSources/978-0-618-61008-2"><bdi>978-0-618-61008-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228123854/https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Elementary+School+Teachers%3A+A+Process+Approach&rft.pages=614&rft.pub=Cengage+Learning&rft.date=2013&rft.isbn=978-0-618-61008-2&rft.au=Mark+A.+Freitag&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG4BVGFiVKG0C%26pg%3DPA614&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Martin2012-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-Martin2012_72-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_E._Martin2012" class="citation book cs1">George E. Martin (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gevlBwAAQBAJ"><i>Transformation Geometry: An Introduction to Symmetry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-5680-9" title="Special:BookSources/978-1-4612-5680-9"><bdi>978-1-4612-5680-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191207041210/https://books.google.com/books?id=gevlBwAAQBAJ">Archived</a> from the original on 7 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transformation+Geometry%3A+An+Introduction+to+Symmetry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4612-5680-9&rft.au=George+E.+Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgevlBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Blacklock2018-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-Blacklock2018_74-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Blacklock2018" class="citation book cs1">Mark Blacklock (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nrNSDwAAQBAJ"><i>The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-875548-7" title="Special:BookSources/978-0-19-875548-7"><bdi>978-0-19-875548-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227145318/https://books.google.com/books?id=nrNSDwAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Emergence+of+the+Fourth+Dimension%3A+Higher+Spatial+Thinking+in+the+Fin+de+Si%C3%A8cle&rft.pub=Oxford+University+Press&rft.date=2018&rft.isbn=978-0-19-875548-7&rft.au=Mark+Blacklock&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnrNSDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Joly1895-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-Joly1895_75-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharles_Jasper_Joly1895" class="citation book cs1">Charles Jasper Joly (1895). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62"><i>Papers</i></a>. The Academy. pp. 62–. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227195202/https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Papers&rft.pages=62-&rft.pub=The+Academy&rft.date=1895&rft.au=Charles+Jasper+Joly&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcOTuAAAAMAAJ%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Temam2013-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-Temam2013_76-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoger_Temam2013" class="citation book cs1">Roger Temam (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367"><i>Infinite-Dimensional Dynamical Systems in Mechanics and Physics</i></a>. Springer Science & Business Media. p. 367. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-0645-3" title="Special:BookSources/978-1-4612-0645-3"><bdi>978-1-4612-0645-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224015857/https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinite-Dimensional+Dynamical+Systems+in+Mechanics+and+Physics&rft.pages=367&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4612-0645-3&rft.au=Roger+Temam&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOB_vBwAAQBAJ%26pg%3DPA367&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-JacobLam1994-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-JacobLam1994_77-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBill_JacobTsit-Yuen_Lam1994" class="citation book cs1">Bill Jacob; Tsit-Yuen Lam (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111"><i>Recent Advances in Real Algebraic Geometry and Quadratic Forms: Proceedings of the RAGSQUAD Year, Berkeley, 1990–1991</i></a>. American Mathematical Soc. p. 111. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5154-8" title="Special:BookSources/978-0-8218-5154-8"><bdi>978-0-8218-5154-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228124040/https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recent+Advances+in+Real+Algebraic+Geometry+and+Quadratic+Forms%3A+Proceedings+of+the+RAGSQUAD+Year%2C+Berkeley%2C+1990%E2%80%931991&rft.pages=111&rft.pub=American+Mathematical+Soc.&rft.date=1994&rft.isbn=978-0-8218-5154-8&rft.au=Bill+Jacob&rft.au=Tsit-Yuen+Lam&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmHwcCAAAQBAJ%26pg%3DPA111&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Stewart2008-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stewart2008_78-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIan_Stewart2008" class="citation book cs1"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Ian Stewart</a> (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6akF1v7Ds3MC"><i>Why Beauty Is Truth: A History of Symmetry</i></a>. Basic Books. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-08237-7" title="Special:BookSources/978-0-465-08237-7"><bdi>978-0-465-08237-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225201454/https://books.google.com/books?id=6akF1v7Ds3MC">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Why+Beauty+Is+Truth%3A+A+History+of+Symmetry&rft.pages=14&rft.pub=Basic+Books&rft.date=2008&rft.isbn=978-0-465-08237-7&rft.au=Ian+Stewart&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6akF1v7Ds3MC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Alexey2009-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-Alexey2009_79-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStakhov_Alexey2009" class="citation book cs1">Stakhov Alexey (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3k7ICgAAQBAJ&pg=PA144"><i>Mathematics Of Harmony: From Euclid To Contemporary Mathematics And Computer Science</i></a>. World Scientific. p. 144. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4472-57-9" title="Special:BookSources/978-981-4472-57-9"><bdi>978-981-4472-57-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191229132952/https://books.google.com/books?id=3k7ICgAAQBAJ&pg=PA144">Archived</a> from the original on 29 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Of+Harmony%3A+From+Euclid+To+Contemporary+Mathematics+And+Computer+Science&rft.pages=144&rft.pub=World+Scientific&rft.date=2009&rft.isbn=978-981-4472-57-9&rft.au=Stakhov+Alexey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3k7ICgAAQBAJ%26pg%3DPA144&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Hahn1998-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hahn1998_80-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWerner_Hahn1998" class="citation book cs1">Werner Hahn (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wzhqDQAAQBAJ"><i>Symmetry as a Developmental Principle in Nature and Art</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-2363-2" title="Special:BookSources/978-981-02-2363-2"><bdi>978-981-02-2363-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200101082827/https://books.google.com/books?id=wzhqDQAAQBAJ">Archived</a> from the original on 1 January 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+as+a+Developmental+Principle+in+Nature+and+Art&rft.pub=World+Scientific&rft.date=1998&rft.isbn=978-981-02-2363-2&rft.au=Werner+Hahn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwzhqDQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Cantwell2002-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cantwell2002_81-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrian_J._Cantwell2002" class="citation book cs1">Brian J. Cantwell (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=76RS2ZQ0UyUC&pg=PR34"><i>Introduction to Symmetry Analysis</i></a>. Cambridge University Press. p. 34. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-43171-2" title="Special:BookSources/978-1-139-43171-2"><bdi>978-1-139-43171-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227012548/https://books.google.com/books?id=76RS2ZQ0UyUC&pg=PR34">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Symmetry+Analysis&rft.pages=34&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-1-139-43171-2&rft.au=Brian+J.+Cantwell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D76RS2ZQ0UyUC%26pg%3DPR34&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-RosenfeldWiebe2013-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-RosenfeldWiebe2013_82-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._RosenfeldBill_Wiebe2013" class="citation book cs1">B. Rosenfeld; Bill Wiebe (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA158"><i>Geometry of Lie Groups</i></a>. Springer Science & Business Media. pp. 158ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-5325-7" title="Special:BookSources/978-1-4757-5325-7"><bdi>978-1-4757-5325-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224193157/https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA158">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Lie+Groups&rft.pages=158ff&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4757-5325-7&rft.au=B.+Rosenfeld&rft.au=Bill+Wiebe&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmIjSBwAAQBAJ%26pg%3DPA158&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Pesic2007-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pesic2007_83-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_Pesic2007" class="citation book cs1">Peter Pesic (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z67x6IOuOUAC"><i>Beyond Geometry: Classic Papers from Riemann to Einstein</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45350-7" title="Special:BookSources/978-0-486-45350-7"><bdi>978-0-486-45350-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183221/https://books.google.com/books?id=Z67x6IOuOUAC">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beyond+Geometry%3A+Classic+Papers+from+Riemann+to+Einstein&rft.pub=Courier+Corporation&rft.date=2007&rft.isbn=978-0-486-45350-7&rft.au=Peter+Pesic&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZ67x6IOuOUAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Kaku2012-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kaku2012_84-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichio_Kaku2012" class="citation book cs1"><a href="/wiki/Michio_Kaku" title="Michio Kaku">Michio Kaku</a> (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151"><i>Strings, Conformal Fields, and Topology: An Introduction</i></a>. Springer Science & Business Media. p. 151. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4684-0397-8" title="Special:BookSources/978-1-4684-0397-8"><bdi>978-1-4684-0397-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224015822/https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Strings%2C+Conformal+Fields%2C+and+Topology%3A+An+Introduction&rft.pages=151&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4684-0397-8&rft.au=Michio+Kaku&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpM8FCAAAQBAJ%26pg%3DPA151&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-BestvinaSageev2014-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-BestvinaSageev2014_85-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMladen_BestvinaMichah_SageevKaren_Vogtmann2014" class="citation book cs1">Mladen Bestvina; Michah Sageev; <a href="/wiki/Karen_Vogtmann" title="Karen Vogtmann">Karen Vogtmann</a> (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132"><i>Geometric Group Theory</i></a>. American Mathematical Soc. p. 132. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-1227-2" title="Special:BookSources/978-1-4704-1227-2"><bdi>978-1-4704-1227-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191229224624/https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132">Archived</a> from the original on 29 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Group+Theory&rft.pages=132&rft.pub=American+Mathematical+Soc.&rft.date=2014&rft.isbn=978-1-4704-1227-2&rft.au=Mladen+Bestvina&rft.au=Michah+Sageev&rft.au=Karen+Vogtmann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRGz1BQAAQBAJ%26pg%3DPA132&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Steeb1996-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-Steeb1996_86-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFW-H._Steeb1996" class="citation book cs1">W-H. Steeb (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rZBIDQAAQBAJ"><i>Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra</i></a>. World Scientific Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-310-503-4" title="Special:BookSources/978-981-310-503-4"><bdi>978-981-310-503-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191226205450/https://books.google.com/books?id=rZBIDQAAQBAJ">Archived</a> from the original on 26 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuous+Symmetries%2C+Lie+Algebras%2C+Differential+Equations+and+Computer+Algebra&rft.pub=World+Scientific+Publishing+Company&rft.date=1996&rft.isbn=978-981-310-503-4&rft.au=W-H.+Steeb&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrZBIDQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Misner2005-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-Misner2005_87-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharles_W._Misner2005" class="citation book cs1"><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Charles W. Misner</a> (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zpGZwmTJZIUC&pg=PA272"><i>Directions in General Relativity: Volume 1: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner</i></a>. Cambridge University Press. p. 272. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-02139-5" title="Special:BookSources/978-0-521-02139-5"><bdi>978-0-521-02139-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191226063925/https://books.google.com/books?id=zpGZwmTJZIUC&pg=PA272">Archived</a> from the original on 26 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Directions+in+General+Relativity%3A+Volume+1%3A+Proceedings+of+the+1993+International+Symposium%2C+Maryland%3A+Papers+in+Honor+of+Charles+Misner&rft.pages=272&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-02139-5&rft.au=Charles+W.+Misner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzpGZwmTJZIUC%26pg%3DPA272&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Dowling1917-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dowling1917_88-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLinnaeus_Wayland_Dowling1917" class="citation book cs1">Linnaeus Wayland Dowling (1917). <a rel="nofollow" class="external text" href="https://archive.org/details/cu31924001523897"><i>Projective Geometry</i></a>. McGraw-Hill book Company, Incorporated. p. <a rel="nofollow" class="external text" href="https://archive.org/details/cu31924001523897/page/n29">10</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Geometry&rft.pages=10&rft.pub=McGraw-Hill+book+Company%2C+Incorporated&rft.date=1917&rft.au=Linnaeus+Wayland+Dowling&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcu31924001523897&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Gierz2006-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gierz2006_89-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFG._Gierz2006" class="citation book cs1">G. Gierz (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2ml6CwAAQBAJ&pg=PA252"><i>Bundles of Topological Vector Spaces and Their Duality</i></a>. Springer. p. 252. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-39437-2" title="Special:BookSources/978-3-540-39437-2"><bdi>978-3-540-39437-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227123430/https://books.google.com/books?id=2ml6CwAAQBAJ&pg=PA252">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bundles+of+Topological+Vector+Spaces+and+Their+Duality&rft.pages=252&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-39437-2&rft.au=G.+Gierz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2ml6CwAAQBAJ%26pg%3DPA252&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-ButtsBrown2012-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-ButtsBrown2012_90-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_E._ButtsJ.R._Brown2012" class="citation book cs1">Robert E. Butts; J.R. Brown (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127"><i>Constructivism and Science: Essays in Recent German Philosophy</i></a>. Springer Science & Business Media. pp. 127–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-009-0959-5" title="Special:BookSources/978-94-009-0959-5"><bdi>978-94-009-0959-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183207/https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Constructivism+and+Science%3A+Essays+in+Recent+German+Philosophy&rft.pages=127-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-94-009-0959-5&rft.au=Robert+E.+Butts&rft.au=J.R.+Brown&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvzTqCAAAQBAJ%26pg%3DPA127&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-91"><span class="mw-cite-backlink"><b><a href="#cite_ref-91">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181"><i>Science</i></a>. Moses King. 1886. pp. 181–. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227013042/https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science&rft.pages=181-&rft.pub=Moses+King&rft.date=1886&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxfNRAQAAMAAJ%26pg%3DPA181&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Abbot2013-92"><span class="mw-cite-backlink"><b><a href="#cite_ref-Abbot2013_92-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFW._Abbot2013" class="citation book cs1">W. Abbot (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6"><i>Practical Geometry and Engineering Graphics: A Textbook for Engineering and Other Students</i></a>. Springer Science & Business Media. pp. 6–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-017-2742-6" title="Special:BookSources/978-94-017-2742-6"><bdi>978-94-017-2742-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225201450/https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Geometry+and+Engineering+Graphics%3A+A+Textbook+for+Engineering+and+Other+Students&rft.pages=6-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-94-017-2742-6&rft.au=W.+Abbot&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1LDsCAAAQBAJ%26pg%3DPP6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-HerseyHersey2001-93"><span class="mw-cite-backlink">^ <a href="#cite_ref-HerseyHersey2001_93-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HerseyHersey2001_93-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HerseyHersey2001_93-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-HerseyHersey2001_93-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_L._Hersey2001" class="citation book cs1">George L. Hersey (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=F1Tl9ok-7_IC"><i>Architecture and Geometry in the Age of the Baroque</i></a>. University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-32783-9" title="Special:BookSources/978-0-226-32783-9"><bdi>978-0-226-32783-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225141623/https://books.google.com/books?id=F1Tl9ok-7_IC">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Architecture+and+Geometry+in+the+Age+of+the+Baroque&rft.pub=University+of+Chicago+Press&rft.date=2001&rft.isbn=978-0-226-32783-9&rft.au=George+L.+Hersey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DF1Tl9ok-7_IC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-VanícekKrakiwsky2015-94"><span class="mw-cite-backlink"><b><a href="#cite_ref-VanícekKrakiwsky2015_94-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._VanícekE.J._Krakiwsky2015" class="citation book cs1">P. Vanícek; E.J. Krakiwsky (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1Mz-BAAAQBAJ"><i>Geodesy: The Concepts</i></a>. Elsevier. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-9079-9" title="Special:BookSources/978-1-4832-9079-9"><bdi>978-1-4832-9079-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191231233050/https://books.google.com/books?id=1Mz-BAAAQBAJ">Archived</a> from the original on 31 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geodesy%3A+The+Concepts&rft.pages=23&rft.pub=Elsevier&rft.date=2015&rft.isbn=978-1-4832-9079-9&rft.au=P.+Van%C3%ADcek&rft.au=E.J.+Krakiwsky&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1Mz-BAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-CummingsMorton2015-95"><span class="mw-cite-backlink"><b><a href="#cite_ref-CummingsMorton2015_95-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRussell_M._CummingsScott_A._MortonWilliam_H._MasonDavid_R._McDaniel2015" class="citation book cs1">Russell M. Cummings; Scott A. Morton; William H. Mason; David R. McDaniel (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449"><i>Applied Computational Aerodynamics</i></a>. Cambridge University Press. p. 449. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-05374-8" title="Special:BookSources/978-1-107-05374-8"><bdi>978-1-107-05374-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183207/https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Computational+Aerodynamics&rft.pages=449&rft.pub=Cambridge+University+Press&rft.date=2015&rft.isbn=978-1-107-05374-8&rft.au=Russell+M.+Cummings&rft.au=Scott+A.+Morton&rft.au=William+H.+Mason&rft.au=David+R.+McDaniel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgwzUBwAAQBAJ%26pg%3DPA449&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Williams1998-96"><span class="mw-cite-backlink"><b><a href="#cite_ref-Williams1998_96-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoy_Williams1998" class="citation book cs1">Roy Williams (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yNzf7OKGLxIC"><i>Geometry of Navigation</i></a>. Horwood Pub. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-898563-46-4" title="Special:BookSources/978-1-898563-46-4"><bdi>978-1-898563-46-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191207041213/https://books.google.com/books?id=yNzf7OKGLxIC">Archived</a> from the original on 7 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">20 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Navigation&rft.pub=Horwood+Pub.&rft.date=1998&rft.isbn=978-1-898563-46-4&rft.au=Roy+Williams&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyNzf7OKGLxIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Schmidt,_W._2002-97"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schmidt,_W._2002_97-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidtHouangCogan2002" class="citation journal cs1">Schmidt, W.; Houang, R.; Cogan, Leland S. (2002). <a rel="nofollow" class="external text" href="https://www.nifdi.org/research/journal-of-di/volume-4-no-1-winter-2004/454-a-coherent-curriculum-the-case-of-mathematics/file.html">"A Coherent Curriculum: The Case of Mathematics"</a>. <i>The American Educator</i>. <b>26</b> (2): 10–26. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118964353">118964353</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Educator&rft.atitle=A+Coherent+Curriculum%3A+The+Case+of+Mathematics.&rft.volume=26&rft.issue=2&rft.pages=10-26&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118964353%23id-name%3DS2CID&rft.aulast=Schmidt&rft.aufirst=W.&rft.au=Houang%2C+R.&rft.au=Cogan%2C+Leland+S.&rft_id=https%3A%2F%2Fwww.nifdi.org%2Fresearch%2Fjournal-of-di%2Fvolume-4-no-1-winter-2004%2F454-a-coherent-curriculum-the-case-of-mathematics%2Ffile.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Walschap2015-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-Walschap2015_98-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerard_Walschap2015" class="citation book cs1">Gerard Walschap (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cXPyCQAAQBAJ"><i>Multivariable Calculus and Differential Geometry</i></a>. De Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-036954-0" title="Special:BookSources/978-3-11-036954-0"><bdi>978-3-11-036954-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227012551/https://books.google.com/books?id=cXPyCQAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multivariable+Calculus+and+Differential+Geometry&rft.pub=De+Gruyter&rft.date=2015&rft.isbn=978-3-11-036954-0&rft.au=Gerard+Walschap&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcXPyCQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Flanders2012-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-Flanders2012_99-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarley_Flanders2012" class="citation book cs1">Harley Flanders (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=U_GLN1eOKaMC"><i>Differential Forms with Applications to the Physical Sciences</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13961-6" title="Special:BookSources/978-0-486-13961-6"><bdi>978-0-486-13961-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183207/https://books.google.com/books?id=U_GLN1eOKaMC">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Forms+with+Applications+to+the+Physical+Sciences&rft.pub=Courier+Corporation&rft.date=2012&rft.isbn=978-0-486-13961-6&rft.au=Harley+Flanders&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DU_GLN1eOKaMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-MarriottSalmon2000-100"><span class="mw-cite-backlink"><b><a href="#cite_ref-MarriottSalmon2000_100-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_MarriottMark_Salmon2000" class="citation book cs1">Paul Marriott; Mark Salmon (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1Jjm4I5tqkUC"><i>Applications of Differential Geometry to Econometrics</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-65116-5" title="Special:BookSources/978-0-521-65116-5"><bdi>978-0-521-65116-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183207/https://books.google.com/books?id=1Jjm4I5tqkUC">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Differential+Geometry+to+Econometrics&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-65116-5&rft.au=Paul+Marriott&rft.au=Mark+Salmon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1Jjm4I5tqkUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-HePetoukhov2011-101"><span class="mw-cite-backlink"><b><a href="#cite_ref-HePetoukhov2011_101-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthew_HeSergey_Petoukhov2011" class="citation book cs1">Matthew He; Sergey Petoukhov (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106"><i>Mathematics of Bioinformatics: Theory, Methods and Applications</i></a>. John Wiley & Sons. p. 106. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-09952-0" title="Special:BookSources/978-1-118-09952-0"><bdi>978-1-118-09952-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227163605/https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+of+Bioinformatics%3A+Theory%2C+Methods+and+Applications&rft.pages=106&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.isbn=978-1-118-09952-0&rft.au=Matthew+He&rft.au=Sergey+Petoukhov&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSkov-LJ1mmQC%26pg%3DPA106&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Dirac2016-102"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dirac2016_102-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP.A.M._Dirac2016" class="citation book cs1">P.A.M. Dirac (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qkWPDAAAQBAJ"><i>General Theory of Relativity</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-8419-3" title="Special:BookSources/978-1-4008-8419-3"><bdi>978-1-4008-8419-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191226205400/https://books.google.com/books?id=qkWPDAAAQBAJ">Archived</a> from the original on 26 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Theory+of+Relativity&rft.pub=Princeton+University+Press&rft.date=2016&rft.isbn=978-1-4008-8419-3&rft.au=P.A.M.+Dirac&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqkWPDAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-AyJost2017-103"><span class="mw-cite-backlink"><b><a href="#cite_ref-AyJost2017_103-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNihat_AyJürgen_JostHông_Vân_LêLorenz_Schwachhöfer2017" class="citation book cs1">Nihat Ay; Jürgen Jost; Hông Vân Lê; Lorenz Schwachhöfer (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185"><i>Information Geometry</i></a>. Springer. p. 185. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-56478-4" title="Special:BookSources/978-3-319-56478-4"><bdi>978-3-319-56478-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224015858/https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">23 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Geometry&rft.pages=185&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-319-56478-4&rft.au=Nihat+Ay&rft.au=J%C3%BCrgen+Jost&rft.au=H%C3%B4ng+V%C3%A2n+L%C3%AA&rft.au=Lorenz+Schwachh%C3%B6fer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpLsyDwAAQBAJ%26pg%3DPA185&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Crossley2011-104"><span class="mw-cite-backlink"><b><a href="#cite_ref-Crossley2011_104-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin_D._Crossley2011" class="citation book cs1">Martin D. Crossley (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QhCgVrLHlLgC"><i>Essential Topology</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-782-7" title="Special:BookSources/978-1-85233-782-7"><bdi>978-1-85233-782-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228094221/https://books.google.com/books?id=QhCgVrLHlLgC">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Topology&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-1-85233-782-7&rft.au=Martin+D.+Crossley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQhCgVrLHlLgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-NashSen1988-105"><span class="mw-cite-backlink"><b><a href="#cite_ref-NashSen1988_105-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharles_NashSiddhartha_Sen1988" class="citation book cs1">Charles Nash; Siddhartha Sen (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nnnNCgAAQBAJ"><i>Topology and Geometry for Physicists</i></a>. Elsevier. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-057085-3" title="Special:BookSources/978-0-08-057085-3"><bdi>978-0-08-057085-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191226215609/https://books.google.com/books?id=nnnNCgAAQBAJ">Archived</a> from the original on 26 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology+and+Geometry+for+Physicists&rft.pages=1&rft.pub=Elsevier&rft.date=1988&rft.isbn=978-0-08-057085-3&rft.au=Charles+Nash&rft.au=Siddhartha+Sen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnnnNCgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Martin1996-106"><span class="mw-cite-backlink"><b><a href="#cite_ref-Martin1996_106-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_E._Martin1996" class="citation book cs1">George E. Martin (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KW4EwONsQJgC"><i>Transformation Geometry: An Introduction to Symmetry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90636-2" title="Special:BookSources/978-0-387-90636-2"><bdi>978-0-387-90636-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191222024915/https://books.google.com/books?id=KW4EwONsQJgC">Archived</a> from the original on 22 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transformation+Geometry%3A+An+Introduction+to+Symmetry&rft.pub=Springer+Science+%26+Business+Media&rft.date=1996&rft.isbn=978-0-387-90636-2&rft.au=George+E.+Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKW4EwONsQJgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-May1999-107"><span class="mw-cite-backlink"><b><a href="#cite_ref-May1999_107-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._P._May1999" class="citation book cs1">J. P. May (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g8SG03R1bpgC"><i>A Concise Course in Algebraic Topology</i></a>. University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-51183-2" title="Special:BookSources/978-0-226-51183-2"><bdi>978-0-226-51183-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191223144107/https://books.google.com/books?id=g8SG03R1bpgC">Archived</a> from the original on 23 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Course+in+Algebraic+Topology&rft.pub=University+of+Chicago+Press&rft.date=1999&rft.isbn=978-0-226-51183-2&rft.au=J.+P.+May&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg8SG03R1bpgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-AHartshorne2013-108"><span class="mw-cite-backlink"><b><a href="#cite_ref-AHartshorne2013_108-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobin_Hartshorne2013" class="citation book cs1">Robin Hartshorne (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7z4mBQAAQBAJ"><i>Algebraic Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-3849-0" title="Special:BookSources/978-1-4757-3849-0"><bdi>978-1-4757-3849-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227163607/https://books.google.com/books?id=7z4mBQAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4757-3849-0&rft.au=Robin+Hartshorne&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7z4mBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Dieudonne1985-109"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dieudonne1985_109-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dieudonne1985_109-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean_Dieudonné1985" class="citation book cs1">Jean Dieudonné (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_uhlf38jOrgC"><i>History of Algebraic Geometry</i></a>. Translated by Judith D. Sally. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-99371-8" title="Special:BookSources/978-0-412-99371-8"><bdi>978-0-412-99371-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225090149/https://books.google.com/books?id=_uhlf38jOrgC">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Algebraic+Geometry&rft.pub=CRC+Press&rft.date=1985&rft.isbn=978-0-412-99371-8&rft.au=Jean+Dieudonn%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_uhlf38jOrgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-CarlsonCarlson2006-110"><span class="mw-cite-backlink"><b><a href="#cite_ref-CarlsonCarlson2006_110-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_CarlsonJames_A._CarlsonArthur_JaffeAndrew_Wiles2006" class="citation book cs1">James Carlson; James A. Carlson; Arthur Jaffe; Andrew Wiles (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7wJIPJ80RdUC"><i>The Millennium Prize Problems</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3679-8" title="Special:BookSources/978-0-8218-3679-8"><bdi>978-0-8218-3679-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160530210333/https://books.google.com/books?id=7wJIPJ80RdUC">Archived</a> from the original on 30 May 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Millennium+Prize+Problems&rft.pub=American+Mathematical+Soc.&rft.date=2006&rft.isbn=978-0-8218-3679-8&rft.au=James+Carlson&rft.au=James+A.+Carlson&rft.au=Arthur+Jaffe&rft.au=Andrew+Wiles&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7wJIPJ80RdUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-HoweLauter2017-111"><span class="mw-cite-backlink"><b><a href="#cite_ref-HoweLauter2017_111-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEverett_W._HoweKristin_E._LauterJudy_L._Walker2017" class="citation book cs1">Everett W. Howe; <a href="/wiki/Kristin_Lauter" title="Kristin Lauter">Kristin E. Lauter</a>; <a href="/wiki/Judy_L._Walker" title="Judy L. Walker">Judy L. Walker</a> (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bPM-DwAAQBAJ"><i>Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-63931-4" title="Special:BookSources/978-3-319-63931-4"><bdi>978-3-319-63931-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227195203/https://books.google.com/books?id=bPM-DwAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry+for+Coding+Theory+and+Cryptography%3A+IPAM%2C+Los+Angeles%2C+CA%2C+February+2016&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-319-63931-4&rft.au=Everett+W.+Howe&rft.au=Kristin+E.+Lauter&rft.au=Judy+L.+Walker&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbPM-DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-MarinoThaddeus2008-112"><span class="mw-cite-backlink"><b><a href="#cite_ref-MarinoThaddeus2008_112-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarcos_MarinoMichael_ThaddeusRavi_Vakil2008" class="citation book cs1">Marcos Marino; Michael Thaddeus; Ravi Vakil (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mb1qCQAAQBAJ"><i>Enumerative Invariants in Algebraic Geometry and String Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6–11, 2005</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-79814-9" title="Special:BookSources/978-3-540-79814-9"><bdi>978-3-540-79814-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227041441/https://books.google.com/books?id=mb1qCQAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enumerative+Invariants+in+Algebraic+Geometry+and+String+Theory%3A+Lectures+given+at+the+C.I.M.E.+Summer+School+held+in+Cetraro%2C+Italy%2C+June+6%E2%80%9311%2C+2005&rft.pub=Springer&rft.date=2008&rft.isbn=978-3-540-79814-9&rft.au=Marcos+Marino&rft.au=Michael+Thaddeus&rft.au=Ravi+Vakil&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dmb1qCQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-113"><span class="mw-cite-backlink"><b><a href="#cite_ref-113">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuybrechts2005" class="citation book cs1">Huybrechts, Daniel (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eZPCfJlHkXMC"><i>Complex geometry : an introduction</i></a>. Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540266877" title="Special:BookSources/9783540266877"><bdi>9783540266877</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/209857590">209857590</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145147/https://books.google.com/books?id=eZPCfJlHkXMC">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">10 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+geometry+%3A+an+introduction&rft.place=Berlin&rft.pub=Springer&rft.date=2005&rft_id=info%3Aoclcnum%2F209857590&rft.isbn=9783540266877&rft.aulast=Huybrechts&rft.aufirst=Daniel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeZPCfJlHkXMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-114"><span class="mw-cite-backlink"><b><a href="#cite_ref-114">^</a></b></span> <span class="reference-text">Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.</span> </li> <li id="cite_note-115"><span class="mw-cite-backlink"><b><a href="#cite_ref-115">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWells2008" class="citation book cs1">Wells, R. O. Jr. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aZXAs9Vu14cC"><i>Differential analysis on complex manifolds</i></a>. Graduate Texts in Mathematics. Vol. 65. O. García-Prada (3rd ed.). New York: Springer-Verlag. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-73892-5">10.1007/978-0-387-73892-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387738918" title="Special:BookSources/9780387738918"><bdi>9780387738918</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/233971394">233971394</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145230/https://books.google.com/books?id=aZXAs9Vu14cC">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+analysis+on+complex+manifolds&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2008&rft_id=info%3Aoclcnum%2F233971394&rft_id=info%3Adoi%2F10.1007%2F978-0-387-73892-5&rft.isbn=9780387738918&rft.aulast=Wells&rft.aufirst=R.+O.+Jr.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaZXAs9Vu14cC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-116"><span class="mw-cite-backlink"><b><a href="#cite_ref-116">^</a></b></span> <span class="reference-text"> Hori, K., Thomas, R., Katz, S., Vafa, C., Pandharipande, R., Klemm, A., ... & Zaslow, E. (2003). Mirror symmetry (Vol. 1). American Mathematical Soc.</span> </li> <li id="cite_note-117"><span class="mw-cite-backlink"><b><a href="#cite_ref-117">^</a></b></span> <span class="reference-text">Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.</span> </li> <li id="cite_note-118"><span class="mw-cite-backlink"><b><a href="#cite_ref-118">^</a></b></span> <span class="reference-text">Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.</span> </li> <li id="cite_note-119"><span class="mw-cite-backlink"><b><a href="#cite_ref-119">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonaldson2011" class="citation book cs1"><a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Donaldson, S. K.</a> (2011). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/861200296"><i>Riemann surfaces</i></a>. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-154584-9" title="Special:BookSources/978-0-19-154584-9"><bdi>978-0-19-154584-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/861200296">861200296</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301145222/https://www.worldcat.org/title/861200296">Archived</a> from the original on 1 March 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann+surfaces&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2011&rft_id=info%3Aoclcnum%2F861200296&rft.isbn=978-0-19-154584-9&rft.aulast=Donaldson&rft.aufirst=S.+K.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F861200296&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-120"><span class="mw-cite-backlink"><b><a href="#cite_ref-120">^</a></b></span> <span class="reference-text"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, J. P.</a> (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.</span> </li> <li id="cite_note-121"><span class="mw-cite-backlink"><b><a href="#cite_ref-121">^</a></b></span> <span class="reference-text"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, J. P.</a> (1956). Géométrie algébrique et géométrie analytique. In Annales de l'Institut Fourier (vol. 6, pp. 1–42).</span> </li> <li id="cite_note-Matoušek2013-122"><span class="mw-cite-backlink"><b><a href="#cite_ref-Matoušek2013_122-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiří_Matoušek2013" class="citation book cs1"><a href="/wiki/Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Jiří Matoušek</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K0fhBwAAQBAJ"><i>Lectures on Discrete Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-0039-7" title="Special:BookSources/978-1-4613-0039-7"><bdi>978-1-4613-0039-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227013417/https://books.google.com/books?id=K0fhBwAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Discrete+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4613-0039-7&rft.au=Ji%C5%99%C3%AD+Matou%C5%A1ek&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK0fhBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Zong2006-123"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zong2006_123-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChuanming_Zong2006" class="citation book cs1">Chuanming Zong (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ola6htFUQ1IC"><i>The Cube – A Window to Convex and Discrete Geometry</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-85535-8" title="Special:BookSources/978-0-521-85535-8"><bdi>978-0-521-85535-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191223150837/https://books.google.com/books?id=Ola6htFUQ1IC">Archived</a> from the original on 23 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cube+%E2%80%93+A+Window+to+Convex+and+Discrete+Geometry&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-85535-8&rft.au=Chuanming+Zong&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOla6htFUQ1IC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Gruber2007-124"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gruber2007_124-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_M._Gruber2007" class="citation book cs1">Peter M. Gruber (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bSZKAAAAQBAJ"><i>Convex and Discrete Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-71133-9" title="Special:BookSources/978-3-540-71133-9"><bdi>978-3-540-71133-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224175343/https://books.google.com/books?id=bSZKAAAAQBAJ">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+and+Discrete+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-3-540-71133-9&rft.au=Peter+M.+Gruber&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbSZKAAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-DevadossO'Rourke2011-125"><span class="mw-cite-backlink"><b><a href="#cite_ref-DevadossO'Rourke2011_125-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSatyan_L._DevadossJoseph_O'Rourke2011" class="citation book cs1"><a href="/wiki/Satyan_Devadoss" title="Satyan Devadoss">Satyan L. Devadoss</a>; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">Joseph O'Rourke</a> (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=InJL6iAaIQQC"><i>Discrete and Computational Geometry</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-3898-1" title="Special:BookSources/978-1-4008-3898-1"><bdi>978-1-4008-3898-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227194659/https://books.google.com/books?id=InJL6iAaIQQC">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+and+Computational+Geometry&rft.pub=Princeton+University+Press&rft.date=2011&rft.isbn=978-1-4008-3898-1&rft.au=Satyan+L.+Devadoss&rft.au=Joseph+O%27Rourke&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DInJL6iAaIQQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Bezdek2010-126"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bezdek2010_126-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKároly_Bezdek2010" class="citation book cs1"><a href="/wiki/K%C3%A1roly_Bezdek" title="Károly Bezdek">Károly Bezdek</a> (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Tov0d9VMOfMC"><i>Classical Topics in Discrete Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-0600-7" title="Special:BookSources/978-1-4419-0600-7"><bdi>978-1-4419-0600-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228051643/https://books.google.com/books?id=Tov0d9VMOfMC">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Topics+in+Discrete+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2010&rft.isbn=978-1-4419-0600-7&rft.au=K%C3%A1roly+Bezdek&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTov0d9VMOfMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-PreparataShamos2012-127"><span class="mw-cite-backlink"><b><a href="#cite_ref-PreparataShamos2012_127-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranco_P._PreparataMichael_I._Shamos2012" class="citation book cs1"><a href="/wiki/Franco_P._Preparata" title="Franco P. Preparata">Franco P. Preparata</a>; <a href="/wiki/Michael_Ian_Shamos" title="Michael Ian Shamos">Michael I. Shamos</a> (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_p3eBwAAQBAJ"><i>Computational Geometry: An Introduction</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-1098-6" title="Special:BookSources/978-1-4612-1098-6"><bdi>978-1-4612-1098-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228093733/https://books.google.com/books?id=_p3eBwAAQBAJ">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Geometry%3A+An+Introduction&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4612-1098-6&rft.au=Franco+P.+Preparata&rft.au=Michael+I.+Shamos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_p3eBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-GuYau2008-128"><span class="mw-cite-backlink"><b><a href="#cite_ref-GuYau2008_128-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXianfeng_David_GuShing-Tung_Yau2008" class="citation book cs1">Xianfeng David Gu; Shing-Tung Yau (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4FDvAAAAMAAJ"><i>Computational Conformal Geometry</i></a>. International Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57146-171-1" title="Special:BookSources/978-1-57146-171-1"><bdi>978-1-57146-171-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224054942/https://books.google.com/books?id=4FDvAAAAMAAJ">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Conformal+Geometry&rft.pub=International+Press&rft.date=2008&rft.isbn=978-1-57146-171-1&rft.au=Xianfeng+David+Gu&rft.au=Shing-Tung+Yau&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4FDvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Löh2017-129"><span class="mw-cite-backlink">^ <a href="#cite_ref-Löh2017_129-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Löh2017_129-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClara_Löh2017" class="citation book cs1">Clara Löh (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1AxEDwAAQBAJ"><i>Geometric Group Theory: An Introduction</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-72254-2" title="Special:BookSources/978-3-319-72254-2"><bdi>978-3-319-72254-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191229132923/https://books.google.com/books?id=1AxEDwAAQBAJ">Archived</a> from the original on 29 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Group+Theory%3A+An+Introduction&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-319-72254-2&rft.au=Clara+L%C3%B6h&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1AxEDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-MorganTian2014-130"><span class="mw-cite-backlink"><b><a href="#cite_ref-MorganTian2014_130-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_MorganGang_Tian2014" class="citation book cs1">John Morgan; Gang Tian (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Qv2cAwAAQBAJ"><i>The Geometrization Conjecture</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5201-9" title="Special:BookSources/978-0-8218-5201-9"><bdi>978-0-8218-5201-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224030537/https://books.google.com/books?id=Qv2cAwAAQBAJ">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometrization+Conjecture&rft.pub=American+Mathematical+Soc.&rft.date=2014&rft.isbn=978-0-8218-5201-9&rft.au=John+Morgan&rft.au=Gang+Tian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQv2cAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Wise2012-131"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wise2012_131-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaniel_T._Wise2012" class="citation book cs1">Daniel T. Wise (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GsTW5oQhRPkC"><i>From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8800-1" title="Special:BookSources/978-0-8218-8800-1"><bdi>978-0-8218-8800-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228115647/https://books.google.com/books?id=GsTW5oQhRPkC">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Riches+to+Raags%3A+3-Manifolds%2C+Right-Angled+Artin+Groups%2C+and+Cubical+Geometry%3A+3-manifolds%2C+Right-angled+Artin+Groups%2C+and+Cubical+Geometry&rft.pub=American+Mathematical+Soc.&rft.date=2012&rft.isbn=978-0-8218-8800-1&rft.au=Daniel+T.+Wise&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGsTW5oQhRPkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Meurant2014-132"><span class="mw-cite-backlink">^ <a href="#cite_ref-Meurant2014_132-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Meurant2014_132-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerard_Meurant2014" class="citation book cs1">Gerard Meurant (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M2viBQAAQBAJ"><i>Handbook of Convex Geometry</i></a>. Elsevier Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-093439-6" title="Special:BookSources/978-0-08-093439-6"><bdi>978-0-08-093439-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210901183208/https://books.google.com/books?id=M2viBQAAQBAJ">Archived</a> from the original on 1 September 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Convex+Geometry&rft.pub=Elsevier+Science&rft.date=2014&rft.isbn=978-0-08-093439-6&rft.au=Gerard+Meurant&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DM2viBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Richter-Gebert2011-133"><span class="mw-cite-backlink"><b><a href="#cite_ref-Richter-Gebert2011_133-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJürgen_Richter-Gebert2011" class="citation book cs1">Jürgen Richter-Gebert (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=F_NP8Kub2XYC"><i>Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-17286-1" title="Special:BookSources/978-3-642-17286-1"><bdi>978-3-642-17286-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191229224621/https://books.google.com/books?id=F_NP8Kub2XYC">Archived</a> from the original on 29 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Perspectives+on+Projective+Geometry%3A+A+Guided+Tour+Through+Real+and+Complex+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-3-642-17286-1&rft.au=J%C3%BCrgen+Richter-Gebert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DF_NP8Kub2XYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Elam2001-134"><span class="mw-cite-backlink"><b><a href="#cite_ref-Elam2001_134-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimberly_Elam2001" class="citation book cs1">Kimberly Elam (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JXIEz2XYnp8C"><i>Geometry of Design: Studies in Proportion and Composition</i></a>. Princeton Architectural Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56898-249-6" title="Special:BookSources/978-1-56898-249-6"><bdi>978-1-56898-249-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191231135913/https://books.google.com/books?id=JXIEz2XYnp8C">Archived</a> from the original on 31 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Design%3A+Studies+in+Proportion+and+Composition&rft.pub=Princeton+Architectural+Press&rft.date=2001&rft.isbn=978-1-56898-249-6&rft.au=Kimberly+Elam&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJXIEz2XYnp8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Guigar2004-135"><span class="mw-cite-backlink"><b><a href="#cite_ref-Guigar2004_135-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrad_J._Guigar2004" class="citation book cs1">Brad J. Guigar (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82"><i>The Everything Cartooning Book: Create Unique And Inspired Cartoons For Fun And Profit</i></a>. Adams Media. pp. 82–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4405-2305-2" title="Special:BookSources/978-1-4405-2305-2"><bdi>978-1-4405-2305-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227032210/https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Everything+Cartooning+Book%3A+Create+Unique+And+Inspired+Cartoons+For+Fun+And+Profit&rft.pages=82-&rft.pub=Adams+Media&rft.date=2004&rft.isbn=978-1-4405-2305-2&rft.au=Brad+J.+Guigar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7gftDQAAQBAJ%26pg%3DPT82&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Livio2008-136"><span class="mw-cite-backlink"><b><a href="#cite_ref-Livio2008_136-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMario_Livio2008" class="citation book cs1">Mario Livio (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bUARfgWRH14C&pg=PA166"><i>The Golden Ratio: The Story of PHI, the World's Most Astonishing Number</i></a>. Crown/Archetype. p. 166. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-307-48552-6" title="Special:BookSources/978-0-307-48552-6"><bdi>978-0-307-48552-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191230093236/https://books.google.com/books?id=bUARfgWRH14C&pg=PA166">Archived</a> from the original on 30 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Golden+Ratio%3A+The+Story+of+PHI%2C+the+World%27s+Most+Astonishing+Number&rft.pages=166&rft.pub=Crown%2FArchetype&rft.date=2008&rft.isbn=978-0-307-48552-6&rft.au=Mario+Livio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbUARfgWRH14C%26pg%3DPA166&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-EmmerSchattschneider2007-137"><span class="mw-cite-backlink"><b><a href="#cite_ref-EmmerSchattschneider2007_137-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichele_EmmerDoris_Schattschneider2007" class="citation book cs1">Michele Emmer; <a href="/wiki/Doris_Schattschneider" title="Doris Schattschneider">Doris Schattschneider</a> (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107"><i>M. C. Escher's Legacy: A Centennial Celebration</i></a>. Springer. p. 107. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-28849-7" title="Special:BookSources/978-3-540-28849-7"><bdi>978-3-540-28849-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191222200130/https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107">Archived</a> from the original on 22 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=M.+C.+Escher%27s+Legacy%3A+A+Centennial+Celebration&rft.pages=107&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-28849-7&rft.au=Michele+Emmer&rft.au=Doris+Schattschneider&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5DDyBwAAQBAJ%26pg%3DPA107&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-CapitoloSchwab2004-138"><span class="mw-cite-backlink"><b><a href="#cite_ref-CapitoloSchwab2004_138-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_CapitoloKen_Schwab2004" class="citation book cs1">Robert Capitolo; Ken Schwab (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/drawingcourse1010000capi"><i>Drawing Course 101</i></a></span>. Sterling Publishing Company, Inc. p. <a rel="nofollow" class="external text" href="https://archive.org/details/drawingcourse1010000capi/page/22">22</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4027-0383-6" title="Special:BookSources/978-1-4027-0383-6"><bdi>978-1-4027-0383-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Drawing+Course+101&rft.pages=22&rft.pub=Sterling+Publishing+Company%2C+Inc.&rft.date=2004&rft.isbn=978-1-4027-0383-6&rft.au=Robert+Capitolo&rft.au=Ken+Schwab&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdrawingcourse1010000capi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Gelineau2011-139"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gelineau2011_139-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhyllis_Gelineau2011" class="citation book cs1">Phyllis Gelineau (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55"><i>Integrating the Arts Across the Elementary School Curriculum</i></a>. Cengage Learning. p. 55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-111-30126-2" title="Special:BookSources/978-1-111-30126-2"><bdi>978-1-111-30126-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191207041800/https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55">Archived</a> from the original on 7 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integrating+the+Arts+Across+the+Elementary+School+Curriculum&rft.pages=55&rft.pub=Cengage+Learning&rft.date=2011&rft.isbn=978-1-111-30126-2&rft.au=Phyllis+Gelineau&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1Ib0mUl_VhwC%26pg%3DPA55&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-CeccatoHesselgren2016-140"><span class="mw-cite-backlink"><b><a href="#cite_ref-CeccatoHesselgren2016_140-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCristiano_CeccatoLars_HesselgrenMark_PaulyHelmut_Pottmann,_Johannes_Wallner2016" class="citation book cs1">Cristiano Ceccato; Lars Hesselgren; Mark Pauly; Helmut Pottmann, Johannes Wallner (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6"><i>Advances in Architectural Geometry 2010</i></a>. Birkhäuser. p. 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-99043-371-3" title="Special:BookSources/978-3-99043-371-3"><bdi>978-3-99043-371-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191225201452/https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6">Archived</a> from the original on 25 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advances+in+Architectural+Geometry+2010&rft.pages=6&rft.pub=Birkh%C3%A4user&rft.date=2016&rft.isbn=978-3-99043-371-3&rft.au=Cristiano+Ceccato&rft.au=Lars+Hesselgren&rft.au=Mark+Pauly&rft.au=Helmut+Pottmann%2C+Johannes+Wallner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq45sDwAAQBAJ%26pg%3DPA6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Pottmann2007-141"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pottmann2007_141-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelmut_Pottmann2007" class="citation book cs1">Helmut Pottmann (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bIceAQAAIAAJ"><i>Architectural geometry</i></a>. Bentley Institute Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-934493-04-5" title="Special:BookSources/978-1-934493-04-5"><bdi>978-1-934493-04-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224030536/https://books.google.com/books?id=bIceAQAAIAAJ">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Architectural+geometry&rft.pub=Bentley+Institute+Press&rft.date=2007&rft.isbn=978-1-934493-04-5&rft.au=Helmut+Pottmann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbIceAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-MoffettFazio2003-142"><span class="mw-cite-backlink"><b><a href="#cite_ref-MoffettFazio2003_142-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarian_MoffettMichael_W._FazioLawrence_Wodehouse2003" class="citation book cs1">Marian Moffett; Michael W. Fazio; Lawrence Wodehouse (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IFMohetegAcC&pg=PT371"><i>A World History of Architecture</i></a>. Laurence King Publishing. p. 371. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85669-371-4" title="Special:BookSources/978-1-85669-371-4"><bdi>978-1-85669-371-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227145458/https://books.google.com/books?id=IFMohetegAcC&pg=PT371">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+World+History+of+Architecture&rft.pages=371&rft.pub=Laurence+King+Publishing&rft.date=2003&rft.isbn=978-1-85669-371-4&rft.au=Marian+Moffett&rft.au=Michael+W.+Fazio&rft.au=Lawrence+Wodehouse&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIFMohetegAcC%26pg%3DPT371&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-GreenGreen1985-143"><span class="mw-cite-backlink"><b><a href="#cite_ref-GreenGreen1985_143-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobin_M._GreenRobin_Michael_Green1985" class="citation book cs1">Robin M. Green; Robin Michael Green (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1"><i>Spherical Astronomy</i></a>. Cambridge University Press. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-31779-5" title="Special:BookSources/978-0-521-31779-5"><bdi>978-0-521-31779-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191221211420/https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1">Archived</a> from the original on 21 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spherical+Astronomy&rft.pages=1&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=978-0-521-31779-5&rft.au=Robin+M.+Green&rft.au=Robin+Michael+Green&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwOpaUFQFwTwC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Alekseevskiĭ2008-144"><span class="mw-cite-backlink"><b><a href="#cite_ref-Alekseevskiĭ2008_144-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDmitriĭ_Vladimirovich_Alekseevskiĭ2008" class="citation book cs1">Dmitriĭ Vladimirovich Alekseevskiĭ (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K6-TgxMKu4QC"><i>Recent Developments in Pseudo-Riemannian Geometry</i></a>. European Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-051-7" title="Special:BookSources/978-3-03719-051-7"><bdi>978-3-03719-051-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228115649/https://books.google.com/books?id=K6-TgxMKu4QC">Archived</a> from the original on 28 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recent+Developments+in+Pseudo-Riemannian+Geometry&rft.pub=European+Mathematical+Society&rft.date=2008&rft.isbn=978-3-03719-051-7&rft.au=Dmitri%C4%AD+Vladimirovich+Alekseevski%C4%AD&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK6-TgxMKu4QC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-YauNadis2010-145"><span class="mw-cite-backlink"><b><a href="#cite_ref-YauNadis2010_145-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShing-Tung_YauSteve_Nadis2010" class="citation book cs1">Shing-Tung Yau; Steve Nadis (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M40Ytp8Os_gC"><i>The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions</i></a>. Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-02266-3" title="Special:BookSources/978-0-465-02266-3"><bdi>978-0-465-02266-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224015855/https://books.google.com/books?id=M40Ytp8Os_gC">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Shape+of+Inner+Space%3A+String+Theory+and+the+Geometry+of+the+Universe%27s+Hidden+Dimensions&rft.pub=Basic+Books&rft.date=2010&rft.isbn=978-0-465-02266-3&rft.au=Shing-Tung+Yau&rft.au=Steve+Nadis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DM40Ytp8Os_gC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-146"><span class="mw-cite-backlink"><b><a href="#cite_ref-146">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBengtssonŻyczkowski2017" class="citation book cs1">Bengtsson, Ingemar; <a href="/wiki/Karol_%C5%BByczkowski" title="Karol Życzkowski">Życzkowski, Karol</a> (2017). <i>Geometry of Quantum States: An Introduction to Quantum Entanglement</i> (2nd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-02625-4" title="Special:BookSources/978-1-107-02625-4"><bdi>978-1-107-02625-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1004572791">1004572791</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Quantum+States%3A+An+Introduction+to+Quantum+Entanglement&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2017&rft_id=info%3Aoclcnum%2F1004572791&rft.isbn=978-1-107-02625-4&rft.aulast=Bengtsson&rft.aufirst=Ingemar&rft.au=%C5%BByczkowski%2C+Karol&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-FlandersPrice2014-147"><span class="mw-cite-backlink"><b><a href="#cite_ref-FlandersPrice2014_147-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarley_FlandersJustin_J._Price2014" class="citation book cs1">Harley Flanders; Justin J. Price (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5abiBQAAQBAJ"><i>Calculus with Analytic Geometry</i></a>. Elsevier Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-6240-6" title="Special:BookSources/978-1-4832-6240-6"><bdi>978-1-4832-6240-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191224175037/https://books.google.com/books?id=5abiBQAAQBAJ">Archived</a> from the original on 24 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+with+Analytic+Geometry&rft.pub=Elsevier+Science&rft.date=2014&rft.isbn=978-1-4832-6240-6&rft.au=Harley+Flanders&rft.au=Justin+J.+Price&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5abiBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-RogawskiAdams2015-148"><span class="mw-cite-backlink"><b><a href="#cite_ref-RogawskiAdams2015_148-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJon_RogawskiColin_Adams2015" class="citation book cs1">Jon Rogawski; Colin Adams (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OWeZBgAAQBAJ"><i>Calculus</i></a>. W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4641-7499-5" title="Special:BookSources/978-1-4641-7499-5"><bdi>978-1-4641-7499-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200101083409/https://books.google.com/books?id=OWeZBgAAQBAJ">Archived</a> from the original on 1 January 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.pub=W.+H.+Freeman&rft.date=2015&rft.isbn=978-1-4641-7499-5&rft.au=Jon+Rogawski&rft.au=Colin+Adams&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOWeZBgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Lozano-Robledo2019-149"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lozano-Robledo2019_149-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFÁlvaro_Lozano-Robledo2019" class="citation book cs1">Álvaro Lozano-Robledo (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ESiODwAAQBAJ"><i>Number Theory and Geometry: An Introduction to Arithmetic Geometry</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-5016-8" title="Special:BookSources/978-1-4704-5016-8"><bdi>978-1-4704-5016-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227145316/https://books.google.com/books?id=ESiODwAAQBAJ">Archived</a> from the original on 27 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+and+Geometry%3A+An+Introduction+to+Arithmetic+Geometry&rft.pub=American+Mathematical+Soc.&rft.date=2019&rft.isbn=978-1-4704-5016-8&rft.au=%C3%81lvaro+Lozano-Robledo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DESiODwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-Sangalli2009-150"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sangalli2009_150-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArturo_Sangalli2009" class="citation book cs1">Arturo Sangalli (2009). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/pythagorasreveng0000sang"><i>Pythagoras' Revenge: A Mathematical Mystery</i></a></span>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/pythagorasreveng0000sang/page/57">57</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-04955-7" title="Special:BookSources/978-0-691-04955-7"><bdi>978-0-691-04955-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pythagoras%27+Revenge%3A+A+Mathematical+Mystery&rft.pages=57&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-04955-7&rft.au=Arturo+Sangalli&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpythagorasreveng0000sang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> <li id="cite_note-CornellSilverman2013-151"><span class="mw-cite-backlink"><b><a href="#cite_ref-CornellSilverman2013_151-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGary_CornellJoseph_H._SilvermanGlenn_Stevens2013" class="citation book cs1">Gary Cornell; Joseph H. Silverman; Glenn Stevens (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jD3TBwAAQBAJ"><i>Modular Forms and Fermat's Last Theorem</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-1974-3" title="Special:BookSources/978-1-4612-1974-3"><bdi>978-1-4612-1974-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191230181409/https://books.google.com/books?id=jD3TBwAAQBAJ">Archived</a> from the original on 30 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">25 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modular+Forms+and+Fermat%27s+Last+Theorem&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4612-1974-3&rft.au=Gary+Cornell&rft.au=Joseph+H.+Silverman&rft.au=Glenn+Stevens&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjD3TBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1991" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, C.B.</a> (1991) [1989]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye"><i>A History of Mathematics</i></a></span> (Second edition, revised by <a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Uta C. Merzbach</a> ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.place=New+York&rft.edition=Second+edition%2C+revised+by+Uta+C.+Merzbach&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=C.B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooke2005" class="citation book cs1">Cooke, Roger (2005). <i>The History of Mathematics</i>. New York: Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-44459-6" title="Special:BookSources/978-0-471-44459-6"><bdi>978-0-471-44459-6</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+History+of+Mathematics&rft.place=New+York&rft.pub=Wiley-Interscience&rft.date=2005&rft.isbn=978-0-471-44459-6&rft.aulast=Cooke&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayashi2003" class="citation book cs1">Hayashi, Takao (2003). "Indian Mathematics". In Grattan-Guinness, Ivor (ed.). <i>Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences</i>. Vol. 1. Baltimore, MD: The <a href="/wiki/Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>. pp. 118–130. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-7396-6" title="Special:BookSources/978-0-8018-7396-6"><bdi>978-0-8018-7396-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Indian+Mathematics&rft.btitle=Companion+Encyclopedia+of+the+History+and+Philosophy+of+the+Mathematical+Sciences&rft.place=Baltimore%2C+MD&rft.pages=118-130&rft.pub=The+Johns+Hopkins+University+Press&rft.date=2003&rft.isbn=978-0-8018-7396-6&rft.aulast=Hayashi&rft.aufirst=Takao&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayashi2005" class="citation book cs1">Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.). <i>The Blackwell Companion to Hinduism</i>. Oxford: <a href="/wiki/Basil_Blackwell" title="Basil Blackwell">Basil Blackwell</a>. pp. 360–375. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4051-3251-0" title="Special:BookSources/978-1-4051-3251-0"><bdi>978-1-4051-3251-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Indian+Mathematics&rft.btitle=The+Blackwell+Companion+to+Hinduism&rft.place=Oxford&rft.pages=360-375&rft.pub=Basil+Blackwell&rft.date=2005&rft.isbn=978-1-4051-3251-0&rft.aulast=Hayashi&rft.aufirst=Takao&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Jay_Kappraff" title="Jay Kappraff">Jay Kappraff</a> (2014). <a rel="nofollow" class="external text" href="http://www.worldscientific.com/worldscibooks/10.1142/8952"><i>A Participatory Approach to Modern Geometry</i></a>. World Scientific Publishing. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F8952">10.1142/8952</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4556-70-5" title="Special:BookSources/978-981-4556-70-5"><bdi>978-981-4556-70-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1364.00004">1364.00004</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Participatory+Approach+to+Modern+Geometry&rft.pub=World+Scientific+Publishing&rft.date=2014&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1364.00004%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1142%2F8952&rft.isbn=978-981-4556-70-5&rft.au=Jay+Kappraff&rft_id=http%3A%2F%2Fwww.worldscientific.com%2Fworldscibooks%2F10.1142%2F8952&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikolai_I._Lobachevsky2010" class="citation book cs1">Nikolai I. Lobachevsky (2010). <i>Pangeometry</i>. Heritage of European Mathematics Series. Vol. 4. translator and editor: A. Papadopoulos. European Mathematical Society.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pangeometry&rft.series=Heritage+of+European+Mathematics+Series&rft.pub=European+Mathematical+Society&rft.date=2010&rft.au=Nikolai+I.+Lobachevsky&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Leonard_Mlodinow" title="Leonard Mlodinow">Leonard Mlodinow</a> (2002). <i>Euclid's Window – The Story of Geometry from Parallel Lines to Hyperspace</i> (UK ed.). Allen Lane. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7139-9634-0" title="Special:BookSources/978-0-7139-9634-0"><bdi>978-0-7139-9634-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euclid%27s+Window+%E2%80%93+The+Story+of+Geometry+from+Parallel+Lines+to+Hyperspace&rft.edition=UK&rft.pub=Allen+Lane&rft.date=2002&rft.isbn=978-0-7139-9634-0&rft.au=Leonard+Mlodinow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></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:r1250146164">.mw-parser-output .sister-box .side-box-abovebelow{padding:0.75em 0;text-align:center}.mw-parser-output .sister-box .side-box-abovebelow>b{display:block}.mw-parser-output .sister-box .side-box-text>ul{border-top:1px solid #aaa;padding:0.75em 0;width:217px;margin:0 auto}.mw-parser-output .sister-box .side-box-text>ul>li{min-height:31px}.mw-parser-output .sister-logo{display:inline-block;width:31px;line-height:31px;vertical-align:middle;text-align:center}.mw-parser-output .sister-link{display:inline-block;margin-left:4px;width:182px;vertical-align:middle}@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-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style><div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> <b>Geometry</b> at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects"><span id="sister-projects">sister projects</span></a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/41px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/54px-Wiktionary-logo-v2.svg.png 2x" data-file-width="391" data-file-height="391" /></span></span></span><span class="sister-link"><a href="https://en.wiktionary.org/wiki/Special:Search/Geometry" class="extiw" title="wikt:Special:Search/Geometry">Definitions</a> from Wiktionary</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span><span class="sister-link"><a href="https://commons.wikimedia.org/wiki/Category:Geometry" class="extiw" title="c:Category:Geometry">Media</a> from Commons</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/27px-Wikinews-logo.svg.png" decoding="async" width="27" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/41px-Wikinews-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/54px-Wikinews-logo.svg.png 2x" data-file-width="759" data-file-height="415" /></span></span></span><span class="sister-link"><a href="https://en.wikinews.org/wiki/Special:Search/Geometry" class="extiw" title="n:Special:Search/Geometry">News</a> from Wikinews</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" decoding="async" width="23" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/35px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/46px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></span><span class="sister-link"><a href="https://en.wikiquote.org/wiki/Geometry" class="extiw" title="q:Geometry">Quotations</a> from Wikiquote</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/26px-Wikisource-logo.svg.png" decoding="async" width="26" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/39px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/51px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></span><span class="sister-link"><a href="https://en.wikisource.org/wiki/Special:Search/Geometry" class="extiw" title="s:Special:Search/Geometry">Texts</a> from Wikisource</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></span><span class="sister-link"><a href="https://en.wikibooks.org/wiki/Geometry" class="extiw" title="b:Geometry">Textbooks</a> from Wikibooks</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></span><span class="sister-link"><a href="https://en.wikiversity.org/wiki/Geometry" class="extiw" title="v:Geometry">Resources</a> from Wikiversity</span></li></ul></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><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 screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">Wikibooks has more on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Special:Search/Geometry" class="extiw" title="wikibooks:Special:Search/Geometry">Geometry</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><div class="side-box metadata side-box-right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> <a href="/wiki/Wikipedia:The_Wikipedia_Library" title="Wikipedia:The Wikipedia Library">Library resources</a> about <br /> <b>Geometry</b> <hr /></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Geometry">Resources in your library</a></li> </ul></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Geometry"><a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Geometry">"Geometry" </a></span>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. Vol. 11 (11th ed.). 1911. pp. 675–736.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Geometry&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=675-736&rft.edition=11th&rft.date=1911&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometry" class="Z3988"></span></li> <li>A <a href="https://en.wikiversity.org/wiki/Geometry" class="extiw" title="v:Geometry">geometry</a> course from <a href="https://en.wikiversity.org/wiki/" class="extiw" title="v:">Wikiversity</a></li> <li><a rel="nofollow" class="external text" href="http://www.8foxes.com/"><i>Unusual Geometry Problems</i></a></li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/library/topics/geometry/"><i>The Math Forum</i> – Geometry</a> <ul><li><a rel="nofollow" class="external text" href="http://mathforum.org/geometry/k12.geometry.html"><i>The Math Forum</i> – K–12 Geometry</a></li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/geometry/coll.geometry.html"><i>The Math Forum</i> – College Geometry</a></li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/advanced/geom.html"><i>The Math Forum</i> – Advanced Geometry</a></li></ul></li> <li><a rel="nofollow" class="external text" href="http://precedings.nature.com/documents/2153/version/1/">Nature Precedings – <i>Pegs and Ropes Geometry at Stonehenge</i></a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060906203141/http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html"><i>The Mathematical Atlas</i> – Geometric Areas of Mathematics</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071004174210/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618">"4000 Years of Geometry"</a>, lecture by Robin Wilson given at <a href="/wiki/Gresham_College" title="Gresham College">Gresham College</a>, 3 October 2007 (available for MP3 and MP4 download as well as a text file) <ul><li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/geometry-finitism/">Finitism in Geometry</a> at the Stanford Encyclopedia of Philosophy</li></ul></li> <li><a rel="nofollow" class="external text" href="http://www.ics.uci.edu/~eppstein/junkyard/topic.html">The Geometry Junkyard</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com">Interactive geometry reference with hundreds of applets</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090321024112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm">Dynamic Geometry Sketches (with some Student Explorations)</a></li> <li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/?video=ca-geometry--area--pythagorean-theorem#california-standards-test-geometry">Geometry classes</a> at <a href="/wiki/Khan_Academy" title="Khan Academy">Khan Academy</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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.navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Geometry" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Geometry" title="Template:Geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Geometry" title="Template talk:Geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geometry" title="Special:EditPage/Template:Geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Geometry" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Geometry</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_geometry" title="History of geometry">History</a> <ul><li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">Timeline</a></li></ul></li> <li><a href="/wiki/Lists_of_geometry_topics" class="mw-redirect" title="Lists of geometry topics">Lists</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean <br /> geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Combinatorial</a></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane geometry</a> <ul><li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li> <li><a href="/wiki/Polyform" title="Polyform">Polyform</a></li></ul></li> <li><a href="/wiki/Solid_geometry" title="Solid geometry">Solid geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean <br /> geometry</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Shape" title="Shape">Shape</a> <ul><li><a href="/wiki/Lists_of_shapes" title="Lists of shapes">Lists</a></li></ul></li> <li><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">List of geometry topics</a></li> <li><a href="/wiki/List_of_differential_geometry_topics" title="List of differential geometry topics">List of differential geometry topics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Geometry" title="Category:Geometry">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a 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0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output 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id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q8087#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q8087#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, 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1 Template:General_geometry"," 4.56% 78.918 1 Template:Short_description"," 4.40% 76.165 4 Template:Cite_web"," 4.06% 70.329 1 Template:Excerpt"]},"scribunto":{"limitreport-timeusage":{"value":"1.125","limit":"10.000"},"limitreport-memusage":{"value":27356435,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAhlfors1979\"] = 1,\n [\"CITEREFArturo_Sangalli2009\"] = 1,\n [\"CITEREFAudun_Holme2010\"] = 1,\n [\"CITEREFB._RosenfeldBill_Wiebe2013\"] = 1,\n [\"CITEREFBengtssonŻyczkowski2017\"] = 1,\n [\"CITEREFBill_JacobTsit-Yuen_Lam1994\"] = 1,\n [\"CITEREFBoyer1991\"] = 1,\n [\"CITEREFBrad_J._Guigar2004\"] = 1,\n [\"CITEREFBrian_J._Cantwell2002\"] = 1,\n [\"CITEREFC._H._Edwards_Jr.2012\"] = 1,\n [\"CITEREFC._R._Wylie2011\"] = 1,\n [\"CITEREFCarl_B._Boyer2012\"] = 1,\n [\"CITEREFCarmo1976\"] = 1,\n [\"CITEREFCharles_Jasper_Joly1895\"] = 1,\n [\"CITEREFCharles_NashSiddhartha_Sen1988\"] = 1,\n [\"CITEREFCharles_W._Misner2005\"] = 1,\n [\"CITEREFChisholm1911\"] = 1,\n [\"CITEREFChuanming_Zong2006\"] = 1,\n [\"CITEREFClara_Löh2017\"] = 1,\n [\"CITEREFClark1985\"] = 1,\n [\"CITEREFCooke2005\"] = 1,\n [\"CITEREFCristiano_CeccatoLars_HesselgrenMark_PaulyHelmut_Pottmann,_Johannes_Wallner2016\"] = 1,\n [\"CITEREFDaniel_T._Wise2012\"] = 1,\n [\"CITEREFDavid_Berlinski2014\"] = 1,\n [\"CITEREFDepuydt1998\"] = 1,\n [\"CITEREFDmitriĭ_Vladimirovich_Alekseevskiĭ2008\"] = 1,\n [\"CITEREFDonaldson2011\"] = 1,\n [\"CITEREFEduardo_Bayro-Corrochano2018\"] = 1,\n [\"CITEREFEverett_W._HoweKristin_E._LauterJudy_L._Walker2017\"] = 1,\n [\"CITEREFFrancis_Buekenhout1995\"] = 1,\n [\"CITEREFFranco_P._PreparataMichael_I._Shamos2012\"] = 1,\n [\"CITEREFFriberg1981\"] = 1,\n [\"CITEREFG._Gierz2006\"] = 1,\n [\"CITEREFGary_CornellJoseph_H._SilvermanGlenn_Stevens2013\"] = 1,\n [\"CITEREFGelʹfand2001\"] = 1,\n [\"CITEREFGeorge_E._Martin1996\"] = 1,\n [\"CITEREFGeorge_E._Martin2012\"] = 1,\n [\"CITEREFGeorge_L._Hersey2001\"] = 1,\n [\"CITEREFGerard_Meurant2014\"] = 1,\n [\"CITEREFGerard_Walschap2015\"] = 1,\n [\"CITEREFGerla,_G.1995\"] = 1,\n [\"CITEREFGilbert_Strang1991\"] = 1,\n [\"CITEREFH._S._Bear2002\"] = 1,\n [\"CITEREFHarley_Flanders2012\"] = 1,\n [\"CITEREFHarley_FlandersJustin_J._Price2014\"] = 1,\n [\"CITEREFHayashi2003\"] = 1,\n [\"CITEREFHayashi2005\"] = 1,\n [\"CITEREFHelmut_Pottmann2007\"] = 1,\n [\"CITEREFHuybrechts2005\"] = 1,\n [\"CITEREFI._M._Yaglom2012\"] = 1,\n [\"CITEREFIan_Stewart2008\"] = 1,\n [\"CITEREFJ._P._May1999\"] = 1,\n [\"CITEREFJames_CarlsonJames_A._CarlsonArthur_JaffeAndrew_Wiles2006\"] = 1,\n [\"CITEREFJames_R._Choike1980\"] = 1,\n [\"CITEREFJames_W._Cannon2017\"] = 1,\n [\"CITEREFJean_Dieudonné1985\"] = 1,\n [\"CITEREFJeremy_Gray2011\"] = 1,\n [\"CITEREFJiří_Matoušek2013\"] = 1,\n [\"CITEREFJohn_Casey1885\"] = 1,\n [\"CITEREFJohn_MorganGang_Tian2014\"] = 1,\n [\"CITEREFJon_RogawskiColin_Adams2015\"] = 1,\n [\"CITEREFJost2002\"] = 1,\n [\"CITEREFJudith_V._FieldJeremy_Gray2012\"] = 1,\n [\"CITEREFJürgen_Richter-Gebert2011\"] = 1,\n [\"CITEREFKimberly_Elam2001\"] = 1,\n [\"CITEREFKurt_Von_Fritz1945\"] = 1,\n [\"CITEREFKároly_Bezdek2010\"] = 1,\n [\"CITEREFLinnaeus_Wayland_Dowling1917\"] = 1,\n [\"CITEREFMarcos_MarinoMichael_ThaddeusRavi_Vakil2008\"] = 1,\n [\"CITEREFMarian_MoffettMichael_W._FazioLawrence_Wodehouse2003\"] = 1,\n [\"CITEREFMario_Livio2008\"] = 1,\n [\"CITEREFMark_A._Freitag2013\"] = 1,\n [\"CITEREFMark_Blacklock2018\"] = 1,\n [\"CITEREFMartin_D._Crossley2011\"] = 1,\n [\"CITEREFMatthew_HeSergey_Petoukhov2011\"] = 1,\n [\"CITEREFMichele_EmmerDoris_Schattschneider2007\"] = 1,\n [\"CITEREFMichio_Kaku2012\"] = 1,\n [\"CITEREFMladen_BestvinaMichah_SageevKaren_Vogtmann2014\"] = 1,\n [\"CITEREFMorris_Kline1990\"] = 1,\n [\"CITEREFMumford1999\"] = 1,\n [\"CITEREFMunkres2000\"] = 1,\n [\"CITEREFNeugebauer1969\"] = 1,\n [\"CITEREFNihat_AyJürgen_JostHông_Vân_LêLorenz_Schwachhöfer2017\"] = 1,\n [\"CITEREFNikolai_I._Lobachevsky2010\"] = 1,\n [\"CITEREFO\u0026#039;Connor,_J.J.Robertson,_E.F.1996\"] = 1,\n [\"CITEREFOssendrijver2016\"] = 1,\n [\"CITEREFP.A.M._Dirac2016\"] = 1,\n [\"CITEREFP._VanícekE.J._Krakiwsky2015\"] = 1,\n [\"CITEREFPat_HerbstTaro_FujitaStefan_HalverscheidMichael_Weiss2017\"] = 1,\n [\"CITEREFPaul_MarriottMark_Salmon2000\"] = 1,\n [\"CITEREFPeter_M._Gruber2007\"] = 1,\n [\"CITEREFPeter_Pesic2007\"] = 1,\n [\"CITEREFPhyllis_Gelineau2011\"] = 1,\n [\"CITEREFRobert_CapitoloKen_Schwab2004\"] = 1,\n [\"CITEREFRobert_E._ButtsJ.R._Brown2012\"] = 1,\n [\"CITEREFRobin_Hartshorne2013\"] = 2,\n [\"CITEREFRobin_M._GreenRobin_Michael_Green1985\"] = 1,\n [\"CITEREFRoger_Temam2013\"] = 1,\n [\"CITEREFRoy_Williams1998\"] = 1,\n [\"CITEREFRussell_M._CummingsScott_A._MortonWilliam_H._MasonDavid_R._McDaniel2015\"] = 1,\n [\"CITEREFRāshid1994\"] = 1,\n [\"CITEREFSatyan_L._DevadossJoseph_O\u0026#039;Rourke2011\"] = 1,\n [\"CITEREFSchmidtHouangCogan2002\"] = 1,\n [\"CITEREFShing-Tung_YauSteve_Nadis2010\"] = 1,\n [\"CITEREFShlomo_Libeskind2008\"] = 1,\n [\"CITEREFSlayman1998\"] = 1,\n [\"CITEREFStaal1999\"] = 1,\n [\"CITEREFStakhov_Alexey2009\"] = 1,\n [\"CITEREFSteven_A._Treese2018\"] = 1,\n [\"CITEREFSzmielew1983\"] = 1,\n [\"CITEREFTabak2014\"] = 1,\n [\"CITEREFTerence_Tao2011\"] = 1,\n [\"CITEREFVictor_J._Katz2000\"] = 1,\n [\"CITEREFVincenzo_De_Risi2015\"] = 1,\n [\"CITEREFW-H._Steeb1996\"] = 1,\n [\"CITEREFW._Abbot2013\"] = 1,\n [\"CITEREFWald1984\"] = 1,\n [\"CITEREFWalter_A._Meyer2006\"] = 1,\n [\"CITEREFWells2008\"] = 1,\n [\"CITEREFWerner_Hahn1998\"] = 1,\n [\"CITEREFXianfeng_David_GuShing-Tung_Yau2008\"] = 1,\n [\"CITEREFÁlvaro_Lozano-Robledo2019\"] = 1,\n [\"Measures\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 1,\n [\"=\"] = 4,\n [\"Anchor\"] = 1,\n [\"Areas of mathematics\"] = 1,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 1,\n [\"Broader\"] = 1,\n [\"Circa\"] = 6,\n [\"Cite EB1911\"] = 1,\n [\"Cite book\"] = 116,\n [\"Cite journal\"] = 7,\n [\"Cite web\"] = 4,\n [\"Efn\"] = 3,\n [\"Etymology\"] = 2,\n [\"Excerpt\"] = 1,\n [\"General geometry\"] = 1,\n [\"Geometry\"] = 1,\n [\"Grc-transl\"] = 3,\n [\"Harv\"] = 10,\n [\"IAST\"] = 1,\n [\"ISBN\"] = 7,\n [\"Lang\"] = 1,\n [\"Library resources box\"] = 1,\n [\"MacTutor Biography\"] = 3,\n [\"Main\"] = 28,\n [\"Main category\"] = 1,\n [\"Math topics TOC\"] = 1,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 4,\n [\"Other uses\"] = 1,\n [\"Portal\"] = 1,\n [\"Pp-move\"] = 1,\n [\"Pp-semi-indef\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"See also\"] = 2,\n [\"Short description\"] = 1,\n [\"Sister project links\"] = 1,\n [\"SpringerEOM\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Wikibooks\"] = 1,\n [\"Wikt-lang\"] = 3,\n}\narticle_whitelist = table#1 {\n}\ntable#1 {\n [\"size\"] = 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