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class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Circles_and_disks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circles_and_disks"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Circles and disks</span> </div> </a> <ul id="toc-Circles_and_disks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hypercycles_and_horocycles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hypercycles_and_horocycles"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Hypercycles and horocycles</span> </div> </a> <ul id="toc-Hypercycles_and_horocycles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Triangles</span> </div> </a> <ul id="toc-Triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regular_apeirogon_and_pseudogon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regular_apeirogon_and_pseudogon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Regular apeirogon and pseudogon</span> </div> </a> <ul id="toc-Regular_apeirogon_and_pseudogon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tessellations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tessellations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Tessellations</span> </div> </a> <ul id="toc-Tessellations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Standardized_Gaussian_curvature" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Standardized_Gaussian_curvature"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Standardized Gaussian curvature</span> </div> </a> <button aria-controls="toc-Standardized_Gaussian_curvature-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 Standardized Gaussian curvature subsection</span> </button> <ul id="toc-Standardized_Gaussian_curvature-sublist" class="vector-toc-list"> <li id="toc-Cartesian-like_coordinate_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian-like_coordinate_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Cartesian-like coordinate systems</span> </div> </a> <ul id="toc-Cartesian-like_coordinate_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Distance</span> </div> </a> <ul id="toc-Distance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <button aria-controls="toc-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 History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-19th-century_developments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#19th-century_developments"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>19th-century developments</span> </div> </a> <ul id="toc-19th-century_developments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Philosophical_consequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Philosophical_consequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Philosophical consequences</span> </div> </a> <ul id="toc-Philosophical_consequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry_of_the_universe_(spatial_dimensions_only)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_of_the_universe_(spatial_dimensions_only)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Geometry of the universe (spatial dimensions only)</span> </div> </a> <ul id="toc-Geometry_of_the_universe_(spatial_dimensions_only)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry_of_the_universe_(special_relativity)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_of_the_universe_(special_relativity)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Geometry of the universe (special relativity)</span> </div> </a> <ul id="toc-Geometry_of_the_universe_(special_relativity)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physical_realizations_of_the_hyperbolic_plane" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Physical_realizations_of_the_hyperbolic_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Physical realizations of the hyperbolic plane</span> </div> </a> <ul id="toc-Physical_realizations_of_the_hyperbolic_plane-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Models_of_the_hyperbolic_plane" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Models_of_the_hyperbolic_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Models of the hyperbolic plane</span> </div> </a> <button aria-controls="toc-Models_of_the_hyperbolic_plane-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 Models of the hyperbolic plane subsection</span> </button> <ul id="toc-Models_of_the_hyperbolic_plane-sublist" class="vector-toc-list"> <li id="toc-The_Beltrami–Klein_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Beltrami–Klein_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>The Beltrami–Klein model</span> </div> </a> <ul id="toc-The_Beltrami–Klein_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Poincaré_disk_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Poincaré_disk_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>The Poincaré disk model</span> </div> </a> <ul id="toc-The_Poincaré_disk_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Poincaré_half-plane_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Poincaré_half-plane_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>The Poincaré half-plane model</span> </div> </a> <ul id="toc-The_Poincaré_half-plane_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_hyperboloid_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_hyperboloid_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>The hyperboloid model</span> </div> </a> <ul id="toc-The_hyperboloid_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_hemisphere_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_hemisphere_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>The hemisphere model</span> </div> </a> <ul id="toc-The_hemisphere_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_between_the_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_between_the_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Connection between the models</span> </div> </a> <ul id="toc-Connection_between_the_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_models_of_hyperbolic_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_models_of_hyperbolic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Other models of hyperbolic geometry</span> </div> </a> <ul id="toc-Other_models_of_hyperbolic_geometry-sublist" class="vector-toc-list"> <li id="toc-The_Gans_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Gans_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7.1</span> <span>The Gans model</span> </div> </a> <ul id="toc-The_Gans_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_conformal_square_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_conformal_square_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7.2</span> <span>The conformal square model</span> </div> </a> <ul id="toc-The_conformal_square_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_band_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_band_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7.3</span> <span>The band model</span> </div> </a> <ul id="toc-The_band_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Isometries_of_the_hyperbolic_plane" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Isometries_of_the_hyperbolic_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Isometries of the hyperbolic plane</span> </div> </a> <ul id="toc-Isometries_of_the_hyperbolic_plane-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperbolic_geometry_in_art" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Hyperbolic_geometry_in_art"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Hyperbolic geometry in art</span> </div> </a> <ul id="toc-Hyperbolic_geometry_in_art-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Higher dimensions</span> </div> </a> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homogeneous_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Homogeneous_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Homogeneous structure</span> </div> </a> <ul id="toc-Homogeneous_structure-sublist" class="vector-toc-list"> </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">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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Available in 46 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-46" 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">46 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Hyperbolische_Geometrie" title="Hyperbolische Geometrie – Alemannic" lang="gsw" hreflang="gsw" data-title="Hyperbolische Geometrie" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B2%D8%A7%D8%A6%D8%AF%D9%8A%D8%A9" title="هندسة زائدية – Arabic" lang="ar" hreflang="ar" data-title="هندسة زائدية" 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_hiperb%C3%B3lica" title="Xeometría hiperbólica – Asturian" lang="ast" hreflang="ast" data-title="Xeometría hiperbólica" 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/Loba%C3%A7evski_h%C9%99nd%C9%99s%C9%99si" title="Lobaçevski həndəsəsi – Azerbaijani" lang="az" hreflang="az" data-title="Lobaçevski həndəsəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Хиперболична геометрия – Bulgarian" lang="bg" hreflang="bg" data-title="Хиперболична 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_hiperb%C3%B2lica" title="Geometria hiperbòlica – Catalan" lang="ca" hreflang="ca" data-title="Geometria hiperbòlica" 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%9B%D0%BE%D0%B1%D0%B0%D1%87%D0%B5%D0%B2%D1%81%D0%BA%D0%B8%D0%B9_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B9%C4%95" title="Лобачевский геометрийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Лобачевский 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/Hyperbolick%C3%A1_geometrie" title="Hyperbolická geometrie – Czech" lang="cs" hreflang="cs" data-title="Hyperbolická 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_hyperbolig" title="Geometreg hyperbolig – Welsh" lang="cy" hreflang="cy" data-title="Geometreg hyperbolig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hyperbolische_Geometrie" title="Hyperbolische Geometrie – German" lang="de" hreflang="de" data-title="Hyperbolische Geometrie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%B5%CF%81%CE%B2%CE%BF%CE%BB%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Υπερβολική γεωμετρία – Greek" lang="el" hreflang="el" data-title="Υπερβολική 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_hiperb%C3%B3lica" title="Geometría hiperbólica – Spanish" lang="es" hreflang="es" data-title="Geometría hiperbólica" 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/Hiperbola_geometrio" title="Hiperbola geometrio – Esperanto" lang="eo" hreflang="eo" data-title="Hiperbola geometrio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</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_%D9%87%D8%B0%D9%84%D9%88%D9%84%D9%88%DB%8C" title="هندسه هذلولوی – Persian" lang="fa" hreflang="fa" data-title="هندسه هذلولوی" 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-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie_hyperbolique" title="Géométrie hyperbolique – French" lang="fr" hreflang="fr" data-title="Géométrie hyperbolique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EA%B3%A1%EA%B8%B0%ED%95%98%ED%95%99" title="쌍곡기하학 – Korean" lang="ko" hreflang="ko" data-title="쌍곡기하학" 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%BC%D5%B8%D5%A2%D5%A1%D5%B9%D6%87%D5%BD%D5%AF%D5%B8%D6%82_%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="Լոբաչևսկու երկրաչափություն – Armenian" lang="hy" hreflang="hy" data-title="Լոբաչևսկու երկրաչափություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Hiperbolala_geometrio" title="Hiperbolala geometrio – Ido" lang="io" hreflang="io" data-title="Hiperbolala geometrio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geometri_hiperbolik" title="Geometri hiperbolik – Indonesian" lang="id" hreflang="id" data-title="Geometri hiperbolik" 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_iperbolica" title="Geometria iperbolica – Italian" lang="it" hreflang="it" data-title="Geometria iperbolica" 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%94%D7%99%D7%A4%D7%A8%D7%91%D7%95%D7%9C%D7%99%D7%AA" title="גאומטריה היפרבולית – Hebrew" lang="he" hreflang="he" data-title="גאומטריה היפרבולית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Hiperbolin%C4%97_geometrija" title="Hiperbolinė geometrija – Lithuanian" lang="lt" hreflang="lt" data-title="Hiperbolinė geometrija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hiperbolikus_geometria" title="Hiperbolikus geometria – Hungarian" lang="hu" hreflang="hu" data-title="Hiperbolikus geometria" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hyperbolische_meetkunde" title="Hyperbolische meetkunde – Dutch" lang="nl" hreflang="nl" data-title="Hyperbolische 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/%E5%8F%8C%E6%9B%B2%E5%B9%BE%E4%BD%95%E5%AD%A6" title="双曲幾何学 – Japanese" lang="ja" hreflang="ja" data-title="双曲幾何学" 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/Hyperbolsk_geometri" title="Hyperbolsk geometri – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hyperbolsk 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/Hyperbolsk_geometri" title="Hyperbolsk geometri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Hyperbolsk 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Lobachevskiy_geometriyasi" title="Lobachevskiy geometriyasi – Uzbek" lang="uz" hreflang="uz" data-title="Lobachevskiy 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Geometria_hiperboliczna" title="Geometria hiperboliczna – Polish" lang="pl" hreflang="pl" data-title="Geometria hiperboliczna" 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_hiperb%C3%B3lica" title="Geometria hiperbólica – Portuguese" lang="pt" hreflang="pt" data-title="Geometria hiperbólica" 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_hiperbolic%C4%83" title="Geometrie hiperbolică – Romanian" lang="ro" hreflang="ro" data-title="Geometrie hiperbolică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D0%9B%D0%BE%D0%B1%D0%B0%D1%87%D0%B5%D0%B2%D1%81%D0%BA%D0%BE%D0%B3%D0%BE" title="Геометрия Лобачевского – Russian" lang="ru" hreflang="ru" data-title="Геометрия Лобачевского" 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_e_Lloba%C3%A7evskit" title="Gjeometria e Llobaçevskit – Albanian" lang="sq" hreflang="sq" data-title="Gjeometria e Llobaçevskit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hyperbolic_geometry" title="Hyperbolic geometry – Simple English" lang="en-simple" hreflang="en-simple" data-title="Hyperbolic 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hiperboli%C4%8Dna_geometrija" title="Hiperbolična geometrija – Slovenian" lang="sl" hreflang="sl" data-title="Hiperbolična geometrija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Хиперболичка геометрија – Serbian" lang="sr" hreflang="sr" data-title="Хиперболичка 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/Hiperboli%C4%8Dka_geometrija" title="Hiperbolička geometrija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Hiperbolička 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/Hyperbolinen_geometria" title="Hyperbolinen geometria – Finnish" lang="fi" hreflang="fi" data-title="Hyperbolinen 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/Hyperbolisk_geometri" title="Hyperbolisk geometri – Swedish" lang="sv" hreflang="sv" data-title="Hyperbolisk geometri" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</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_%D0%9B%D0%BE%D0%B1%D0%B0%D1%87%D0%B5%D0%B2%D1%81%D0%BA%D0%B8%D0%B9" title="Ҳандасаи Лобачевский – Tajik" lang="tg" hreflang="tg" data-title="Ҳандасаи Лобачевский" 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/Hiperbolik_geometri" title="Hiperbolik geometri – Turkish" lang="tr" hreflang="tr" data-title="Hiperbolik geometri" 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%93%D1%96%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D1%96%D1%87%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Гіперболічна геометрія – Ukrainian" lang="uk" hreflang="uk" data-title="Гіперболічна 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_hyperbol" title="Hình học hyperbol – Vietnamese" lang="vi" hreflang="vi" data-title="Hình học hyperbol" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E5%87%A0%E4%BD%95" title="双曲几何 – Wu" lang="wuu" hreflang="wuu" data-title="双曲几何" 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 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Hyperbolic+geometry%22">"Hyperbolic geometry"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Hyperbolic+geometry%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Hyperbolic+geometry%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Hyperbolic+geometry%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Hyperbolic+geometry%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Hyperbolic+geometry%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">June 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of non-Euclidean geometry</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Hyperbolic_(disambiguation)" class="mw-redirect mw-disambig" title="Hyperbolic (disambiguation)">Hyperbolic (disambiguation)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Hyperbolic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Hyperbolic.svg/199px-Hyperbolic.svg.png" decoding="async" width="199" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Hyperbolic.svg/299px-Hyperbolic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Hyperbolic.svg/398px-Hyperbolic.svg.png 2x" data-file-width="199" data-file-height="199" /></a><figcaption>Lines through a given point <i>P</i> and asymptotic to line <i>R</i></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output 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href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size notpageimage" typeof="mw:File/Frameless"><a href="/wiki/File:Stereographic_projection_in_3D.svg" class="mw-file-description"><img alt="Stereographic projection from the top of a sphere onto a plane beneath it" 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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> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hyperbolic_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/250px-Hyperbolic_triangle.svg.png" decoding="async" width="250" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/375px-Hyperbolic_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/500px-Hyperbolic_triangle.svg.png 2x" data-file-width="809" data-file-height="559" /></a><figcaption>A triangle immersed in a saddle-shape plane (a <a href="/wiki/Hyperbolic_paraboloid" class="mw-redirect" title="Hyperbolic paraboloid">hyperbolic paraboloid</a>), along with two diverging ultra-parallel lines</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>hyperbolic geometry</b> (also called <b>Lobachevskian geometry</b> or <b><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a>–<a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevskian</a> geometry</b>) is a <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>. The <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is replaced with: </p> <dl><dd>For any given line <i>R</i> and point <i>P</i> not on <i>R</i>, in the plane containing both line <i>R</i> and point <i>P</i> there are at least two distinct lines through <i>P</i> that do not intersect <i>R</i>.</dd></dl> <p>(Compare the above with <a href="/wiki/Playfair%27s_axiom" title="Playfair's axiom">Playfair's axiom</a>, the modern version of <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>.) </p><p>The <b>hyperbolic plane</b> is a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> where every point is a <a href="/wiki/Saddle_point" title="Saddle point">saddle point</a>. Hyperbolic plane <a href="/wiki/Geometry" title="Geometry">geometry</a> is also the geometry of <a href="/wiki/Pseudosphere" title="Pseudosphere">pseudospherical surfaces</a>, surfaces with a constant negative <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>. <a href="/wiki/Saddle_surface" class="mw-redirect" title="Saddle surface">Saddle surfaces</a> have negative Gaussian curvature in at least some regions, where they <a href="/wiki/Local_property" title="Local property">locally</a> resemble the hyperbolic plane. </p><p>The <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a> of hyperbolic geometry provides a representation of <a href="/wiki/Event_(relativity)" title="Event (relativity)">events</a> one temporal unit into the future in <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, the basis of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. Each of these events corresponds to a <a href="/wiki/Rapidity" title="Rapidity">rapidity</a> in some direction. </p><p>When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> finally gave the subject the name <b>hyperbolic geometry</b> to include it in the now rarely used sequence <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a> (<a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a>), parabolic geometry (<a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>), and hyperbolic geometry. In the <a href="/wiki/Post-Soviet_states" title="Post-Soviet states">former Soviet Union</a>, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer <a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Nikolai Lobachevsky</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Relation_to_Euclidean_geometry">Relation to Euclidean geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=2" title="Edit section: Relation to Euclidean geometry"><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>Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only <a href="/wiki/Axiom" title="Axiom">axiomatic</a> difference is the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. When the parallel postulate is removed from Euclidean geometry the resulting geometry is <a href="/wiki/Absolute_geometry" title="Absolute geometry">absolute geometry</a>. There are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's <i>Elements</i> prove the existence of parallel/non-intersecting lines. </p><p>This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the <a href="/wiki/Angle_of_parallelism" title="Angle of parallelism">angle of parallelism</a>, hyperbolic geometry has an <a href="/wiki/Absolute_scale" title="Absolute scale">absolute scale</a>, a relation between distance and angle measurements. </p> <div class="mw-heading mw-heading3"><h3 id="Lines">Lines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=3" title="Edit section: Lines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended. </p><p>Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a>. </p><p>When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines. </p><p>These properties are all independent of the <a href="#Models_of_the_hyperbolic_plane">model</a> used, even if the lines may look radically different. </p> <div class="mw-heading mw-heading4"><h4 id="Non-intersecting_/_parallel_lines"><span id="Non-intersecting_.2F_parallel_lines"></span>Non-intersecting / parallel lines</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=4" title="Edit section: Non-intersecting / parallel lines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Hyperbolic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Hyperbolic.svg/199px-Hyperbolic.svg.png" decoding="async" width="199" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Hyperbolic.svg/299px-Hyperbolic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Hyperbolic.svg/398px-Hyperbolic.svg.png 2x" data-file-width="199" data-file-height="199" /></a><figcaption>Lines through a given point <i>P</i> and asymptotic to line <i>R</i></figcaption></figure> <p>Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>: </p> <dl><dd>For any line <i>R</i> and any point <i>P</i> which does not lie on <i>R</i>, in the plane containing line <i>R</i> and point <i>P</i> there are at least two distinct lines through <i>P</i> that do not intersect <i>R</i>.</dd></dl> <p>This implies that there are through <i>P</i> an infinite number of coplanar lines that do not intersect <i>R</i>. </p><p>These non-intersecting lines are divided into two classes: </p> <ul><li>Two of the lines (<i>x</i> and <i>y</i> in the diagram) are <a href="/wiki/Limiting_parallel" title="Limiting parallel">limiting parallels</a> (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the <a href="/wiki/Ideal_point" title="Ideal point">ideal points</a> at the "ends" of <i>R</i>, asymptotically approaching <i>R</i>, always getting closer to <i>R</i>, but never meeting it.</li> <li>All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called <i>ultraparallel</i>, <i>diverging parallel</i> or sometimes <i>non-intersecting.</i></li></ul> <p>Some geometers simply use the phrase "<i>parallel</i> lines" to mean "<i>limiting parallel</i> lines", with <i>ultraparallel</i> lines meaning just <i>non-intersecting</i>. </p><p>These <a href="/wiki/Limiting_parallel" title="Limiting parallel">limiting parallels</a> make an angle <i>θ</i> with <i>PB</i>; this angle depends only on the <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> of the plane and the distance <i>PB</i> and is called the <a href="/wiki/Angle_of_parallelism" title="Angle of parallelism">angle of parallelism</a>. </p><p>For ultraparallel lines, the <a href="/wiki/Ultraparallel_theorem" title="Ultraparallel theorem">ultraparallel theorem</a> states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines. </p> <div class="mw-heading mw-heading3"><h3 id="Circles_and_disks">Circles and disks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=5" title="Edit section: Circles and disks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In hyperbolic geometry, the circumference of a circle of radius <i>r</i> is greater than <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\pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e811131a9c6c5f45e6657e0fc506e7e2a37f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.176ex;" alt="{\displaystyle 2\pi r}"></span>. </p><p>Let <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 R={\frac {1}{\sqrt {-K}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mo>−</mo> <mi>K</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {1}{\sqrt {-K}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1071e39b083e9cc93033a995203e1931eea083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.509ex; height:6.176ex;" alt="{\displaystyle R={\frac {1}{\sqrt {-K}}}}"></span>, where <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is the <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> of the plane. In hyperbolic geometry, <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is negative, so the square root is of a positive number. </p><p>Then the circumference of a circle of radius <i>r</i> is equal to: </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 2\pi R\sinh {\frac {r}{R}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>R</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi R\sinh {\frac {r}{R}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6030cb8e083443cee7474a56e777978def54c6a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.815ex; height:4.843ex;" alt="{\displaystyle 2\pi R\sinh {\frac {r}{R}}\,.}"></span></dd></dl> <p>And the area of the enclosed disk is: </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 4\pi R^{2}\sinh ^{2}{\frac {r}{2R}}=2\pi R^{2}\left(\cosh {\frac {r}{R}}-1\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>R</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi R^{2}\sinh ^{2}{\frac {r}{2R}}=2\pi R^{2}\left(\cosh {\frac {r}{R}}-1\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bb2e8dbf5a4797a2fad2d30382536e25cec82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.44ex; height:5.009ex;" alt="{\displaystyle 4\pi R^{2}\sinh ^{2}{\frac {r}{2R}}=2\pi R^{2}\left(\cosh {\frac {r}{R}}-1\right)\,.}"></span></dd></dl> <p>Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than <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\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span>, though it can be made arbitrarily close by selecting a small enough circle. </p><p>If the Gaussian curvature of the plane is −1 then the <a href="/wiki/Geodesic_curvature" title="Geodesic curvature">geodesic curvature</a> of a circle of radius <i>r</i> is: <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 {\frac {1}{\tanh(r)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\tanh(r)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611696f696b457b48fbd5ce7d780fc69c9e4cead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.346ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\tanh(r)}}}"></span><sup id="cite_ref-auto_1-0" class="reference"><a href="#cite_note-auto-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hypercycles_and_horocycles">Hypercycles and horocycles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=6" title="Edit section: Hypercycles and horocycles"><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:Hyperbolic_pseudogon_example0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Hyperbolic_pseudogon_example0.png/220px-Hyperbolic_pseudogon_example0.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Hyperbolic_pseudogon_example0.png/330px-Hyperbolic_pseudogon_example0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Hyperbolic_pseudogon_example0.png/440px-Hyperbolic_pseudogon_example0.png 2x" data-file-width="2518" data-file-height="2518" /></a><figcaption>Hypercycle and pseudogon in the <a href="/wiki/Poincare_disk_model" class="mw-redirect" title="Poincare disk model">Poincare disk model</a> </figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Hypercycle_(hyperbolic_geometry)" class="mw-redirect" title="Hypercycle (hyperbolic geometry)">Hypercycle (hyperbolic geometry)</a> and <a href="/wiki/Horocycle" title="Horocycle">horocycle</a></div> <p>In hyperbolic geometry, there is no line whose points are all equidistant from another line. Instead, the points that are all the same distance from a given line lie on a curve called a <a href="/wiki/Hypercycle_(hyperbolic_geometry)" class="mw-redirect" title="Hypercycle (hyperbolic geometry)">hypercycle</a>. </p><p>Another special curve is the <a href="/wiki/Horocycle" title="Horocycle">horocycle</a>, whose <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> radii (<a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> lines) are all <a href="/wiki/Limiting_parallel" title="Limiting parallel">limiting parallel</a> to each other (all converge asymptotically in one direction to the same <a href="/wiki/Ideal_point" title="Ideal point">ideal point</a>, the centre of the horocycle). </p><p>Through every pair of points there are two horocycles. The centres of the horocycles are the <a href="/wiki/Ideal_point" title="Ideal point">ideal points</a> of the <a href="/wiki/Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisector</a> of the line-segment between them. </p><p>Given any three distinct points, they all lie on either a line, hypercycle, <a href="/wiki/Horocycle" title="Horocycle">horocycle</a>, or circle. </p><p>The <b>length</b> of a line-segment is the shortest length between two points. </p><p>The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of the arc horocycle, connecting the same two points. </p><p>The lengths of the arcs of both horocycles connecting two points are equal, and are longer than the arclength of any hypercycle connecting the points and shorter than the arc of any circle connecting the two points. </p><p>If the Gaussian curvature of the plane is −1, then the <a href="/wiki/Geodesic_curvature" title="Geodesic curvature">geodesic curvature</a> of a horocycle is 1 and that of a hypercycle is between 0 and 1.<sup id="cite_ref-auto_1-1" class="reference"><a href="#cite_note-auto-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Triangles">Triangles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=7" title="Edit section: Triangles"><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/Hyperbolic_triangle" title="Hyperbolic triangle">Hyperbolic triangle</a></div> <p>Unlike Euclidean triangles, where the angles always add up to π <a href="/wiki/Radian" title="Radian">radians</a> (180°, a <a href="/wiki/Straight_angle" class="mw-redirect" title="Straight angle">straight angle</a>), in hyperbolic space the sum of the angles of a triangle is always strictly less than π radians (180°). The difference is called the <a href="/wiki/Angular_defect" title="Angular defect">defect</a>. Generally, the defect of a convex hyperbolic polygon with <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> sides is its angle sum subtracted from <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-2)\cdot 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-2)\cdot 180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1bf4f86a416611a8b88e8761935b0278ace822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.428ex; height:2.843ex;" alt="{\displaystyle (n-2)\cdot 180^{\circ }}"></span>. </p><p>The area of a hyperbolic triangle is given by its defect in radians multiplied by <i>R</i><sup>2</sup>, which is also true for all convex hyperbolic polygons.<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> Therefore all hyperbolic triangles have an area less than or equal to <i>R</i><sup>2</sup>π. The area of a hyperbolic <a href="/wiki/Ideal_triangle" title="Ideal triangle">ideal triangle</a> in which all three angles are 0° is equal to this maximum. </p><p>As in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, each hyperbolic triangle has an <a href="/wiki/Incircle" class="mw-redirect" title="Incircle">incircle</a>. In hyperbolic space, if all three of its vertices lie on a <a href="/wiki/Horocycle" title="Horocycle">horocycle</a> or <a href="/wiki/Hypercycle_(hyperbolic_geometry)" class="mw-redirect" title="Hypercycle (hyperbolic geometry)">hypercycle</a>, then the triangle has no <a href="/wiki/Circumscribed_circle" title="Circumscribed circle">circumscribed circle</a>. </p><p>As in <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical</a> and <a href="/wiki/Elliptical_geometry" class="mw-redirect" title="Elliptical geometry">elliptical geometry</a>, in hyperbolic geometry if two triangles are similar, they must be congruent. </p> <div class="mw-heading mw-heading3"><h3 id="Regular_apeirogon_and_pseudogon">Regular apeirogon and pseudogon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=8" title="Edit section: Regular apeirogon and pseudogon"><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:Hyperbolic_apeirogon_example.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Hyperbolic_apeirogon_example.png/220px-Hyperbolic_apeirogon_example.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Hyperbolic_apeirogon_example.png/330px-Hyperbolic_apeirogon_example.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Hyperbolic_apeirogon_example.png/440px-Hyperbolic_apeirogon_example.png 2x" data-file-width="800" data-file-height="798" /></a><figcaption>An <a href="/wiki/Apeirogon" title="Apeirogon">apeirogon</a> and circumscribed <a href="/wiki/Horocycle" title="Horocycle">horocycle</a> in the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Apeirogon#Hyperbolic_geometry" title="Apeirogon">Apeirogon § Hyperbolic geometry</a></div> <p>Special polygons in hyperbolic geometry are the regular <a href="/wiki/Apeirogon" title="Apeirogon">apeirogon</a> and <b>pseudogon</b> <a href="/wiki/Uniform_polygon" class="mw-redirect" title="Uniform polygon">uniform polygons</a> with an infinite number of sides. </p><p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180° and the apeirogon approaches a straight line. </p><p>However, in hyperbolic geometry, a regular apeirogon or pseudogon has sides of any length (i.e., it remains a polygon with noticeable sides). </p><p>The side and angle <a href="/wiki/Bisection" title="Bisection">bisectors</a> will, depending on the side length and the angle between the sides, be limiting or diverging parallel. If the bisectors are limiting parallel then it is an apeirogon and can be inscribed and circumscribed by concentric <a href="/wiki/Horocycle" title="Horocycle">horocycles</a>. </p><p>If the bisectors are diverging parallel then it is a pseudogon and can be inscribed and circumscribed by <a href="/wiki/Hypercycle_(geometry)" title="Hypercycle (geometry)">hypercycles</a> (all vertices are the same distance of a line, the axis, also the midpoint of the side segments are all equidistant to the same axis). </p> <div class="mw-heading mw-heading3"><h3 id="Tessellations">Tessellations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=9" title="Edit section: Tessellations"><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/Uniform_tilings_in_hyperbolic_plane" title="Uniform tilings in hyperbolic plane">Uniform tilings in hyperbolic plane</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Regular_hyperbolic_tiling" class="mw-redirect" title="Regular hyperbolic tiling">Regular hyperbolic tiling</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rhombitriheptagonal_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Rhombitriheptagonal_tiling.svg/220px-Rhombitriheptagonal_tiling.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Rhombitriheptagonal_tiling.svg/330px-Rhombitriheptagonal_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Rhombitriheptagonal_tiling.svg/440px-Rhombitriheptagonal_tiling.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a><figcaption><a href="/wiki/Rhombitriheptagonal_tiling" title="Rhombitriheptagonal tiling">Rhombitriheptagonal tiling</a> of the hyperbolic plane, seen in the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a> </figcaption></figure> <p>Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> as <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>. </p><p>There are an infinite number of uniform tilings based on the <a href="/wiki/Schwarz_triangles" class="mw-redirect" title="Schwarz triangles">Schwarz triangles</a> (<i>p</i> <i>q</i> <i>r</i>) where 1/<i>p</i> + 1/<i>q</i> + 1/<i>r</i> < 1, where <i>p</i>, <i>q</i>, <i>r</i> are each orders of reflection symmetry at three points of the <a href="/wiki/Fundamental_domain_triangle" class="mw-redirect" title="Fundamental domain triangle">fundamental domain triangle</a>, the symmetry group is a hyperbolic <a href="/wiki/Triangle_group" title="Triangle group">triangle group</a>. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.<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> <div class="mw-heading mw-heading2"><h2 id="Standardized_Gaussian_curvature">Standardized Gaussian curvature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=10" title="Edit section: Standardized Gaussian curvature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Though hyperbolic geometry applies for any surface with a constant negative <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>, it is usual to assume a scale in which the curvature <i>K</i> is −1. </p><p>This results in some formulas becoming simpler. Some examples are: </p> <ul><li>The area of a triangle is equal to its angle defect in <a href="/wiki/Radian" title="Radian">radians</a>.</li> <li>The area of a horocyclic sector is equal to the length of its horocyclic arc.</li> <li>An arc of a <a href="/wiki/Horocycle" title="Horocycle">horocycle</a> so that a line that is tangent at one endpoint is <a href="/wiki/Limiting_parallel" title="Limiting parallel">limiting parallel</a> to the radius through the other endpoint has a length of 1.<sup id="cite_ref-Sommerville2005_4-0" class="reference"><a href="#cite_note-Sommerville2005-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>The ratio of the arc lengths between two radii of two concentric <a href="/wiki/Horocycle" title="Horocycle">horocycles</a> where the horocycles are a distance 1 apart is <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><i>e</i></a> : 1.<sup id="cite_ref-Sommerville2005_4-1" class="reference"><a href="#cite_note-Sommerville2005-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Cartesian-like_coordinate_systems">Cartesian-like coordinate systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=11" title="Edit section: Cartesian-like coordinate systems"><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/Coordinate_systems_for_the_hyperbolic_plane" title="Coordinate systems for the hyperbolic plane">Coordinate systems for the hyperbolic plane</a></div> <p>Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes it to rotate when it returns to the origin; etc. </p><p>There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the <i>x</i>-axis) and after that many choices exist. </p><p>The Lobachevsky coordinates <i>x</i> and <i>y</i> are found by dropping a perpendicular onto the <i>x</i>-axis. <i>x</i> will be the label of the foot of the perpendicular. <i>y</i> will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other). </p><p>Another coordinate system measures the distance from the point to the <a href="/wiki/Horocycle" title="Horocycle">horocycle</a> through the origin centered around <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 (0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de77e40eb7e2582eef8a5a1da1bc027b7d9a8d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:2.843ex;" alt="{\displaystyle (0,+\infty )}"></span> and the length along this horocycle.<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><p>Other coordinate systems use the Klein model or the Poincaré disk model described below, and take the Euclidean coordinates as hyperbolic. </p> <div class="mw-heading mw-heading3"><h3 id="Distance">Distance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=12" title="Edit section: Distance"><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/Coordinate_systems_for_the_hyperbolic_plane#Polar_coordinate_system" title="Coordinate systems for the hyperbolic plane">Coordinate systems for the hyperbolic plane § Polar coordinate system</a></div> <p>A Cartesian-like<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. (September 2022)">citation needed</span></a></i>]</sup> coordinate system (<i>x, y</i>) on the oriented hyperbolic plane is constructed as follows. Choose a line in the hyperbolic plane together with an orientation and an origin <i>o</i> on this line. Then: </p> <ul><li>the <i>x</i>-coordinate of a point is the signed distance of its projection onto the line (the foot of the perpendicular segment to the line from that point) to the origin;</li> <li>the <i>y</i>-coordinate is the signed <a href="/wiki/Distance_from_a_point_to_a_line" title="Distance from a point to a line">distance</a> from the point to the line, with the sign according to whether the point is on the positive or negative side of the oriented line.</li></ul> <p>The distance between two points represented by (<i>x_i, y_i</i>), <i>i=1,2</i> in this coordinate system is<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. (December 2018)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dist} (\langle x_{1},y_{1}\rangle ,\langle x_{2},y_{2}\rangle )=\operatorname {arcosh} \left(\cosh y_{1}\cosh(x_{2}-x_{1})\cosh y_{2}-\sinh y_{1}\sinh y_{2}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dist</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arcosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cosh</mi> <mo>⁡</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>sinh</mi> <mo>⁡</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sinh</mi> <mo>⁡</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dist} (\langle x_{1},y_{1}\rangle ,\langle x_{2},y_{2}\rangle )=\operatorname {arcosh} \left(\cosh y_{1}\cosh(x_{2}-x_{1})\cosh y_{2}-\sinh y_{1}\sinh y_{2}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eedf69ec11c340ab0b2b82303fb67beac51f379" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.113ex; height:2.843ex;" alt="{\displaystyle \operatorname {dist} (\langle x_{1},y_{1}\rangle ,\langle x_{2},y_{2}\rangle )=\operatorname {arcosh} \left(\cosh y_{1}\cosh(x_{2}-x_{1})\cosh y_{2}-\sinh y_{1}\sinh y_{2}\right)\,.}"></span> </p><p>This formula can be derived from the formulas about <a href="/wiki/Hyperbolic_triangle" title="Hyperbolic triangle">hyperbolic triangles</a>. </p><p>The corresponding metric tensor field is: <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 (\mathrm {d} s)^{2}=\cosh ^{2}y\,(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>cosh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {d} s)^{2}=\cosh ^{2}y\,(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f658469c6075ba6a1e1b345f6b4c0c45f1b22461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.37ex; height:3.176ex;" alt="{\displaystyle (\mathrm {d} s)^{2}=\cosh ^{2}y\,(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}"></span>. </p><p>In this coordinate system, straight lines take one of these forms ((<i>x</i>, <i>y</i>) is a point on the line; <i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>, <i>A</i>, and <i>α</i> are parameters): </p><p>ultraparallel to the <i>x</i>-axis </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 \tanh(y)=\tanh(y_{0})\cosh(x-x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh(y)=\tanh(y_{0})\cosh(x-x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff6cc442687715fe9023fe266a043ddba8a379a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.524ex; height:2.843ex;" alt="{\displaystyle \tanh(y)=\tanh(y_{0})\cosh(x-x_{0})}"></span></dd></dl> <p>asymptotically parallel on the negative side </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 \tanh(y)=A\exp(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh(y)=A\exp(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb61b813a77a339b42a2ad561134dba706a074b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.537ex; height:2.843ex;" alt="{\displaystyle \tanh(y)=A\exp(x)}"></span></dd></dl> <p>asymptotically parallel on the positive side </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 \tanh(y)=A\exp(-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh(y)=A\exp(-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096177bb52f9b86b107058184685ee883fac2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.345ex; height:2.843ex;" alt="{\displaystyle \tanh(y)=A\exp(-x)}"></span></dd></dl> <p>intersecting perpendicularly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span></dd></dl> <p>intersecting at an angle <i>α</i> </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 \tanh(y)=\tan(\alpha )\sinh(x-x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh(y)=\tan(\alpha )\sinh(x-x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdf99aff1377a4212d13e49781d6b16f28f10d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.27ex; height:2.843ex;" alt="{\displaystyle \tanh(y)=\tan(\alpha )\sinh(x-x_{0})}"></span></dd></dl> <p>Generally, these equations will only hold in a bounded domain (of <i>x</i> values). At the edge of that domain, the value of <i>y</i> blows up to ±infinity. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=13" title="Edit section: 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/Non-euclidean_Geometry" class="mw-redirect" title="Non-euclidean Geometry">Non-euclidean Geometry</a></div> <p>Since the publication of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a> circa 300BC, many <a href="/wiki/Geometers" class="mw-redirect" title="Geometers">geometers</a> tried to prove the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. Some tried to prove it by <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">assuming its negation and trying to derive a contradiction</a>. Foremost among these were <a href="/wiki/Proclus" title="Proclus">Proclus</a>, <a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a> (Alhacen), <a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Omar Khayyám</a>,<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> <a href="/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%AB" class="mw-redirect" title="Nasīr al-Dīn al-Tūsī">Nasīr al-Dīn al-Tūsī</a>, <a href="/wiki/Witelo" class="mw-redirect" title="Witelo">Witelo</a>, <a href="/wiki/Gersonides" title="Gersonides">Gersonides</a>, <a href="/wiki/Abner_of_Burgos" title="Abner of Burgos">Alfonso</a>, and later <a href="/wiki/Giovanni_Gerolamo_Saccheri" class="mw-redirect" title="Giovanni Gerolamo Saccheri">Giovanni Gerolamo Saccheri</a>, <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a>, <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a>, and <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a>.<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> Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry. </p><p>The theorems of Alhacen, Khayyam and al-Tūsī on <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilaterals</a>, including the <a href="/wiki/Ibn_al-Haytham%E2%80%93Lambert_quadrilateral" class="mw-redirect" title="Ibn al-Haytham–Lambert quadrilateral">Ibn al-Haytham–Lambert quadrilateral</a> and <a href="/wiki/Khayyam%E2%80%93Saccheri_quadrilateral" class="mw-redirect" title="Khayyam–Saccheri quadrilateral">Khayyam–Saccheri quadrilateral</a>, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.<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> </p><p>In the 18th century, <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> introduced the <a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">hyperbolic functions</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and computed the area of a <a href="/wiki/Hyperbolic_triangle" title="Hyperbolic triangle">hyperbolic triangle</a>.<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> <div class="mw-heading mw-heading3"><h3 id="19th-century_developments">19th-century developments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=14" title="Edit section: 19th-century developments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 19th century, hyperbolic geometry was explored extensively by <a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Nikolai Lobachevsky</a>, <a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">János Bolyai</a>, <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> and <a href="/wiki/Franz_Taurinus" title="Franz Taurinus">Franz Taurinus</a>. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.<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><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><p>Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "<a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>"<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> causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self-consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832. </p><p>In 1868, <a href="/wiki/Eugenio_Beltrami" title="Eugenio Beltrami">Eugenio Beltrami</a> provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> Euclidean geometry was. </p><p>The term "hyperbolic geometry" was introduced by <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> in 1871.<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> Klein followed an initiative of <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> to use the transformations of <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> to produce <a href="/wiki/Isometries" class="mw-redirect" title="Isometries">isometries</a>. The idea used a <a href="/wiki/Conic_section" title="Conic section">conic section</a> or <a href="/wiki/Quadric" title="Quadric">quadric</a> to define a region, and used <a href="/wiki/Cross_ratio" class="mw-redirect" title="Cross ratio">cross ratio</a> to define a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>. The projective transformations that leave the conic section or quadric <a href="/wiki/Invariant_(mathematics)#Invariant_set" title="Invariant (mathematics)">stable</a> are the isometries. "Klein showed that if the <a href="/wiki/Cayley_absolute" class="mw-redirect" title="Cayley absolute">Cayley absolute</a> is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."<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="Philosophical_consequences">Philosophical consequences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=15" title="Edit section: Philosophical consequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The discovery of hyperbolic geometry had important <a href="/wiki/Philosophical" class="mw-redirect" title="Philosophical">philosophical</a> consequences. Before its discovery many philosophers (such as <a href="/wiki/Hobbes" class="mw-redirect" title="Hobbes">Hobbes</a> and <a href="/wiki/Spinoza" class="mw-redirect" title="Spinoza">Spinoza</a>) viewed philosophical rigor in terms of the "geometrical method", referring to the method of reasoning used in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>. </p><p><a href="/wiki/Kant" class="mw-redirect" title="Kant">Kant</a> in <a href="/wiki/Critique_of_Pure_Reason#Space_and_time" title="Critique of Pure Reason"><i>Critique of Pure Reason</i></a> concluded that space (in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.<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> </p><p>It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the <a href="/wiki/Boeotia" title="Boeotia">Boeotians</a>" (stereotyped as dullards by the ancient Athenians<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>), which would ruin his status as <i>princeps mathematicorum</i> (Latin, "the Prince of Mathematicians").<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> The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in <a href="/wiki/Mathematical_rigour" class="mw-redirect" title="Mathematical rigour">mathematical rigour</a>, <a href="/wiki/Analytical_philosophy" class="mw-redirect" title="Analytical philosophy">analytical philosophy</a> and <a href="/wiki/Logic" title="Logic">logic</a>. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry. </p> <div class="mw-heading mw-heading3"><h3 id="Geometry_of_the_universe_(spatial_dimensions_only)"><span id="Geometry_of_the_universe_.28spatial_dimensions_only.29"></span>Geometry of the universe (spatial dimensions only)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=16" title="Edit section: Geometry of the universe (spatial dimensions only)"><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/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of space and time</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Shape_of_the_universe#Curvature_of_the_universe" title="Shape of the universe">Shape of the universe § Curvature of the universe</a></div> <p>Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature? </p><p>Lobachevsky had already tried to measure the curvature of the universe by measuring the <a href="/wiki/Parallax" title="Parallax">parallax</a> of <a href="/wiki/Sirius" title="Sirius">Sirius</a> and treating Sirius as the ideal point of an <a href="/wiki/Angle_of_parallelism" title="Angle of parallelism">angle of parallelism</a>. He realized that his measurements were <a href="/wiki/Margin_of_error" title="Margin of error">not precise enough</a> to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the <a href="#Standardized_Gaussian_curvature">absolute length</a> is at least one million times the diameter of <a href="/wiki/Earth%27s_orbit" title="Earth's orbit">Earth's orbit</a> (<span class="nowrap"><span data-sort-value="7017299195741400000♠"></span>2<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span> <a href="/wiki/Astronomical_unit" title="Astronomical unit">AU</a></span>, 10 <a href="/wiki/Parsec" title="Parsec">parsec</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> Some argue that his measurements were methodologically flawed.<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> </p><p><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>, with his <a href="/wiki/Sphere-world" title="Sphere-world">sphere-world</a> <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a>, came to the conclusion that everyday experience does not necessarily rule out other geometries. </p><p>The <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a> gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy.<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-heading3"><h3 id="Geometry_of_the_universe_(special_relativity)"><span id="Geometry_of_the_universe_.28special_relativity.29"></span>Geometry of the universe (special relativity)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=17" title="Edit section: Geometry of the universe (special relativity)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a> places space and time on equal footing, so that one considers the geometry of a unified <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> instead of considering space and time separately.<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><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> <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski geometry</a> replaces <a href="/wiki/Galilean_geometry" class="mw-redirect" title="Galilean geometry">Galilean geometry</a> (which is the 3-dimensional Euclidean space with time of <a href="/wiki/Galilean_relativity" class="mw-redirect" title="Galilean relativity">Galilean relativity</a>).<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><p>In relativity, rather than Euclidean, elliptic and hyperbolic geometry, the appropriate geometries to consider are <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, <a href="/wiki/De_Sitter_space" title="De Sitter space">de Sitter space</a> and <a href="/wiki/Anti-de_Sitter_space" title="Anti-de Sitter space">anti-de Sitter space</a>,<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><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> corresponding to zero, positive and negative curvature respectively. </p><p>Hyperbolic geometry enters special relativity through <a href="/wiki/Rapidity" title="Rapidity">rapidity</a>, which stands in for <a href="/wiki/Velocity" title="Velocity">velocity</a>, and is expressed by a <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a>. The study of this velocity geometry has been called <a href="/wiki/Non-Euclidean_geometry#Kinematic_geometries" title="Non-Euclidean geometry">kinematic geometry</a>. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities).<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-heading2"><h2 id="Physical_realizations_of_the_hyperbolic_plane">Physical realizations of the hyperbolic plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=18" title="Edit section: Physical realizations of the hyperbolic plane"><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:Crochet_hyperbolic_kelp.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Crochet_hyperbolic_kelp.jpg/220px-Crochet_hyperbolic_kelp.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Crochet_hyperbolic_kelp.jpg/330px-Crochet_hyperbolic_kelp.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Crochet_hyperbolic_kelp.jpg/440px-Crochet_hyperbolic_kelp.jpg 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>A collection of crocheted hyperbolic planes, in imitation of a coral reef, by <a href="/wiki/Institute_For_Figuring" title="Institute For Figuring">Institute For Figuring</a></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hyperbolicsoccerball.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Hyperbolicsoccerball.jpg/220px-Hyperbolicsoccerball.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/22/Hyperbolicsoccerball.jpg 1.5x" data-file-width="320" data-file-height="240" /></a><figcaption>The "hyperbolic soccerball", a paper model which approximates (part of) the hyperbolic plane as a <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a> approximates the sphere.</figcaption></figure> <p>There exist various <a href="/wiki/Pseudosphere" title="Pseudosphere">pseudospheres</a> in Euclidean space that have a finite area of constant negative Gaussian curvature. </p><p>By <a href="/wiki/Hilbert%27s_theorem_(differential_geometry)" title="Hilbert's theorem (differential geometry)">Hilbert's theorem</a>, one cannot isometrically <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immerse</a> a complete hyperbolic plane (a complete regular surface of constant negative <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>) in a 3-D Euclidean space. </p><p>Other useful <a href="#Models_of_the_hyperbolic_plane">models</a> of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the <a href="/wiki/Pseudosphere" title="Pseudosphere">pseudosphere</a> is due to <a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a>. </p><p>The art of <a href="/wiki/Crochet" title="Crochet">crochet</a> has been <a href="/wiki/Mathematics_and_fiber_arts#Knitting_and_crochet" title="Mathematics and fiber arts">used</a> to demonstrate hyperbolic planes, the first such demonstration having been made by <a href="/wiki/Daina_Taimi%C5%86a" title="Daina Taimiņa">Daina Taimiņa</a>.<sup id="cite_ref-hyperbolicspace_28-0" class="reference"><a href="#cite_note-hyperbolicspace-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "<a href="/wiki/Hyperbolic_soccerball" class="mw-redirect" title="Hyperbolic soccerball">hyperbolic soccerball</a>" (more precisely, a <a href="/wiki/Truncated_order-7_triangular_tiling" title="Truncated order-7 triangular tiling">truncated order-7 triangular tiling</a>).<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><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> </p><p>Instructions on how to make a hyperbolic quilt, designed by <a href="/wiki/Helaman_Ferguson" title="Helaman Ferguson">Helaman Ferguson</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> have been made available by <a href="/wiki/Jeffrey_Weeks_(mathematician)" title="Jeffrey Weeks (mathematician)">Jeff Weeks</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> </p> <div class="mw-heading mw-heading2"><h2 id="Models_of_the_hyperbolic_plane">Models of the hyperbolic plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=19" title="Edit section: Models of the hyperbolic plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Various <a href="/wiki/Pseudosphere" title="Pseudosphere">pseudospheres</a> – surfaces with constant negative Gaussian curvature – can be embedded in 3-D space under the standard Euclidean metric, and so can be made into tangible models. Of these, the <a href="/wiki/Pseudosphere#Tractroid" title="Pseudosphere">tractoid</a> (or pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using a <a href="/wiki/Cone" title="Cone">cone</a> or <a href="/wiki/Cylinder" title="Cylinder">cylinder</a> as a model of the Euclidean plane. However, the entire hyperbolic plane cannot be embedded into Euclidean space in this way, and various other models are more convenient for abstractly exploring hyperbolic geometry. </p><p>There are four <a href="/wiki/Mathematical_model" title="Mathematical model">models</a> commonly used for hyperbolic geometry: the <a href="/wiki/Klein_model" class="mw-redirect" title="Klein model">Klein model</a>, the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a>, the <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a>, and the Lorentz or <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a>. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models of hyperbolic space by <a href="/wiki/Eugenio_Beltrami" title="Eugenio Beltrami">Beltrami</a>, not by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> or <a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a>. All these models are extendable to more dimensions. </p> <div class="mw-heading mw-heading3"><h3 id="The_Beltrami–Klein_model"><span id="The_Beltrami.E2.80.93Klein_model"></span>The Beltrami–Klein model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=20" title="Edit section: The Beltrami–Klein model"><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/Beltrami%E2%80%93Klein_model" title="Beltrami–Klein model">Beltrami–Klein model</a></div> <p>The <a href="/wiki/Beltrami%E2%80%93Klein_model" title="Beltrami–Klein model">Beltrami–Klein model</a>, also known as the projective disk model, Klein disk model and <a href="/wiki/Klein_model" class="mw-redirect" title="Klein model">Klein model</a>, is named after <a href="/wiki/Eugenio_Beltrami" title="Eugenio Beltrami">Eugenio Beltrami</a> and <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>. </p><p>For the two dimensions this model uses the interior of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> for the complete hyperbolic <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>, and the <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chords</a> of this circle are the hyperbolic lines. </p><p>For higher dimensions this model uses the interior of the <a href="/wiki/Unit_ball" class="mw-redirect" title="Unit ball">unit ball</a>, and the <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chords</a> of this <i>n</i>-ball are the hyperbolic lines. </p> <ul><li>This model has the advantage that lines are straight, but the disadvantage that <a href="/wiki/Angle" title="Angle">angles</a> are distorted (the mapping is not <a href="/wiki/Conformal_map" title="Conformal map">conformal</a>), and also circles are not represented as circles.</li> <li>The distance in this model is half the logarithm of the <a href="/wiki/Cross-ratio" title="Cross-ratio">cross-ratio</a>, which was introduced by <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> in <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="The_Poincaré_disk_model"><span id="The_Poincar.C3.A9_disk_model"></span>The Poincaré disk model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=21" title="Edit section: The Poincaré disk model"><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:Hyperbolic_tiling_omnitruncated_3-7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hyperbolic_tiling_omnitruncated_3-7.png/220px-Hyperbolic_tiling_omnitruncated_3-7.png" decoding="async" width="220" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hyperbolic_tiling_omnitruncated_3-7.png/330px-Hyperbolic_tiling_omnitruncated_3-7.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hyperbolic_tiling_omnitruncated_3-7.png/440px-Hyperbolic_tiling_omnitruncated_3-7.png 2x" data-file-width="1447" data-file-height="1388" /></a><figcaption>Poincaré disk model with <a href="/wiki/Truncated_triheptagonal_tiling" title="Truncated triheptagonal tiling">truncated triheptagonal tiling</a></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a></div> <p>The <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a>, also known as the conformal disk model, also employs the interior of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>, but lines are represented by arcs of circles that are <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to the boundary circle, plus diameters of the boundary circle. </p> <ul><li>This model preserves angles, and is thereby <a href="/wiki/Conformal_map" title="Conformal map">conformal</a>. All isometries within this model are therefore <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>.</li> <li>Circles entirely within the disk remain circles although the Euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle.</li> <li><a href="/wiki/Horocycle" title="Horocycle">Horocycles</a> are circles within the disk which are <a href="/wiki/Tangent" title="Tangent">tangent</a> to the boundary circle, minus the point of contact.</li> <li><a href="/wiki/Hypercycle_(hyperbolic_geometry)" class="mw-redirect" title="Hypercycle (hyperbolic geometry)">Hypercycles</a> are open-ended chords and circular arcs within the disc that terminate on the boundary circle at non-orthogonal angles.</li></ul> <div class="mw-heading mw-heading3"><h3 id="The_Poincaré_half-plane_model"><span id="The_Poincar.C3.A9_half-plane_model"></span>The Poincaré half-plane model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=22" title="Edit section: The Poincaré half-plane model"><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/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a></div> <p>The <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a> takes one-half of the Euclidean plane, bounded by a line <i>B</i> of the plane, to be a model of the hyperbolic plane. The line <i>B</i> is not included in the model. </p><p>The Euclidean plane may be taken to be a plane with the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> and the <a href="/wiki/X-axis" class="mw-redirect" title="X-axis">x-axis</a> is taken as line <i>B</i> and the half plane is the upper half (<i>y</i> > 0 ) of this plane. </p> <ul><li>Hyperbolic lines are then either half-circles orthogonal to <i>B</i> or rays perpendicular to <i>B</i>.</li> <li>The length of an interval on a ray is given by <a href="/wiki/Logarithmic_measure" class="mw-redirect" title="Logarithmic measure">logarithmic measure</a> so it is invariant under a <a href="/wiki/Homothetic_transformation" class="mw-redirect" title="Homothetic transformation">homothetic transformation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto (\lambda x,\lambda y),\quad \lambda >0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>x</mi> <mo>,</mo> <mi>λ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>λ</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto (\lambda x,\lambda y),\quad \lambda >0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894ebded73c0a97dbb48c04905f2e70b1f335fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.601ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto (\lambda x,\lambda y),\quad \lambda >0.}"></span></li> <li>Like the Poincaré disk model, this model preserves angles, and is thus <a href="/wiki/Conformal_map" title="Conformal map">conformal</a>. All isometries within this model are therefore <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> of the plane.</li> <li>The half-plane model is the limit of the Poincaré disk model whose boundary is tangent to <i>B</i> at the same point while the radius of the disk model goes to infinity.</li></ul> <div class="mw-heading mw-heading3"><h3 id="The_hyperboloid_model">The hyperboloid model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=23" title="Edit section: The hyperboloid model"><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/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a></div> <p>The <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a> or Lorentz model employs a 2-dimensional <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> of revolution (of two sheets, but using one) embedded in 3-dimensional <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>. This model is generally credited to Poincaré, but Reynolds<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> says that <a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a> used this model in 1885 </p> <ul><li>This model has direct application to <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, as Minkowski 3-space is a model for <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>, suppressing one spatial dimension. One can take the hyperboloid to represent the events (positions in spacetime) that various <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertially</a> moving observers, starting from a common event, will reach in a fixed <a href="/wiki/Proper_time" title="Proper time">proper time</a>.</li> <li>The hyperbolic distance between two points on the hyperboloid can then be identified with the relative <a href="/wiki/Rapidity" title="Rapidity">rapidity</a> between the two corresponding observers.</li> <li>The model generalizes directly to an additional dimension: a hyperbolic 3-space three-dimensional hyperbolic geometry relates to Minkowski 4-space.</li></ul> <div class="mw-heading mw-heading3"><h3 id="The_hemisphere_model">The hemisphere model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=24" title="Edit section: The hemisphere model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Sphere#Hemisphere" title="Sphere">hemisphere</a> model is not often used as model by itself, but it functions as a useful tool for visualizing transformations between the other models. </p><p>The hemisphere model uses the upper half of the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}=1,z>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>z</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}=1,z>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3e99a27cc7bad6a904f4c528549ec8376b5163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.715ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=1,z>0.}"></span> </p><p>The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere. </p><p>The hemisphere model is part of a <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>, and different projections give different models of the hyperbolic plane: </p> <ul><li><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> from <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 (0,0,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe7ded50a977a81c8f0f03bee443a0530133932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle (0,0,-1)}"></span> onto the plane <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 z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"></span> projects corresponding points on the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a></li> <li><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> from <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 (0,0,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe7ded50a977a81c8f0f03bee443a0530133932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle (0,0,-1)}"></span> onto the surface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-z^{2}=-1,z>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>z</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-z^{2}=-1,z>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4472861247f27e5163c32b2292d24a6c3959202d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.876ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-z^{2}=-1,z>0}"></span> projects corresponding points on the <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a></li> <li><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> from <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 (-1,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b81cdd04577e120ae2c243d061a5dec0a91469e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle (-1,0,0)}"></span> onto the plane <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"></span> projects corresponding points on the <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a></li> <li><a href="/wiki/Orthographic_projection" title="Orthographic projection">Orthographic projection</a> onto a plane <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 z=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b54d21d572ffbb0e9a2c4541330d1e012ed0823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle z=C}"></span> projects corresponding points on the <a href="/wiki/Beltrami%E2%80%93Klein_model" title="Beltrami–Klein model">Beltrami–Klein model</a>.</li> <li><a href="/wiki/Central_projection" class="mw-redirect" title="Central projection">Central projection</a> from the centre of the sphere onto the plane <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 z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}"></span> projects corresponding points on the <a href="/w/index.php?title=Gans_Model&action=edit&redlink=1" class="new" title="Gans Model (page does not exist)">Gans Model</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Connection_between_the_models">Connection between the models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=25" title="Edit section: Connection between the models"><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:Relation5models.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Relation5models.png/260px-Relation5models.png" decoding="async" width="260" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/73/Relation5models.png 1.5x" data-file-width="299" data-file-height="332" /></a><figcaption>Poincaré disk, hemispherical and hyperboloid models are related by <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a> from −1. <a href="/wiki/Beltrami%E2%80%93Klein_model" title="Beltrami–Klein model">Beltrami–Klein model</a> is <a href="/wiki/Orthographic_projection" title="Orthographic projection">orthographic projection</a> from hemispherical model. <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a> here projected from the hemispherical model by rays from left end of Poincaré disk model.</figcaption></figure> <p>All models essentially describe the same structure. The difference between them is that they represent different <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">coordinate charts</a> laid down on the same <a href="/wiki/Metric_space" title="Metric space">metric space</a>, namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is that it has a constant negative <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>, which is indifferent to the coordinate chart used. The <a href="/wiki/Geodesic" title="Geodesic">geodesics</a> are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry is generally introduced in terms of the geodesics and their intersections on the hyperbolic plane.<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> </p><p>Once we choose a coordinate chart (one of the "models"), we can always <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">embed</a> it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics. </p><p>Since the four models describe the same metric space, each can be transformed into the other. </p><p>See, for example: </p> <ul><li><a href="/wiki/Beltrami%E2%80%93Klein_model#Relation_to_the_hyperboloid_model" title="Beltrami–Klein model">the Beltrami–Klein model's relation to the hyperboloid model</a>,</li> <li><a href="/wiki/Beltrami%E2%80%93Klein_model#Relation_to_the_Poincaré_disk_model" title="Beltrami–Klein model">the Beltrami–Klein model's relation to the Poincaré disk model</a>,</li> <li>and <a href="/wiki/Poincar%C3%A9_disk_model#Relation_to_the_hyperboloid_model" title="Poincaré disk model">the Poincaré disk model's relation to the hyperboloid model</a>.</li></ul> <p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Other_models_of_hyperbolic_geometry">Other models of hyperbolic geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=26" title="Edit section: Other models of hyperbolic geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="The_Gans_model">The Gans model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=27" title="Edit section: The Gans model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1966 David Gans proposed a <a href="/w/index.php?title=Flattened_hyperboloid_model&action=edit&redlink=1" class="new" title="Flattened hyperboloid model (page does not exist)">flattened hyperboloid model</a> in the journal <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>.<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> It is an <a href="/wiki/Orthographic_projection" title="Orthographic projection">orthographic projection</a> of the hyperboloid model onto the xy-plane. This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry. </p> <ul><li>Unlike the Klein or the Poincaré models, this model utilizes the entire <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>.</li> <li>The lines in this model are represented as branches of a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="The_conformal_square_model">The conformal square model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=28" title="Edit section: The conformal square model"><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:Omnitruncated_tiling_on_conformal_square.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Omnitruncated_tiling_on_conformal_square.png/220px-Omnitruncated_tiling_on_conformal_square.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Omnitruncated_tiling_on_conformal_square.png/330px-Omnitruncated_tiling_on_conformal_square.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Omnitruncated_tiling_on_conformal_square.png/440px-Omnitruncated_tiling_on_conformal_square.png 2x" data-file-width="1464" data-file-height="1464" /></a><figcaption>Conformal square model with <a href="/wiki/Truncated_triheptagonal_tiling" title="Truncated triheptagonal tiling">truncated triheptagonal tiling</a></figcaption></figure> <p>The conformal square model of the hyperbolic plane arises from using <a href="/wiki/Schwarz-Christoffel_mapping" class="mw-redirect" title="Schwarz-Christoffel mapping">Schwarz-Christoffel mapping</a> to convert the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk</a> into a square.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> This model has finite extent, like the Poincaré disk. However, all of the points are inside a square. This model is conformal, which makes it suitable for artistic applications. </p> <div class="mw-heading mw-heading4"><h4 id="The_band_model">The band model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=29" title="Edit section: The band model"><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/Band_model" title="Band model">Band model</a></div> <p>The band model employs a portion of the Euclidean plane between two parallel lines.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Distance is preserved along one line through the middle of the band. Assuming the band is given by <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 \{z\in \mathbb {C} :|\operatorname {Im} z|<\pi /2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{z\in \mathbb {C} :|\operatorname {Im} z|<\pi /2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860d51593cce9c44d61d013d264b8c3251fc6f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.556ex; height:2.843ex;" alt="{\displaystyle \{z\in \mathbb {C} :|\operatorname {Im} z|<\pi /2\}}"></span>, the metric is given by <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 |dz|\sec(\operatorname {Im} z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |dz|\sec(\operatorname {Im} z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce681ca746737975a0edac679dc51cdc97724b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.026ex; height:2.843ex;" alt="{\displaystyle |dz|\sec(\operatorname {Im} z)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Isometries_of_the_hyperbolic_plane">Isometries of the hyperbolic plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=30" title="Edit section: Isometries of the hyperbolic plane"><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/Hyperbolic_motion" title="Hyperbolic motion">Hyperbolic motion</a> and <a href="/wiki/Transformation_geometry" title="Transformation geometry">transformation geometry</a></div> <p>Every <a href="/wiki/Isometry" title="Isometry">isometry</a> (<a href="/wiki/Geometric_transformation" title="Geometric transformation">transformation</a> or <a href="/wiki/Motion_(geometry)" title="Motion (geometry)">motion</a>) of the hyperbolic plane to itself can be realized as the composition of at most three <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>. In <i>n</i>-dimensional hyperbolic space, up to <i>n</i>+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.) </p><p>All isometries of the hyperbolic plane can be classified into these classes: </p> <ul><li>Orientation preserving <ul><li>the <a href="/wiki/Identity_function" title="Identity function">identity isometry</a> — nothing moves; zero reflections; zero <a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a>.</li> <li><a href="/wiki/Point_reflection" title="Point reflection">inversion through a point (half turn)</a> — two reflections through mutually perpendicular lines passing through the given point, i.e. a rotation of 180 degrees around the point; two <a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a>.</li> <li><a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">rotation</a> around a normal point — two reflections through lines passing through the given point (includes inversion as a special case); points move on circles around the center; three degrees of freedom.</li> <li>"rotation" around an <a href="/wiki/Ideal_point" title="Ideal point">ideal point</a> (horolation) — two reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom.</li> <li>translation along a straight line — two reflections through lines perpendicular to the given line; points off the given line move along hypercycles; three degrees of freedom.</li></ul></li> <li>Orientation reversing <ul><li>reflection through a line — one reflection; two degrees of freedom.</li> <li>combined reflection through a line and translation along the same line — the reflection and translation commute; three reflections required; three degrees of freedom.<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. (July 2016)">citation needed</span></a></i>]</sup></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Hyperbolic_geometry_in_art">Hyperbolic geometry in art</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=31" title="Edit section: Hyperbolic geometry in art"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>'s famous prints <i><a href="/wiki/Circle_Limit_III" title="Circle Limit III">Circle Limit III</a></i> and <i>Circle Limit IV</i> illustrate the conformal disc model (<a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a>) quite well. The white lines in <i>III</i> are not quite geodesics (they are <a href="/wiki/Hypercycle_(hyperbolic_geometry)" class="mw-redirect" title="Hypercycle (hyperbolic geometry)">hypercycles</a>), but are close to them. It is also possible to see quite plainly the negative <a href="/wiki/Curvature" title="Curvature">curvature</a> of the hyperbolic plane, through its effect on the sum of angles in triangles and squares. </p><p>For example, in <i>Circle Limit III</i> every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is <a href="/wiki/Exponential_growth" title="Exponential growth">exponential growth</a>. In <i>Circle Limit III</i>, for example, one can see that the number of fishes within a distance of <i>n</i> from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius <i>n</i> must rise exponentially in <i>n</i>. </p><p>The art of <a href="/wiki/Crochet" title="Crochet">crochet</a> has <a href="/wiki/Mathematics_and_fiber_arts#Knitting_and_crochet" title="Mathematics and fiber arts">been used</a> to demonstrate hyperbolic planes (pictured above) with the first being made by <a href="/wiki/Daina_Taimi%C5%86a" title="Daina Taimiņa">Daina Taimiņa</a>,<sup id="cite_ref-hyperbolicspace_28-1" class="reference"><a href="#cite_note-hyperbolicspace-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> whose book <i><a href="/wiki/Crocheting_Adventures_with_Hyperbolic_Planes" title="Crocheting Adventures with Hyperbolic Planes">Crocheting Adventures with Hyperbolic Planes</a></i> won the 2009 <a href="/wiki/Bookseller/Diagram_Prize_for_Oddest_Title_of_the_Year" title="Bookseller/Diagram Prize for Oddest Title of the Year">Bookseller/Diagram Prize for Oddest Title of the Year</a>.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/HyperRogue" title="HyperRogue">HyperRogue</a> is a <a href="/wiki/Roguelike" title="Roguelike">roguelike</a> game set on various tilings of the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Higher_dimensions">Higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=32" title="Edit section: Higher dimensions"><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/Hyperbolic_space" title="Hyperbolic space">Hyperbolic space</a></div> <p>Hyperbolic geometry is not limited to 2 dimensions; a hyperbolic geometry exists for every higher number of dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Homogeneous_structure">Homogeneous structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=33" title="Edit section: Homogeneous structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hyperbolic_space" title="Hyperbolic space">Hyperbolic space</a> of dimension <i>n</i> is a special case of a Riemannian <a href="/wiki/Symmetric_space" title="Symmetric space">symmetric space</a> of noncompact type, as it is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the quotient </p> <dl><dd><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 \mathrm {O} (1,n)/(\mathrm {O} (1)\times \mathrm {O} (n)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (1,n)/(\mathrm {O} (1)\times \mathrm {O} (n)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8637748b1dd1f3668dbe448fa6ca9ef46d1c65f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.46ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (1,n)/(\mathrm {O} (1)\times \mathrm {O} (n)).}"></span></dd></dl></dd></dl> <p>The <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> <span class="nowrap">O(1, <i>n</i>)</span> <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> by norm-preserving transformations on <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> <b>R</b><sup>1,<i>n</i></sup>, and it acts <a href="/wiki/Group_action_(mathematics)#Types_of_actions" class="mw-redirect" title="Group action (mathematics)">transitively</a> on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic <i>n</i>-space. The <a href="/wiki/Stabilizer_subgroup" class="mw-redirect" title="Stabilizer subgroup">stabilizer</a> of any particular line is isomorphic to the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">product</a> of the orthogonal groups O(<i>n</i>) and O(1), where O(<i>n</i>) acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebraic</a> terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations. </p><p>In small dimensions, there are exceptional isomorphisms of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms <span class="nowrap">SO<sup>+</sup>(1, 2) ≅ PSL(2, <b>R</b>) ≅ PSU(1, 1)</span> allow one to interpret the upper half plane model as the quotient <span class="nowrap">SL(2, <b>R</b>)/SO(2)</span> and the Poincaré disc model as the quotient <span class="nowrap">SU(1, 1)/U(1)</span>. In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in <span class="nowrap">PGL(2, <b>C</b>)</span> of the respective subspaces of the Riemann sphere. The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. In dimension 3, the fractional linear action of <span class="nowrap">PGL(2, <b>C</b>)</span> on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism <span class="nowrap">O<sup>+</sup>(1, 3) ≅ PGL(2, <b>C</b>)</span>. This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by <a href="/wiki/Unipotent" title="Unipotent">unipotent</a> <a href="/wiki/Triangular_matrix" title="Triangular matrix">upper triangular</a> matrices. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=34" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Band_model" title="Band model">Band model</a></li> <li><a href="/wiki/Constructions_in_hyperbolic_geometry" title="Constructions in hyperbolic geometry">Constructions in hyperbolic geometry</a></li> <li><a href="/wiki/Hjelmslev_transformation" title="Hjelmslev transformation">Hjelmslev transformation</a></li> <li><a href="/wiki/Hyperbolic_3-manifold" title="Hyperbolic 3-manifold">Hyperbolic 3-manifold</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic manifold</a></li> <li><a href="/wiki/Hyperbolic_set" title="Hyperbolic set">Hyperbolic set</a></li> <li><a href="/wiki/Hyperbolic_tree" title="Hyperbolic tree">Hyperbolic tree</a></li> <li><a href="/wiki/Kleinian_group" title="Kleinian group">Kleinian group</a></li> <li><a href="/wiki/Lambert_quadrilateral" title="Lambert quadrilateral">Lambert quadrilateral</a></li> <li><a href="/wiki/Open_universe" class="mw-redirect" title="Open universe">Open universe</a></li> <li><a href="/wiki/Poincar%C3%A9_metric" title="Poincaré metric">Poincaré metric</a></li> <li><a href="/wiki/Saccheri_quadrilateral" title="Saccheri quadrilateral">Saccheri quadrilateral</a></li> <li><a href="/wiki/Systolic_geometry" title="Systolic geometry">Systolic geometry</a></li> <li><a href="/wiki/Uniform_tilings_in_hyperbolic_plane" title="Uniform tilings in hyperbolic plane">Uniform tilings in hyperbolic plane</a></li> <li><a href="/wiki/%CE%94-hyperbolic_space" class="mw-redirect" title="Δ-hyperbolic space">δ-hyperbolic space</a></li></ul> </div> <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=Hyperbolic_geometry&action=edit&section=35" 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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-auto-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-auto_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-auto_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/2430495/88985">"Curvature of curves on the hyperbolic plane"</a>. <i>math <a href="/wiki/Stackexchange" class="mw-redirect" title="Stackexchange">stackexchange</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math+stackexchange&rft.atitle=Curvature+of+curves+on+the+hyperbolic+plane&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F2430495%2F88985&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThorgeirsson2014" class="citation book cs1">Thorgeirsson, Sverrir (2014). <a rel="nofollow" class="external text" href="https://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503"><i>Hyperbolic geometry: history, models, and axioms</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hyperbolic+geometry%3A+history%2C+models%2C+and+axioms&rft.date=2014&rft.aulast=Thorgeirsson&rft.aufirst=Sverrir&rft_id=https%3A%2F%2Furn.kb.se%2Fresolve%3Furn%3Durn%3Anbn%3Ase%3Auu%3Adiva-227503&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHydeRamsden2003" class="citation journal cs1">Hyde, S.T.; Ramsden, S. (2003). "Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings". <i>The European Physical Journal B</i>. <b>31</b> (2): <span class="nowrap">273–</span>284. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003EPJB...31..273H">2003EPJB...31..273H</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.720.5527">10.1.1.720.5527</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjb%2Fe2003-00032-8">10.1140/epjb/e2003-00032-8</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:41146796">41146796</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+European+Physical+Journal+B&rft.atitle=Some+novel+three-dimensional+Euclidean+crystalline+networks+derived+from+two-dimensional+hyperbolic+tilings&rft.volume=31&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E273-%3C%2Fspan%3E284&rft.date=2003&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.720.5527%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A41146796%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1140%2Fepjb%2Fe2003-00032-8&rft_id=info%3Abibcode%2F2003EPJB...31..273H&rft.aulast=Hyde&rft.aufirst=S.T.&rft.au=Ramsden%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-Sommerville2005-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sommerville2005_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sommerville2005_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSommerville2005" class="citation book cs1">Sommerville, D.M.Y. (2005). <i>The elements of non-Euclidean geometry</i> (Unabr. and unaltered republ. ed.). Mineola, N.Y.: Dover Publications. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-44222-5" title="Special:BookSources/0-486-44222-5"><bdi>0-486-44222-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+elements+of+non-Euclidean+geometry&rft.place=Mineola%2C+N.Y.&rft.pages=58&rft.edition=Unabr.+and+unaltered+republ.&rft.pub=Dover+Publications&rft.date=2005&rft.isbn=0-486-44222-5&rft.aulast=Sommerville&rft.aufirst=D.M.Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamsayRichtmyer1995" class="citation book cs1">Ramsay, Arlan; Richtmyer, Robert D. (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams"><i>Introduction to hyperbolic geometry</i></a></span>. New York: Springer-Verlag. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams/page/97">97–103</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387943390" title="Special:BookSources/0387943390"><bdi>0387943390</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+hyperbolic+geometry&rft.place=New+York&rft.pages=97-103&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=0387943390&rft.aulast=Ramsay&rft.aufirst=Arlan&rft.au=Richtmyer%2C+Robert+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontohy0000rams&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">See for instance, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070928084550/http://www.resonancepub.com/omarkhayyam.htm">"Omar Khayyam 1048–1131"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.resonancepub.com/omarkhayyam.htm">the original</a> on 2007-09-28<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-01-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Omar+Khayyam+1048%E2%80%931131&rft_id=http%3A%2F%2Fwww.resonancepub.com%2Fomarkhayyam.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html">"Non-Euclidean Geometry Seminar"</a>. <i>Math.columbia.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">21 January</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math.columbia.edu&rft.atitle=Non-Euclidean+Geometry+Seminar&rft_id=http%3A%2F%2Fwww.math.columbia.edu%2F~pinkham%2Fteaching%2Fseminars%2FNonEuclidean.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+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">Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., <i><a href="/wiki/Encyclopedia_of_the_History_of_Arabic_Science" title="Encyclopedia of the History of Arabic Science">Encyclopedia of the History of Arabic Science</a></i>, Vol. 2, p. 447–494 [470], <a href="/wiki/Routledge" title="Routledge">Routledge</a>, London and New York: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>"Three scientists, Ibn al-Haytham, Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's <i><a href="/wiki/Book_of_Optics" title="Book of Optics">Book of Optics</a></i> (<i>Kitab al-Manazir</i>) – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar <a href="/wiki/Gersonides" title="Gersonides">Levi ben Gerson</a>, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that <i>Pseudo-Tusi's Exposition of Euclid</i> had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."</p></blockquote></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves2012" class="citation cs2">Eves, Howard (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J9QcmFHj8EwC&pg=PA59"><i>Foundations and Fundamental Concepts of Mathematics</i></a>, Courier Dover Publications, p. 59, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486132204" title="Special:BookSources/9780486132204"><bdi>9780486132204</bdi></a>, <q>We also owe to Lambert the first systematic development of the theory of hyperbolic functions and, indeed, our present notation for these functions.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+and+Fundamental+Concepts+of+Mathematics&rft.pages=59&rft.pub=Courier+Dover+Publications&rft.date=2012&rft.isbn=9780486132204&rft.aulast=Eves&rft.aufirst=Howard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJ9QcmFHj8EwC%26pg%3DPA59&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRatcliffe2006" class="citation cs2">Ratcliffe, John (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JV9m8o-ok6YC&pg=PA99"><i>Foundations of Hyperbolic Manifolds</i></a>, Graduate Texts in Mathematics, vol. 149, Springer, p. 99, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387331973" title="Special:BookSources/9780387331973"><bdi>9780387331973</bdi></a>, <q>That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph <i>Theorie der Parallellinien</i>, which was published posthumously in 1786.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Hyperbolic+Manifolds&rft.series=Graduate+Texts+in+Mathematics&rft.pages=99&rft.pub=Springer&rft.date=2006&rft.isbn=9780387331973&rft.aulast=Ratcliffe&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJV9m8o-ok6YC%26pg%3DPA99&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBonola,_R.1912" class="citation book cs1">Bonola, R. (1912). <a rel="nofollow" class="external text" href="https://archive.org/details/noneuclideangeom00bono"><i>Non-Euclidean geometry: A critical and historical study of its development</i></a>. Chicago: Open Court.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-Euclidean+geometry%3A+A+critical+and+historical+study+of+its+development&rft.place=Chicago&rft.pub=Open+Court&rft.date=1912&rft.au=Bonola%2C+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnoneuclideangeom00bono&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenberg2003" class="citation book cs1"><a href="/wiki/Marvin_Greenberg" title="Marvin Greenberg">Greenberg, Marvin Jay</a> (2003). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/euclideannoneucl00gree_304"><i>Euclidean and non-Euclidean geometries: development and history</i></a></span> (3rd ed.). New York: Freeman. p. <a rel="nofollow" class="external text" href="https://archive.org/details/euclideannoneucl00gree_304/page/n194">177</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0716724464" title="Special:BookSources/0716724464"><bdi>0716724464</bdi></a>. <q>Out of nothing I have created a strange new universe. JÁNOS BOLYAI</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euclidean+and+non-Euclidean+geometries%3A+development+and+history&rft.place=New+York&rft.pages=177&rft.edition=3rd&rft.pub=Freeman&rft.date=2003&rft.isbn=0716724464&rft.aulast=Greenberg&rft.aufirst=Marvin+Jay&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feuclideannoneucl00gree_304&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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">Felix Klein, <i>Elementary Mathematics from an Advanced Standpoint: Geometry</i>, Dover, 1948 (reprint of English translation of 3rd Edition, 1940. First edition in German, 1908) pg. 176</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">F. Klein. "Über die sogenannte Nicht-Euklidische Geometrie". <i>Math. Ann.</i> 4, 573–625 (also in <i>Gesammelte Mathematische Abhandlungen</i> 1, 244–350).</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">Rosenfeld, B.A. (1988) <i>A History of Non-Euclidean Geometry</i>, page 236, Springer-Verlag <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/0-387-96458-4" title="Special:BookSources/0-387-96458-4">0-387-96458-4</a></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLucas1984" class="citation book cs1"><a href="/wiki/John_Lucas_(philosopher)" title="John Lucas (philosopher)">Lucas, John Randolph</a> (1984). <i>Space, Time and Causality</i>. Clarendon Press. p. 149. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-875057-9" title="Special:BookSources/0-19-875057-9"><bdi>0-19-875057-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space%2C+Time+and+Causality&rft.pages=149&rft.pub=Clarendon+Press&rft.date=1984&rft.isbn=0-19-875057-9&rft.aulast=Lucas&rft.aufirst=John+Randolph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWood1959" class="citation journal cs1">Wood, Donald (April 1959). "Some Greek stereotypes of other peoples". <i>Race</i>. <b>1</b> (2): <span class="nowrap">65–</span>71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F030639685900100207">10.1177/030639685900100207</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Race&rft.atitle=Some+Greek+stereotypes+of+other+peoples&rft.volume=1&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E65-%3C%2Fspan%3E71&rft.date=1959-04&rft_id=info%3Adoi%2F10.1177%2F030639685900100207&rft.aulast=Wood&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTorretti1978" class="citation book cs1">Torretti, Roberto (1978). <i>Philosophy of Geometry from Riemann to Poincare</i>. Dordrecht Holland: Reidel. p. 255.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Geometry+from+Riemann+to+Poincare&rft.place=Dordrecht+Holland&rft.pages=255&rft.pub=Reidel&rft.date=1978&rft.aulast=Torretti&rft.aufirst=Roberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBonola1955" class="citation book cs1">Bonola, Roberto (1955). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/noneuclideangeom0000bono"><i>Non-Euclidean geometry : a critical and historical study of its developments</i></a></span> (Unabridged and unaltered republ. of the 1. English translation 1912. ed.). New York, NY: Dover. p. <a rel="nofollow" class="external text" href="https://archive.org/details/noneuclideangeom0000bono/page/95">95</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0486600270" title="Special:BookSources/0486600270"><bdi>0486600270</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-Euclidean+geometry+%3A+a+critical+and+historical+study+of+its+developments&rft.place=New+York%2C+NY&rft.pages=95&rft.edition=Unabridged+and+unaltered+republ.+of+the+1.+English+translation+1912.&rft.pub=Dover&rft.date=1955&rft.isbn=0486600270&rft.aulast=Bonola&rft.aufirst=Roberto&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnoneuclideangeom0000bono&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichtmyer1995" class="citation book cs1">Richtmyer, Arlan Ramsay, Robert D. (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams/page/118"><i>Introduction to hyperbolic geometry</i></a>. New York: Springer-Verlag. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams/page/118">118–120</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387943390" title="Special:BookSources/0387943390"><bdi>0387943390</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+hyperbolic+geometry&rft.place=New+York&rft.pages=118-120&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=0387943390&rft.aulast=Richtmyer&rft.aufirst=Arlan+Ramsay%2C+Robert+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontohy0000rams%2Fpage%2F118&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.learner.org/courses/mathilluminated/units/8/textbook/08.php">"Mathematics Illuminated - Unit 8 - 8.8 Geometrization Conjecture"</a>. <i>Learner.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">21 January</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Learner.org&rft.atitle=Mathematics+Illuminated+-+Unit+8+-+8.8+Geometrization+Conjecture&rft_id=http%3A%2F%2Fwww.learner.org%2Fcourses%2Fmathilluminated%2Funits%2F8%2Ftextbook%2F08.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFL._D._LandauE._M._Lifshitz1973" class="citation book cs1">L. D. Landau; E. M. Lifshitz (1973). <i>Classical Theory of Fields</i>. <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a>. Vol. 2 (4th ed.). Butterworth Heinemann. pp. <span class="nowrap">1–</span>4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2768-9" title="Special:BookSources/978-0-7506-2768-9"><bdi>978-0-7506-2768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Theory+of+Fields&rft.series=Course+of+Theoretical+Physics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E4&rft.edition=4th&rft.pub=Butterworth+Heinemann&rft.date=1973&rft.isbn=978-0-7506-2768-9&rft.au=L.+D.+Landau&rft.au=E.+M.+Lifshitz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._P._FeynmanR._B._LeightonM._Sands1963" class="citation book cs1">R. 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Addison Wesley. p. (17-1)–(17-3). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02116-1" title="Special:BookSources/0-201-02116-1"><bdi>0-201-02116-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feynman+Lectures+on+Physics&rft.pages=%2817-1%29-%2817-3%29&rft.pub=Addison+Wesley&rft.date=1963&rft.isbn=0-201-02116-1&rft.au=R.+P.+Feynman&rft.au=R.+B.+Leighton&rft.au=M.+Sands&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._R._ForshawA._G._Smith2008" class="citation book cs1"><a href="/wiki/Jeff_Forshaw" title="Jeff Forshaw">J. R. Forshaw</a>; A. G. Smith (2008). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/dynamicsrelativi00fors"><i>Dynamics and Relativity</i></a></span>. Manchester physics series. Wiley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/dynamicsrelativi00fors/page/n260">246</a>–248. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-01460-8" title="Special:BookSources/978-0-470-01460-8"><bdi>978-0-470-01460-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+and+Relativity&rft.series=Manchester+physics+series&rft.pages=246-248&rft.pub=Wiley&rft.date=2008&rft.isbn=978-0-470-01460-8&rft.au=J.+R.+Forshaw&rft.au=A.+G.+Smith&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdynamicsrelativi00fors&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1">Misner; Thorne; Wheeler (1973). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/gravitation00cwmi"><i>Gravitation</i></a></span>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/gravitation00cwmi/page/n53">21</a>, 758.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pages=21%2C+758&rft.date=1973&rft.au=Misner&rft.au=Thorne&rft.au=Wheeler&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgravitation00cwmi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_K._BeemPaul_EhrlichKevin_Easley1996" class="citation book cs1">John K. 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Retrieved <span class="nowrap">21 January</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Theiff.org&rft.atitle=How+to+Build+your+own+Hyperbolic+Soccer+Ball&rft_id=http%3A%2F%2Fwww.theiff.org%2Fimages%2FIFF_HypSoccerBall.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.tamu.edu/~sottile/research/stories/hyperbolic_football/index.html">"Hyperbolic Football"</a>. <i>Math.tamu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">21 January</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math.tamu.edu&rft.atitle=Hyperbolic+Football&rft_id=http%3A%2F%2Fwww.math.tamu.edu%2F~sottile%2Fresearch%2Fstories%2Fhyperbolic_football%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110711162245/http://www.helasculpt.com/gallery/hyperbolicquilt/">"Helaman Ferguson, Hyperbolic Quilt"</a>. 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(2016). <a rel="nofollow" class="external text" href="http://archive.bridgesmathart.org/2016/bridges2016-179.pdf"><i>The Conformal Hyperbolic Square and Its Ilk</i></a> <span class="cs1-format">(PDF)</span>. Bridges Finland Conference Proceedings.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=The+Conformal+Hyperbolic+Square+and+Its+Ilk&rft.date=2016&rft.aulast=Fong&rft.aufirst=C.&rft_id=http%3A%2F%2Farchive.bridgesmathart.org%2F2016%2Fbridges2016-179.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://matrixeditions.com/TVol1.Chap2.pdf">"2"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="http://matrixeditions.com/TeichmullerVol1.html"><i>Teichmüller theory and applications to geometry, topology, and dynamics</i></a>. Hubbard, John Hamal. Ithaca, NY: Matrix Editions. 2006–2016. p. 25. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780971576629" title="Special:BookSources/9780971576629"><bdi>9780971576629</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/57965863">57965863</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2&rft.btitle=Teichm%C3%BCller+theory+and+applications+to+geometry%2C+topology%2C+and+dynamics&rft.place=Ithaca%2C+NY&rft.pages=25&rft.pub=Matrix+Editions&rft.date=2006%2F2016&rft_id=info%3Aoclcnum%2F57965863&rft.isbn=9780971576629&rft_id=http%3A%2F%2Fmatrixeditions.com%2FTVol1.Chap2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: others (<a href="/wiki/Category:CS1_maint:_others" title="Category:CS1 maint: others">link</a>)</span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloxham2010" class="citation journal cs1">Bloxham, Andy (March 26, 2010). <a rel="nofollow" class="external text" href="https://www.telegraph.co.uk/culture/books/bookprizes/7520047/Crocheting-Adventures-with-Hyperbolic-Planes-wins-oddest-book-title-award.html">"Crocheting Adventures with Hyperbolic Planes wins oddest book title award"</a>. <i><a href="/wiki/The_Daily_Telegraph" title="The Daily Telegraph">The Telegraph</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Telegraph&rft.atitle=Crocheting+Adventures+with+Hyperbolic+Planes+wins+oddest+book+title+award&rft.date=2010-03-26&rft.aulast=Bloxham&rft.aufirst=Andy&rft_id=https%3A%2F%2Fwww.telegraph.co.uk%2Fculture%2Fbooks%2Fbookprizes%2F7520047%2FCrocheting-Adventures-with-Hyperbolic-Planes-wins-oddest-book-title-award.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=36" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A'Campo, Norbert and Papadopoulos, Athanase, (2012) <i>Notes on hyperbolic geometry</i>, in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-105-7" title="Special:BookSources/978-3-03719-105-7">978-3-03719-105-7</a>, DOI 10.4171–105.</li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a>, (1942) <i>Non-Euclidean geometry</i>, University of Toronto Press, Toronto</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFenchel1989" class="citation book cs1"><a href="/wiki/Werner_Fenchel" title="Werner Fenchel">Fenchel, Werner</a> (1989). <i>Elementary geometry in hyperbolic space</i>. De Gruyter Studies in mathematics. Vol. 11. Berlin-New York: Walter de Gruyter & Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+geometry+in+hyperbolic+space&rft.place=Berlin-New+York&rft.series=De+Gruyter+Studies+in+mathematics&rft.pub=Walter+de+Gruyter+%26+Co.&rft.date=1989&rft.aulast=Fenchel&rft.aufirst=Werner&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFenchelNielsen,_Jakob2003" class="citation book cs1"><a href="/wiki/Werner_Fenchel" title="Werner Fenchel">Fenchel, Werner</a>; <a href="/wiki/Jakob_Nielsen_(mathematician)" title="Jakob Nielsen (mathematician)">Nielsen, Jakob</a> (2003). Asmus L. Schmidt (ed.). <i>Discontinuous groups of isometries in the hyperbolic plane</i>. De Gruyter Studies in mathematics. Vol. 29. Berlin: Walter de Gruyter & Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discontinuous+groups+of+isometries+in+the+hyperbolic+plane&rft.place=Berlin&rft.series=De+Gruyter+Studies+in+mathematics&rft.pub=Walter+de+Gruyter+%26+Co.&rft.date=2003&rft.aulast=Fenchel&rft.aufirst=Werner&rft.au=Nielsen%2C+Jakob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></li> <li>Lobachevsky, Nikolai I., (2010) <i>Pangeometry</i>, Edited and translated by Athanase Papadopoulos, Heritage of European Mathematics, Vol. 4. Zürich: European Mathematical Society (EMS). xii, 310~p, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-087-6" title="Special:BookSources/978-3-03719-087-6">978-3-03719-087-6</a>/hbk</li> <li><a href="/wiki/John_Milnor" title="John Milnor">Milnor, John W.</a>, (1982) <i><a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bams/1183548588">Hyperbolic geometry: The first 150 years</a></i>, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.</li> <li>Reynolds, William F., (1993) <i>Hyperbolic Geometry on a Hyperboloid</i>, <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a> 100:442–455.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell1996" class="citation book cs1"><a href="/wiki/John_Stillwell" title="John Stillwell">Stillwell, John</a> (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZQjBXxxQsucC"><i>Sources of hyperbolic geometry</i></a>. History of Mathematics. Vol. 10. Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0529-9" title="Special:BookSources/978-0-8218-0529-9"><bdi>978-0-8218-0529-9</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1402697">1402697</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sources+of+hyperbolic+geometry&rft.place=Providence%2C+R.I.&rft.series=History+of+Mathematics&rft.pub=American+Mathematical+Society&rft.date=1996&rft.isbn=978-0-8218-0529-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1402697%23id-name%3DMR&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZQjBXxxQsucC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></li> <li>Samuels, David, (March 2006) <i>Knit Theory</i> Discover Magazine, volume 27, Number 3.</li> <li>James W. Anderson, <i>Hyperbolic Geometry</i>, Springer 2005, <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/1-85233-934-9" title="Special:BookSources/1-85233-934-9">1-85233-934-9</a></li> <li>James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry (1997) <i><a rel="nofollow" class="external text" href="http://www.msri.org/communications/books/Book31/files/cannon.pdf">Hyperbolic Geometry</a></i>, MSRI Publications, volume 31.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometry&action=edit&section=37" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Hyperbolic_geometry" class="extiw" title="commons:Category:Hyperbolic geometry">Hyperbolic geometry</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html">Javascript freeware for creating sketches in the Poincaré Disk Model of Hyperbolic Geometry</a> University of New Mexico</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=B16YjC9OS0k&mode=user&search=">"The Hyperbolic Geometry Song"</a> A short music video about the basics of Hyperbolic Geometry available at YouTube.</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=Lobachevskii_geometry">"Lobachevskii geometry"</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=Lobachevskii+geometry&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLobachevskii_geometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Gauss–Bolyai–Lobachevsky_Space"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HyperbolicGeometry.html">"Hyperbolic Geometry"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Hyperbolic+Geometry&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHyperbolicGeometry.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140303111006/http://www.geom.uiuc.edu/~crobles/hyperbolic/">More on hyperbolic geometry, including movies and equations for conversion between the different models</a> University of Illinois at Urbana-Champaign</li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0903.3287">Hyperbolic Voronoi diagrams made easy, Frank Nielsen</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStothers2000" class="citation journal cs1">Stothers, Wilson (2000). <a rel="nofollow" class="external text" href="http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html">"Hyperbolic geometry"</a>. <i>maths.gla.ac.uk</i>. <a href="/wiki/University_of_Glasgow" title="University of Glasgow">University of Glasgow</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=maths.gla.ac.uk&rft.atitle=Hyperbolic+geometry&rft.date=2000&rft.aulast=Stothers&rft.aufirst=Wilson&rft_id=http%3A%2F%2Fwww.maths.gla.ac.uk%2F~wws%2Fcabripages%2Fhyperbolic%2Fhyperbolic0.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometry" class="Z3988"></span>, interactive instructional website.</li> <li><a rel="nofollow" class="external text" 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