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id="toc-Notation_and_terminology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notation_and_terminology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Notation and terminology</span> </div> </a> <button aria-controls="toc-Notation_and_terminology-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 Notation and terminology subsection</span> </button> <ul id="toc-Notation_and_terminology-sublist" class="vector-toc-list"> <li id="toc-Naming_of_points_and_figures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Naming_of_points_and_figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Naming of points and figures</span> </div> </a> <ul id="toc-Naming_of_points_and_figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complementary_and_supplementary_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complementary_and_supplementary_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Complementary and supplementary angles</span> </div> </a> <ul id="toc-Complementary_and_supplementary_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_versions_of_Euclid's_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_versions_of_Euclid's_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Modern versions of Euclid's notation</span> </div> </a> <ul id="toc-Modern_versions_of_Euclid's_notation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Some_important_or_well_known_results" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Some_important_or_well_known_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Some important or well known results</span> </div> </a> <button aria-controls="toc-Some_important_or_well_known_results-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 Some important or well known results subsection</span> </button> <ul id="toc-Some_important_or_well_known_results-sublist" class="vector-toc-list"> <li id="toc-Pons_asinorum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pons_asinorum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Pons asinorum</span> </div> </a> <ul id="toc-Pons_asinorum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruence_of_triangles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congruence_of_triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Congruence of triangles</span> </div> </a> <ul id="toc-Congruence_of_triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangle_angle_sum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triangle_angle_sum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Triangle angle sum</span> </div> </a> <ul id="toc-Triangle_angle_sum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pythagorean_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pythagorean_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Pythagorean theorem</span> </div> </a> <ul id="toc-Pythagorean_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thales'_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thales'_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Thales' theorem</span> </div> </a> <ul id="toc-Thales'_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaling_of_area_and_volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling_of_area_and_volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Scaling of area and volume</span> </div> </a> <ul id="toc-Scaling_of_area_and_volume-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-System_of_measurement_and_arithmetic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#System_of_measurement_and_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>System of measurement and arithmetic</span> </div> </a> <ul id="toc-System_of_measurement_and_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_engineering" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In engineering</span> </div> </a> <button aria-controls="toc-In_engineering-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 In engineering subsection</span> </button> <ul id="toc-In_engineering-sublist" class="vector-toc-list"> <li id="toc-Design_and_Analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Design_and_Analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Design and Analysis</span> </div> </a> <ul id="toc-Design_and_Analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Dynamics</span> </div> </a> <ul id="toc-Dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CAD_Systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CAD_Systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>CAD Systems</span> </div> </a> <ul id="toc-CAD_Systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Circuit_Design" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circuit_Design"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Circuit Design</span> </div> </a> <ul id="toc-Circuit_Design-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electromagnetic_and_Fluid_Flow_Fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electromagnetic_and_Fluid_Flow_Fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Electromagnetic and Fluid Flow Fields</span> </div> </a> <ul id="toc-Electromagnetic_and_Fluid_Flow_Fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Controls" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Controls"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Controls</span> </div> </a> <ul id="toc-Controls-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_general_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_general_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other general applications</span> </div> </a> <ul id="toc-Other_general_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Later_history" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Later_history"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Later history</span> </div> </a> <button aria-controls="toc-Later_history-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 Later history subsection</span> </button> <ul id="toc-Later_history-sublist" class="vector-toc-list"> <li id="toc-Archimedes_and_Apollonius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Archimedes_and_Apollonius"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Archimedes and Apollonius</span> </div> </a> <ul id="toc-Archimedes_and_Apollonius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-17th_century:_Descartes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#17th_century:_Descartes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>17th century: Descartes</span> </div> </a> <ul id="toc-17th_century:_Descartes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-18th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#18th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>18th century</span> </div> </a> <ul id="toc-18th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-19th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#19th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>19th century</span> </div> </a> <ul id="toc-19th_century-sublist" class="vector-toc-list"> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4.1</span> <span>Higher dimensions</span> </div> </a> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <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">7.4.2</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-20th_century_and_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#20th_century_and_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>20th century and relativity</span> </div> </a> <ul id="toc-20th_century_and_relativity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-As_a_description_of_the_structure_of_space" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#As_a_description_of_the_structure_of_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>As a description of the structure of space</span> </div> </a> <ul id="toc-As_a_description_of_the_structure_of_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Treatment_of_infinity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Treatment_of_infinity"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Treatment of infinity</span> </div> </a> <button aria-controls="toc-Treatment_of_infinity-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 Treatment of infinity subsection</span> </button> <ul id="toc-Treatment_of_infinity-sublist" class="vector-toc-list"> <li id="toc-Infinite_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Infinite objects</span> </div> </a> <ul id="toc-Infinite_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Infinite processes</span> </div> </a> <ul id="toc-Infinite_processes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logical_basis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Logical_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Logical basis</span> </div> </a> <button aria-controls="toc-Logical_basis-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 Logical basis subsection</span> </button> <ul id="toc-Logical_basis-sublist" class="vector-toc-list"> <li id="toc-Classical_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Classical logic</span> </div> </a> <ul id="toc-Classical_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_standards_of_rigor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_standards_of_rigor"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Modern standards of rigor</span> </div> </a> <ul id="toc-Modern_standards_of_rigor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Axiomatic_formulations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Axiomatic_formulations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Axiomatic formulations</span> </div> </a> <ul id="toc-Axiomatic_formulations-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">11</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-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 See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Classical_theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Classical theorems</span> </div> </a> <ul id="toc-Classical_theorems-sublist" class="vector-toc-list"> </ul> </li> </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">12</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">13</span> <span>References</span> </div> </a> <ul id="toc-References-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">14</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 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class="mw-page-title-main">Euclidean 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 83 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-83" 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">83 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Euklidiese_meetkunde" title="Euklidiese meetkunde – Afrikaans" lang="af" hreflang="af" data-title="Euklidiese meetkunde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A9%E1%8A%AD%E1%88%8A%E1%8B%B3%E1%8B%8A_%E1%8C%82%E1%8B%8E%E1%88%9C%E1%89%B5%E1%88%AA" title="ዩክሊዳዊ ጂዎሜትሪ – Amharic" lang="am" hreflang="am" data-title="ዩክሊዳዊ ጂዎሜትሪ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</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%A5%D9%82%D9%84%D9%8A%D8%AF%D9%8A%D8%A9" title="هندسة إقليدية – Arabic" lang="ar" hreflang="ar" data-title="هندسة إقليدية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Xeometr%C3%ADa_eucl%C3%ADdea" title="Xeometría euclídea – Asturian" lang="ast" hreflang="ast" data-title="Xeometría euclídea" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Evklid_h%C9%99nd%C9%99s%C9%99si" title="Evklid həndəsəsi – Azerbaijani" lang="az" hreflang="az" data-title="Evklid həndəsəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%87%E0%A6%89%E0%A6%95%E0%A7%8D%E0%A6%B2%E0%A6%BF%E0%A6%A1%E0%A7%80%E0%A6%AF%E0%A6%BC_%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="ইউক্লিডীয় জ্যামিতি – Bangla" lang="bn" hreflang="bn" data-title="ইউক্লিডীয় জ্যামিতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Евклид геометрияһы – Bashkir" lang="ba" hreflang="ba" data-title="Евклид геометрияһы" 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%AD%D1%9E%D0%BA%D0%BB%D1%96%D0%B4%D0%B0%D0%B2%D0%B0_%D0%B3%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F" title="Эўклідава геаметрыя – Belarusian" lang="be" hreflang="be" data-title="Эўклідава 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%AD%D1%9E%D0%BA%D0%BB%D1%96%D0%B4%D0%B0%D0%B2%D0%B0_%D0%B3%D0%B5%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F" title="Эўклідава геамэтрыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Эўклідава 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Euclidean_geometry" title="Euclidean geometry – Central Bikol" lang="bcl" hreflang="bcl" data-title="Euclidean geometry" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Евклидова геометрия – Bulgarian" lang="bg" hreflang="bg" data-title="Евклидова геометрия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geometria_euclidiana" title="Geometria euclidiana – Catalan" lang="ca" hreflang="ca" data-title="Geometria euclidiana" 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%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B9%C4%95" title="Евклид геометрийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Евклид геометрийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eukleidovsk%C3%A1_geometrie" title="Eukleidovská geometrie – Czech" lang="cs" hreflang="cs" data-title="Eukleidovská geometrie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Geometreg_Euclidaidd" title="Geometreg Euclidaidd – Welsh" lang="cy" hreflang="cy" data-title="Geometreg Euclidaidd" 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/Euklidisk_geometri" title="Euklidisk geometri – Danish" lang="da" hreflang="da" data-title="Euklidisk geometri" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Euklidische_Geometrie" title="Euklidische Geometrie – German" lang="de" hreflang="de" data-title="Euklidische 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/Eukleidese_geomeetria" title="Eukleidese geomeetria – Estonian" lang="et" hreflang="et" data-title="Eukleidese 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%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Ευκλείδεια γεωμετρία – Greek" lang="el" hreflang="el" data-title="Ευκλείδεια γεωμετρία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Geometr%C3%ADa_euclidiana" title="Geometría euclidiana – Spanish" lang="es" hreflang="es" data-title="Geometría euclidiana" 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/E%C5%ADklida_geometrio" title="Eŭklida geometrio – Esperanto" lang="eo" hreflang="eo" data-title="Eŭklida geometrio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Geometria_euklidear" title="Geometria euklidear – Basque" lang="eu" hreflang="eu" data-title="Geometria euklidear" 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_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="هندسه اقلیدسی – Persian" lang="fa" hreflang="fa" data-title="هندسه اقلیدسی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie_euclidienne" title="Géométrie euclidienne – French" lang="fr" hreflang="fr" data-title="Géométrie euclidienne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Xeometr%C3%ADa_euclidiana" title="Xeometría euclidiana – Galician" lang="gl" hreflang="gl" data-title="Xeometría euclidiana" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Euclidean_Geometry" title="Euclidean Geometry – Kikuyu" lang="ki" hreflang="ki" data-title="Euclidean 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%AF%E0%AB%81%E0%AA%95%E0%AB%8D%E0%AA%B2%E0%AA%BF%E0%AA%A1%E0%AA%BF%E0%AA%AF%E0%AA%A8_%E0%AA%AD%E0%AB%82%E0%AA%AE%E0%AA%BF%E0%AA%A4%E0%AA%BF" title="યુક્લિડિયન ભૂમિતિ – Gujarati" lang="gu" hreflang="gu" data-title="યુક્લિડિયન 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/Euclid_K%C3%AD-h%C3%B2" title="Euclid Kí-hò – Hakka Chinese" lang="hak" hreflang="hak" data-title="Euclid Kí-hò" 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/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B8%B0%ED%95%98%ED%95%99" title="유클리드 기하학 – Korean" lang="ko" hreflang="ko" data-title="유클리드 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%B7%D5%BE%D5%AF%D5%AC%D5%AB%D5%A4%D5%A5%D5%BD%D5%B5%D5%A1%D5%B6_%D5%A5%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="Էվկլիդեսյան երկրաչափություն – Armenian" lang="hy" hreflang="hy" data-title="Էվկլիդեսյան 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%AF%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BF%E0%A4%A1%E0%A5%80%E0%A4%AF_%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="यूक्लिडीय ज्यामिति – Hindi" lang="hi" hreflang="hi" data-title="यूक्लिडीय 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/Euklidska_geometrija" title="Euklidska geometrija – Croatian" lang="hr" hreflang="hr" data-title="Euklidska geometrija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geometri_Euklides" title="Geometri Euklides – Indonesian" lang="id" hreflang="id" data-title="Geometri Euklides" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geometria_euclidea" title="Geometria euclidea – Italian" lang="it" hreflang="it" data-title="Geometria euclidea" 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_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99%D7%AA" title="גאומטריה אוקלידית – Hebrew" lang="he" hreflang="he" data-title="גאומטריה אוקלידית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AF%E0%B3%82%E0%B2%95%E0%B3%8D%E0%B2%B2%E0%B2%BF%E0%B2%A1%E0%B3%80%E0%B2%AF_%E0%B2%9C%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%AE%E0%B2%BF%E0%B2%A4%E0%B2%BF" title="ಯೂಕ್ಲಿಡೀಯ ಜ್ಯಾಮಿತಿ – Kannada" lang="kn" hreflang="kn" data-title="ಯೂಕ್ಲಿಡೀಯ ಜ್ಯಾಮಿತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D1%82%D1%96%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Евклидтік геометрия – Kazakh" lang="kk" hreflang="kk" data-title="Евклидтік геометрия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Евклид геометриясы – Kyrgyz" lang="ky" hreflang="ky" data-title="Евклид геометриясы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Geometria_Euclidea" title="Geometria Euclidea – Latin" lang="la" hreflang="la" data-title="Geometria Euclidea" 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/Eikl%C4%ABda_%C4%A3eometrija" title="Eiklīda ģeometrija – Latvian" lang="lv" hreflang="lv" data-title="Eiklīda ģeometrija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Euklidin%C4%97_geometrija" title="Euklidinė geometrija – Lithuanian" lang="lt" hreflang="lt" data-title="Euklidinė geometrija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Jeometria_euclidal" title="Jeometria euclidal – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Jeometria euclidal" 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-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/euklidi_tamcmaci" title="euklidi tamcmaci – Lojban" lang="jbo" hreflang="jbo" data-title="euklidi tamcmaci" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euklideszi_geometria" title="Euklideszi geometria – Hungarian" lang="hu" hreflang="hu" data-title="Euklideszi 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%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Евклидова геометрија – Macedonian" lang="mk" hreflang="mk" data-title="Евклидова геометрија" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AF%E0%B5%82%E0%B4%95%E0%B5%8D%E0%B4%B2%E0%B4%BF%E0%B4%A1%E0%B4%BF%E0%B4%AF%E0%B5%BB_%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="യൂക്ലിഡിയൻ ജ്യാമിതി – Malayalam" lang="ml" hreflang="ml" data-title="യൂക്ലിഡിയൻ ജ്യാമിതി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Geometri_Euclid" title="Geometri Euclid – Malay" lang="ms" hreflang="ms" data-title="Geometri Euclid" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80" title="Евклидийн геометр – Mongolian" lang="mn" hreflang="mn" data-title="Евклидийн геометр" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Euclidische_meetkunde" title="Euclidische meetkunde – Dutch" lang="nl" hreflang="nl" data-title="Euclidische meetkunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E5%B9%BE%E4%BD%95%E5%AD%A6" title="ユークリッド幾何学 – Japanese" lang="ja" hreflang="ja" data-title="ユークリッド幾何学" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Euklidsk_geometri" title="Euklidsk geometri – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Euklidsk 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/Euklidsk_geometri" title="Euklidsk geometri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Euklidsk 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Geometria_euclidiana" title="Geometria euclidiana – Occitan" lang="oc" hreflang="oc" data-title="Geometria euclidiana" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yevklid_geometriyasi" title="Yevklid geometriyasi – Uzbek" lang="uz" hreflang="uz" data-title="Yevklid geometriyasi" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Geometr%C3%ACa_euclid%C3%A9a" title="Geometrìa euclidéa – Piedmontese" lang="pms" hreflang="pms" data-title="Geometrìa euclidéa" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Geometria_euklidesowa" title="Geometria euklidesowa – Polish" lang="pl" hreflang="pl" data-title="Geometria euklidesowa" 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_euclidiana" title="Geometria euclidiana – Portuguese" lang="pt" hreflang="pt" data-title="Geometria euclidiana" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geometrie_euclidian%C4%83" title="Geometrie euclidiană – Romanian" lang="ro" hreflang="ro" data-title="Geometrie euclidiană" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D1%81%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Евклидовска геометрия – Rusyn" lang="rue" hreflang="rue" data-title="Евклидовска геометрия" 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%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Евклидова геометрия – Russian" lang="ru" hreflang="ru" data-title="Евклидова геометрия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gjeometria_euklidiane" title="Gjeometria euklidiane – Albanian" lang="sq" hreflang="sq" data-title="Gjeometria euklidiane" 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/Giomitr%C3%ACa_euclidea" title="Giomitrìa euclidea – Sicilian" lang="scn" hreflang="scn" data-title="Giomitrìa euclidea" 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%BA%E0%B7%96%E0%B6%9A%E0%B7%8A%E0%B6%BD%E0%B7%92%E0%B6%A9%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%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="යූක්ලිඩියානු ජ්‍යාමිතිය – Sinhala" lang="si" hreflang="si" data-title="යූක්ලිඩියානු 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/Euclidean_geometry" title="Euclidean geometry – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euclidean 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Euklidovsk%C3%A1_geometria" title="Euklidovská geometria – Slovak" lang="sk" hreflang="sk" data-title="Euklidovská 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/Evklidska_geometrija" title="Evklidska geometrija – Slovenian" lang="sl" hreflang="sl" data-title="Evklidska geometrija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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%DB%8C_%D8%A6%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="ئەندازەی ئیقلیدسی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئەندازەی ئیقلیدسی" 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%95%D1%83%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Еуклидова геометрија – Serbian" lang="sr" hreflang="sr" data-title="Еуклидова 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/Euklidova_geometrija" title="Euklidova geometrija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Euklidova 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Euklidinen_geometria" title="Euklidinen geometria – Finnish" lang="fi" hreflang="fi" data-title="Euklidinen 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/Euklidisk_geometri" title="Euklidisk geometri – Swedish" lang="sv" hreflang="sv" data-title="Euklidisk geometri" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AF%E0%AF%82%E0%AE%95%E0%AF%8D%E0%AE%B3%E0%AF%80%E0%AE%9F%E0%AF%8D_%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="யூக்ளீட் வடிவியல் – Tamil" lang="ta" hreflang="ta" data-title="யூக்ளீட் வடிவியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D1%87%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Евклидча геометрия – Tatar" lang="tt" hreflang="tt" data-title="Евклидча 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-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%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%A2%E0%B8%B8%E0%B8%84%E0%B8%A5%E0%B8%B4%E0%B8%94" title="เรขาคณิตแบบยุคลิด – Thai" lang="th" hreflang="th" data-title="เรขาคณิตแบบยุคลิด" 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%D0%B8_%D1%83%D2%9B%D0%BB%D0%B8%D0%B4%D1%83%D1%81%D3%A3" title="Ҳандасаи уқлидусӣ – Tajik" lang="tg" hreflang="tg" data-title="Ҳандасаи уқлидусӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96klid_geometrisi" title="Öklid geometrisi – Turkish" lang="tr" hreflang="tr" data-title="Öklid geometrisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D0%BE%D0%B2%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Евклідова геометрія – Ukrainian" lang="uk" hreflang="uk" data-title="Евклідова геометрія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_h%E1%BB%8Dc_Euclid" title="Hình học Euclid – Vietnamese" lang="vi" hreflang="vi" data-title="Hình học Euclid" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%AD%90%E6%B0%8F%E5%B9%BE%E4%BD%95" title="歐氏幾何 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="歐氏幾何" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何 – Wu" lang="wuu" hreflang="wuu" data-title="欧几里得几何" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AD%90%E5%B9%BE%E9%87%8C%E5%BE%97%E5%B9%BE%E4%BD%95" title="歐幾里得幾何 – Cantonese" lang="yue" hreflang="yue" data-title="歐幾里得幾何" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何 – Chinese" lang="zh" hreflang="zh" data-title="欧几里得几何" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B4%B0%E2%B5%8F%E2%B5%A3%E2%B4%B3%E2%B4%B3%E2%B5%89%E2%B5%9C_%E2%B5%9C%E2%B5%93%E2%B4%BD%E2%B5%8D%E2%B5%89%E2%B4%B7%E2%B5%9C" title="ⵜⴰⵏⵣⴳⴳⵉⵜ ⵜⵓⴽⵍⵉⴷⵜ – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵏⵣⴳⴳⵉⵜ ⵜⵓⴽⵍⵉⴷⵜ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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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 .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_geometry" title="Template:General geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_geometry" title="Template talk:General geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_geometry" title="Special:EditPage/Template:General geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Euclidean geometry</b> is a mathematical system attributed to ancient <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematician</a> <a href="/wiki/Euclid" title="Euclid">Euclid</a>, which he described in his textbook on <a href="/wiki/Geometry" title="Geometry">geometry</a>, <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i>. Euclid's approach consists in assuming a small set of intuitively appealing <a href="/wiki/Axiom" title="Axiom">axioms</a> (postulates) and deducing many other <a href="/wiki/Proposition" title="Proposition">propositions</a> (<a href="/wiki/Theorem" title="Theorem">theorems</a>) from these. Although many of Euclid's results had been stated earlier,<sup id="cite_ref-eves1_19_1-0" class="reference"><a href="#cite_note-eves1_19-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Euclid was the first to organize these propositions into a <a href="/wiki/Logic" title="Logic">logical system</a> in which each result is <i><a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a></i> from axioms and previously proved theorems.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The <i>Elements</i> begins with <b>plane geometry</b>, still taught in <a href="/wiki/Secondary_school" title="Secondary school">secondary school</a> (high school) as the first <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a> and the first examples of <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proofs</a>. It goes on to the <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a> of <a href="/wiki/Three_dimensions" class="mw-redirect" title="Three dimensions">three dimensions</a>. Much of the <i>Elements</i> states results of what are now called <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Number_theory" title="Number theory">number theory</a>, explained in geometrical language.<sup id="cite_ref-eves1_19_1-1" class="reference"><a href="#cite_note-eves1_19-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For more than two thousand years, the adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with the possible exception of the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other <a href="/wiki/Self-consistent" class="mw-redirect" title="Self-consistent">self-consistent</a> <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometries</a> are known, the first ones having been discovered in the early 19th century. An implication of <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> is that physical space itself is not Euclidean, and <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Euclidean space</a> is a good approximation for it only over short distances (relative to the strength of the <a href="/wiki/Gravity" title="Gravity">gravitational field</a>).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Euclidean geometry is an example of <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast to <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, introduced almost 2,000 years later by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, which uses <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> to express geometric properties by means of <a href="/wiki/Algebraic_formula" class="mw-redirect" title="Algebraic formula">algebraic formulas</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_Elements">The <i>Elements</i></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=1" title="Edit section: The Elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a></div> <p>The <i>Elements</i> is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. </p><p>There are 13 books in the <i>Elements</i>: </p><p>Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) </p><p>Books V and VII–X deal with <a href="/wiki/Number_theory" title="Number theory">number theory</a>, with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as <a href="/wiki/Prime_numbers" class="mw-redirect" title="Prime numbers">prime numbers</a> and <a href="/wiki/Rational_number" title="Rational number">rational</a> and <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> are introduced. It is proved that there are infinitely many prime numbers. </p><p>Books XI–XIII concern <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a>. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The <a href="/wiki/Platonic_solid" title="Platonic solid">platonic solids</a> are constructed. </p> <div class="mw-heading mw-heading3"><h3 id="Axioms">Axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=2" title="Edit section: Axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" 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>The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.</figcaption></figure> <p>Euclidean geometry is an <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a>, in which all <a href="/wiki/Theorem" title="Theorem">theorems</a> ("true statements") are derived from a small number of simple axioms. Until the advent of <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality.<sup id="cite_ref-Wolfe_4-0" class="reference"><a href="#cite_note-Wolfe-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Near the beginning of the first book of the <i>Elements</i>, Euclid gives five <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a> (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>Let the following be postulated:</dd></dl> <ol><li>To draw a <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> from any <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> to any point.</li> <li>To produce (extend) a <a href="/wiki/Line_segment" title="Line segment">finite straight line</a> continuously in a straight line.</li> <li>To describe a <a href="/wiki/Circle" title="Circle">circle</a> with any centre and distance (radius).</li> <li>That all <a href="/wiki/Right_angle" title="Right angle">right angles</a> are equal to one another.</li> <li>[The <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.</li></ol> <p>Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique. </p><p>The <i>Elements</i> also include the following five "common notions": </p> <ol><li>Things that are equal to the same thing are also equal to one another (the <a href="/wiki/Transitive_property" class="mw-redirect" title="Transitive property">transitive property</a> of a <a href="/wiki/Euclidean_relation" title="Euclidean relation">Euclidean relation</a>).</li> <li>If equals are added to equals, then the wholes are equal (Addition property of equality).</li> <li>If equals are subtracted from equals, then the differences are equal (subtraction property of equality).</li> <li>Things that coincide with one another are equal to one another (reflexive property).</li> <li>The whole is greater than the part.</li></ol> <p>Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Modern <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">treatments</a> use more extensive and complete sets of axioms. </p> <div class="mw-heading mw-heading3"><h3 id="Parallel_postulate">Parallel postulate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=3" title="Edit section: Parallel postulate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a></div> <p>To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the <i>Elements</i>: his first 28 propositions are those that can be proved without it. </p><p>Many alternative axioms can be formulated which are <a href="/wiki/Logical_equivalence" title="Logical equivalence">logically equivalent</a> to the parallel postulate (in the context of the other axioms). For example, <a href="/wiki/Playfair%27s_axiom" title="Playfair's axiom">Playfair's axiom</a> states: </p> <dl><dd>In a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, through a point not on a given straight line, at most one line can be drawn that never meets the given line.</dd></dl> <p>The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Euclid-proof.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Euclid-proof.svg/220px-Euclid-proof.svg.png" decoding="async" width="220" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Euclid-proof.svg/330px-Euclid-proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Euclid-proof.svg/440px-Euclid-proof.svg.png 2x" data-file-width="445" data-file-height="573" /></a><figcaption>A proof from Euclid's <i>Elements</i> that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Methods_of_proof">Methods of proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=4" title="Edit section: Methods of proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclidean Geometry is <i><a href="/wiki/Constructive_proof" title="Constructive proof">constructive</a></i>. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and an unmarked straightedge</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as <a href="/wiki/Set_theory" title="Set theory">set theory</a>, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.<sup id="cite_ref-set_theory_9-0" class="reference"><a href="#cite_note-set_theory-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Strictly speaking, the lines on paper are <i><a href="/wiki/Scientific_modelling" title="Scientific modelling">models</a></i> of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider <a href="/wiki/Non-constructive_proof" class="mw-redirect" title="Non-constructive proof">nonconstructive proofs</a> just as sound as constructive ones, they are often considered less <a href="/wiki/Mathematical_beauty" title="Mathematical beauty">elegant</a>, intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring a statement such as "Find the greatest common measure of ..."<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Euclid often used <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notation_and_terminology">Notation and terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=5" title="Edit section: Notation and terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Naming_of_points_and_figures">Naming of points and figures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=6" title="Edit section: Naming of points and figures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C. </p> <div class="mw-heading mw-heading3"><h3 id="Complementary_and_supplementary_angles">Complementary and supplementary angles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=7" title="Edit section: Complementary and supplementary angles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Angles whose sum is a right angle are called <a href="/wiki/Complementary_angles" class="mw-redirect" title="Complementary angles">complementary</a>. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. </p><p>Angles whose sum is a straight angle are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a>. Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite. </p> <div class="mw-heading mw-heading3"><h3 id="Modern_versions_of_Euclid's_notation"><span id="Modern_versions_of_Euclid.27s_notation"></span>Modern versions of Euclid's notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=8" title="Edit section: Modern versions of Euclid's notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In modern terminology, angles would normally be measured in <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> or <a href="/wiki/Radian" title="Radian">radians</a>. </p><p>Modern school textbooks often define separate figures called <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> (infinite), <a href="/wiki/Line_(mathematics)#Ray" class="mw-redirect" title="Line (mathematics)">rays</a> (semi-infinite), and <a href="/wiki/Line_segment" title="Line segment">line segments</a> (of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. </p> <div class="mw-heading mw-heading2"><h2 id="Some_important_or_well_known_results">Some important or well known results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=9" title="Edit section: Some important or well known results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional" style="max-width: 652px;"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pons_asinorum_dzmanto.png" class="mw-file-description" title="The pons asinorum or bridge of asses theorem states that in an isosceles triangle, α = β and γ = δ."><img alt="The pons asinorum or bridge of asses theorem states that in an isosceles triangle, α = β and γ = δ." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Pons_asinorum_dzmanto.png/107px-Pons_asinorum_dzmanto.png" decoding="async" width="107" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Pons_asinorum_dzmanto.png/160px-Pons_asinorum_dzmanto.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Pons_asinorum_dzmanto.png/214px-Pons_asinorum_dzmanto.png 2x" data-file-width="352" data-file-height="395" /></a></span></div> <div class="gallerytext">The <i><a href="/wiki/Pons_asinorum" title="Pons asinorum">pons asinorum</a></i> or <i>bridge of asses theorem</i> states that in an isosceles triangle, α = β and γ = δ.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Sum_of_angles_of_triangle_dzmanto.png" class="mw-file-description" title="The triangle angle sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees."><img alt="The triangle angle sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Sum_of_angles_of_triangle_dzmanto.png/120px-Sum_of_angles_of_triangle_dzmanto.png" decoding="async" width="120" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Sum_of_angles_of_triangle_dzmanto.png/180px-Sum_of_angles_of_triangle_dzmanto.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Sum_of_angles_of_triangle_dzmanto.png/240px-Sum_of_angles_of_triangle_dzmanto.png 2x" data-file-width="261" data-file-height="159" /></a></span></div> <div class="gallerytext">The <i>triangle angle sum theorem</i> states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pythagorean.svg" class="mw-file-description" title="The Pythagorean theorem states that the sum of the areas of the two squares on the legs (a and b) of a right triangle equals the area of the square on the hypotenuse (c)."><img alt="The Pythagorean theorem states that the sum of the areas of the two squares on the legs (a and b) of a right triangle equals the area of the square on the hypotenuse (c)." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/120px-Pythagorean.svg.png" decoding="async" width="120" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/180px-Pythagorean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/240px-Pythagorean.svg.png 2x" data-file-width="512" data-file-height="466" /></a></span></div> <div class="gallerytext">The <i><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></i> states that the sum of the areas of the two squares on the legs (<i>a</i> and <i>b</i>) of a right triangle equals the area of the square on the hypotenuse (<i>c</i>).</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Thales%27_Theorem_Simple.svg" class="mw-file-description" title="Thales' theorem states that if AC is a diameter, then the angle at B is a right angle."><img alt="Thales' theorem states that if AC is a diameter, then the angle at B is a right angle." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Thales%27_Theorem_Simple.svg/120px-Thales%27_Theorem_Simple.svg.png" decoding="async" width="120" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Thales%27_Theorem_Simple.svg/180px-Thales%27_Theorem_Simple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Thales%27_Theorem_Simple.svg/240px-Thales%27_Theorem_Simple.svg.png 2x" data-file-width="200" data-file-height="175" /></a></span></div> <div class="gallerytext"><i><a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a></i> states that if AC is a diameter, then the angle at B is a right angle.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Pons_asinorum">Pons asinorum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=10" title="Edit section: Pons asinorum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Pons_asinorum" title="Pons asinorum">pons asinorum</a> (<i>bridge of asses</i>) states that <i>in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another</i>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Its name may be attributed to its frequent role as the first real test in the <i>Elements</i> of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Congruence_of_triangles">Congruence of triangles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=11" title="Edit section: Congruence of triangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Congruent_triangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Congruent_triangles.svg/220px-Congruent_triangles.svg.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Congruent_triangles.svg/330px-Congruent_triangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Congruent_triangles.svg/440px-Congruent_triangles.svg.png 2x" data-file-width="300" data-file-height="375" /></a><figcaption>Congruence of triangles is determined by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle.</figcaption></figure> <p>Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. </p> <div class="mw-heading mw-heading3"><h3 id="Triangle_angle_sum">Triangle angle sum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=12" title="Edit section: Triangle angle sum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sum of the angles of a triangle is equal to a straight angle (180 degrees).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one <a href="/wiki/Obtuse_angle" class="mw-redirect" title="Obtuse angle">obtuse</a> or <a href="/wiki/Right_angle" title="Right angle">right angle</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Pythagorean_theorem">Pythagorean theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=13" title="Edit section: Pythagorean theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The celebrated <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). </p> <div class="mw-heading mw-heading3"><h3 id="Thales'_theorem"><span id="Thales.27_theorem"></span>Thales' theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=14" title="Edit section: Thales' theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a>, named after <a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a> states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after the manner of Euclid Book III, Prop. 31.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Scaling_of_area_and_volume">Scaling of area and volume</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=15" title="Edit section: Scaling of area and volume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, <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\propto L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∝</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\propto L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fce05b97628a7f099db68a7decbda56329848e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.479ex; height:2.676ex;" alt="{\displaystyle A\propto L^{2}}"></span>, and the volume of a solid to the cube, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\propto L^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∝</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\propto L^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167f9fddefe70029c097f25fc39660c214247bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.523ex; height:2.676ex;" alt="{\displaystyle V\propto L^{3}}"></span>. Euclid proved these results in various special cases such as the area of a circle<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and the volume of a parallelepipedal solid.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> who proved that a sphere has 2/3 the volume of the circumscribing cylinder.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="System_of_measurement_and_arithmetic">System of measurement and arithmetic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=16" title="Edit section: System of measurement and arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclidean geometry has two fundamental types of measurements: <a href="/wiki/Angle" title="Angle">angle</a> and <a href="/wiki/Euclidean_distance" title="Euclidean distance">distance</a>. The angle scale is absolute, and Euclid uses the <a href="/wiki/Right_angle" title="Right angle">right angle</a> as his basic unit, so that, for example, a 45-<a href="/wiki/Degree_(angle)" title="Degree (angle)">degree</a> angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. </p><p>Measurements of <a href="/wiki/Area_(geometry)" class="mw-redirect" title="Area (geometry)">area</a> and <a href="/wiki/Volume" title="Volume">volume</a> are derived from distances. For example, a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Congruentie.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Congruentie.svg/220px-Congruentie.svg.png" decoding="async" width="220" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Congruentie.svg/330px-Congruentie.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Congruentie.svg/440px-Congruentie.svg.png 2x" data-file-width="594" data-file-height="188" /></a><figcaption>An example of congruence. The two figures on the left are congruent, while the third is <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, like <a href="/wiki/Distance" title="Distance">distance</a> and <a href="/wiki/Angle" title="Angle">angles</a>. The latter sort of properties are called <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariants</a> and studying them is the essence of geometry.</figcaption></figure> <p>Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term "<a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a>" refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a>. <a href="/wiki/Corresponding_sides_and_corresponding_angles" title="Corresponding sides and corresponding angles">Corresponding angles</a> in a pair of similar shapes are equal and <a href="/wiki/Corresponding_sides" class="mw-redirect" title="Corresponding sides">corresponding sides</a> are in proportion to each other. </p> <div class="mw-heading mw-heading2"><h2 id="In_engineering">In engineering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=17" title="Edit section: In engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Design_and_Analysis">Design and Analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=18" title="Edit section: Design and Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Stress Analysis</b>: <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress Analysis</a> - Euclidean geometry is pivotal in determining <a href="/wiki/Stress%E2%80%93strain_analysis" title="Stress–strain analysis">stress distribution</a> in mechanical components, which is essential for ensuring <a href="/wiki/Structural_integrity" class="mw-redirect" title="Structural integrity">structural integrity</a> and <a href="/wiki/Durability" title="Durability">durability</a>.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Different-types-of-mechanical-stress_EN.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Different-types-of-mechanical-stress_EN.svg/220px-Different-types-of-mechanical-stress_EN.svg.png" decoding="async" width="220" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Different-types-of-mechanical-stress_EN.svg/330px-Different-types-of-mechanical-stress_EN.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Different-types-of-mechanical-stress_EN.svg/440px-Different-types-of-mechanical-stress_EN.svg.png 2x" data-file-width="612" data-file-height="337" /></a><figcaption>Mechanical Stress</figcaption></figure> <ul><li><b>Gear Design</b>: <a href="/wiki/Gear" title="Gear">Gear</a> - The design of gears, a crucial element in many <a href="/wiki/Mechanical_system" class="mw-redirect" title="Mechanical system">mechanical systems</a>, relies heavily on Euclidean geometry to ensure proper tooth shape and <a href="/wiki/Engagement" title="Engagement">engagement</a> for efficient <a href="/wiki/Power_transmission" title="Power transmission">power transmission</a>.</li></ul> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gear-kegelzahnrad.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Gear-kegelzahnrad.svg/150px-Gear-kegelzahnrad.svg.png" decoding="async" width="150" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Gear-kegelzahnrad.svg/225px-Gear-kegelzahnrad.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Gear-kegelzahnrad.svg/300px-Gear-kegelzahnrad.svg.png 2x" data-file-width="2100" data-file-height="1235" /></a><figcaption>Gear</figcaption></figure> <ul><li><b>Heat Exchanger Design</b>: <a href="/wiki/Heat_exchanger" title="Heat exchanger">Heat exchanger</a> - In <a href="/wiki/Thermal_engineering" title="Thermal engineering">thermal engineering</a>, Euclidean geometry is used to design <a href="/wiki/Heat_exchangers" class="mw-redirect" title="Heat exchangers">heat exchangers</a>, where the geometric configuration greatly influences <a href="/wiki/Thermal_efficiency" title="Thermal efficiency">thermal efficiency</a>. See <a href="/wiki/Shell-and-tube_heat_exchanger" title="Shell-and-tube heat exchanger">shell-and-tube heat exchangers</a> and <a href="/wiki/Plate_heat_exchanger" title="Plate heat exchanger">plate heat exchangers</a> for more details.</li></ul> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:U-tube_heat_exchanger.svg" class="mw-file-description" title="U-Tube Shell and Tube Heat Exchanger"><img alt="U-Tube Shell and Tube Heat Exchanger" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/U-tube_heat_exchanger.svg/400px-U-tube_heat_exchanger.svg.png" decoding="async" width="400" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/U-tube_heat_exchanger.svg/600px-U-tube_heat_exchanger.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/U-tube_heat_exchanger.svg/800px-U-tube_heat_exchanger.svg.png 2x" data-file-width="730" data-file-height="482" /></a><figcaption>U-Tube Shell and Tube Heat Exchanger</figcaption></figure> <ul><li><b>Lens Design</b>: <a href="/wiki/Lens" title="Lens">Lens</a> - In optical engineering, Euclidean geometry is critical in the design of lenses, where precise geometric shapes determine the <a href="/wiki/Focus_(optics)" title="Focus (optics)">focusing</a> properties. <a href="/wiki/Geometric_optics" class="mw-redirect" title="Geometric optics">Geometric optics</a> analyzes the focusing of <a href="/wiki/Light" title="Light">light</a> by <a href="/wiki/Lens" title="Lens">lenses</a> and <a href="/wiki/Mirrors" class="mw-redirect" title="Mirrors">mirrors</a>.</li></ul> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Lenses_en.svg" class="mw-file-description"><img alt="Types of lenses" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Lenses_en.svg/450px-Lenses_en.svg.png" decoding="async" width="450" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Lenses_en.svg/675px-Lenses_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Lenses_en.svg/900px-Lenses_en.svg.png 2x" data-file-width="567" data-file-height="265" /></a><figcaption>Types of Lenses</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Dynamics">Dynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=19" title="Edit section: Dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Vibration Analysis</b>: <a href="/wiki/Vibration" title="Vibration">Vibration</a> - Euclidean geometry is essential in analyzing and understanding the <a href="/wiki/Vibrations" class="mw-redirect" title="Vibrations">vibrations</a> in <a href="/wiki/Mechanical_systems" class="mw-redirect" title="Mechanical systems">mechanical systems</a>, aiding in the design of systems that can withstand or utilize these <a href="/wiki/Vibrations" class="mw-redirect" title="Vibrations">vibrations</a> effectively.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Drum_vibration_mode21.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Drum_vibration_mode21.gif/220px-Drum_vibration_mode21.gif" decoding="async" width="220" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6e/Drum_vibration_mode21.gif 1.5x" data-file-width="248" data-file-height="130" /></a><figcaption>Vibration - <a href="/wiki/Oscillations" class="mw-redirect" title="Oscillations">Oscillations</a></figcaption></figure> <ul><li><b>Wing Design</b>: <a href="/wiki/Aerodynamics" title="Aerodynamics">Aircraft Wing Design</a> - The application of Euclidean geometry in <a href="/wiki/Aerodynamics" title="Aerodynamics">aerodynamics</a> is evident in <a href="/wiki/Aircraft" title="Aircraft">aircraft</a> <a href="/wiki/Airfoil" title="Airfoil">wing design</a>, <a href="/wiki/Airfoil" title="Airfoil">airfoils</a>, and <a href="/wiki/Hydrofoil" title="Hydrofoil">hydrofoils</a> where geometric shape directly impacts <a href="/wiki/Lift_(force)" title="Lift (force)">lift</a> and <a href="/wiki/Drag_(physics)" title="Drag (physics)">drag</a> characteristics.</li></ul> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Wing_profile_nomenclature.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wing_profile_nomenclature.svg/400px-Wing_profile_nomenclature.svg.png" decoding="async" width="400" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wing_profile_nomenclature.svg/600px-Wing_profile_nomenclature.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wing_profile_nomenclature.svg/800px-Wing_profile_nomenclature.svg.png 2x" data-file-width="1022" data-file-height="433" /></a><figcaption>Airfoil Nomenclature</figcaption></figure> <ul><li><b>Satellite Orbits</b>: <a href="/wiki/Orbit" title="Orbit">Satellite Orbits</a> - Euclidean geometry helps in calculating and predicting the <a href="/wiki/Orbit" title="Orbit">orbits</a> of <a href="/wiki/Satellites" class="mw-redirect" title="Satellites">satellites</a>, essential for successful <a href="/wiki/List_of_Space_Shuttle_missions" title="List of Space Shuttle missions">space missions</a> and <a href="/wiki/Satellite" title="Satellite">satellite</a> operations. Also see <a href="/wiki/Astrodynamics" class="mw-redirect" title="Astrodynamics">astrodynamics</a>, <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, and <a href="/wiki/Elliptic_orbit" title="Elliptic orbit">elliptic orbit</a>.</li></ul> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Animation_of_Orbital_eccentricity.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Animation_of_Orbital_eccentricity.gif/250px-Animation_of_Orbital_eccentricity.gif" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Animation_of_Orbital_eccentricity.gif/375px-Animation_of_Orbital_eccentricity.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Animation_of_Orbital_eccentricity.gif/500px-Animation_of_Orbital_eccentricity.gif 2x" data-file-width="560" data-file-height="420" /></a><figcaption>Animation of Orbit by Eccentricity</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="CAD_Systems">CAD Systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=20" title="Edit section: CAD Systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>3D Modeling</b>: In <a href="/wiki/Computer-aided_design" title="Computer-aided design">CAD (computer-aided design)</a> systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs before <a href="/wiki/Manufacturing" title="Manufacturing">manufacturing</a>.</li></ul> <ul><li><b>Design and Manufacturing</b>: Much of <a href="/wiki/Computer-aided_manufacturing" title="Computer-aided manufacturing">CAM (computer-aided manufacturing)</a> relies on Euclidean geometry. The design geometry in CAD/CAM typically consists of shapes bounded by <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planes</a>, <a href="/wiki/Cylinder" title="Cylinder">cylinders</a>, <a href="/wiki/Cone" title="Cone">cones</a>, <a href="/wiki/Torus" title="Torus">tori</a>, and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, from <a href="/wiki/Car" title="Car">cars</a> and <a href="/wiki/Airplane" title="Airplane">airplanes</a> to <a href="/wiki/Ship" title="Ship">ships</a> and <a href="/wiki/Smartphone" title="Smartphone">smartphones</a>.</li></ul> <ul><li><b>Evolution of Drafting Practices</b>: Historically, advanced Euclidean geometry, including theorems like <a href="/wiki/Pascal%27s_theorem" title="Pascal's theorem">Pascal's theorem</a> and <a href="/wiki/Brianchon%27s_theorem" title="Brianchon's theorem">Brianchon's theorem</a>, was integral to drafting practices. However, with the advent of modern CAD systems, such in-depth knowledge of these theorems is less necessary in contemporary design and manufacturing processes.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_CAD_software" title="History of CAD software">History of CAD software</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cad_crank.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Cad_crank.jpg/150px-Cad_crank.jpg" decoding="async" width="150" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Cad_crank.jpg/225px-Cad_crank.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Cad_crank.jpg/300px-Cad_crank.jpg 2x" data-file-width="460" data-file-height="400" /></a><figcaption>3D CAD Model</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Circuit_Design">Circuit Design</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=21" title="Edit section: Circuit Design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>PCB Layouts</b>: <a href="/wiki/Printed_circuit_board" title="Printed circuit board">Printed Circuit Board (PCB) Design</a> utilizes Euclidean geometry for the efficient placement and routing of components, ensuring functionality while <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimizing space</a>. Efficient layout of electronic components on PCBs is critical for minimizing <a href="/wiki/Electrical_interference" class="mw-redirect" title="Electrical interference">signal interference</a> and optimizing <a href="/wiki/Electric_circuit" class="mw-redirect" title="Electric circuit">circuit performance</a>.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SEG_DVD_430_-_Printed_circuit_board-4276.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/SEG_DVD_430_-_Printed_circuit_board-4276.jpg/220px-SEG_DVD_430_-_Printed_circuit_board-4276.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/SEG_DVD_430_-_Printed_circuit_board-4276.jpg/330px-SEG_DVD_430_-_Printed_circuit_board-4276.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/SEG_DVD_430_-_Printed_circuit_board-4276.jpg/440px-SEG_DVD_430_-_Printed_circuit_board-4276.jpg 2x" data-file-width="4605" data-file-height="3454" /></a><figcaption>PCB of a DVD Player</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Electromagnetic_and_Fluid_Flow_Fields">Electromagnetic and Fluid Flow Fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=22" title="Edit section: Electromagnetic and Fluid Flow Fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Antenna Design</b>: <a href="/wiki/Antenna_(radio)" title="Antenna (radio)">Antenna Design</a> - Euclidean <a href="/wiki/Antenna_types" title="Antenna types">geometry of antennas</a> helps in designing antennas, where the spatial arrangement and dimensions directly affect antenna and <a href="/wiki/Antenna_array" title="Antenna array">array</a> performance in transmitting and receiving <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a>.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg/220px-Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg/330px-Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg/440px-Canberra_Deep_Dish_Communications_Complex_-_GPN-2000-000502.jpg 2x" data-file-width="2526" data-file-height="2394" /></a><figcaption>NASA <a href="/wiki/Cassegrain_antenna" title="Cassegrain antenna">Cassegrain</a>, Extremely high gain ~70 dBi.</figcaption></figure> <ul><li><b>Field Theory</b>: <a href="/wiki/Potential_flow" title="Potential flow">Complex Potential Flow</a> - In the study of <a href="/wiki/Inviscid_flow" title="Inviscid flow">inviscid flow</a> fields and <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a>, Euclidean geometry aids in visualizing and solving <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a> problems. This is essential for understanding <a href="/wiki/Fluid_velocity" class="mw-redirect" title="Fluid velocity">fluid velocity</a> field and <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> interactions in three-dimensional space. The relationship of which is characterized by an <a href="/wiki/Irrotational" class="mw-redirect" title="Irrotational">irrotational</a> <a href="/wiki/Solenoidal_field" class="mw-redirect" title="Solenoidal field">solenoidal field</a> or a <a href="/wiki/Conservative_vector_field" title="Conservative vector field">conservative vector field</a>.</li></ul> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Inviscid_flow_around_a_cylinder.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Inviscid_flow_around_a_cylinder.gif/200px-Inviscid_flow_around_a_cylinder.gif" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Inviscid_flow_around_a_cylinder.gif/300px-Inviscid_flow_around_a_cylinder.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Inviscid_flow_around_a_cylinder.gif/400px-Inviscid_flow_around_a_cylinder.gif 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Potential Flow Around a Source without <a href="/wiki/Circulation_(physics)" title="Circulation (physics)">Circulation</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Controls">Controls</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=23" title="Edit section: Controls"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Control System Analysis</b>: <a href="/wiki/Control_theory" title="Control theory">Control Systems</a> - The application of Euclidean geometry in <a href="/wiki/Control_theory" title="Control theory">control theory</a> helps in the analysis and design of <a href="/wiki/Control_systems" class="mw-redirect" title="Control systems">control systems</a>, particularly in understanding and <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimizing</a> system stability and <a href="/wiki/Request%E2%80%93response" title="Request–response">response</a>.</li></ul> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ideal_feedback_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ideal_feedback_model.svg/220px-Ideal_feedback_model.svg.png" decoding="async" width="220" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ideal_feedback_model.svg/330px-Ideal_feedback_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ideal_feedback_model.svg/440px-Ideal_feedback_model.svg.png 2x" data-file-width="733" data-file-height="269" /></a><figcaption> Basic feedback loop.</figcaption></figure> <ul><li><b>Calculation Tools</b>: <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> - Euclidean geometry is integral in using Jacobian matrices for transformations and control systems in both <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical</a> and <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a> fields, providing insights into system behavior and properties. The Jacobian serves as a linearized <a href="/wiki/Design_matrix" title="Design matrix">design matrix</a> in statistical <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> and <a href="/wiki/Curve_fitting" title="Curve fitting">curve fitting</a>; see <a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">non-linear least squares</a>. The Jacobian is also used in <a href="/wiki/Random_matrices" class="mw-redirect" title="Random matrices">random matrices</a>, <a href="/wiki/Moment_(physics)" title="Moment (physics)">moment</a>, <a href="/wiki/Statistics" title="Statistics">statistics</a>, and <a href="/wiki/Diagnostic" class="mw-redirect" title="Diagnostic">diagnostics</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_general_applications">Other general applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=24" title="Edit section: Other general applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here. </p> <ul class="gallery mw-gallery-traditional" style="max-width: 489px;"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Us_land_survey_officer.jpg" class="mw-file-description" title="A surveyor uses a level"><img alt="A surveyor uses a level" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Us_land_survey_officer.jpg/120px-Us_land_survey_officer.jpg" decoding="async" width="120" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Us_land_survey_officer.jpg/180px-Us_land_survey_officer.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Us_land_survey_officer.jpg/240px-Us_land_survey_officer.jpg 2x" data-file-width="500" data-file-height="328" /></a></span></div> <div class="gallerytext">A surveyor uses a <a href="/wiki/Dumpy_level" class="mw-redirect" title="Dumpy level">level</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Ambersweet_oranges.jpg" class="mw-file-description" title="Sphere packing applies to a stack of oranges."><img alt="Sphere packing applies to a stack of oranges." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Ambersweet_oranges.jpg/106px-Ambersweet_oranges.jpg" decoding="async" width="106" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Ambersweet_oranges.jpg/160px-Ambersweet_oranges.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Ambersweet_oranges.jpg/213px-Ambersweet_oranges.jpg 2x" data-file-width="1999" data-file-height="2254" /></a></span></div> <div class="gallerytext"><a href="/wiki/Sphere_packing" title="Sphere packing">Sphere packing</a> applies to a stack of <a href="/wiki/Orange_(fruit)" title="Orange (fruit)">oranges</a>.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Parabola_with_focus_and_arbitrary_line.svg" class="mw-file-description" title="A parabolic mirror brings parallel rays of light to a focus."><img alt="A parabolic mirror brings parallel rays of light to a focus." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Parabola_with_focus_and_arbitrary_line.svg/120px-Parabola_with_focus_and_arbitrary_line.svg.png" decoding="async" width="120" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Parabola_with_focus_and_arbitrary_line.svg/180px-Parabola_with_focus_and_arbitrary_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Parabola_with_focus_and_arbitrary_line.svg/240px-Parabola_with_focus_and_arbitrary_line.svg.png 2x" data-file-width="880" data-file-height="600" /></a></span></div> <div class="gallerytext">A parabolic mirror brings parallel rays of light to a focus.</div> </li> </ul> <p>As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is <a href="/wiki/Surveying" title="Surveying">surveying</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In addition it has been used in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> and the <a href="/wiki/Visual_perception#Cognitive_and_computational_approaches" title="Visual perception">cognitive and computational approaches to visual perception of objects</a>. Certain practical results from Euclidean geometry (such as the right-angle property of the 3-4-5 triangle) were used long before they were proved formally.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as <a href="/wiki/Gunter%27s_chain" title="Gunter's chain">Gunter's chain</a>, and angles using graduated circles and, later, the <a href="/wiki/Theodolite" title="Theodolite">theodolite</a>. </p><p>An application of Euclidean solid geometry is the <a href="/wiki/Packing_problem" class="mw-redirect" title="Packing problem">determination of packing arrangements</a>, such as the problem of finding the most efficient <a href="/wiki/Sphere_packing" title="Sphere packing">packing of spheres</a> in n dimensions. This problem has applications in <a href="/wiki/Error_detection_and_correction" title="Error detection and correction">error detection and correction</a>. </p> <ul class="gallery mw-gallery-traditional" style="max-width: 489px;"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Damascus_Khan_asad_Pacha_cropped.jpg" class="mw-file-description" title="Geometry is used in art and architecture."><img alt="Geometry is used in art and architecture." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Damascus_Khan_asad_Pacha_cropped.jpg/120px-Damascus_Khan_asad_Pacha_cropped.jpg" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Damascus_Khan_asad_Pacha_cropped.jpg/180px-Damascus_Khan_asad_Pacha_cropped.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Damascus_Khan_asad_Pacha_cropped.jpg/240px-Damascus_Khan_asad_Pacha_cropped.jpg 2x" data-file-width="850" data-file-height="827" /></a></span></div> <div class="gallerytext">Geometry is used in art and architecture.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Water_tower_cropped.jpg" class="mw-file-description" title="The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry."><img alt="The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Water_tower_cropped.jpg/105px-Water_tower_cropped.jpg" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Water_tower_cropped.jpg/157px-Water_tower_cropped.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Water_tower_cropped.jpg/209px-Water_tower_cropped.jpg 2x" data-file-width="657" data-file-height="753" /></a></span></div> <div class="gallerytext">The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Origami_crane_cropped.jpg" class="mw-file-description" title="Geometry can be used to design origami."><img alt="Geometry can be used to design origami." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Origami_crane_cropped.jpg/120px-Origami_crane_cropped.jpg" decoding="async" width="120" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Origami_crane_cropped.jpg/180px-Origami_crane_cropped.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Origami_crane_cropped.jpg/240px-Origami_crane_cropped.jpg 2x" data-file-width="930" data-file-height="776" /></a></span></div> <div class="gallerytext">Geometry can be used to design origami.</div> </li> </ul> <p>Geometry is used extensively in <a href="/wiki/Architecture" title="Architecture">architecture</a>. </p><p>Geometry can be used to design <a href="/wiki/Origami" title="Origami">origami</a>. Some <a href="/wiki/Compass_and_straightedge_constructions#Impossible_constructions" class="mw-redirect" title="Compass and straightedge constructions">classical construction problems of geometry</a> are impossible using <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>, but can be <a href="/wiki/Mathematics_of_paper_folding" title="Mathematics of paper folding">solved using origami</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Later_history">Later history<span class="anchor" id="History"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=25" title="Edit section: Later history"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_geometry" title="History of geometry">History of geometry</a> and <a href="/wiki/Non-Euclidean_geometry#History" title="Non-Euclidean geometry">Non-Euclidean geometry § History</a></div> <div class="mw-heading mw-heading3"><h3 id="Archimedes_and_Apollonius">Archimedes and Apollonius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=26" title="Edit section: Archimedes and Apollonius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes_sphere_and_cylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/220px-Archimedes_sphere_and_cylinder.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/330px-Archimedes_sphere_and_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/440px-Archimedes_sphere_and_cylinder.svg.png 2x" data-file-width="426" data-file-height="461" /></a><figcaption>A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.</figcaption></figure> <p><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 287 BCE</span> – c.<span style="white-space:nowrap;"> 212 BCE</span>), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a> of finite numbers. </p><p><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 240 BCE</span> – c.<span style="white-space:nowrap;"> 190 BCE</span>) is mainly known for his investigation of conic sections. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg/220px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg" decoding="async" width="220" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg/330px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg/440px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg 2x" data-file-width="817" data-file-height="1000" /></a><figcaption>René Descartes. Portrait after <a href="/wiki/Frans_Hals" title="Frans Hals">Frans Hals</a>, 1648.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="17th_century:_Descartes">17th century: Descartes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=27" title="Edit section: 17th century: Descartes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (1596–1650) developed <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, an alternative method for formalizing geometry which focused on turning geometry into algebra.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>In this approach, a point on a plane is represented by its <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian</a> (<i>x</i>, <i>y</i>) coordinates, a line is represented by its equation, and so on. </p><p>In Euclid's original approach, the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. </p><p>The equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |PQ|={\sqrt {(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |PQ|={\sqrt {(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/583bcd10678b3be4752d1ff41ecb51606ecf895a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:33.792ex; height:4.843ex;" alt="{\displaystyle |PQ|={\sqrt {(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}}}\,}"></span></dd></dl> <p>defining the distance between two points <i>P</i> = (<i>p<sub>x</sub></i>, <i>p<sub>y</sub></i>) and <i>Q</i> = (<i>q<sub>x</sub></i>, <i>q<sub>y</sub></i>) is then known as the <i>Euclidean <a href="/wiki/Metric_space" title="Metric space">metric</a></i>, and other metrics define <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometries</a>. </p><p>In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., <i>y</i> = 2<i>x</i> + 1 (a line), or <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 7 (a circle). </p><p>Also in the 17th century, <a href="/wiki/Girard_Desargues" title="Girard Desargues">Girard Desargues</a>, motivated by the theory of <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a>, introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Squaring_the_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/220px-Squaring_the_circle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/330px-Squaring_the_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/440px-Squaring_the_circle.svg.png 2x" data-file-width="281" data-file-height="281" /></a><figcaption>Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="18th_century">18th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=28" title="Edit section: 18th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of <a href="/wiki/Trisecting_an_angle" class="mw-redirect" title="Trisecting an angle">trisecting an angle</a> with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until <a href="/wiki/Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include <a href="/wiki/Doubling_the_cube" title="Doubling the cube">doubling the cube</a> and <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a>. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> while doubling a cube requires the solution of a third-order equation. </p><p><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> discussed a generalization of Euclidean geometry called <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint). </p> <div class="mw-heading mw-heading3"><h3 id="19th_century">19th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=29" title="Edit section: 19th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Comparison_of_geometries.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Comparison_of_geometries.svg/264px-Comparison_of_geometries.svg.png" decoding="async" width="264" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Comparison_of_geometries.svg/396px-Comparison_of_geometries.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Comparison_of_geometries.svg/528px-Comparison_of_geometries.svg.png 2x" data-file-width="512" data-file-height="228" /></a><figcaption>Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions</figcaption></figure> <p>In the early 19th century, <a href="/wiki/Lazare_Carnot" title="Lazare Carnot">Carnot</a> and <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a> systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Higher_dimensions">Higher dimensions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=30" title="Edit section: Higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 1840s <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> developed the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>, and <a href="/wiki/John_T._Graves" title="John T. Graves">John T. Graves</a> and <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> the <a href="/wiki/Octonion" title="Octonion">octonions</a>. These are <a href="/wiki/Normed_algebra" title="Normed algebra">normed algebras</a> which extend the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>. Later it was understood that the quaternions are also a Euclidean geometric system with four real Cartesian coordinates.<sup id="cite_ref-FOOTNOTEStillwell200118–21_28-0" class="reference"><a href="#cite_note-FOOTNOTEStillwell200118–21-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Cayley used quaternions to study <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space" title="Rotations in 4-dimensional Euclidean space">rotations in 4-dimensional Euclidean space</a>.<sup id="cite_ref-FOOTNOTEPerez-GraciaThomas2017_29-0" class="reference"><a href="#cite_note-FOOTNOTEPerez-GraciaThomas2017-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>At mid-century <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> developed the general concept of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, extending Euclidean geometry to <a href="/wiki/Ludwig_Schl%C3%A4fli#Higher_dimensions" title="Ludwig Schläfli">higher dimensions</a>. He defined <i>polyschemes</i>, later called <a href="/wiki/Polytope" title="Polytope">polytopes</a>, which are the <a href="/wiki/Four-dimensional_space#Dimensional_analogy" title="Four-dimensional space">higher-dimensional analogues</a> of <a href="/wiki/Polygon" title="Polygon">polygons</a> and <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a>. He developed their theory and discovered all the regular polytopes, i.e. the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional analogues of regular polygons and <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a>. He found there are six <a href="/wiki/Regular_4-polytopes" class="mw-redirect" title="Regular 4-polytopes">regular convex polytopes in dimension four</a>, and three in all higher dimensions. </p> <table class="wikitable mw-collapsible mw-collapsed" style="white-space:nowrap;text-align:center;"> <tbody><tr> <th colspan="8"><a href="/wiki/Regular_4-polytopes" class="mw-redirect" title="Regular 4-polytopes">Regular convex 4-polytopes</a> </th></tr> <tr> <th style="text-align:right;"><a href="/wiki/Coxeter_group" title="Coxeter group">Symmetry group</a> </th> <td><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">A<sub>4</sub></a> </td> <td colspan="2"><a href="/wiki/Hyperoctahedral_group" title="Hyperoctahedral group">B<sub>4</sub></a> </td> <td><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a> </td> <td colspan="2"><a href="/wiki/H4_polytope" title="H4 polytope">H<sub>4</sub></a> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Name </th> <td style="vertical-align:top;"><a href="/wiki/5-cell" title="5-cell">5-cell</a><br /> <p>Hyper-<a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a><br /> 5-point </p> </td> <td style="vertical-align:top;"><a href="/wiki/16-cell" title="16-cell">16-cell</a><br /> <p>Hyper-<a href="/wiki/Octahedron" title="Octahedron">octahedron</a><br /> 8-point </p> </td> <td style="vertical-align:top;"><a href="/wiki/8-cell" class="mw-redirect" title="8-cell">8-cell</a><br /> <p>Hyper-<a href="/wiki/Cube" title="Cube">cube</a><br /> 16-point </p> </td> <td style="vertical-align:top;"><a href="/wiki/24-cell" title="24-cell">24-cell</a><br /> <p><br />24-point </p> </td> <td style="vertical-align:top;"><a href="/wiki/600-cell" title="600-cell">600-cell</a><br /> <p>Hyper-<a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a><br /> 120-point </p> </td> <td style="vertical-align:top;"><a href="/wiki/120-cell" title="120-cell">120-cell</a><br /> <p>Hyper-<a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a><br /> 600-point </p> </td></tr> <tr> <th style="text-align:right;"><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <td>{3, 3, 3} </td> <td>{3, 3, 4} </td> <td>{4, 3, 3} </td> <td>{3, 4, 3} </td> <td>{3, 3, 5} </td> <td>{5, 3, 3} </td></tr> <tr> <th style="text-align:right;"><a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter mirrors</a> </th> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td></tr> <tr> <th style="text-align:right;">Mirror dihedrals </th> <td><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">𝝅</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Graph </th> <td><span typeof="mw:File"><a href="/wiki/File:4-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/120px-4-simplex_t0.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/180px-4-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/240px-4-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:4-cube_t3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/120px-4-cube_t3.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/180px-4-cube_t3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/240px-4-cube_t3.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:4-cube_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/120px-4-cube_t0.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/180px-4-cube_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/240px-4-cube_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:24-cell_t0_F4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/120px-24-cell_t0_F4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/180px-24-cell_t0_F4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/240px-24-cell_t0_F4.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:600-cell_graph_H4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/120px-600-cell_graph_H4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/180px-600-cell_graph_H4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/240px-600-cell_graph_H4.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:120-cell_graph_H4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/120px-120-cell_graph_H4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/180px-120-cell_graph_H4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/240px-120-cell_graph_H4.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td></tr> <tr> <th style="text-align:right;">Vertices </th> <td>5 tetrahedral </td> <td>8 octahedral </td> <td>16 tetrahedral </td> <td>24 cubical </td> <td>120 icosahedral </td> <td>600 tetrahedral </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/120-cell#Chords" title="120-cell">Edges</a> </th> <td>10 triangular </td> <td>24 square </td> <td>32 triangular </td> <td>96 triangular </td> <td>720 pentagonal </td> <td>1200 triangular </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Faces </th> <td>10 triangles </td> <td>32 triangles </td> <td>24 squares </td> <td>96 triangles </td> <td>1200 triangles </td> <td>720 pentagons </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Cells </th> <td>5 tetrahedra </td> <td>16 tetrahedra </td> <td>8 cubes </td> <td>24 octahedra </td> <td>600 tetrahedra </td> <td>120 dodecahedra </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/600-cell#Clifford_parallel_cell_rings" title="600-cell">Tori</a> </th> <td>1 <a href="/wiki/5-cell#Boerdijk–Coxeter_helix" title="5-cell">5-tetrahedron</a> </td> <td>2 <a href="/wiki/16-cell#Helical_construction" title="16-cell">8-tetrahedron</a> </td> <td>2 <a href="/wiki/8-cell#Construction" class="mw-redirect" title="8-cell">4-cube</a> </td> <td>4 <a href="/wiki/24-cell#Cell_rings" title="24-cell">6-octahedron</a> </td> <td>20 <a href="/wiki/600-cell#Boerdijk–Coxeter_helix_rings" title="600-cell">30-tetrahedron</a> </td> <td>12 <a href="/wiki/120-cell#Intertwining_rings" title="120-cell">10-dodecahedron</a> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Inscribed </th> <td>120 in 120-cell </td> <td>675 in 120-cell </td> <td>2 16-cells </td> <td>3 8-cells </td> <td>25 24-cells </td> <td>10 600-cells </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/Great_circle" title="Great circle">Great polygons</a> </th> <td> </td> <td>2 <a href="/wiki/16-cell#Coordinates" title="16-cell">squares</a> x 3 </td> <td>4 rectangles x 4 </td> <td>4 <a href="/wiki/24-cell#Hexagons" title="24-cell">hexagons</a> x 4 </td> <td>12 <a href="/wiki/600-cell#Geodesics" title="600-cell">decagons</a> x 6 </td> <td>100 <a href="/wiki/120-cell#Chords" title="120-cell">irregular hexagons</a> x 4 </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygons</a> </th> <td>1 <a href="/wiki/5-cell#Boerdijk–Coxeter_helix" title="5-cell">pentagon</a> x 2 </td> <td>1 <a href="/wiki/16-cell#Helical_construction" title="16-cell">octagon</a> x 3 </td> <td>2 <a href="/wiki/Octagon#Skew_octagon" title="Octagon">octagons</a> x 4 </td> <td>2 <a href="/wiki/Dodecagon#Skew_dodecagon" title="Dodecagon">dodecagons</a> x 4 </td> <td>4 <a href="/wiki/30-gon#Petrie_polygons" class="mw-redirect" title="30-gon">30-gons</a> x 6 </td> <td>20 <a href="/wiki/30-gon#Petrie_polygons" class="mw-redirect" title="30-gon">30-gons</a> x 4 </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Long radius </th> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Edge length </th> <td><small><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 {\sqrt {\tfrac {5}{2}}}\approx 1.581}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>1.581</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7891ac9b4af750cc3e54f5f21486f4b396729a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}"></span></small> </td> <td><small><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 {\sqrt {2}}\approx 1.414}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.414</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1.414}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fafaf1870e0527a631858dcc0c7579b34ff297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.493ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\approx 1.414}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><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 {\tfrac {1}{\phi }}\approx 0.618}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>ϕ</mi> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.618</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1e8a92bb277903a43ca3ae567ea64505fc32eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.211ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}"></span></small> </td> <td><small><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 {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.270</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29afe3cfc54031633a59df0637d1a3f7f71bf684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.233ex; height:4.343ex;" alt="{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}"></span></small> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Short radius </th> <td><small><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 {\tfrac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4825cd2a1ca51dfc4d53042434f6d3733370a57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}}"></span></small> </td> <td><small><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 {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span></small> </td> <td><small><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 {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span></small> </td> <td><small><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 {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.707</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4dc0f6527d5ac3f56ed3e68bc66e0b2f6a2bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"></span></small> </td> <td><small><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 {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.926</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05c35ecaf7f1d828adb58f78720535d06a3c177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.366ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"></span></small> </td> <td><small><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 {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.926</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05c35ecaf7f1d828adb58f78720535d06a3c177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.366ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"></span></small> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Area </th> <td><small><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 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>10.825</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaa7c7d04842630db48b33b9d74c338be70c6b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.894ex; height:4.843ex;" alt="{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}"></span></small> </td> <td><small><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 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>32</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>27.713</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b95c593579e89ea2c8e3ebb2207bed8433a02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.027ex; height:4.843ex;" alt="{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}"></span></small> </td> <td><small><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 24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be92101c8b0277e66fdefeef1ccdd7788e88ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 24}"></span></small> </td> <td><small><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 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>96</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>16</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>41.569</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b99126f57cc5b6592e4ef72670abbdf84068c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.849ex; height:4.843ex;" alt="{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}"></span></small> </td> <td><small><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 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mrow> <mn>4</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>198.48</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5688be271d95c5759354e1e5459c0fc1f8cbfc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.839ex; height:5.009ex;" alt="{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}"></span></small> </td> <td><small><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 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>720</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>25</mn> <mo>+</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </msqrt> <mrow> <mn>8</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>90.366</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14322accb342db5a422b3b14329f5e4576ef6e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.35ex; height:7.509ex;" alt="{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}"></span></small> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Volume </th> <td><small><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 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>24</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>2.329</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8253a425495bc8bc6c197e38e71d1eb1515a37d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.569ex; height:4.843ex;" alt="{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}"></span></small> </td> <td><small><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 16\left({\tfrac {1}{3}}\right)\approx 5.333}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>5.333</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4c8bcc6a19368c79cc7db6ebad8bb5b10d5cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.541ex; height:4.843ex;" alt="{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}"></span></small> </td> <td><small><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 8}"></span></small> </td> <td><small><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 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>11.314</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd86a479242f86673a77d371d02474fd1211994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.072ex; height:4.843ex;" alt="{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}"></span></small> </td> <td><small><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 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>600</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mrow> <mn>12</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>16.693</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6510bf43702967fc6602d15435caea9989169b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.499ex; height:5.009ex;" alt="{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}"></span></small> </td> <td><small><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 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>120</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>15</mn> <mo>+</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> </msqrt> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>18.118</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e90aaf6322085e2c0dd2bf8ca33e054d4c46592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.979ex; height:5.009ex;" alt="{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}"></span></small> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">4-Content </th> <td><small><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 {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>24</mn> </mfrac> </mstyle> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.146</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd8259f9d61b844367d627adf4da0edae4aef88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.279ex; height:5.343ex;" alt="{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}"></span></small> </td> <td><small><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 {\tfrac {2}{3}}\approx 0.667}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.667</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}\approx 0.667}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f729b57d66245544a3f57b8d2b0a31a0a9a883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.053ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}\approx 0.667}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></small> </td> <td><small><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span></small> </td> <td><small><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 {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Short</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Vol</mtext> </mrow> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>3.863</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04209037c0378ec46376aba5e33615134ace5ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.956ex; height:3.676ex;" alt="{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}"></span></small> </td> <td><small><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 {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Short</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Vol</mtext> </mrow> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>4.193</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f33beed2b51250ff3c61ead8b4214092c74e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.956ex; height:3.676ex;" alt="{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}"></span></small> </td></tr></tbody></table> <p>Schläfli performed this work in relative obscurity and it was published in full only posthumously in 1901. It had little influence until it was rediscovered and <a href="/wiki/Regular_polytopes_(book)" class="mw-redirect" title="Regular polytopes (book)">fully documented in 1948</a> by <a href="/wiki/H.S.M._Coxeter" class="mw-redirect" title="H.S.M. Coxeter">H.S.M. Coxeter</a>. </p><p>In 1878 <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">William Kingdon Clifford</a> introduced what is now termed <a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebra</a>, unifying Hamilton's quaternions with <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>'s algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions. The <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a> on the surface of the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a cylinder is "flat"). </p> <div class="mw-heading mw-heading4"><h4 id="Non-Euclidean_geometry">Non-Euclidean geometry</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=31" title="Edit section: Non-Euclidean geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></div> <p>The century's most influential development in geometry occurred when, around 1830, <a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">János Bolyai</a> and <a href="/wiki/Nikolai_Ivanovich_Lobachevsky" class="mw-redirect" title="Nikolai Ivanovich Lobachevsky">Nikolai Ivanovich Lobachevsky</a> separately published work on <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, in which the parallel postulate is not valid.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates. </p><p>In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the <i>Elements</i>. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the <i>Elements,</i> shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third <a href="https://en.wiktionary.org/wiki/vertex" class="extiw" title="wikt:vertex">vertex</a>. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the <a href="/wiki/Real_number#Completeness" title="Real number">completeness</a> property of the real numbers. Starting with <a href="/wiki/Moritz_Pasch" title="Moritz Pasch">Moritz Pasch</a> in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert</a>,<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">George Birkhoff</a>,<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski</a>.<sup id="cite_ref-Tarski_1951_33-0" class="reference"><a href="#cite_note-Tarski_1951-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="20th_century_and_relativity">20th century and relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=32" title="Edit section: 20th century and relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:1919_eclipse_negative.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/1919_eclipse_negative.jpg/220px-1919_eclipse_negative.jpg" decoding="async" width="220" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/1919_eclipse_negative.jpg/330px-1919_eclipse_negative.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/1919_eclipse_negative.jpg/440px-1919_eclipse_negative.jpg 2x" data-file-width="700" data-file-height="899" /></a><figcaption>A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar <a href="/wiki/Eclipse" title="Eclipse">eclipse</a>. The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.</figcaption></figure> <p><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein's</a> theory of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> involves a four-dimensional <a href="/wiki/Space-time" class="mw-redirect" title="Space-time">space-time</a>, the <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, which is <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a>. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> cannot be proved, are also useful for describing the physical world. </p><p>However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, for which the geometry of the space part of space-time is not Euclidean geometry.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the <a href="/wiki/Global_Positioning_System" title="Global Positioning System">GPS</a> system.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="As_a_description_of_the_structure_of_space">As a description of the structure of space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=33" title="Edit section: As a description of the structure of space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclid believed that his <a href="/wiki/Axioms" class="mw-redirect" title="Axioms">axioms</a> were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms,<sup id="cite_ref-Trudeau_36-0" class="reference"><a href="#cite_note-Trudeau-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called <i>Euclidean motions</i>, which include translations, reflections and rotations of figures.<sup id="cite_ref-Euclidean_Motion_37-0" class="reference"><a href="#cite_note-Euclidean_Motion-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> and figures may be moved to any location while maintaining <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a>; and postulate 5 (the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>) that space is flat (has no <a href="/wiki/Intrinsic_curvature" class="mw-redirect" title="Intrinsic curvature">intrinsic curvature</a>).<sup id="cite_ref-Penrose_38-0" class="reference"><a href="#cite_note-Penrose-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>As discussed above, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> significantly modifies this view. </p><p>The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite<sup id="cite_ref-Heath,_p._200_39-0" class="reference"><a href="#cite_note-Heath,_p._200-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> (see below) and what its <a href="/wiki/Topology" title="Topology">topology</a> is. Modern, more rigorous reformulations of the system<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a <a href="/wiki/Torus" title="Torus">torus</a> for two-dimensional Euclidean geometry). </p> <div class="mw-heading mw-heading2"><h2 id="Treatment_of_infinity">Treatment of infinity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=34" title="Edit section: Treatment of infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Infinite_objects">Infinite objects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=35" title="Edit section: Infinite objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "<a href="/wiki/Infinity" title="Infinity">infinite</a> lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is infinite.<sup id="cite_ref-Heath,_p._200_39-1" class="reference"><a href="#cite_note-Heath,_p._200-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>The notion of <a href="/wiki/Infinitesimals" class="mw-redirect" title="Infinitesimals">infinitesimal quantities</a> had previously been discussed extensively by the <a href="/wiki/Eleatic_School" class="mw-redirect" title="Eleatic School">Eleatic School</a>, but nobody had been able to put them on a firm logical basis, with paradoxes such as <a href="/wiki/Zeno%27s_paradox" class="mw-redirect" title="Zeno's paradox">Zeno's paradox</a> occurring that had not been resolved to universal satisfaction. Euclid used the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> rather than infinitesimals.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>Later ancient commentators, such as <a href="/wiki/Proclus" title="Proclus">Proclus</a> (410–485 CE), treated many questions about infinity as issues demanding proof and, e.g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>At the turn of the 20th century, <a href="/wiki/Otto_Stolz" title="Otto Stolz">Otto Stolz</a>, <a href="/wiki/Paul_du_Bois-Reymond" title="Paul du Bois-Reymond">Paul du Bois-Reymond</a>, <a href="/wiki/Giuseppe_Veronese" title="Giuseppe Veronese">Giuseppe Veronese</a>, and others produced controversial work on <a href="/wiki/Archimedean_property" title="Archimedean property">non-Archimedean</a> models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>–<a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a> sense.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Fifty years later, <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a> provided a rigorous logical foundation for Veronese's work.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Infinite_processes">Infinite processes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=36" title="Edit section: Infinite processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ancient geometers may have considered the parallel postulate – that two parallel lines do not ever intersect – less certain than the others because it makes a statement about infinitely remote regions of space, and so cannot be physically verified.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p>The modern formulation of <a href="/wiki/Proof_by_induction" class="mw-redirect" title="Proof by induction">proof by induction</a> was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>Supposed paradoxes involving infinite series, such as <a href="/wiki/Zeno%27s_paradox" class="mw-redirect" title="Zeno's paradox">Zeno's paradox</a>, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> in IX.35 without commenting on the possibility of letting the number of terms become infinite. </p> <div class="mw-heading mw-heading2"><h2 id="Logical_basis">Logical basis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=37" title="Edit section: Logical basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Euclidean_geometry&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">June 2010</span>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a>, <a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a>, and <a href="/wiki/Real_closed_field" title="Real closed field">Real closed field</a></div> <div class="mw-heading mw-heading3"><h3 id="Classical_logic">Classical logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=38" title="Edit section: Classical logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclid frequently used the method of <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a>, and therefore the traditional presentation of Euclidean geometry assumes <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, in which every proposition is either true or false, i.e., for any proposition P, the proposition "P or not P" is automatically true. </p> <div class="mw-heading mw-heading3"><h3 id="Modern_standards_of_rigor">Modern standards of rigor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=39" title="Edit section: Modern standards of rigor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries.<sup id="cite_ref-Smith_47-0" class="reference"><a href="#cite_note-Smith-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> The role of <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notions</a>, or undefined concepts, was clearly put forward by <a href="/wiki/Alessandro_Padoa" title="Alessandro Padoa">Alessandro Padoa</a> of the <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a> delegation at the 1900 Paris conference:<sup id="cite_ref-Smith_47-1" class="reference"><a href="#cite_note-Smith-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-revue_48-0" class="reference"><a href="#cite_note-revue-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> <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></p><blockquote class="templatequote"><p>...when we begin to formulate the theory, we can imagine that the undefined symbols are <i>completely devoid of meaning</i> and that the unproved propositions are simply <i>conditions</i> imposed upon the undefined symbols. </p><p>Then, the <i>system of ideas</i> that we have initially chosen is simply <i>one interpretation</i> of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind by <i>another interpretation</i>.. that satisfies the conditions... </p><p><i>Logical</i> questions thus become completely independent of <i>empirical</i> or <i>psychological</i> questions... </p><p> The system of undefined symbols can then be regarded as the <i>abstraction</i> obtained from the <i>specialized theories</i> that result when...the system of undefined symbols is successively replaced by each of the interpretations...</p><div class="templatequotecite">— <cite>Padoa, <i>Essai d'une théorie algébrique des nombre entiers, avec une Introduction logique à une théorie déductive quelconque</i></cite></div></blockquote> <p>That is, mathematics is context-independent knowledge within a hierarchical framework. As said by <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>:<sup id="cite_ref-Newman_49-0" class="reference"><a href="#cite_note-Newman-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If our hypothesis is about <i>anything</i>, and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.</p><div class="templatequotecite">— <cite>Bertrand Russell, <i>Mathematics and the metaphysicians</i></cite></div></blockquote> <p>Such foundational approaches range between <a href="/wiki/Foundationalism" title="Foundationalism">foundationalism</a> and <a href="/wiki/Formalism_(mathematics)" class="mw-redirect" title="Formalism (mathematics)">formalism</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Axiomatic_formulations">Axiomatic formulations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=40" title="Edit section: Axiomatic formulations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Geometry is the science of correct reasoning on incorrect figures.</p><div class="templatequotecite">— <cite><a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a>, <i>How to Solve It</i>, p. 208</cite></div></blockquote> <ul><li>Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time.<sup id="cite_ref-Russell_50-0" class="reference"><a href="#cite_note-Russell-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a>.</li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a>: Hilbert's axioms had the goal of identifying a <i>simple</i> and <i>complete</i> set of <i>independent</i> axioms from which the most important geometric theorems could be deduced. The outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel postulate.</li> <li><a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">Birkhoff's axioms</a>: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>.<sup id="cite_ref-Brikhoff_51-0" class="reference"><a href="#cite_note-Brikhoff-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Smith2_52-0" class="reference"><a href="#cite_note-Smith2-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Moise_53-0" class="reference"><a href="#cite_note-Moise-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> The notions of <i>angle</i> and <i>distance</i> become primitive concepts.<sup id="cite_ref-Silvester_54-0" class="reference"><a href="#cite_note-Silvester-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a>: <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> (1902–1983) and his students defined <i>elementary</i> Euclidean geometry as the geometry that can be expressed in <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> and does not depend on <a href="/wiki/Set_theory" title="Set theory">set theory</a> for its logical basis,<sup id="cite_ref-Tarski0_55-0" class="reference"><a href="#cite_note-Tarski0-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> in contrast to Hilbert's axioms, which involve point sets.<sup id="cite_ref-Simmons_56-0" class="reference"><a href="#cite_note-Simmons-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">sense</a>: there is an algorithm that, for every proposition, can be shown either true or false.<sup id="cite_ref-Tarski_1951_33-1" class="reference"><a href="#cite_note-Tarski_1951-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> (This does not violate <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's theorem</a>, because Euclidean geometry cannot describe a sufficient amount of <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">arithmetic</a> for the theorem to apply.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup>) This is equivalent to the decidability of <a href="/wiki/Real_closed_fields" class="mw-redirect" title="Real closed fields">real closed fields</a>, of which elementary Euclidean geometry is a model.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=41" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Absolute_geometry" title="Absolute geometry">Absolute geometry</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic geometry</a></li> <li><a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">Birkhoff's axioms</a></li> <li><a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence geometry</a></li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">List of interactive geometry software</a></li> <li><a href="/wiki/Metric_space" title="Metric space">Metric space</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></li> <li><a href="/wiki/Ordered_geometry" title="Ordered geometry">Ordered geometry</a></li> <li><a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Classical_theorems">Classical theorems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=42" title="Edit section: Classical theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a></li> <li><a href="/wiki/Butterfly_theorem" title="Butterfly theorem">Butterfly theorem</a></li> <li><a href="/wiki/Ceva%27s_theorem" title="Ceva's theorem">Ceva's theorem</a></li> <li><a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a></li> <li><a href="/wiki/Menelaus%27_theorem" class="mw-redirect" title="Menelaus' theorem">Menelaus' theorem</a></li> <li><a href="/wiki/Nine-point_circle" title="Nine-point circle">Nine-point circle</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=43" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 35em;"> <ol class="references"> <li id="cite_note-eves1_19-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-eves1_19_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-eves1_19_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFEves1963">Eves 1963</a>, p. 19.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFEves1963">Eves 1963</a>, p. 10.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Misner, Thorne, and Wheeler (1973), p. 47.</span> </li> <li id="cite_note-Wolfe-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wolfe_4-0">^</a></b></span> <span class="reference-text">The assumptions of Euclid are discussed from a modern perspective in <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHarold_E._Wolfe2007" class="citation book cs1">Harold E. Wolfe (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VPHn3MutWhQC&pg=PA9"><i>Introduction to Non-Euclidean Geometry</i></a>. Mill Press. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4067-1852-2" title="Special:BookSources/978-1-4067-1852-2"><bdi>978-1-4067-1852-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Non-Euclidean+Geometry&rft.pages=9&rft.pub=Mill+Press&rft.date=2007&rft.isbn=978-1-4067-1852-2&rft.au=Harold+E.+Wolfe&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVPHn3MutWhQC%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">tr. Heath, pp. 195–202.</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="CITEREFVenema2006" class="citation cs2">Venema, Gerard A. (2006), <i>Foundations of Geometry</i>, Prentice-Hall, p. 8, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-143700-5" title="Special:BookSources/978-0-13-143700-5"><bdi>978-0-13-143700-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Geometry&rft.pages=8&rft.pub=Prentice-Hall&rft.date=2006&rft.isbn=978-0-13-143700-5&rft.aulast=Venema&rft.aufirst=Gerard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" 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="CITEREFFlorence_P._Lewis1920" class="citation cs2">Florence P. Lewis (Jan 1920), "History of the Parallel Postulate", <i>The American Mathematical Monthly</i>, <b>27</b> (1), The American Mathematical Monthly, Vol. 27, No. 1: 16–23, <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%2F2973238">10.2307/2973238</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/2973238">2973238</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+Mathematical+Monthly&rft.atitle=History+of+the+Parallel+Postulate&rft.volume=27&rft.issue=1&rft.pages=16-23&rft.date=1920-01&rft_id=info%3Adoi%2F10.2307%2F2973238&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2973238%23id-name%3DJSTOR&rft.au=Florence+P.+Lewis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Ball, p. 56.</span> </li> <li id="cite_note-set_theory-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-set_theory_9-0">^</a></b></span> <span class="reference-text">Within Euclid's assumptions, it is quite easy to give a formula for area of triangles and squares. However, in a more general context like set theory, it is not as easy to prove that the area of a square is the sum of areas of its pieces, for example. See <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> and <a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a>.</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="CITEREFDaniel_Shanks2002" class="citation book cs1">Daniel Shanks (2002). <i>Solved and Unsolved Problems in Number Theory</i>. American Mathematical Society.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solved+and+Unsolved+Problems+in+Number+Theory&rft.pub=American+Mathematical+Society&rft.date=2002&rft.au=Daniel+Shanks&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" 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">Euclid, book I, proposition 5, tr. Heath, p. 251.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Ignoring the alleged difficulty of Book I, Proposition 5, <a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Sir Thomas L. Heath</a> mentions another interpretation. This rests on the resemblance of the figure's lower straight lines to a steeply inclined bridge that could be crossed by an ass but not by a horse: "But there is another view (as I have learnt lately) which is more complimentary to the ass. It is that, the figure of the proposition being like that of a trestle bridge, with a ramp at each end which is more practicable the flatter the figure is drawn, the bridge is such that, while a horse could not surmount the ramp, an ass could; in other words, the term is meant to refer to the sure-footedness of the ass rather than to any want of intelligence on his part." (in "Excursis II", volume 1 of Heath's translation of <i>The Thirteen Books of the Elements</i>).</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">Euclid, book I, proposition 32.</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">Heath, p. 135. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=drnY3Vjix3kC&pg=PA135">Extract of page 135</a>.</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">Heath, p. 318.</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">Euclid, book XII, proposition 2.</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">Euclid, book XI, proposition 33.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Ball, p. 66.</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">Ball, p. 5.</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">Eves, vol. 1, p. 5; Mlodinow, p. 7.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTom_Hull" class="citation web cs1">Tom Hull. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190618084941/http://mars.wne.edu/~thull/omfiles/geoconst.html">"Origami and Geometric Constructions"</a>. Archived from <a rel="nofollow" class="external text" href="http://mars.wne.edu/~thull/omfiles/geoconst.html">the original</a> on 2019-06-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-12-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Origami+and+Geometric+Constructions&rft.au=Tom+Hull&rft_id=http%3A%2F%2Fmars.wne.edu%2F~thull%2Fomfiles%2Fgeoconst.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></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">Eves, p. 27.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Ball, pp. 268ff.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Eves (1963).</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Hofstadter 1979, p. 91.</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">Theorem 120, Elements of Abstract Algebra, Allan Clark, Dover, <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/0-486-64725-0" title="Special:BookSources/0-486-64725-0">0-486-64725-0</a>.</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">Eves (1963), p. 64.</span> </li> <li id="cite_note-FOOTNOTEStillwell200118–21-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStillwell200118–21_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStillwell2001">Stillwell 2001</a>, p. 18–21; In four-dimensional Euclidean geometry, a <a href="/wiki/Quaternion" title="Quaternion">quaternion</a> is simply a (w, x, y, z) Cartesian coordinate. <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> did not see them as such when he <a href="/wiki/History_of_quaternions" title="History of quaternions">discovered the quaternions</a>. <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Schläfli</a> would be the first to consider <a href="/wiki/4-dimensional_space" class="mw-redirect" title="4-dimensional space">four-dimensional Euclidean space</a>, publishing his discovery of the regular <a href="/wiki/Polyscheme" class="mw-redirect" title="Polyscheme">polyschemes</a> in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space. Although he described a quaternion as an <i>ordered four-element multiple of real numbers</i>, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.</span> </li> <li id="cite_note-FOOTNOTEPerez-GraciaThomas2017-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPerez-GraciaThomas2017_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPerez-GraciaThomas2017">Perez-Gracia & Thomas 2017</a>; "It is actually Cayley whom we must thank for the correct development of quaternions as a representation of rotations."</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">Ball, p. 485.</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">* <a href="/wiki/Howard_Eves" title="Howard Eves">Howard Eves</a>, 1997 (1958). <i>Foundations and Fundamental Concepts of Mathematics</i>. Dover.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)", Annals of Mathematics 33.</span> </li> <li id="cite_note-Tarski_1951-33"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tarski_1951_33-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Tarski_1951_33-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Tarski (1951).</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Misner, Thorne, and Wheeler (1973), p. 191.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Rizos, Chris. <a href="/wiki/University_of_New_South_Wales" title="University of New South Wales">University of New South Wales</a>. <a rel="nofollow" class="external text" href="http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm">GPS Satellite Signals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100612004027/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm">Archived</a> 2010-06-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 1999.</span> </li> <li id="cite_note-Trudeau-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-Trudeau_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_J._Trudeau2008" class="citation book cs1">Richard J. Trudeau (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YRB4VBCLB3IC&pg=PA39">"Euclid's axioms"</a>. <i>The Non-Euclidean Revolution</i>. Birkhäuser. pp. 39 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4782-7" title="Special:BookSources/978-0-8176-4782-7"><bdi>978-0-8176-4782-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Euclid%27s+axioms&rft.btitle=The+Non-Euclidean+Revolution&rft.pages=39+%27%27ff%27%27&rft.pub=Birkh%C3%A4user&rft.date=2008&rft.isbn=978-0-8176-4782-7&rft.au=Richard+J.+Trudeau&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYRB4VBCLB3IC%26pg%3DPA39&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Euclidean_Motion-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Euclidean_Motion_37-0">^</a></b></span> <span class="reference-text"> See, for example: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuciano_da_Fontoura_CostaRoberto_Marcondes_Cesar2001" class="citation book cs1">Luciano da Fontoura Costa; Roberto Marcondes Cesar (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x_wiWedtc0cC&pg=PA314"><i>Shape analysis and classification: theory and practice</i></a>. CRC Press. p. 314. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8493-3493-4" title="Special:BookSources/0-8493-3493-4"><bdi>0-8493-3493-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Shape+analysis+and+classification%3A+theory+and+practice&rft.pages=314&rft.pub=CRC+Press&rft.date=2001&rft.isbn=0-8493-3493-4&rft.au=Luciano+da+Fontoura+Costa&rft.au=Roberto+Marcondes+Cesar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx_wiWedtc0cC%26pg%3DPA314&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelmut_PottmannJohannes_Wallner2010" class="citation book cs1">Helmut Pottmann; Johannes Wallner (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3Mk2JIJKsGwC&pg=PA60"><i>Computational Line Geometry</i></a>. Springer. p. 60. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-04017-7" title="Special:BookSources/978-3-642-04017-7"><bdi>978-3-642-04017-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Line+Geometry&rft.pages=60&rft.pub=Springer&rft.date=2010&rft.isbn=978-3-642-04017-7&rft.au=Helmut+Pottmann&rft.au=Johannes+Wallner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3Mk2JIJKsGwC%26pg%3DPA60&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span> The <i>group of motions</i> underlie the metric notions of geometry. See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelix_Klein2004" class="citation book cs1">Felix Klein (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fj-ryrSBuxAC&pg=PA167"><i>Elementary Mathematics from an Advanced Standpoint: Geometry</i></a> (Reprint of 1939 Macmillan Company ed.). Courier Dover. p. 167. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43481-8" title="Special:BookSources/0-486-43481-8"><bdi>0-486-43481-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Mathematics+from+an+Advanced+Standpoint%3A+Geometry&rft.pages=167&rft.edition=Reprint+of+1939+Macmillan+Company&rft.pub=Courier+Dover&rft.date=2004&rft.isbn=0-486-43481-8&rft.au=Felix+Klein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dfj-ryrSBuxAC%26pg%3DPA167&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Penrose-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-Penrose_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoger_Penrose2007" class="citation book cs1">Roger Penrose (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=coahAAAACAAJ&q=editions:cYahAAAACAAJ"><i>The Road to Reality: A Complete Guide to the Laws of the Universe</i></a>. Vintage Books. p. 29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-679-77631-4" title="Special:BookSources/978-0-679-77631-4"><bdi>978-0-679-77631-4</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+Road+to+Reality%3A+A+Complete+Guide+to+the+Laws+of+the+Universe&rft.pages=29&rft.pub=Vintage+Books&rft.date=2007&rft.isbn=978-0-679-77631-4&rft.au=Roger+Penrose&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcoahAAAACAAJ%26q%3Deditions%3AcYahAAAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Heath,_p._200-39"><span class="mw-cite-backlink">^ <a href="#cite_ref-Heath,_p._200_39-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Heath,_p._200_39-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Heath, p. 200.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">e.g., Tarski (1951).</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">Ball, p. 31.</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">Heath, p. 268.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Giuseppe Veronese, On Non-Archimedean Geometry, 1908. English translation in Real Numbers, Generalizations of the Reals, and Theories of Continua, ed. Philip Ehrlich, Kluwer, 1994.</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">Robinson, Abraham (1966). Non-standard analysis.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Nagel and Newman, 1958, p. 9.</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">Cajori (1918), p. 197.</span> </li> <li id="cite_note-Smith-47"><span class="mw-cite-backlink">^ <a href="#cite_ref-Smith_47-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Smith_47-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A detailed discussion can be found in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_T._Smith2000" class="citation book cs1">James T. Smith (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mWpWplOVQ6MC&pg=RA1-PA19">"Chapter 2: Foundations"</a>. <i>Methods of geometry</i>. Wiley. pp. 19 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-25183-6" title="Special:BookSources/0-471-25183-6"><bdi>0-471-25183-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2%3A+Foundations&rft.btitle=Methods+of+geometry&rft.pages=19+%27%27ff%27%27&rft.pub=Wiley&rft.date=2000&rft.isbn=0-471-25183-6&rft.au=James+T.+Smith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmWpWplOVQ6MC%26pg%3DRA1-PA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-revue-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-revue_48-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSociété_française_de_philosophie1900" class="citation book cs1">Société française de philosophie (1900). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4aoLAAAAIAAJ&pg=PA592"><i>Revue de métaphysique et de morale, Volume 8</i></a>. Hachette. p. 592.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Revue+de+m%C3%A9taphysique+et+de+morale%2C+Volume+8&rft.pages=592&rft.pub=Hachette&rft.date=1900&rft.au=Soci%C3%A9t%C3%A9+fran%C3%A7aise+de+philosophie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4aoLAAAAIAAJ%26pg%3DPA592&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Newman-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-Newman_49-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBertrand_Russell2000" class="citation book cs1">Bertrand Russell (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_b2ShqRj8YMC&pg=PA1577">"Mathematics and the metaphysicians"</a>. In James Roy Newman (ed.). <i>The world of mathematics</i>. Vol. 3 (Reprint of Simon and Schuster 1956 ed.). Courier Dover Publications. p. 1577. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-41151-6" title="Special:BookSources/0-486-41151-6"><bdi>0-486-41151-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+and+the+metaphysicians&rft.btitle=The+world+of+mathematics&rft.pages=1577&rft.edition=Reprint+of+Simon+and+Schuster+1956&rft.pub=Courier+Dover+Publications&rft.date=2000&rft.isbn=0-486-41151-6&rft.au=Bertrand+Russell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_b2ShqRj8YMC%26pg%3DPA1577&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Russell-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-Russell_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBertrand_Russell1897" class="citation book cs1">Bertrand Russell (1897). "Introduction". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NecGAAAAYAAJ&pg=PA1"><i>An essay on the foundations of geometry</i></a>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction&rft.btitle=An+essay+on+the+foundations+of+geometry&rft.pub=Cambridge+University+Press&rft.date=1897&rft.au=Bertrand+Russell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNecGAAAAYAAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Brikhoff-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brikhoff_51-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_David_BirkhoffRalph_Beatley1999" class="citation book cs1">George David Birkhoff; Ralph Beatley (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TB6xYdomdjQC&pg=PA38">"Chapter 2: The five fundamental principles"</a>. <i>Basic Geometry</i> (3rd ed.). AMS Bookstore. pp. 38 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2101-6" title="Special:BookSources/0-8218-2101-6"><bdi>0-8218-2101-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2%3A+The+five+fundamental+principles&rft.btitle=Basic+Geometry&rft.pages=38+%27%27ff%27%27&rft.edition=3rd&rft.pub=AMS+Bookstore&rft.date=1999&rft.isbn=0-8218-2101-6&rft.au=George+David+Birkhoff&rft.au=Ralph+Beatley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTB6xYdomdjQC%26pg%3DPA38&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Smith2-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-Smith2_52-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_T._Smith2000" class="citation book cs1">James T. Smith (10 January 2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mWpWplOVQ6MC&pg=RA1-PA84">"Chapter 3: Elementary Euclidean Geometry"</a>. <i>Cited work</i>. John Wiley & Sons. pp. 84 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471251835" title="Special:BookSources/9780471251835"><bdi>9780471251835</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+3%3A+Elementary+Euclidean+Geometry&rft.btitle=Cited+work&rft.pages=84+%27%27ff%27%27&rft.pub=John+Wiley+%26+Sons&rft.date=2000-01-10&rft.isbn=9780471251835&rft.au=James+T.+Smith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmWpWplOVQ6MC%26pg%3DRA1-PA84&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Moise-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-Moise_53-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwin_E._Moise1990" class="citation book cs1">Edwin E. Moise (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3UjvAAAAMAAJ&q=Birkhoff"><i>Elementary geometry from an advanced standpoint</i></a> (3rd ed.). Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-50867-2" title="Special:BookSources/0-201-50867-2"><bdi>0-201-50867-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+geometry+from+an+advanced+standpoint&rft.edition=3rd&rft.pub=Addison%E2%80%93Wesley&rft.date=1990&rft.isbn=0-201-50867-2&rft.au=Edwin+E.+Moise&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3UjvAAAAMAAJ%26q%3DBirkhoff&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Silvester-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-Silvester_54-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_R._Silvester2001" class="citation book cs1">John R. Silvester (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VtH_QG6scSUC&pg=PA5">"§1.4 Hilbert and Birkhoff"</a>. <i>Geometry: ancient and modern</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850825-5" title="Special:BookSources/0-19-850825-5"><bdi>0-19-850825-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A71.4+Hilbert+and+Birkhoff&rft.btitle=Geometry%3A+ancient+and+modern&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=0-19-850825-5&rft.au=John+R.+Silvester&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVtH_QG6scSUC%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Tarski0-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tarski0_55-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlfred_Tarski2007" class="citation book cs1">Alfred Tarski (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eVVKtnKzfnUC&pg=PA16">"What is elementary geometry"</a>. In Leon Henkin; Patrick Suppes; Alfred Tarski (eds.). <i>Studies in Logic and the Foundations of Mathematics – The Axiomatic Method with Special Reference to Geometry and Physics</i> (Proceedings of International Symposium at Berkeley 1957–8; Reprint ed.). Brouwer Press. p. 16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4067-5355-4" title="Special:BookSources/978-1-4067-5355-4"><bdi>978-1-4067-5355-4</bdi></a>. <q>We regard as elementary that part of Euclidean geometry which can be formulated and established without the help of any set-theoretical devices</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=What+is+elementary+geometry&rft.btitle=Studies+in+Logic+and+the+Foundations+of+Mathematics+%E2%80%93+The+Axiomatic+Method+with+Special+Reference+to+Geometry+and+Physics&rft.pages=16&rft.edition=Proceedings+of+International+Symposium+at+Berkeley+1957%E2%80%938%3B+Reprint&rft.pub=Brouwer+Press&rft.date=2007&rft.isbn=978-1-4067-5355-4&rft.au=Alfred+Tarski&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeVVKtnKzfnUC%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Simmons-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-Simmons_56-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeith_Simmons2009" class="citation book cs1">Keith Simmons (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K5dU9bEKencC&pg=PA574">"Tarski's logic"</a>. In Dov M. Gabbay; John Woods (eds.). <i>Logic from Russell to Church</i>. Elsevier. p. 574. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51620-6" title="Special:BookSources/978-0-444-51620-6"><bdi>978-0-444-51620-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tarski%27s+logic&rft.btitle=Logic+from+Russell+to+Church&rft.pages=574&rft.pub=Elsevier&rft.date=2009&rft.isbn=978-0-444-51620-6&rft.au=Keith+Simmons&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK5dU9bEKencC%26pg%3DPA574&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" 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">Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters. <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/1-56881-238-8" title="Special:BookSources/1-56881-238-8">1-56881-238-8</a>. Pp. 25–26.</span> </li> </ol></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-lower-alpha"> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_geometry&action=edit&section=44" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall1960" class="citation book cs1"><a href="/wiki/W._W._Rouse_Ball" title="W. W. Rouse Ball">Ball, W. W. Rouse</a> (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi0000ball/page/50"><i>A Short Account of the History of Mathematics</i></a> (4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908] ed.). New York: Dover Publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi0000ball/page/50">50–62</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-20630-0" title="Special:BookSources/0-486-20630-0"><bdi>0-486-20630-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Short+Account+of+the+History+of+Mathematics&rft.place=New+York&rft.pages=50-62&rft.edition=4th+ed.+%5BReprint.+Original+publication%3A+London%3A+Macmillan+%26+Co.%2C+1908%5D&rft.pub=Dover+Publications&rft.date=1960&rft.isbn=0-486-20630-0&rft.aulast=Ball&rft.aufirst=W.+W.+Rouse&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshortaccountofhi0000ball%2Fpage%2F50&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1961" class="citation book cs1"><a href="/wiki/H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">Coxeter, H. S. M.</a> (1961). <i>Introduction to Geometry</i>. New York: Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Geometry&rft.place=New+York&rft.pub=Wiley&rft.date=1961&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1963" class="citation book cs1">Eves, Howard (1963). <i>A Survey of Geometry (Volume One)</i>. Allyn and Bacon.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Geometry+%28Volume+One%29&rft.pub=Allyn+and+Bacon&rft.date=1963&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1956" class="citation book cs1"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, Thomas L.</a> (1956). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/thirteenbooksofe00eucl"><i>The Thirteen Books of Euclid's Elements</i></a></span> (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.place=New+York&rft.edition=2nd+ed.+%5BFacsimile.+Original+publication%3A+Cambridge+University+Press%2C+1925%5D&rft.pub=Dover+Publications&rft.date=1956&rft.aulast=Heath&rft.aufirst=Thomas+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthirteenbooksofe00eucl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span> In 3 vols.: vol. 1 <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/0-486-60088-2" title="Special:BookSources/0-486-60088-2">0-486-60088-2</a>, vol. 2 <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/0-486-60089-0" title="Special:BookSources/0-486-60089-0">0-486-60089-0</a>, vol. 3 <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/0-486-60090-4" title="Special:BookSources/0-486-60090-4">0-486-60090-4</a>. Heath's authoritative translation of Euclid's Elements, plus his extensive historical research and detailed commentary throughout the text.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1"><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner, Charles W.</a>; <a href="/wiki/Kip_S._Thorne" class="mw-redirect" title="Kip S. Thorne">Thorne, Kip S.</a>; <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (1973). <i><a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Gravitation</a></i>. W. H. Freeman.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pub=W.+H.+Freeman&rft.date=1973&rft.aulast=Misner&rft.aufirst=Charles+W.&rft.au=Thorne%2C+Kip+S.&rft.au=Wheeler%2C+John+Archibald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMlodinow2001" class="citation book cs1">Mlodinow (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/euclidswindowsto00mlod"><i>Euclid's Window</i></a></span>. The Free Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780684865232" title="Special:BookSources/9780684865232"><bdi>9780684865232</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&rft.pub=The+Free+Press&rft.date=2001&rft.isbn=9780684865232&rft.au=Mlodinow&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feuclidswindowsto00mlod&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagel,_E.Newman,_J._R.1958" class="citation book cs1">Nagel, E.; Newman, J. R. (1958). <a rel="nofollow" class="external text" href="https://archive.org/details/gdelsproof00nage"><i>Gödel's Proof</i></a>. New York University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%B6del%27s+Proof&rft.pub=New+York+University+Press&rft.date=1958&rft.au=Nagel%2C+E.&rft.au=Newman%2C+J.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgdelsproof00nage&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarski1951" class="citation book cs1"><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a> (1951). <i>A Decision Method for Elementary Algebra and Geometry</i>. Univ. of California Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Decision+Method+for+Elementary+Algebra+and+Geometry&rft.pub=Univ.+of+California+Press&rft.date=1951&rft.aulast=Tarski&rft.aufirst=Alfred&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell2001" class="citation journal cs1">Stillwell, John (January 2001). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200101/fea-stillwell.pdf">"The Story of the 120-Cell"</a> <span class="cs1-format">(PDF)</span>. <i>Notices of the AMS</i>. <b>48</b> (1): 17–25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=The+Story+of+the+120-Cell&rft.volume=48&rft.issue=1&rft.pages=17-25&rft.date=2001-01&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200101%2Ffea-stillwell.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerez-GraciaThomas2017" class="citation journal cs1">Perez-Gracia, Alba; Thomas, Federico (2017). <a rel="nofollow" class="external text" href="https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf">"On Cayley's Factorization of 4D Rotations and Applications"</a> <span class="cs1-format">(PDF)</span>. <i>Adv. Appl. Clifford Algebras</i>. <b>27</b>: 523–538. <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%2Fs00006-016-0683-9">10.1007/s00006-016-0683-9</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2117%2F113067">2117/113067</a></span>. <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:12350382">12350382</a>.</cite><span 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Euclidean+geometry&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DEuclidean_geometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Plane_trigonometry">"Plane trigonometry"</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>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Plane+trigonometry&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPlane_trigonometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+geometry" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www-math.mit.edu/~kedlaya/geometryunbound">Kiran Kedlaya, <i>Geometry Unbound</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111026045325/http://www-math.mit.edu/~kedlaya/geometryunbound/">Archived</a> 2011-10-26 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (a treatment using analytic geometry; PDF format, GFDL licensed)</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 .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output 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.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="Ancient_Greek_mathematics" style="padding:3px"><table class="nowraplinks 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:Ancient_Greek_mathematics" title="Template:Ancient Greek mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ancient_Greek_mathematics" title="Template talk:Ancient Greek mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ancient_Greek_mathematics" title="Special:EditPage/Template:Ancient Greek mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Ancient_Greek_mathematics" style="font-size:114%;margin:0 4em"><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greek mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_Greek_mathematicians" title="List of Greek mathematicians">Mathematicians</a><br /><a href="/wiki/Timeline_of_ancient_Greek_mathematicians" title="Timeline of ancient Greek mathematicians">(timeline)</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anaxagoras" title="Anaxagoras">Anaxagoras</a></li> <li><a href="/wiki/Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius</a></li> <li><a href="/wiki/Archytas" title="Archytas">Archytas</a></li> <li><a href="/wiki/Aristaeus_the_Elder" title="Aristaeus the Elder">Aristaeus the Elder</a></li> <li><a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a></li> <li><a href="/wiki/Aristotle" title="Aristotle">Aristotle</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/Autolycus_of_Pitane" title="Autolycus of Pitane">Autolycus</a></li> <li><a href="/wiki/Bion_of_Abdera" title="Bion of Abdera">Bion</a></li> <li><a href="/wiki/Bryson_of_Heraclea" title="Bryson of Heraclea">Bryson</a></li> <li><a href="/wiki/Callippus" title="Callippus">Callippus</a></li> <li><a href="/wiki/Carpus_of_Antioch" title="Carpus of Antioch">Carpus</a></li> <li><a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a></li> <li><a href="/wiki/Cleomedes" title="Cleomedes">Cleomedes</a></li> <li><a href="/wiki/Conon_of_Samos" title="Conon of Samos">Conon</a></li> <li><a href="/wiki/Ctesibius" title="Ctesibius">Ctesibius</a></li> <li><a href="/wiki/Democritus" title="Democritus">Democritus</a></li> <li><a href="/wiki/Dicaearchus" title="Dicaearchus">Dicaearchus</a></li> <li><a href="/wiki/Diocles_(mathematician)" title="Diocles (mathematician)">Diocles</a></li> <li><a href="/wiki/Diophantus" title="Diophantus">Diophantus</a></li> <li><a href="/wiki/Dinostratus" title="Dinostratus">Dinostratus</a></li> <li><a href="/wiki/Dionysodorus" title="Dionysodorus">Dionysodorus</a></li> <li><a href="/wiki/Domninus_of_Larissa" title="Domninus of Larissa">Domninus</a></li> <li><a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a></li> <li><a href="/wiki/Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a></li> <li><a href="/wiki/Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius</a></li> <li><a href="/wiki/Geminus" title="Geminus">Geminus</a></li> <li><a href="/wiki/Heliodorus_of_Larissa" title="Heliodorus of Larissa">Heliodorus</a></li> <li><a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Heron</a></li> <li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Hippasus" title="Hippasus">Hippasus</a></li> <li><a href="/wiki/Hippias" title="Hippias">Hippias</a></li> <li><a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates</a></li> <li><a href="/wiki/Hypatia" title="Hypatia">Hypatia</a></li> <li><a href="/wiki/Hypsicles" title="Hypsicles">Hypsicles</a></li> <li><a href="/wiki/Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a></li> <li><a href="/wiki/Leon_(mathematician)" title="Leon (mathematician)">Leon</a></li> <li><a href="/wiki/Marinus_of_Neapolis" title="Marinus of Neapolis">Marinus</a></li> <li><a href="/wiki/Menaechmus" title="Menaechmus">Menaechmus</a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus</a></li> <li><a href="/wiki/Metrodorus_(grammarian)" title="Metrodorus (grammarian)">Metrodorus</a></li> <li><a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a></li> <li><a href="/wiki/Nicomedes_(mathematician)" title="Nicomedes (mathematician)">Nicomedes</a></li> <li><a href="/wiki/Nicoteles_of_Cyrene" title="Nicoteles of Cyrene">Nicoteles</a></li> <li><a href="/wiki/Oenopides" title="Oenopides">Oenopides</a></li> <li><a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a></li> <li><a href="/wiki/Perseus_(geometer)" title="Perseus (geometer)">Perseus</a></li> <li><a href="/wiki/Philolaus" title="Philolaus">Philolaus</a></li> <li><a href="/wiki/Philon" title="Philon">Philon</a></li> <li><a href="/wiki/Philonides_of_Laodicea" title="Philonides of Laodicea">Philonides</a></li> <li><a href="/wiki/Plato" title="Plato">Plato</a></li> <li><a href="/wiki/Porphyry_(philosopher)" title="Porphyry (philosopher)">Porphyry</a></li> <li><a href="/wiki/Posidonius" title="Posidonius">Posidonius</a></li> <li><a href="/wiki/Proclus" title="Proclus">Proclus</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Serenus_of_Antino%C3%B6polis" title="Serenus of Antinoöpolis">Serenus </a></li> <li><a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a></li> <li><a href="/wiki/Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes</a></li> <li><a href="/wiki/Sporus_of_Nicaea" title="Sporus of Nicaea">Sporus</a></li> <li><a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales</a></li> <li><a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a></li> <li><a href="/wiki/Theano_(philosopher)" title="Theano (philosopher)">Theano</a></li> <li><a href="/wiki/Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a></li> <li><a href="/wiki/Theodosius_of_Bithynia" title="Theodosius of Bithynia">Theodosius</a></li> <li><a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a></li> <li><a href="/wiki/Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a></li> <li><a href="/wiki/Thymaridas" title="Thymaridas">Thymaridas</a></li> <li><a href="/wiki/Xenocrates" title="Xenocrates">Xenocrates</a></li> <li><a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a></li> <li><a href="/wiki/Zeno_of_Sidon" title="Zeno of Sidon">Zeno of Sidon</a></li> <li><a href="/wiki/Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Treatises</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Almagest" title="Almagest">Almagest</a></i></li> <li><a href="/wiki/Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li> <li><i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i></li> <li><a href="/wiki/Apollonius_of_Perga#Conics" title="Apollonius of Perga"><i>Conics</i> <span style="font-size:85%;">(Apollonius)</span></a></li> <li><i><a href="/wiki/Catoptrics" title="Catoptrics">Catoptrics</a></i></li> <li><a href="/wiki/Data_(Euclid)" class="mw-redirect" title="Data (Euclid)"><i>Data</i> <span style="font-size:85%;">(Euclid)</span></a></li> <li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i> <span style="font-size:85%;">(Euclid)</span></a></li> <li><i><a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i></li> <li><i><a href="/wiki/On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></i></li> <li><a href="/wiki/On_the_Sizes_and_Distances_(Aristarchus)" title="On the Sizes and Distances (Aristarchus)"><i>On the Sizes and Distances</i> <span style="font-size:85%;">(Aristarchus)</span></a></li> <li><a href="/wiki/On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)"><i>On Sizes and Distances</i> <span style="font-size:85%;">(Hipparchus)</span></a></li> <li><a href="/wiki/Autolycus_of_Pitane" title="Autolycus of Pitane"><i>On the Moving Sphere</i> <span style="font-size:85%;">(Autolycus)</span></a></li> <li><a href="/wiki/Euclid%27s_Optics" title="Euclid's Optics"><i>Optics</i> <span style="font-size:85%;">(Euclid)</span></a></li> <li><i><a href="/wiki/On_Spirals" title="On Spirals">On Spirals</a></i></li> <li><i><a href="/wiki/On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i></li> <li><i><a href="/wiki/Ostomachion" title="Ostomachion">Ostomachion</a></i></li> <li><i><a href="/wiki/Planisphaerium" title="Planisphaerium">Planisphaerium</a></i></li> <li><a href="/wiki/Theodosius%27_Spherics" title="Theodosius' Spherics"><i>Spherics</i> <span style="font-size:85%;">(Theodosius)</span></a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria"><i>Spherics</i> <span style="font-size:85%;">(Menelaus)</span></a></li> <li><i><a href="/wiki/The_Quadrature_of_the_Parabola" class="mw-redirect" title="The Quadrature of the Parabola">The Quadrature of the Parabola</a></i></li> <li><i><a href="/wiki/The_Sand_Reckoner" title="The Sand Reckoner">The Sand Reckoner</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Problems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a> <ul><li><a href="/wiki/Angle_trisection" title="Angle trisection">Angle trisection</a></li> <li><a href="/wiki/Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li> <li><a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li></ul></li> <li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Problem of Apollonius</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts<br />and definitions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a> <ul><li><a href="/wiki/Central_angle" title="Central angle">Central</a></li> <li><a href="/wiki/Inscribed_angle" title="Inscribed angle">Inscribed</a></li></ul></li> <li><a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a></li></ul></li> <li><a href="/wiki/Chord_(geometry)" title="Chord (geometry)">Chord</a></li> <li><a href="/wiki/Circles_of_Apollonius" title="Circles of Apollonius">Circles of Apollonius</a> <ul><li><a href="/wiki/Apollonian_circles" title="Apollonian circles">Apollonian circles</a></li> <li><a href="/wiki/Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li></ul></li> <li><a href="/wiki/Circumscribed_circle" title="Circumscribed circle">Circumscribed circle</a></li> <li><a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">Commensurability</a></li> <li><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a></li> <li><a href="https://en.wikiquote.org/wiki/Doctrine_of_proportion_(mathematics)" class="extiw" title="wikiquote:Doctrine of proportion (mathematics)">Doctrine of proportionality</a></li> <li><a class="mw-selflink selflink">Euclidean geometry</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a></li> <li><a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">Incircle and excircles of a triangle</a></li> <li><a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">Method of exhaustion</a></li> <li><a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li> <li><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a></li> <li><a href="/wiki/Lune_of_Hippocrates" title="Lune of Hippocrates">Lune of Hippocrates</a></li> <li><a href="/wiki/Quadratrix_of_Hippias" title="Quadratrix of Hippias">Quadratrix of Hippias</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a></li> <li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass construction</a></li> <li><a href="/wiki/Triangle_center" title="Triangle center">Triangle center</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">In <a href="/wiki/Euclid%27s_elements" class="mw-redirect" title="Euclid's elements"><i>Elements</i></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/Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a></li> <li><a href="/wiki/Exterior_angle_theorem" title="Exterior angle theorem">Exterior angle theorem</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li> <li><a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a></li> <li><a href="/wiki/Geometric_mean_theorem" title="Geometric mean theorem">Geometric mean theorem</a></li> <li><a href="/wiki/Greek_geometric_algebra" class="mw-redirect" title="Greek geometric algebra">Greek geometric algebra</a></li> <li><a href="/wiki/Hinge_theorem" title="Hinge theorem">Hinge theorem</a></li> <li><a href="/wiki/Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">Inscribed angle theorem</a></li> <li><a href="/wiki/Intercept_theorem" title="Intercept theorem">Intercept theorem</a></li> <li><a href="/wiki/Intersecting_chords_theorem" title="Intersecting chords theorem">Intersecting chords theorem</a></li> <li><a href="/wiki/Intersecting_secants_theorem" title="Intersecting secants theorem">Intersecting secants theorem</a></li> <li><a href="/wiki/Law_of_cosines" title="Law of cosines">Law of cosines</a></li> <li><a href="/wiki/Pons_asinorum" title="Pons asinorum">Pons asinorum</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Tangent-secant_theorem" class="mw-redirect" title="Tangent-secant theorem">Tangent-secant theorem</a></li> <li><a href="/wiki/Thales%27s_theorem" title="Thales's theorem">Thales's theorem</a></li> <li><a href="/wiki/Theorem_of_the_gnomon" title="Theorem of the gnomon">Theorem of the gnomon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Apollonius_of_Tyana" title="Apollonius of Tyana">Apollonius</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/Apollonius%27s_theorem" title="Apollonius's theorem">Apollonius's theorem</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/Aristarchus%27s_inequality" title="Aristarchus's inequality">Aristarchus's inequality</a></li> <li><a href="/wiki/Crossbar_theorem" title="Crossbar theorem">Crossbar theorem</a></li> <li><a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a></li> <li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Law_of_sines" title="Law of sines">Law of sines</a></li> <li><a href="/wiki/Menelaus%27s_theorem" title="Menelaus's theorem">Menelaus's theorem</a></li> <li><a href="/wiki/Pappus%27s_area_theorem" title="Pappus's area theorem">Pappus's area theorem</a></li> <li><a href="/wiki/Diophantus_II.VIII" title="Diophantus II.VIII">Problem II.8 of <i>Arithmetica</i></a></li> <li><a href="/wiki/Ptolemy%27s_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a></li> <li><a href="/wiki/Ptolemy%27s_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a></li> <li><a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a></li> <li><a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyrene,_Libya" title="Cyrene, Libya">Cyrene</a></li> <li><a href="/wiki/Musaeum" class="mw-redirect" title="Musaeum">Mouseion of Alexandria</a></li> <li><a href="/wiki/Platonic_Academy" title="Platonic Academy">Platonic Academy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ancient_Greek_astronomy" title="Ancient Greek astronomy">Ancient Greek astronomy</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic numerals</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a></li> <li><a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translations of the 12th century</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Neusis_construction" title="Neusis construction">Neusis construction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">History of</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a></i> <ul><li>by <a href="/wiki/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Thomas Heath</a></li></ul></li> <li><a href="/wiki/History_of_algebra" title="History of algebra">algebra</a> <ul><li><a href="/wiki/Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li> <li><a href="/wiki/History_of_arithmetic" class="mw-redirect" title="History of arithmetic">arithmetic</a> <ul><li><a href="/wiki/Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">calculus</a> <ul><li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">geometry</a> <ul><li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li> <li><a href="/wiki/History_of_logic" title="History of logic">logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_mathematics" title="History of mathematics">mathematics</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">timeline</a></li></ul></li> <li><a href="/wiki/History_of_numbers" class="mw-redirect" title="History of numbers">numbers</a> <ul><li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">prehistoric counting</a></li></ul></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">numeral systems</a> <ul><li><a href="/wiki/List_of_numeral_systems" title="List of numeral systems">list</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other cultures</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/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Arabian/Islamic</a></li> <li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Egyptian</a></li> <li><a href="/wiki/Mathematics_of_the_Incas" title="Mathematics of the Incas">Incan</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian</a></li> <li><a href="/wiki/Japanese_mathematics" title="Japanese mathematics">Japanese</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/16px-Parthenon_from_west.jpg" decoding="async" width="16" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/24px-Parthenon_from_west.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/32px-Parthenon_from_west.jpg 2x" data-file-width="2048" data-file-height="1536" /></span></span> </span><a href="/wiki/Portal:Ancient_Greece" title="Portal:Ancient Greece">Ancient Greece portal</a></b> •  <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" 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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 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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 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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 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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 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