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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convex_polyhedra" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Convex_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Convex polyhedra</span> </div> </a> <ul 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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Topological classification</span> </div> </a> <ul id="toc-Topological_classification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Duality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Duality</span> </div> </a> <ul id="toc-Duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vertex_figures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vertex_figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Vertex figures</span> </div> </a> <ul id="toc-Vertex_figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_area_and_distances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surface_area_and_distances"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Surface area and distances</span> </div> </a> <ul id="toc-Surface_area_and_distances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Volume</span> </div> </a> <ul id="toc-Volume-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dehn_invariant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dehn_invariant"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Dehn invariant</span> </div> </a> <ul id="toc-Dehn_invariant-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symmetries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Symmetries</span> </div> </a> <button aria-controls="toc-Symmetries-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 Symmetries subsection</span> </button> <ul id="toc-Symmetries-sublist" class="vector-toc-list"> <li id="toc-Regular_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regular_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Regular polyhedra</span> </div> </a> <ul id="toc-Regular_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniform_polyhedra_and_their_duals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniform_polyhedra_and_their_duals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Uniform polyhedra and their duals</span> </div> </a> <ul id="toc-Uniform_polyhedra_and_their_duals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isohedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isohedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Isohedra</span> </div> </a> <ul id="toc-Isohedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Symmetry groups</span> </div> </a> <ul id="toc-Symmetry_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_important_families_of_polyhedra" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_important_families_of_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other important families of polyhedra</span> </div> </a> <button aria-controls="toc-Other_important_families_of_polyhedra-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 Other important families of polyhedra subsection</span> </button> <ul id="toc-Other_important_families_of_polyhedra-sublist" class="vector-toc-list"> <li id="toc-Zonohedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zonohedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Zonohedra</span> </div> </a> <ul id="toc-Zonohedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Space-filling_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Space-filling_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Space-filling polyhedra</span> </div> </a> <ul id="toc-Space-filling_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lattice_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lattice_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Lattice polyhedra</span> </div> </a> <ul id="toc-Lattice_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flexible_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flexible_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Flexible polyhedra</span> </div> </a> <ul id="toc-Flexible_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compounds"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Compounds</span> </div> </a> <ul id="toc-Compounds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthogonal_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthogonal_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Orthogonal polyhedra</span> </div> </a> <ul id="toc-Orthogonal_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Embedded_regular_maps_with_planar_faces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Embedded_regular_maps_with_planar_faces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Embedded regular maps with planar faces</span> </div> </a> <ul id="toc-Embedded_regular_maps_with_planar_faces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalisations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalisations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalisations</span> </div> </a> <button aria-controls="toc-Generalisations-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 Generalisations subsection</span> </button> <ul id="toc-Generalisations-sublist" class="vector-toc-list"> <li id="toc-Apeirohedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Apeirohedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Apeirohedra</span> </div> </a> <ul id="toc-Apeirohedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Complex polyhedra</span> </div> </a> <ul id="toc-Complex_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curved_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curved_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Curved polyhedra</span> </div> </a> <ul id="toc-Curved_polyhedra-sublist" class="vector-toc-list"> <li id="toc-Spherical_polyhedra" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spherical_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.1</span> <span>Spherical polyhedra</span> </div> </a> <ul id="toc-Spherical_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curved_spacefilling_polyhedra" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Curved_spacefilling_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.2</span> <span>Curved spacefilling polyhedra</span> </div> </a> <ul id="toc-Curved_spacefilling_polyhedra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ideal_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Ideal polyhedra</span> </div> </a> <ul id="toc-Ideal_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skeletons_and_polyhedra_as_graphs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skeletons_and_polyhedra_as_graphs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Skeletons and polyhedra as graphs</span> </div> </a> <ul id="toc-Skeletons_and_polyhedra_as_graphs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Alternative_usages" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Alternative_usages"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Alternative usages</span> </div> </a> <button aria-controls="toc-Alternative_usages-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 Alternative usages subsection</span> </button> <ul id="toc-Alternative_usages-sublist" class="vector-toc-list"> <li id="toc-Higher-dimensional_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher-dimensional_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Higher-dimensional polyhedra</span> </div> </a> <ul id="toc-Higher-dimensional_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Topological polyhedra</span> </div> </a> <ul id="toc-Topological_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abstract_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Abstract polyhedra</span> </div> </a> <ul id="toc-Abstract_polyhedra-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">8</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-Before_the_Greeks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Before_the_Greeks"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Before the Greeks</span> </div> </a> <ul id="toc-Before_the_Greeks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ancient_Greece" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient_Greece"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Ancient Greece</span> </div> </a> <ul id="toc-Ancient_Greece-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ancient_China" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient_China"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Ancient China</span> </div> </a> <ul id="toc-Ancient_China-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medieval_Islam" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Medieval_Islam"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Medieval Islam</span> </div> </a> <ul id="toc-Medieval_Islam-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Renaissance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Renaissance"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span>Renaissance</span> </div> </a> <ul id="toc-Renaissance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-17th–19th_centuries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#17th–19th_centuries"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.6</span> <span>17th–19th centuries</span> </div> </a> <ul id="toc-17th–19th_centuries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-20th–21st_centuries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#20th–21st_centuries"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.7</span> <span>20th–21st centuries</span> </div> </a> <ul id="toc-20th–21st_centuries-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_nature" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_nature"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>In nature</span> </div> </a> <ul id="toc-In_nature-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-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-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 External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-General_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>General theory</span> </div> </a> <ul id="toc-General_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lists_and_databases_of_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lists_and_databases_of_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.2</span> <span>Lists and databases of polyhedra</span> </div> </a> <ul id="toc-Lists_and_databases_of_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_software" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_software"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.3</span> <span>Free software</span> </div> </a> <ul id="toc-Free_software-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resources_for_making_physical_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resources_for_making_physical_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.4</span> <span>Resources for making physical models</span> </div> </a> <ul id="toc-Resources_for_making_physical_models-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Polyhedron</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 71 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-71" 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">71 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B9%D8%AF%D8%AF_%D8%A7%D9%84%D8%B3%D8%B7%D9%88%D8%AD" title="متعدد السطوح – Arabic" lang="ar" hreflang="ar" data-title="متعدد السطوح" 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/Poliedru" title="Poliedru – Asturian" lang="ast" hreflang="ast" data-title="Poliedru" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B9%E0%A7%81%E0%A6%A4%E0%A6%B2%E0%A6%95" title="বহুতলক – Bangla" lang="bn" hreflang="bn" data-title="বহুতলক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/To-bi%C4%81n-th%C3%A9" title="To-biān-thé – Minnan" lang="nan" hreflang="nan" data-title="To-biān-thé" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BF%D2%A1%D1%8B%D1%80" title="Күпҡыр – Bashkir" lang="ba" hreflang="ba" data-title="Күпҡыр" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%81%D1%82%D0%B5%D0%BD" title="Многостен – Bulgarian" lang="bg" hreflang="bg" data-title="Многостен" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Poliedar" title="Poliedar – Bosnian" lang="bs" hreflang="bs" data-title="Poliedar" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Pol%C3%ADedre" title="Políedre – Catalan" lang="ca" hreflang="ca" data-title="Políedre" 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%9D%D1%83%D0%BC%D0%B0%D0%B9%D1%85%D1%8B%D1%81%D0%B0%D0%BA%D0%BB%C4%83%D1%85" title="Нумайхысаклăх – Chuvash" lang="cv" hreflang="cv" data-title="Нумайхысаклăх" 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/Mnohost%C4%9Bn" title="Mnohostěn – Czech" lang="cs" hreflang="cs" data-title="Mnohostěn" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhoozhinji" title="Nhoozhinji – Shona" lang="sn" hreflang="sn" data-title="Nhoozhinji" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Polyhedron" title="Polyhedron – Welsh" lang="cy" hreflang="cy" data-title="Polyhedron" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Polyeder" title="Polyeder – Danish" lang="da" hreflang="da" data-title="Polyeder" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Polyeder" title="Polyeder – German" lang="de" hreflang="de" data-title="Polyeder" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hulktahukas" title="Hulktahukas – Estonian" lang="et" hreflang="et" data-title="Hulktahukas" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CF%8D%CE%B5%CE%B4%CF%81%CE%BF" title="Πολύεδρο – Greek" lang="el" hreflang="el" data-title="Πολύεδρο" 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/Poliedro" title="Poliedro – Spanish" lang="es" hreflang="es" data-title="Poliedro" 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/Pluredro" title="Pluredro – Esperanto" lang="eo" hreflang="eo" data-title="Pluredro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Poliedro" title="Poliedro – Basque" lang="eu" hreflang="eu" data-title="Poliedro" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%AF%D9%88%D8%AC%D9%87%DB%8C" title="چندوجهی – Persian" lang="fa" hreflang="fa" data-title="چندوجهی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Poly%C3%A8dre" title="Polyèdre – French" lang="fr" hreflang="fr" data-title="Polyèdre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Polaih%C3%A9adr%C3%A1n" title="Polaihéadrán – Irish" lang="ga" hreflang="ga" data-title="Polaihéadrán" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Poliedro" title="Poliedro – Galician" lang="gl" hreflang="gl" data-title="Poliedro" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EB%A9%B4%EC%B2%B4" title="다면체 – Korean" lang="ko" hreflang="ko" data-title="다면체" 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%B2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B6%D5%AB%D5%BD%D5%BF" title="Բազմանիստ – Armenian" lang="hy" hreflang="hy" data-title="Բազմանիստ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B9%E0%A5%81%E0%A4%AB%E0%A4%B2%E0%A4%95" title="बहुफलक – Hindi" lang="hi" hreflang="hi" data-title="बहुफलक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Poliedar" title="Poliedar – Croatian" lang="hr" hreflang="hr" data-title="Poliedar" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Poliedro" title="Poliedro – Ido" lang="io" hreflang="io" data-title="Poliedro" 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/Polihedron" title="Polihedron – Indonesian" lang="id" hreflang="id" data-title="Polihedron" 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 badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://it.wikipedia.org/wiki/Poliedro" title="Poliedro – Italian" lang="it" hreflang="it" data-title="Poliedro" 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%A4%D7%90%D7%95%D7%9F" title="פאון – Hebrew" lang="he" hreflang="he" data-title="פאון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B2%B9%E0%B3%81%E0%B2%AB%E0%B2%B2%E0%B2%95" title="ಬಹುಫಲಕ – Kannada" lang="kn" hreflang="kn" data-title="ಬಹುಫಲಕ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%90%E1%83%9A%E1%83%AC%E1%83%90%E1%83%AE%E1%83%9C%E1%83%90%E1%83%92%E1%83%90" title="მრავალწახნაგა – Georgian" lang="ka" hreflang="ka" data-title="მრავალწახნაგა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BF%D0%B6%D0%B0%D2%9B" title="Көпжақ – Kazakh" lang="kk" hreflang="kk" data-title="Көпжақ" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Poly%C3%A8d" title="Polyèd – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Polyèd" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BF_%D0%BA%D0%B0%D0%BF%D1%82%D0%B0%D0%BB%D0%B4%D1%83%D1%83%D0%BB%D0%B0%D1%80" title="Көп капталдуулар – Kyrgyz" lang="ky" hreflang="ky" data-title="Көп капталдуулар" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Daudzskaldnis" title="Daudzskaldnis – Latvian" lang="lv" hreflang="lv" data-title="Daudzskaldnis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Briaunainis" title="Briaunainis – Lithuanian" lang="lt" hreflang="lt" data-title="Briaunainis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Poliedro" title="Poliedro – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Poliedro" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Poli%C3%A9der" title="Poliéder – Hungarian" lang="hu" hreflang="hu" data-title="Poliéder" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%B5%D0%B4%D0%B0%D1%80" title="Полиедар – Macedonian" lang="mk" hreflang="mk" data-title="Полиедар" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B4%B9%E0%B5%81%E0%B4%AB%E0%B4%B2%E0%B4%95%E0%B4%82" title="ബഹുഫലകം – Malayalam" lang="ml" hreflang="ml" data-title="ബഹുഫലകം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Veelvlak" title="Veelvlak – Dutch" lang="nl" hreflang="nl" data-title="Veelvlak" 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%A4%9A%E9%9D%A2%E4%BD%93" title="多面体 – Japanese" lang="ja" hreflang="ja" data-title="多面体" 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/Polyeder" title="Polyeder – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Polyeder" 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/Polyeder" title="Polyeder – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Polyeder" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Poliedr" title="Poliedr – Piedmontese" lang="pms" hreflang="pms" data-title="Poliedr" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wielo%C5%9Bcian" title="Wielościan – Polish" lang="pl" hreflang="pl" data-title="Wielościan" 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/Poliedro" title="Poliedro – Portuguese" lang="pt" hreflang="pt" data-title="Poliedro" 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/Poliedru" title="Poliedru – Romanian" lang="ro" hreflang="ro" data-title="Poliedru" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Achka_t%27asla" title="Achka t'asla – Quechua" lang="qu" hreflang="qu" data-title="Achka t'asla" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%B3%D1%80%D0%B0%D0%BD%D0%BD%D0%B8%D0%BA" title="Многогранник – Russian" lang="ru" hreflang="ru" data-title="Многогранник" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Polyhedron" title="Polyhedron – Simple English" lang="en-simple" hreflang="en-simple" data-title="Polyhedron" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%BE%D9%88%D9%84%D9%8A_%DA%BE%D9%8A%DA%8A%D8%B1%D8%A7%D9%86" title="پولي ھيڊران – Sindhi" lang="sd" hreflang="sd" data-title="پولي ھيڊران" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mnohosten" title="Mnohosten – Slovak" lang="sk" hreflang="sk" data-title="Mnohosten" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Polieder" title="Polieder – Slovenian" lang="sl" hreflang="sl" data-title="Polieder" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%B1%DB%95%DA%95%D9%88%D9%88" title="فرەڕوو – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فرەڕوو" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%B5%D0%B4%D0%B0%D1%80" title="Полиедар – Serbian" lang="sr" hreflang="sr" data-title="Полиедар" 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/Poliedar" title="Poliedar – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Poliedar" 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/Monitahokas" title="Monitahokas – Finnish" lang="fi" hreflang="fi" data-title="Monitahokas" 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<ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Polyhedra" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q172937" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance 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free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Three-dimensional shape with flat faces, straight edges, and sharp corners</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/Polyhedron_(disambiguation)" class="mw-disambig" title="Polyhedron (disambiguation)">Polyhedron (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Polyhedra" redirects here. Not to be confused with <a href="/wiki/Polyhedra_(software)" title="Polyhedra (software)">Polyhedra (software)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><caption class="infobox-title">Examples of polyhedra</caption><tbody><tr><td colspan="2" class="infobox-image"><style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="multiimageinner" style="width:342px;max-width:342px;border:none"><div class="trow"><div class="tsingle" style="width:178px;max-width:178px"><div style="height:166px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetrahedron.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/176px-Tetrahedron.jpg" decoding="async" width="176" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/264px-Tetrahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/352px-Tetrahedron.jpg 2x" data-file-width="643" data-file-height="607" /></a></span></div><div><a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">Regular tetrahedron</a><br />(<a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>)</div></div><div class="tsingle" style="width:160px;max-width:160px"><div style="height:166px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Small_stellated_dodecahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/158px-Small_stellated_dodecahedron.png" decoding="async" width="158" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/237px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/316px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953" /></a></span></div><div><a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">Small stellated dodecahedron</a><br />(<a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedron</a>)</div></div></div><div class="trow"><div class="tsingle" style="width:169px;max-width:169px"><div style="height:167px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Icosidodecahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/167px-Icosidodecahedron.png" decoding="async" width="167" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/251px-Icosidodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/334px-Icosidodecahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div><div><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a><br />(<a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solid</a>)</div></div><div class="tsingle" style="width:169px;max-width:169px"><div style="height:167px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Great_cubicuboctahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Great_cubicuboctahedron.png/167px-Great_cubicuboctahedron.png" decoding="async" width="167" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Great_cubicuboctahedron.png/251px-Great_cubicuboctahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Great_cubicuboctahedron.png/334px-Great_cubicuboctahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div><div><a href="/wiki/Great_cubicuboctahedron" title="Great cubicuboctahedron">Great cubicuboctahedron</a><br />(<a href="/wiki/Uniform_star-polyhedron" class="mw-redirect" title="Uniform star-polyhedron">Uniform star-polyhedron</a>)</div></div></div><div class="trow"><div class="tsingle" style="width:139px;max-width:139px"><div style="height:136px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Rhombic_triacontahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Rhombic_triacontahedron.png/137px-Rhombic_triacontahedron.png" decoding="async" width="137" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Rhombic_triacontahedron.png/206px-Rhombic_triacontahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Rhombic_triacontahedron.png/274px-Rhombic_triacontahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div><div><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">Rhombic triacontahedron</a><br />(<a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solid</a>)</div></div><div class="tsingle" style="width:199px;max-width:199px"><div style="height:136px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Hexagonal_torus.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hexagonal_torus.svg/197px-Hexagonal_torus.svg.png" decoding="async" width="197" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hexagonal_torus.svg/296px-Hexagonal_torus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hexagonal_torus.svg/394px-Hexagonal_torus.svg.png 2x" data-file-width="830" data-file-height="576" /></a></span></div><div>A <a href="/wiki/Toroidal_polyhedron" title="Toroidal polyhedron">toroidal polyhedron</a></div></div></div></div></div></td></tr><tr><th scope="row" class="infobox-label">Definition</th><td class="infobox-data">A three-dimensional example of the more general <a href="/wiki/Polytope" title="Polytope">polytope</a> in any number of dimensions</td></tr><tr><td colspan="2" class="infobox-navbar"><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 .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist 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li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><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="/w/index.php?title=Template:Polyhedron&action=edit&redlink=1" class="new" title="Template:Polyhedron (page does not exist)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Polyhedron&action=edit&redlink=1" class="new" title="Template talk:Polyhedron (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polyhedron" title="Special:EditPage/Template:Polyhedron"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>polyhedron</b> (<abbr title="plural form">pl.</abbr>: <b>polyhedra</b> or <b>polyhedrons</b>; from <a href="/wiki/Greek_language" title="Greek language">Greek</a> <i> </i><a href="https://en.wiktionary.org/wiki/%CF%80%CE%BF%CE%BB%CF%8D%CF%82" class="extiw" title="wikt:πολύς">πολύ</a><i> <span class="nowrap">(poly-)</span> </i> 'many' and <i> </i><a href="https://en.wiktionary.org/wiki/%E1%BC%95%CE%B4%CF%81%CE%B1" class="extiw" title="wikt:ἕδρα">ἕδρον</a><i> <span class="nowrap">(-hedron)</span> </i> 'base, seat') is a <a href="/wiki/Three-dimensional_figure" class="mw-redirect" title="Three-dimensional figure">three-dimensional figure</a> with flat <a href="/wiki/Polygon" title="Polygon">polygonal</a> <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>, straight <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a> and sharp corners or <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>. </p><p>A <i>convex polyhedron</i> is a polyhedron that bounds a <a href="/wiki/Convex_set" title="Convex set">convex set</a>. Every convex polyhedron can be constructed as the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. <a href="/wiki/Cube" title="Cube">Cubes</a> and <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a> are examples of convex polyhedra. </p><p>A polyhedron is a generalization of a 2-dimensional <a href="/wiki/Polygon" title="Polygon">polygon</a> and a 3-dimensional specialization of a <a href="/wiki/Polytope" title="Polytope">polytope</a>, a more general concept in any number of <a href="/wiki/Dimension" title="Dimension">dimensions</a>. </p> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">Convex polyhedra</a> are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,<sup id="cite_ref-lakatos_1-0" class="reference"><a href="#cite_note-lakatos-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> some more rigorous than others, and there is no universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the <a href="/wiki/Star_polyhedron" title="Star polyhedron">self-crossing polyhedra</a>) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not <a href="/wiki/Manifold" title="Manifold">manifolds</a>). As <a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a> observed, </p> <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>"The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ... the writers failed to define what are the polyhedra".<sup id="cite_ref-sin_2-0" class="reference"><a href="#cite_note-sin-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> (corner points), <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a> (line segments connecting certain pairs of vertices), <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> (two-dimensional <a href="/wiki/Polygon" title="Polygon">polygons</a>), and that it sometimes can be said to have a particular three-dimensional interior <a href="/wiki/Volume" title="Volume">volume</a>. One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its <a href="/wiki/Incidence_geometry" title="Incidence geometry">incidence geometry</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><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> or that it is a solid formed as the union of finitely many convex polyhedra.<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> Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. The faces of such a polyhedron can be defined as the <a href="/wiki/Connected_space" title="Connected space">connected components</a> of the parts of the boundary within each of the planes that cover it, and the edges and vertices as the line segments and points where the faces meet. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, whose faces may not form <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygons</a>, and some of whose edges may belong to more than two faces.<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></li> <li>Definitions based on the idea of a bounding surface rather than a solid are also common.<sup id="cite_ref-cromwell_8-0" class="reference"><a href="#cite_note-cromwell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For instance, <a href="#CITEREFO'Rourke1993">O'Rourke (1993)</a> defines a polyhedron as a union of <a href="/wiki/Convex_polygon" title="Convex polygon">convex polygons</a> (its faces), arranged in space so that the intersection of any two polygons is a shared vertex or edge or the <a href="/wiki/Empty_set" title="Empty set">empty set</a> and so that their union is a <a href="/wiki/Manifold" title="Manifold">manifold</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> If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angles</a> between them. Somewhat more generally, Grünbaum defines an <i>acoptic polyhedron</i> to be a collection of simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each.<sup id="cite_ref-acoptic_10-0" class="reference"><a href="#cite_note-acoptic-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Cromwell's <i><a href="/wiki/Polyhedra_(book)" title="Polyhedra (book)">Polyhedra</a></i> gives a similar definition but without the restriction of at least three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra.<sup id="cite_ref-cromwell_8-1" class="reference"><a href="#cite_note-cromwell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">topological disks</a> (the faces) whose pairwise intersections are required to be points (vertices), topological arcs (edges), or the empty set. However, there exist topological polyhedra (even with all faces triangles) that cannot be realized as acoptic polyhedra.<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></li> <li>One modern approach is based on the theory of <a href="/wiki/Abstract_polyhedron" class="mw-redirect" title="Abstract polyhedron">abstract polyhedra</a>. These can be defined as <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element (in this partial order) when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order (representing the empty set) and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart (that is, between each face and the bottom element, and between the top element and each vertex) have the same structure as the abstract representation of a polygon, then these partially ordered sets carry exactly the same information as a topological polyhedron.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="the 11-cell and 57-cell are valid abstract polytopes but not valid topological polytopes; the latter approach assumes simple balls but the former does not. How can these be said to carry the "same" information? (May 2023)">citation needed</span></a></i>]</sup> However, these requirements are often relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment.<sup id="cite_ref-bursta_12-0" class="reference"><a href="#cite_note-bursta-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> (This means that each edge contains two vertices and belongs to two faces, and that each vertex on a face belongs to two edges of that face.) Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra. A <i>realization</i> of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron.<sup id="cite_ref-grunbaum-same_13-0" class="reference"><a href="#cite_note-grunbaum-same-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Realizations that omit the requirement of face planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered.<sup id="cite_ref-bursta_12-1" class="reference"><a href="#cite_note-bursta-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Unlike the solid-based and surface-based definitions, this works perfectly well for star polyhedra. However, without additional restrictions, this definition allows <a href="/wiki/Degeneracy_(mathematics)" title="Degeneracy (mathematics)">degenerate</a> or unfaithful polyhedra (for instance, by mapping all vertices to a single point) and the question of how to constrain realizations to avoid these degeneracies has not been settled.</li></ul> <p>In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general <a href="/wiki/Polytope" title="Polytope">polytope</a> in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a <a href="/wiki/4-polytope" title="4-polytope">4-polytope</a> has a four-dimensional body and an additional set of three-dimensional "cells". However, some of the literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many <a href="/wiki/Half-space_(geometry)" title="Half-space (geometry)">half-spaces</a>, and a polytope to be a bounded polyhedron.<sup id="cite_ref-polytope-bounded-1_14-0" class="reference"><a href="#cite_note-polytope-bounded-1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-polytope-bounded-2_15-0" class="reference"><a href="#cite_note-polytope-bounded-2-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The remainder of this article considers only three-dimensional polyhedra. </p> <div class="mw-heading mw-heading2"><h2 id="Convex_polyhedra">Convex polyhedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=2" title="Edit section: Convex polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:134px;max-width:134px"><div class="thumbimage" style="height:161px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Hexagonal_pyramid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Hexagonal_pyramid.png/132px-Hexagonal_pyramid.png" decoding="async" width="132" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Hexagonal_pyramid.png/198px-Hexagonal_pyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Hexagonal_pyramid.png/264px-Hexagonal_pyramid.png 2x" data-file-width="665" data-file-height="811" /></a></span></div></div><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:161px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Afgeknotte_driezijdige_piramide.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Afgeknotte_driezijdige_piramide.png/152px-Afgeknotte_driezijdige_piramide.png" decoding="async" width="152" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Afgeknotte_driezijdige_piramide.png/228px-Afgeknotte_driezijdige_piramide.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Afgeknotte_driezijdige_piramide.png/304px-Afgeknotte_driezijdige_piramide.png 2x" data-file-width="424" data-file-height="451" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:137px;max-width:137px"><div class="thumbimage" style="height:139px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Triakisicosahedron.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/135px-Triakisicosahedron.jpg" decoding="async" width="135" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/203px-Triakisicosahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/270px-Triakisicosahedron.jpg 2x" data-file-width="819" data-file-height="849" /></a></span></div></div><div class="tsingle" style="width:151px;max-width:151px"><div class="thumbimage" style="height:139px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Triaugmented_triangular_prism_(symmetric_view).svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Triaugmented_triangular_prism_%28symmetric_view%29.svg/149px-Triaugmented_triangular_prism_%28symmetric_view%29.svg.png" decoding="async" width="149" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Triaugmented_triangular_prism_%28symmetric_view%29.svg/224px-Triaugmented_triangular_prism_%28symmetric_view%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Triaugmented_triangular_prism_%28symmetric_view%29.svg/298px-Triaugmented_triangular_prism_%28symmetric_view%29.svg.png 2x" data-file-width="440" data-file-height="413" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Top left to bottom right: <a href="/wiki/Hexagonal_pyramid" title="Hexagonal pyramid">hexagonal pyramid</a> as the family of <a href="/wiki/Prismatoid" title="Prismatoid">prismatoids</a>, <a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a> as the family of <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a>, <a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a> as the family of <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a>, and <a href="/wiki/Triaugmented_triangular_prism" title="Triaugmented triangular prism">triaugmented triangular prism</a> as the family of both <a href="/wiki/Deltahedron" title="Deltahedron">deltahedrons</a> and <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a>. All of these classes are convex polyhedrons.</div></div></div></div> <p>A <i>convex polyhedron</i> is a polyhedron that forms a <a href="/wiki/Convex_set" title="Convex set">convex set</a> as a solid. That being said, it is a three-dimensional solid whose every line segment connects two of its points lies its <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a> or on its <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a>; none of its faces are <a href="/wiki/Coplanar" class="mw-redirect" title="Coplanar">coplanar</a> (they do not share the same plane) and none of its edges are <a href="/wiki/Colinearity" class="mw-redirect" title="Colinearity">collinear</a> (they are not segments of the same line).<sup id="cite_ref-by_16-0" class="reference"><a href="#cite_note-by-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-litchenberg_17-0" class="reference"><a href="#cite_note-litchenberg-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> A convex polyhedron can also be defined as a bounded intersection of finitely many <a href="/wiki/Half-space_(geometry)" title="Half-space (geometry)">half-spaces</a>, or as the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of finitely many points, in either case, restricted to intersections or hulls that have nonzero volume.<sup id="cite_ref-polytope-bounded-1_14-1" class="reference"><a href="#cite_note-polytope-bounded-1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-polytope-bounded-2_15-1" class="reference"><a href="#cite_note-polytope-bounded-2-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Important classes of convex polyhedra include the family of <a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a>, the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>, the <a href="/wiki/Archimedean_polyhedron" class="mw-redirect" title="Archimedean polyhedron">Archimedean solids</a> and their duals the <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a>, and the regular polygonal faces polyhedron. The prismatoids are the polyhedron whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles.<sup id="cite_ref-prismatoid_18-0" class="reference"><a href="#cite_note-prismatoid-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Examples of prismatoids are <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a>, <a href="/wiki/Wedge_(geometry)" title="Wedge (geometry)">wedges</a>, <a href="/wiki/Parallelipiped" class="mw-redirect" title="Parallelipiped">parallelipipeds</a>, <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a>, <a href="/wiki/Antiprism" title="Antiprism">antiprisms</a>, <a href="/wiki/Cupola" title="Cupola">cupolas</a>, and <a href="/wiki/Frustum" title="Frustum">frustums</a>. The Platonic solids are the five ancientness polyhedrons—<a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">tetrahedron</a>, <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">octahedron</a>, <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a>, <a href="/wiki/Cube" title="Cube">cube</a>, and <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a>—classified by <a href="/wiki/Plato" title="Plato">Plato</a> in his <a href="/wiki/Timaeus_(dialogue)" title="Timaeus (dialogue)"><i>Timaeus</i></a> whose connecting four <a href="/wiki/Classical_element" title="Classical element">classical elements</a> of nature.<sup id="cite_ref-FOOTNOTECromwell199751&ndash;52_19-0" class="reference"><a href="#cite_note-FOOTNOTECromwell199751&ndash;52-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The Archimedean solids are the class of thirteen polyhedrons whose faces are all regular polygons and whose vertices are symmetric to each other;<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> their dual polyhedrons are <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a>.<sup id="cite_ref-diudea_22-0" class="reference"><a href="#cite_note-diudea-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The class of regular polygonal faces polyhedron are the <a href="/wiki/Deltahedron" title="Deltahedron">deltahedron</a> (whose faces are all <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a> and <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a> (whose faces are arbitrary regular polygons).<sup id="cite_ref-cundy_23-0" class="reference"><a href="#cite_note-cundy-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-berman_24-0" class="reference"><a href="#cite_note-berman-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The convex polyhedron can be categorized into <a href="/wiki/Elementary_polyhedron" class="mw-redirect" title="Elementary polyhedron">elementary polyhedron</a> or composite polyhedron. An elementary polyhedron is a convex regular-faced polyhedron that cannot be produced into two or more polyhedrons by slicing it with a plane.<sup id="cite_ref-FOOTNOTEHartshorne2000[httpsbooksgooglecombooksidEJCSL9S6la0CpgPA464_464]_25-0" class="reference"><a href="#cite_note-FOOTNOTEHartshorne2000[httpsbooksgooglecombooksidEJCSL9S6la0CpgPA464_464]-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Quite opposite to a composite polyhedron, it can be alternatively defined as a polyhedron that can be constructed by attaching more elementary polyhedrons. For example, <a href="/wiki/Triaugmented_triangular_prism" title="Triaugmented triangular prism">triaugmented triangular prism</a> is a composite polyhedron since it can be constructed by attaching three <a href="/wiki/Equilateral_square_pyramid" class="mw-redirect" title="Equilateral square pyramid">equilateral square pyramids</a> onto the square faces of a <a href="/wiki/Triangular_prism" title="Triangular prism">triangular prism</a>; the square pyramids and the triangular prism are elementary.<sup id="cite_ref-timofeenko-2010_26-0" class="reference"><a href="#cite_note-timofeenko-2010-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Midsphere.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Midsphere.png/170px-Midsphere.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Midsphere.png/255px-Midsphere.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Midsphere.png/340px-Midsphere.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>A canonical polyhedron</figcaption></figure> <p>A <a href="/wiki/Midsphere" title="Midsphere">midsphere</a> of a convex polyhedron is a sphere tangent to every edge of a polyhedron, an intermediate sphere in radius between the <a href="/wiki/Insphere" class="mw-redirect" title="Insphere">insphere</a> and <a href="/wiki/Circumsphere" class="mw-redirect" title="Circumsphere">circumsphere</a>, for polyhedra for which all three of these spheres exist. Every convex polyhedron is combinatorially equivalent to a <i>canonical polyhedron</i>, a polyhedron that has a midsphere whose center coincides with the <a href="/wiki/Centroid" title="Centroid">centroid</a> of the polyhedron. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by the combinatorial structure of the given polyhedron.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Some polyhedrons do not have the property of convexity, and they are called <i>non-convex polyhedrons</i>. Such polyhedrons are <a href="/wiki/Star_polyhedron" title="Star polyhedron">star polyhedrons</a> and <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedrons</a>, which constructed by either <a href="/wiki/Stellation" title="Stellation">stellation</a> (process of extending the faces—within their planes—so that they meet) or <a href="/wiki/Faceting" title="Faceting">faceting</a> (whose process of removing parts of a polyhedron to create new faces—or facets—without creating any new vertices).<sup id="cite_ref-bridge_28-0" class="reference"><a href="#cite_note-bridge-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a <i><a href="/wiki/Face_(geometry)" title="Face (geometry)">face</a></i>.<sup id="cite_ref-bridge_28-1" class="reference"><a href="#cite_note-bridge-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron. </p> <div class="mw-heading mw-heading2"><h2 id="Characteristics">Characteristics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=3" title="Edit section: Characteristics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Number_of_faces">Number of faces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=4" title="Edit section: Number of faces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> is a polyhedron with four faces, a <a href="/wiki/Pentahedron" title="Pentahedron">pentahedron</a> is a polyhedron with five faces, a <a href="/wiki/Hexahedron" title="Hexahedron">hexahedron</a> is a polyhedron with six faces, etc.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> For a complete list of the Greek numeral prefixes see <a href="/wiki/Numeral_prefix#Table_of_number_prefixes_in_English" title="Numeral prefix">Numeral prefix § Table of number prefixes in English</a>, in the column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (8-sided polyhedra), dodecahedra (12-sided polyhedra), and icosahedra (20-sided polyhedra) are sometimes used without additional qualification to refer to the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>, and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Topological_classification">Topological classification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=5" title="Edit section: Topological classification"><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/Toroidal_polyhedron" title="Toroidal polyhedron">Toroidal polyhedron</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tetrahemihexahedron_rotation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Tetrahemihexahedron_rotation.gif/220px-Tetrahemihexahedron_rotation.gif" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Tetrahemihexahedron_rotation.gif/330px-Tetrahemihexahedron_rotation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Tetrahemihexahedron_rotation.gif/440px-Tetrahemihexahedron_rotation.gif 2x" data-file-width="733" data-file-height="698" /></a><figcaption>The <a href="/wiki/Tetrahemihexahedron" title="Tetrahemihexahedron">tetrahemihexahedron</a>, a non-orientable self-intersecting polyhedron with four triangular faces (red) and three square faces (yellow). As with a <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> or <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a>, a continuous path along the surface of this polyhedron can reach the point on the opposite side of the surface from its starting point, making it impossible to separate the surface into an inside and an outside. (Topologically, this polyhedron is a <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>.)</figcaption></figure> <p>Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a <a href="/wiki/Convex_polytope" title="Convex polytope">convex polyhedron</a> paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are <a href="/wiki/Orientability" title="Orientability">orientable</a>. The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as the <a href="/wiki/Tetrahemihexahedron" title="Tetrahemihexahedron">tetrahemihexahedron</a>, it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours. In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it is orientable or non-orientable by considering a topological <a href="/wiki/Cell_complex" class="mw-redirect" title="Cell complex">cell complex</a> with the same incidences between its vertices, edges, and faces.<sup id="cite_ref-ringel_32-0" class="reference"><a href="#cite_note-ringel-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>A more subtle distinction between polyhedron surfaces is given by their <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a>, which combines the numbers of vertices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, edges <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, and faces <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> of a polyhedron into a single number <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 \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"></span> defined by the formula </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 \chi =V-E+F.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <mi>V</mi> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi =V-E+F.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e322cd3b5333d260c4ace1e89991893921cc370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.153ex; height:2.509ex;" alt="{\displaystyle \chi =V-E+F.\ }"></span></dd></dl> <p>The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For a convex polyhedron, or more generally any simply connected polyhedron with surface a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of <a href="/wiki/Toroid" title="Toroid">toroidal</a> holes, handles or <a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">cross-caps</a> in the surface and will be less than 2.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed <a href="/wiki/Toroid" title="Toroid">toroid</a> and the <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> both have <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 \chi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2a52c29ba6859766c02e88299b3114d010e3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.716ex; height:2.509ex;" alt="{\displaystyle \chi =0}"></span>, with the first being orientable and the other not.<sup id="cite_ref-ringel_32-1" class="reference"><a href="#cite_note-ringel-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a <a href="/wiki/Manifold" title="Manifold">manifold</a>. This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along a shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). For polyhedra defined in these ways, the <a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">classification of manifolds</a> implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere.<sup id="cite_ref-ringel_32-2" class="reference"><a href="#cite_note-ringel-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Toroidal_polyhedron" title="Toroidal polyhedron">toroidal polyhedron</a> is a polyhedron whose <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> is less than or equal to 0, or equivalently whose <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> is 1 or greater. Topologically, the surfaces of such polyhedra are <a href="/wiki/Torus" title="Torus">torus</a> surfaces having one or more holes through the middle.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Duality">Duality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=6" title="Edit section: Duality"><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/Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dual_Cube-Octahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/180px-Dual_Cube-Octahedron.svg.png" decoding="async" width="180" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/270px-Dual_Cube-Octahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/360px-Dual_Cube-Octahedron.svg.png 2x" data-file-width="744" data-file-height="749" /></a><figcaption>The octahedron is dual to the cube</figcaption></figure> <p>For every convex polyhedron, there exists a dual polyhedron having </p> <ul><li>faces in place of the original's vertices and vice versa, and</li> <li>the same number of edges.</li></ul> <p>The dual of a convex polyhedron can be obtained by the process of <a href="/wiki/Dual_polyhedron#Polar_reciprocation" title="Dual polyhedron">polar reciprocation</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>Abstract polyhedra also have duals, obtained by reversing the <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> defining the polyhedron to obtain its <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dual or opposite order</a>.<sup id="cite_ref-grunbaum-same_13-1" class="reference"><a href="#cite_note-grunbaum-same-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> These have the same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition.<sup id="cite_ref-acoptic_10-1" class="reference"><a href="#cite_note-acoptic-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Vertex_figures">Vertex figures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=7" title="Edit section: Vertex figures"><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/Vertex_figure" title="Vertex figure">Vertex figure</a></div> <p>For every vertex one can define a <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a>, which describes the local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a vertex.<sup id="cite_ref-cromwell_8-2" class="reference"><a href="#cite_note-cromwell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex,<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in <a href="/wiki/Convex_position" title="Convex position">convex position</a>, this slice can be chosen as any plane separating the vertex from the other vertices.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center;<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it is common instead to slice the polyhedron by a small sphere centered at the vertex.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes. </p> <div class="mw-heading mw-heading3"><h3 id="Surface_area_and_distances">Surface area and distances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=8" title="Edit section: Surface area and distances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Surface_area" title="Surface area">surface area</a> of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which the area of a face is well-defined. The <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By <a href="/wiki/Alexandrov%27s_uniqueness_theorem" title="Alexandrov's uniqueness theorem">Alexandrov's uniqueness theorem</a>, every convex polyhedron is uniquely determined by the <a href="/wiki/Metric_space" title="Metric space">metric space</a> of geodesic distances on its surface. However, non-convex polyhedra can have the same surface distances as each other, or the same as certain convex polyhedra.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Volume">Volume</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=9" title="Edit section: Volume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polyhedral solids have an associated quantity called <a href="/wiki/Volume" title="Volume">volume</a> that measures how much space they occupy. Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepipeds</a> can easily be expressed in terms of their edge lengths or other coordinates. (See <a href="/wiki/Volume#Formulas" title="Volume">Volume § Volume formulas</a> for a list that includes many of these formulas.) </p><p>Volumes of more complicated polyhedra may not have simple formulas. Volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by <a href="/wiki/Point-set_triangulation" title="Point-set triangulation">triangulation</a>). For example, the <a href="/wiki/Platonic_solid#Radii,_area,_and_volume" title="Platonic solid">volume of a regular polyhedron</a> can be computed by dividing it into congruent <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a>, with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex. </p><p>In general, it can be derived from the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a> that the volume of a polyhedral solid is given by <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 {\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>area</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57163b6827e62c51d639c81c07e9ba7230bf8f22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.394ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,}"></span> where the sum is over faces <span class="texhtml mvar" style="font-style:italic;">F</span> of the polyhedron, <span class="texhtml"><i>Q</i><sub><i>F</i></sub></span> is an arbitrary point on face <span class="texhtml mvar" style="font-style:italic;">F</span>, <span class="texhtml"><i>N</i><sub><i>F</i></sub></span> is the <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> perpendicular to <span class="texhtml mvar" style="font-style:italic;">F</span> pointing outside the solid, and the multiplication dot is the <a href="/wiki/Dot_product" title="Dot product">dot product</a>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> to determine the volume in these cases.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Dehn_invariant">Dehn invariant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=10" title="Edit section: Dehn invariant"><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/Dehn_invariant" title="Dehn invariant">Dehn invariant</a></div> <p>In two dimensions, the <a href="/wiki/Bolyai%E2%80%93Gerwien_theorem" class="mw-redirect" title="Bolyai–Gerwien theorem">Bolyai–Gerwien theorem</a> asserts that any polygon may be transformed into any other polygon of the same area by <a href="/wiki/Dissection_problem" title="Dissection problem">cutting it up into finitely many polygonal pieces and rearranging them</a>. The analogous question for polyhedra was the subject of <a href="/wiki/Hilbert%27s_third_problem" title="Hilbert's third problem">Hilbert's third problem</a>. <a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a> solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the <a href="/wiki/Dehn_invariant" title="Dehn invariant">Dehn invariant</a>, such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The Dehn invariant is not a number, but a <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vector</a> in an infinite-dimensional vector space, determined from the lengths and <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angles</a> of a polyhedron's edges.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p><p>Another of Hilbert's problems, <a href="/wiki/Hilbert%27s_18th_problem" class="mw-redirect" title="Hilbert's 18th problem">Hilbert's 18th problem</a>, concerns (among other things) polyhedra that <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">tile space</a>. Every such polyhedron must have Dehn invariant zero.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> The Dehn invariant has also been connected to <a href="/wiki/Flexible_polyhedron" title="Flexible polyhedron">flexible polyhedra</a> by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Symmetries">Symmetries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=11" title="Edit section: Symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/280px--Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="280" height="158" data-durationhint="14" data-mwtitle="Revolução_de_poliedros_03.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Revolu%C3%A7%C3%A3o_de_poliedros_03.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm" type="video/webm; codecs="vp8, vorbis"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3c/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm/Revolu%C3%A7%C3%A3o_de_poliedros_03.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>Some polyhedra rotating around a symmetrical axis (at <a href="https://pt.wikipedia.org/wiki/Matemateca_IME-USP" class="extiw" title="pt:Matemateca IME-USP">Matemateca IME-USP</a>)</figcaption></figure> <p>Many of the most studied polyhedra are highly <a href="/wiki/Symmetry" title="Symmetry">symmetrical</a>, that is, their appearance is unchanged by some reflection or rotation of space. Each such symmetry may change the location of a given vertex, face, or edge, but the set of all vertices (likewise faces, edges) is unchanged. The collection of symmetries of a polyhedron is called its <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a>. </p><p>All the elements that can be superimposed on each other by symmetries are said to form a <a href="/wiki/Symmetry_orbit#Orbits_and_stabilizers" class="mw-redirect" title="Symmetry orbit">symmetry orbit</a>. For example, all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be transitive on that orbit. For example, a cube is face-transitive, while a <a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a> has two symmetry orbits of faces. </p><p>The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a>, the most symmetrical geometry is often implied.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2017)">citation needed</span></a></i>]</sup> </p><p>There are several types of highly symmetric polyhedron, classified by which kind of element – faces, edges, or vertices – belong to a single symmetry orbit: </p> <ul><li><a href="/wiki/Regular_polyhedron" title="Regular polyhedron">Regular</a>: vertex-transitive, edge-transitive and face-transitive. (This implies that every face is the same <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>; it also implies that every vertex is regular.)</li> <li><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">Quasi-regular</a>: vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quasi-regular dual is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive.</li> <li><a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">Semi-regular</a>: vertex-transitive but not edge-transitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class.) These polyhedra include the semiregular <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a> and <a href="/wiki/Antiprism" title="Antiprism">antiprisms</a>. A semi-regular dual is face-transitive but not vertex-transitive, and every vertex is regular.</li> <li><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">Uniform</a>: vertex-transitive and every face is a regular polygon, i.e., it is regular, quasi-regular or semi-regular. A uniform dual is face-transitive and has regular vertices, but is not necessarily vertex-transitive.</li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a>: vertex-transitive.</li> <li><a href="/wiki/Isotoxal" class="mw-redirect" title="Isotoxal">Isotoxal</a>: edge-transitive.</li> <li><a href="/wiki/Isohedral" class="mw-redirect" title="Isohedral">Isohedral</a>: face-transitive.</li> <li><a href="/wiki/Noble_polyhedron" title="Noble polyhedron">Noble</a>: face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble.</li></ul> <p>Some classes of polyhedra have only a single main axis of symmetry. These include the <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a>, <a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a>, <a href="/wiki/Trapezohedra" class="mw-redirect" title="Trapezohedra">trapezohedra</a>, <a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupolae</a>, as well as the semiregular prisms and antiprisms. </p> <div class="mw-heading mw-heading3"><h3 id="Regular_polyhedra">Regular polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=12" title="Edit section: Regular polyhedra"><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/Regular_polyhedron" title="Regular polyhedron">Regular polyhedron</a></div> <p>Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra: five convex and four star polyhedra. </p><p>The five convex examples have been known since antiquity and are called the <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a>. These are the triangular pyramid or <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, <a href="/wiki/Cube" title="Cube">cube</a>, <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a> and <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a>: </p> <table class="wikitable" style="margin-left:1em"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Tetrahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Tetrahedron.svg/50px-Tetrahedron.svg.png" decoding="async" width="50" height="47" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Tetrahedron.svg/75px-Tetrahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Tetrahedron.svg/100px-Tetrahedron.svg.png 2x" data-file-width="570" data-file-height="540" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hexahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexahedron.svg/50px-Hexahedron.svg.png" decoding="async" width="50" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexahedron.svg/75px-Hexahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexahedron.svg/100px-Hexahedron.svg.png 2x" data-file-width="540" data-file-height="600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Octahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/50px-Octahedron.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/75px-Octahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/100px-Octahedron.svg.png 2x" data-file-width="840" data-file-height="832" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Dodecahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/50px-Dodecahedron.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/75px-Dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/100px-Dodecahedron.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/50px-Icosahedron.svg.png" decoding="async" width="50" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/75px-Icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/100px-Icosahedron.svg.png 2x" data-file-width="512" data-file-height="492" /></a></span> </td></tr></tbody></table> <p>There are also four regular star polyhedra, known as the <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedra" class="mw-redirect" title="Kepler–Poinsot polyhedra">Kepler–Poinsot polyhedra</a> after their discoverers. </p><p>The dual of a regular polyhedron is also regular. </p> <div class="mw-heading mw-heading3"><h3 id="Uniform_polyhedra_and_their_duals">Uniform polyhedra and their duals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=13" title="Edit section: Uniform polyhedra and their duals"><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_polyhedron" title="Uniform polyhedron">Uniform polyhedron</a></div> <p>Uniform polyhedra are <a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a> and every face is a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>. They may be subdivided into the <a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular</a>, <a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">quasi-regular</a>, or <a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semi-regular</a>, and may be convex or starry. </p><p>The duals of the uniform polyhedra have irregular faces but are <a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>, and every <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a>. </p><p>The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are <a href="/wiki/Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">convex</a> or not. </p> <table class="wikitable"> <tbody><tr> <th> </th> <th>Convex uniform </th> <th>Convex uniform dual </th> <th>Star uniform </th> <th>Star uniform dual </th></tr> <tr> <th><a href="/wiki/Regular_polyhedron" title="Regular polyhedron">Regular</a> </th> <td style="text-align:center;" colspan="2"><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> </td> <td style="text-align:center;" colspan="2"><a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedra</a> </td></tr> <tr style="text-align:center;"> <th><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">Quasiregular</a> </th> <td rowspan="2"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a> </td> <td rowspan="2"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a> </td> <td colspan="2" rowspan="2"><a href="/wiki/Uniform_star_polyhedron" title="Uniform star polyhedron">Uniform star polyhedron</a> </td></tr> <tr style="text-align:center;"> <th rowspan="3"><a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">Semiregular</a> </th></tr> <tr style="text-align:center;"> <td><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">Prisms</a> </td> <td><a href="/wiki/Bipyramid" title="Bipyramid">Bipyramids</a> </td> <td><a href="/wiki/Star_prism" class="mw-redirect" title="Star prism">Star prisms</a> </td> <td><a href="/wiki/Star_bipyramid" class="mw-redirect" title="Star bipyramid">Star bipyramids</a> </td></tr> <tr style="text-align:center;"> <td><a href="/wiki/Antiprism" title="Antiprism">Antiprisms</a> </td> <td><a href="/wiki/Trapezohedron" title="Trapezohedron">Trapezohedra</a> </td> <td><a href="/wiki/Star_antiprism" class="mw-redirect" title="Star antiprism">Star antiprisms</a> </td> <td><a href="/wiki/Star_trapezohedra" class="mw-redirect" title="Star trapezohedra">Star trapezohedra</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Isohedra">Isohedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=14" title="Edit section: Isohedra"><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/Isohedron" class="mw-redirect" title="Isohedron">Isohedron</a></div> <p>An <b>isohedron</b> is a polyhedron with symmetries acting transitively on its faces. Their topology can be represented by a <a href="/wiki/Face_configuration" class="mw-redirect" title="Face configuration">face configuration</a>. All 5 <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a> and 13 <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a> are isohedra, as well as the infinite families of <a href="/wiki/Trapezohedra" class="mw-redirect" title="Trapezohedra">trapezohedra</a> and <a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a>. Some definitions of isohedra allow geometric variations including concave and self-intersecting forms. </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry_groups">Symmetry groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=15" title="Edit section: Symmetry groups"><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:Icosahedral_reflection_domains.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/220px-Icosahedral_reflection_domains.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/330px-Icosahedral_reflection_domains.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/440px-Icosahedral_reflection_domains.png 2x" data-file-width="811" data-file-height="812" /></a><figcaption>Full <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a> divides the sphere into 120 triangular domains.</figcaption></figure> <p>Many of the symmetries or <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a> are named after polyhedra having the associated symmetry. These include: </p> <ul><li><b>T</b> – <b>chiral <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">tetrahedral symmetry</a></b>; the rotation group for a regular <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>; order 12.</li> <li><b>T<sub>d</sub></b> – <b>full <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">tetrahedral symmetry</a></b>; the symmetry group for a regular <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>; order 24.</li> <li><b>T<sub>h</sub></b> – <b><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">pyritohedral symmetry</a></b>; the symmetry of a <a href="/wiki/Pyritohedron" class="mw-redirect" title="Pyritohedron">pyritohedron</a>; order 24.</li> <li><b>O</b> – <b>chiral <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a></b>;the rotation group of the <a href="/wiki/Cube_(geometry)" class="mw-redirect" title="Cube (geometry)">cube</a> and <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>; order 24.</li> <li><b>O<sub>h</sub></b> – <b>full <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a></b>; the symmetry group of the <a href="/wiki/Cube_(geometry)" class="mw-redirect" title="Cube (geometry)">cube</a> and <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>; order 48.</li> <li><b>I</b> – <b>chiral <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a></b>; the rotation group of the <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> and the <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a>; order 60.</li> <li><b>I<sub>h</sub></b> – <b>full <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a></b>; the symmetry group of the <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> and the <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a>; order 120.</li> <li><b>C<sub>nv</sub></b> – <a href="/wiki/Cyclic_symmetries" class="mw-redirect" title="Cyclic symmetries"><i>n</i>-fold pyramidal symmetry</a></li> <li><b>D<sub>nh</sub></b> – <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry"><i>n</i>-fold prismatic symmetry</a></li> <li><b>D<sub>nv</sub></b> – <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry"><i>n</i>-fold antiprismatic symmetry</a>.</li></ul> <p>Those with <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a> symmetry do not have <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> and hence have two enantiomorphous forms which are reflections of each other. Examples include the <a href="/wiki/Snub_cuboctahedron" class="mw-redirect" title="Snub cuboctahedron">snub cuboctahedron</a> and <a href="/wiki/Snub_icosidodecahedron" class="mw-redirect" title="Snub icosidodecahedron">snub icosidodecahedron</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_important_families_of_polyhedra">Other important families of polyhedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=16" title="Edit section: Other important families of polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Zonohedra">Zonohedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=17" title="Edit section: Zonohedra"><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/Zonohedron" title="Zonohedron">Zonohedron</a></div> <p>A zonohedron is a convex polyhedron in which every face is a <a href="/wiki/Polygon" title="Polygon">polygon</a> that is symmetric under <a href="/wiki/Rotation" title="Rotation">rotations</a> through 180°. Zonohedra can also be characterized as the <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sums</a> of line segments, and include several important space-filling polyhedra.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Space-filling_polyhedra">Space-filling polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=18" title="Edit section: Space-filling polyhedra"><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/Honeycomb_(geometry)" title="Honeycomb (geometry)">Honeycomb (geometry)</a></div> <p>A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a <a href="#Dehn_invariant">Dehn invariant</a> equal to zero. Some honeycombs involve more than one kind of polyhedron. </p> <div class="mw-heading mw-heading3"><h3 id="Lattice_polyhedra">Lattice polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=19" title="Edit section: Lattice polyhedra"><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/Ehrhart_polynomial" title="Ehrhart polynomial">Ehrhart polynomial</a></div> <p>A convex polyhedron in which all vertices have integer coordinates is called a <a href="/wiki/Convex_lattice_polytope" class="mw-redirect" title="Convex lattice polytope">lattice polyhedron</a> or <a href="/wiki/Integral_polyhedron" class="mw-redirect" title="Integral polyhedron">integral polyhedron</a>. The Ehrhart polynomial of a lattice polyhedron counts how many points with <a href="/wiki/Integer" title="Integer">integer</a> coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> There is a far-reaching equivalence between lattice polyhedra and certain <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> called <a href="/wiki/Toric_variety" title="Toric variety">toric varieties</a>.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> This was used by Stanley to prove the <a href="/wiki/Dehn%E2%80%93Sommerville_equations" title="Dehn–Sommerville equations">Dehn–Sommerville equations</a> for <a href="/wiki/Simplicial_polytope" title="Simplicial polytope">simplicial polytopes</a>.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Flexible_polyhedra">Flexible polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=20" title="Edit section: Flexible polyhedra"><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/Flexible_polyhedron" title="Flexible polyhedron">Flexible polyhedron</a></div> <p>It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By <a href="/wiki/Cauchy%27s_theorem_(geometry)" title="Cauchy's theorem (geometry)">Cauchy's rigidity theorem</a>, flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result is known as the bellows theorem.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Compounds">Compounds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=21" title="Edit section: Compounds"><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/Polyhedral_compound" class="mw-redirect" title="Polyhedral compound">Polyhedral compound</a></div> <p>A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the <a href="/wiki/List_of_Wenninger_polyhedron_models" title="List of Wenninger polyhedron models">list of Wenninger polyhedron models</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Orthogonal_polyhedra">Orthogonal polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=22" title="Edit section: Orthogonal polyhedra"><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:Soma_cube_figures.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Soma_cube_figures.svg/220px-Soma_cube_figures.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Soma_cube_figures.svg/330px-Soma_cube_figures.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Soma_cube_figures.svg/440px-Soma_cube_figures.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Some orthogonal polyhedra made of <a href="/wiki/Soma_cube" title="Soma cube">Soma cube</a> pieces, themselves <a href="/wiki/Polycube" title="Polycube">polycubes</a></figcaption></figure> <p>An orthogonal polyhedron is one all of whose edges are parallel to axes of a Cartesian coordinate system. This implies that all faces meet at <a href="/wiki/Right_angle" title="Right angle">right angles</a>, but this condition is weaker: <a href="/wiki/Jessen%27s_icosahedron" title="Jessen's icosahedron">Jessen's icosahedron</a> has faces meeting at right angles, but does not have axis-parallel edges. </p><p>Aside from the <a href="/wiki/Rectangular_cuboid" title="Rectangular cuboid">rectangular cuboids</a>, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as <a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">rectilinear polygons</a>. Orthogonal polyhedra are used in <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a <a href="/wiki/Polygonal_net" class="mw-redirect" title="Polygonal net">polygonal net</a>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Polycube" title="Polycube">Polycubes</a> are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar <a href="/wiki/Polyomino" title="Polyomino">polyominoes</a>.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Embedded_regular_maps_with_planar_faces">Embedded regular maps with planar faces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=23" title="Edit section: Embedded regular maps with planar faces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="height:187px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Heawood_map_on_a_hexagon.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Heawood_map_on_a_hexagon.svg/180px-Heawood_map_on_a_hexagon.svg.png" decoding="async" width="180" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Heawood_map_on_a_hexagon.svg/270px-Heawood_map_on_a_hexagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Heawood_map_on_a_hexagon.svg/360px-Heawood_map_on_a_hexagon.svg.png 2x" data-file-width="2804" data-file-height="2914" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Heawood_graph" title="Heawood graph">Heawood map</a>, a regular map on a topological torus formed by gluing opposite edges of the outer hexagon</div></div><div class="tsingle" style="width:206px;max-width:206px"><div class="thumbimage" style="height:187px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Szilassi_polyhedron.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Szilassi_polyhedron.svg/204px-Szilassi_polyhedron.svg.png" decoding="async" width="204" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Szilassi_polyhedron.svg/306px-Szilassi_polyhedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Szilassi_polyhedron.svg/408px-Szilassi_polyhedron.svg.png 2x" data-file-width="337" data-file-height="309" /></a></span></div><div class="thumbcaption">The Szilassi polyhedron, a polyhedron realizing the Heawood map</div></div></div></div></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/Regular_map_(graph_theory)" title="Regular map (graph theory)">Regular map (graph theory)</a></div> <p><a href="/wiki/Regular_map_(graph_theory)" title="Regular map (graph theory)">Regular maps</a> are flag transitive abstract 2-manifolds and they have been studied already in the nineteenth century. In some cases they have geometric realizations. An example is the <a href="/wiki/Szilassi_polyhedron" title="Szilassi polyhedron">Szilassi polyhedron</a>, a toroidal polyhedron that realizes the <a href="/wiki/Heawood_graph" title="Heawood graph">Heawood map</a>. In this case, the polyhedron is much less symmetric than the underlying map, but in some cases it is possible for self-crossing polyhedra to realize some or all of the symmetries of a regular map. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Generalisations">Generalisations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=24" title="Edit section: Generalisations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra. </p> <div class="mw-heading mw-heading3"><h3 id="Apeirohedra">Apeirohedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=25" title="Edit section: Apeirohedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A classical polyhedral surface has a finite number of faces, joined in pairs along edges. The <a href="/wiki/Apeirohedron" class="mw-redirect" title="Apeirohedron">apeirohedra</a> form a related class of objects with infinitely many faces. Examples of apeirohedra include: </p> <ul><li>tilings or <a href="/wiki/Tessellation" title="Tessellation">tessellations</a> of the plane, and</li> <li>sponge-like structures called <a href="/wiki/Skew_apeirohedron" title="Skew apeirohedron">infinite skew polyhedra</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Complex_polyhedra">Complex polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=26" title="Edit section: Complex polyhedra"><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/Complex_polytope" title="Complex polytope">Complex polytope</a></div> <p>There are objects called complex polyhedra, for which the underlying space is a <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> rather than real Euclidean space. Precise definitions exist only for the regular complex polyhedra, whose symmetry groups are <a href="/wiki/Complex_reflection_group" title="Complex reflection group">complex reflection groups</a>. The complex polyhedra are mathematically more closely related to <a href="/wiki/Configuration_(polytope)" title="Configuration (polytope)">configurations</a> than to real polyhedra.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Curved_polyhedra">Curved polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=27" title="Edit section: Curved polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some fields of study allow polyhedra to have curved faces and edges. Curved faces can allow <a href="/wiki/Digon" title="Digon">digonal</a> faces to exist with a positive area. </p> <div class="mw-heading mw-heading4"><h4 id="Spherical_polyhedra">Spherical polyhedra</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=28" title="Edit section: Spherical polyhedra"><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/Spherical_polyhedron" title="Spherical polyhedron">Spherical polyhedron</a></div> <p>When the surface of a sphere is divided by finitely many <a href="/wiki/Great_arc" class="mw-redirect" title="Great arc">great arcs</a> (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the <a href="/wiki/Hosohedron" title="Hosohedron">hosohedra</a>) have no flat-faced analogue.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Curved_spacefilling_polyhedra">Curved spacefilling polyhedra</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=29" title="Edit section: Curved spacefilling polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If faces are allowed to be concave as well as convex, adjacent faces may be made to meet together with no gap. Some of these curved polyhedra can pack together to fill space. Two important types are: </p> <ul><li>Bubbles in froths and foams, such as <a href="/wiki/Weaire%E2%80%93Phelan_structure" title="Weaire–Phelan structure">Weaire-Phelan bubbles</a>.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup></li> <li>Forms used in architecture.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ideal_polyhedra">Ideal polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=30" title="Edit section: Ideal polyhedra"><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/Ideal_polyhedron" title="Ideal polyhedron">Ideal polyhedron</a></div> <p>Convex polyhedra can be defined in three-dimensional <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a> in the same way as in Euclidean space, as the <a href="/wiki/Convex_hull" title="Convex hull">convex hulls</a> of finite sets of points. However, in hyperbolic space, it is also possible to consider <a href="/wiki/Ideal_point" title="Ideal point">ideal points</a> as well as the points that lie within the space. An <a href="/wiki/Ideal_polyhedron" title="Ideal polyhedron">ideal polyhedron</a> is the convex hull of a finite set of ideal points. Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space. </p> <div class="mw-heading mw-heading3"><h3 id="Skeletons_and_polyhedra_as_graphs">Skeletons and polyhedra as graphs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=31" title="Edit section: Skeletons and polyhedra as graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By forgetting the face structure, any polyhedron gives rise to a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>, called its <a href="/wiki/N-skeleton" title="N-skeleton">skeleton</a>, with corresponding vertices and edges. Such figures have a long history: <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> devised frame models of the regular solids, which he drew for <a href="/wiki/Pacioli" class="mw-redirect" title="Pacioli">Pacioli</a>'s book <i>Divina Proportione</i>, and similar <a href="/wiki/Wire-frame_model" title="Wire-frame model">wire-frame</a> polyhedra appear in <a href="/wiki/M.C._Escher" class="mw-redirect" title="M.C. Escher">M.C. Escher</a>'s print <a href="/wiki/Stars_(M._C._Escher)" title="Stars (M. C. Escher)"><i>Stars</i></a>.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> One highlight of this approach is <a href="/wiki/Steinitz%27s_theorem" title="Steinitz's theorem">Steinitz's theorem</a>, which gives a purely graph-theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a <a href="/wiki/Vertex_connectivity" class="mw-redirect" title="Vertex connectivity">3-connected</a> <a href="/wiki/Planar_graph" title="Planar graph">planar graph</a>, and every 3-connected planar graph is the skeleton of some convex polyhedron. </p><p>An early idea of <a href="#Abstract_polyhedra">abstract polyhedra</a> was developed in <a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a>'s study of "hollow-faced polyhedra." Grünbaum defined faces to be cyclically ordered sets of vertices, and allowed them to be <a href="/wiki/Skew_polygon" title="Skew polygon">skew</a> as well as planar.<sup id="cite_ref-sin_2-1" class="reference"><a href="#cite_note-sin-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The graph perspective allows one to apply <a href="/wiki/Glossary_of_graph_theory" title="Glossary of graph theory">graph terminology</a> and properties to polyhedra. For example, the tetrahedron and <a href="/wiki/Cs%C3%A1sz%C3%A1r_polyhedron" title="Császár polyhedron">Császár polyhedron</a> are the only known polyhedra whose skeletons are <a href="/wiki/Complete_graph" title="Complete graph">complete graphs</a> (K<sub>4</sub>), and various symmetry restrictions on polyhedra give rise to skeletons that are <a href="/wiki/Symmetric_graph" title="Symmetric graph">symmetric graphs</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_usages">Alternative usages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=32" title="Edit section: Alternative usages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the latter half of the twentieth century, various mathematical constructs have been found to have properties also present in traditional polyhedra. Rather than confining the term "polyhedron" to describe a three-dimensional polytope, it has been adopted to describe various related but distinct kinds of structure. </p> <div class="mw-heading mw-heading3"><h3 id="Higher-dimensional_polyhedra">Higher-dimensional polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=33" title="Edit section: Higher-dimensional polyhedra"><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/N-dimensional_polyhedron" title="N-dimensional polyhedron">n-dimensional polyhedron</a></div> <p>A polyhedron has been defined as a set of points in <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Affine_space" title="Affine space">affine</a> (or <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a>) space of any dimension <i>n</i> that has flat sides. It may alternatively be defined as the intersection of finitely many <a href="/wiki/Half-space_(geometry)" title="Half-space (geometry)">half-spaces</a>. Unlike a conventional polyhedron, it may be bounded or unbounded. In this meaning, a <a href="/wiki/Polytope" title="Polytope">polytope</a> is a bounded polyhedron.<sup id="cite_ref-polytope-bounded-1_14-2" class="reference"><a href="#cite_note-polytope-bounded-1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-polytope-bounded-2_15-2" class="reference"><a href="#cite_note-polytope-bounded-2-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra in this way provides a geometric perspective for problems in <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a>.<sup id="cite_ref-:0_60-0" class="reference"><a href="#cite_note-:0-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 9">: 9 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Topological_polyhedra">Topological polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=34" title="Edit section: Topological polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A topological polytope is a topological space given along with a specific decomposition into shapes that are topologically equivalent to <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytopes</a> and that are attached to each other in a regular way. </p><p>Such a figure is called <i>simplicial</i> if each of its regions is a <a href="/wiki/Simplex" title="Simplex">simplex</a>, i.e. in an <i>n</i>-dimensional space each region has <i>n</i>+1 vertices. The dual of a simplicial polytope is called <i>simple</i>. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an <i>n</i>-dimensional cube. </p> <div class="mw-heading mw-heading3"><h3 id="Abstract_polyhedra">Abstract polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=35" title="Edit section: Abstract polyhedra"><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/Abstract_polytope" title="Abstract polytope">Abstract polytope</a></div> <p>An <a href="/wiki/Abstract_polytope" title="Abstract polytope">abstract polytope</a> is a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset) of elements whose partial ordering obeys certain rules of incidence (connectivity) and ranking. The elements of the set correspond to the vertices, edges, faces and so on of the polytope: vertices have rank 0, edges rank 1, etc. with the partially ordered ranking corresponding to the dimensionality of the geometric elements. The empty set, required by set theory, has a rank of −1 and is sometimes said to correspond to the null polytope. An abstract polyhedron is an abstract polytope having the following ranking: </p> <ul><li>rank 3: The maximal element, sometimes identified with the body.</li> <li>rank 2: The <a href="/wiki/Face_(geometry)" title="Face (geometry)">polygonal faces</a>.</li> <li>rank 1: The <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>.</li> <li>rank 0: the <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>.</li> <li>rank −1: The empty set, sometimes identified with the <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="null_polytope"></span><span id="null_polytope"></span><span id="nullitope"></span><span class="vanchor-text"><i>null polytope</i> or <i>nullitope</i></span></span>.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup></li></ul> <p>Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset as described above. </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=Polyhedron&action=edit&section=36" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Before_the_Greeks">Before the Greeks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=37" title="Edit section: Before the Greeks"><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:Papyrus_moscow_4676-problem_14_part_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Papyrus_moscow_4676-problem_14_part_1.jpg/220px-Papyrus_moscow_4676-problem_14_part_1.jpg" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Papyrus_moscow_4676-problem_14_part_1.jpg/330px-Papyrus_moscow_4676-problem_14_part_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Papyrus_moscow_4676-problem_14_part_1.jpg/440px-Papyrus_moscow_4676-problem_14_part_1.jpg 2x" data-file-width="1024" data-file-height="842" /></a><figcaption>Problem 14 of the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>, on calculating the volume of a <a href="/wiki/Frustum" title="Frustum">frustum</a></figcaption></figure> <p>Polyhedra appeared in early <a href="/wiki/Architecture" title="Architecture">architectural forms</a> such as cubes and cuboids, with the earliest four-sided <a href="/wiki/Egyptian_pyramids" title="Egyptian pyramids">Egyptian pyramids</a> dating from the <a href="/wiki/27th_century_BC" title="27th century BC">27th century BC</a>.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a> from approximately 1800–1650 BC includes an early written study of polyhedra and their volumes (specifically, the volume of a <a href="/wiki/Frustum" title="Frustum">frustum</a>).<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> The mathematics of the <a href="/wiki/Old_Babylonian_Empire" title="Old Babylonian Empire">Old Babylonian Empire</a>, from roughly the same time period as the Moscow Papyrus, also included calculations of the volumes of <a href="/wiki/Cuboid" title="Cuboid">cuboids</a> (and of non-polyhedral <a href="/wiki/Cylinder" title="Cylinder">cylinders</a>), and calculations of the height of such a shape needed to attain a given volume.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Etruscan_civilization" title="Etruscan civilization">Etruscans</a> preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery of an <a href="/wiki/Etruscan_civilization" title="Etruscan civilization">Etruscan</a> <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a> made of <a href="/wiki/Soapstone" title="Soapstone">soapstone</a> on <a href="/wiki/Monte_Loffa" title="Monte Loffa">Monte Loffa</a>. Its faces were marked with different designs, suggesting to some scholars that it may have been used as a gaming die.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ancient_Greece">Ancient Greece</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=38" title="Edit section: Ancient Greece"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ancient Greek mathematicians discovered and studied the <a href="/wiki/Regular_polyhedron#History" title="Regular polyhedron">convex regular polyhedra</a>, which came to be known as the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>. Their first written description is in the <i><a href="/wiki/Timaeus_(dialogue)" title="Timaeus (dialogue)">Timaeus</a></i> of <a href="/wiki/Plato" title="Plato">Plato</a> (circa 360 BC), which associates four of them with the <a href="/wiki/Four_elements" class="mw-redirect" title="Four elements">four elements</a> and the fifth to the overall shape of the universe. A more mathematical treatment of these five polyhedra was written soon after in the <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i> of <a href="/wiki/Euclid" title="Euclid">Euclid</a>. An early commentator on Euclid (possibly <a href="/wiki/Geminus" title="Geminus">Geminus</a>) writes that the attribution of these shapes to Plato is incorrect: <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> knew the <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, <a href="/wiki/Cube" title="Cube">cube</a>, and <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a>, and <a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a> (circa 417 BC) discovered the other two, the <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> and <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a>.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> Later, <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> expanded his study to the <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">convex uniform polyhedra</a> which now bear his name. His original work is lost and his solids come down to us through <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a>.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ancient_China">Ancient China</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=39" title="Edit section: Ancient China"><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:14-sided_Chinese_dice_from_warring_states_period.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/14-sided_Chinese_dice_from_warring_states_period.jpg/170px-14-sided_Chinese_dice_from_warring_states_period.jpg" decoding="async" width="170" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/14-sided_Chinese_dice_from_warring_states_period.jpg/255px-14-sided_Chinese_dice_from_warring_states_period.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/14-sided_Chinese_dice_from_warring_states_period.jpg/340px-14-sided_Chinese_dice_from_warring_states_period.jpg 2x" data-file-width="1189" data-file-height="1173" /></a><figcaption>14-sided die from the <a href="/wiki/Warring_States_period" title="Warring States period">Warring States period</a></figcaption></figure> <p>Both cubical dice and 14-sided dice in the shape of a <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a> in China have been dated back as early as the <a href="/wiki/Warring_States_period" title="Warring States period">Warring States period</a>.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p><p>By 236 AD, <a href="/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a> was describing the dissection of the cube into its characteristic tetrahedron (<a href="/wiki/Orthoscheme" class="mw-redirect" title="Orthoscheme">orthoscheme</a>) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Medieval_Islam">Medieval Islam</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=40" title="Edit section: Medieval Islam"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Mathematics in medieval Islam</a>).<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> The 9th century scholar <a href="/wiki/Thabit_ibn_Qurra" class="mw-redirect" title="Thabit ibn Qurra">Thabit ibn Qurra</a> included the calculation of volumes in his studies,<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> and wrote a work on the <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a>. Then in the 10th century <a href="/wiki/Ab%C5%ABl_Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%AB" class="mw-redirect" title="Abūl Wafā' Būzjānī">Abu'l Wafa</a> described the convex regular and quasiregular spherical polyhedra.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Renaissance">Renaissance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=41" title="Edit section: Renaissance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:217px;max-width:217px"><div class="thumbimage" style="height:179px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Pacioli.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/215px-Pacioli.jpg" decoding="async" width="215" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/323px-Pacioli.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/430px-Pacioli.jpg 2x" data-file-width="1500" data-file-height="1250" /></a></span></div><div class="thumbcaption"><i><a href="/wiki/Portrait_of_Luca_Pacioli" title="Portrait of Luca Pacioli">Doppio ritratto</a></i>, attributed to <a href="/wiki/Jacopo_de%27_Barbari" title="Jacopo de' Barbari">Jacopo de' Barbari</a>, depicting <a href="/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a> and a student studying a glass <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a> half-filled with water.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup></div></div><div class="tsingle" style="width:171px;max-width:171px"><div class="thumbimage" style="height:179px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Leonardo_polyhedra.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Leonardo_polyhedra.png/169px-Leonardo_polyhedra.png" decoding="async" width="169" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Leonardo_polyhedra.png/254px-Leonardo_polyhedra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Leonardo_polyhedra.png/338px-Leonardo_polyhedra.png 2x" data-file-width="1000" data-file-height="1063" /></a></span></div><div class="thumbcaption">A skeletal polyhedron (specifically, a <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a>) drawn by <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> to illustrate a book by <a href="/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a></div></div></div></div></div> <p>As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a>.<sup id="cite_ref-polyhedrists_74-0" class="reference"><a href="#cite_note-polyhedrists-74"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Toroidal_polyhedron" title="Toroidal polyhedron">Toroidal polyhedra</a>, made of wood and used to support headgear, became a common exercise in perspective drawing, and were depicted in <a href="/wiki/Marquetry" title="Marquetry">marquetry</a> panels of the period as a symbol of geometry.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a> wrote about constructing perspective views of polyhedra, and rediscovered many of the Archimedean solids. <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> illustrated skeletal models of several polyhedra for a book by <a href="/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a>,<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> with text largely plagiarized from della Francesca.<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Polyhedral_net" class="mw-redirect" title="Polyhedral net">Polyhedral nets</a> make an appearance in the work of <a href="/wiki/Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a>.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> </p><p>Several works from this time investigate star polyhedra, and other elaborations of the basic Platonic forms. A marble tarsia in the floor of <a href="/wiki/St._Mark%27s_Basilica" class="mw-redirect" title="St. Mark's Basilica">St. Mark's Basilica</a>, Venice, designed by <a href="/wiki/Paolo_Uccello" title="Paolo Uccello">Paolo Uccello</a>, depicts a stellated dodecahedron.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> As the Renaissance spread beyond Italy, later artists such as <a href="/wiki/Wenzel_Jamnitzer" title="Wenzel Jamnitzer">Wenzel Jamnitzer</a>, Dürer and others also depicted polyhedra of increasing complexity, many of them novel, in imaginative etchings.<sup id="cite_ref-polyhedrists_74-1" class="reference"><a href="#cite_note-polyhedrists-74"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> (1571–1630) used <a href="/wiki/Star_polygon" title="Star polygon">star polygons</a>, typically <a href="/wiki/Pentagram" title="Pentagram">pentagrams</a>, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polyhedra must be convex.<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p><p>In the same period, <a href="/wiki/Euler%27s_polyhedral_formula" class="mw-redirect" title="Euler's polyhedral formula">Euler's polyhedral formula</a>, a <a href="/wiki/Linear_equation" title="Linear equation">linear equation</a> relating the numbers of vertices, edges, and faces of a polyhedron, was stated for the Platonic solids in 1537 in an unpublished manuscript by <a href="/wiki/Francesco_Maurolico" title="Francesco Maurolico">Francesco Maurolico</a>.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="17th–19th_centuries"><span id="17th.E2.80.9319th_centuries"></span>17th–19th centuries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=42" title="Edit section: 17th–19th centuries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, in around 1630, wrote his book <i><a href="/wiki/De_solidorum_elementis" class="mw-redirect" title="De solidorum elementis">De solidorum elementis</a></i> studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations. The work was lost, and not rediscovered until the 19th century. One of its contributions was <a href="/wiki/Descartes%27_theorem_on_total_angular_defect" class="mw-redirect" title="Descartes' theorem on total angular defect">Descartes' theorem on total angular defect</a>, which is closely related to Euler's polyhedral formula.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, for whom the formula is named, introduced it in 1758 for convex polyhedra more generally, albeit with an incorrect proof.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> Euler's work (together with his earlier solution to the puzzle of the <a href="/wiki/Seven_Bridges_of_K%C3%B6nigsberg" title="Seven Bridges of Königsberg">Seven Bridges of Königsberg</a>) became the foundation of the new field of <a href="/wiki/Topology" title="Topology">topology</a>.<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>, <a href="/wiki/Enrico_Betti" title="Enrico Betti">Enrico Betti</a>, <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>, and others.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> </p><p>In the early 19th century, <a href="/wiki/Louis_Poinsot" title="Louis Poinsot">Louis Poinsot</a> extended Kepler's work, and discovered the remaining two regular star polyhedra. Soon after, <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> proved Poinsot's list complete, subject to an unstated assumption that the sequence of vertices and edges of each polygonal side cannot admit repetitions (an assumption that had been considered but rejected in the earlier work of A. F. L. Meister).<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> They became known as the <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedra" class="mw-redirect" title="Kepler–Poinsot polyhedra">Kepler–Poinsot polyhedra</a>, and their usual names were given by <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a>.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> Meanwhile, the discovery of higher dimensions in the early 19th century led <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> by 1853 to the idea of higher-dimensional polytopes.<sup id="cite_ref-FOOTNOTECoxeter1947141–143_88-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1947141–143-88"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> Additionally, in the late 19th century, Russian crystallographer <a href="/wiki/Evgraf_Fedorov" title="Evgraf Fedorov">Evgraf Fedorov</a> completed the classification of <a href="/wiki/Parallelohedron" title="Parallelohedron">parallelohedra</a>, convex polyhedra that tile space by translations.<sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="20th–21st_centuries"><span id="20th.E2.80.9321st_centuries"></span>20th–21st centuries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=43" title="Edit section: 20th–21st centuries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematics in the 20th century dawned with <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">Hilbert's problems</a>, one of which, <a href="/wiki/Hilbert%27s_third_problem" title="Hilbert's third problem">Hilbert's third problem</a>, concerned polyhedra and their <a href="/wiki/Dissection_problem" title="Dissection problem">dissections</a>. It was quickly solved by Hilbert's student <a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a>, introducing the <a href="/wiki/Dehn_invariant" title="Dehn invariant">Dehn invariant</a> of polyhedra.<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Steinitz%27s_theorem" title="Steinitz's theorem">Steinitz's theorem</a>, published by <a href="/wiki/Ernst_Steinitz" title="Ernst Steinitz">Ernst Steinitz</a> in 1992, characterized the graphs of convex polyhedra, bringing modern ideas from <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> into the study of polyhedra.<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> </p><p>The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called <a href="/wiki/Stellation" title="Stellation">stellation</a>. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by <a href="/wiki/H.S.M._Coxeter" class="mw-redirect" title="H.S.M. Coxeter">H.S.M. Coxeter</a> and others in 1938, with the now famous paper <i>The 59 icosahedra</i>.<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> Coxeter's analysis signalled a rebirth of interest in geometry. Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the <a href="/wiki/Regular_skew_polyhedron" title="Regular skew polyhedron">regular skew polyhedra</a> and to develop the theory of <a href="/wiki/Complex_polytope" title="Complex polytope">complex polyhedra</a> first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of geometry.<sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> </p><p>In the second part of the twentieth century, both <a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a> and <a href="/wiki/Imre_Lakatos" title="Imre Lakatos">Imre Lakatos</a> pointed out the tendency among mathematicians to define a "polyhedron" in different and sometimes incompatible ways to suit the needs of the moment.<sup id="cite_ref-lakatos_1-1" class="reference"><a href="#cite_note-lakatos-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sin_2-2" class="reference"><a href="#cite_note-sin-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In a series of papers, Grünbaum broadened the accepted definition of a polyhedron, discovering many new <a href="/wiki/Regular_polyhedron#History" title="Regular polyhedron">regular polyhedra</a>. At the close of the 20th century these latter ideas merged with other work on incidence complexes to create the modern idea of an abstract polyhedron (as an abstract 3-polytope), notably presented by McMullen and Schulte.<sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> </p><p>Polyhedra make a frequent appearance in modern <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>, <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, and <a href="/wiki/Geometric_design" title="Geometric design">geometric design</a> with topics including the reconstruction of polyhedral surfaces or <a href="/wiki/Polygon_mesh" title="Polygon mesh">surface meshes</a> from scattered data points,<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> geodesics on polyhedral surfaces,<sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Visibility_(geometry)" title="Visibility (geometry)">visibility</a> and illumination in polyhedral scenes,<sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Polycube" title="Polycube">polycubes</a> and other non-convex polyhedra with axis-parallel sides,<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> algorithmic forms of Steinitz's theorem,<sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> and the still-unsolved problem of the existence of polyhedral nets for convex polyhedra.<sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_nature">In nature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=44" title="Edit section: In nature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For natural occurrences of regular polyhedra, see <a href="/wiki/Regular_polyhedron#Regular_polyhedra_in_nature" title="Regular polyhedron">Regular polyhedron § Regular polyhedra in nature</a>. </p><p>Irregular polyhedra appear in nature as <a href="/wiki/Crystal" title="Crystal">crystals</a>. </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=Polyhedron&action=edit&section=45" 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: 18em;"> <ul><li><a href="/wiki/Extension_of_a_polyhedron" class="mw-redirect" title="Extension of a polyhedron">Extension of a polyhedron</a></li> <li><a href="/wiki/Goldberg_polyhedron" title="Goldberg polyhedron">Goldberg polyhedron</a></li> <li><a href="/wiki/List_of_books_about_polyhedra" title="List of books about polyhedra">List of books about polyhedra</a></li> <li><a href="/wiki/List_of_convex_regular-faced_polyhedra" class="mw-redirect" title="List of convex regular-faced polyhedra">List of convex regular-faced polyhedra</a></li> <li><a href="/wiki/List_of_small_polyhedra_by_vertex_count" title="List of small polyhedra by vertex count">List of small polyhedra by vertex count</a></li> <li><a href="/wiki/Near-miss_Johnson_solid" title="Near-miss Johnson solid">Near-miss Johnson solid</a></li> <li><a href="/wiki/Polyhedron_model" title="Polyhedron model">Polyhedron model</a></li> <li><a href="/wiki/Polyhedral_combinatorics" title="Polyhedral combinatorics">Polyhedral combinatorics</a></li> <li><a href="/wiki/Polyhedral_group" title="Polyhedral group">Polyhedral group</a></li> <li><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">Polyhedral number</a></li> <li><a href="/wiki/Polyhedral_skeletal_electron_pair_theory" title="Polyhedral skeletal electron pair theory">Polyhedral skeletal electron pair theory</a></li> <li><a href="/wiki/Polyhedral_space" title="Polyhedral space">Polyhedral space</a></li> <li><a href="/wiki/Polyhedral_symbol" title="Polyhedral symbol">Polyhedral symbol</a></li> <li><a href="/wiki/Polyhedral_terrain" title="Polyhedral terrain">Polyhedral terrain</a></li> <li><a href="/wiki/Polytope_model" title="Polytope model">Polytope model</a></li> <li><a href="/wiki/Schlegel_diagram" title="Schlegel diagram">Schlegel diagram</a></li> <li><a href="/wiki/Lists_of_shapes" title="Lists of shapes">Lists of shapes</a></li> <li><a href="/wiki/Stella_(software)" title="Stella (software)">Stella (software)</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=Polyhedron&action=edit&section=46" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">The Archimedean solids once had fourteenth solid known as <a href="/wiki/Pseudorhombicuboctahedron" class="mw-redirect" title="Pseudorhombicuboctahedron">pseudorhombicuboctahedron</a>, mistakenly constructing <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a>. However, it was debarred for having no <a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a> property, which included it to the Johnson solid instead.<sup id="cite_ref-14th-archimedean_20-0" class="reference"><a href="#cite_note-14th-archimedean-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=47" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-lakatos-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-lakatos_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lakatos_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 id="CITEREFLakatos2015" class="citation cs2"><a href="/wiki/Imre_Lakatos" title="Imre Lakatos">Lakatos, Imre</a> (2015) [1976], Worrall, John; Zahar, Elie (eds.), <a href="/wiki/Proofs_and_Refutations" title="Proofs and Refutations"><i>Proofs and Refutations: The logic of mathematical discovery</i></a>, Cambridge Philosophy Classics, Cambridge: Cambridge University Press, p. 16, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9781316286425">10.1017/CBO9781316286425</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-53405-6" title="Special:BookSources/978-1-107-53405-6"><bdi>978-1-107-53405-6</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=3469698">3469698</a>, <q>definitions are frequently proposed and argued about</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Refutations%3A+The+logic+of+mathematical+discovery&rft.place=Cambridge&rft.series=Cambridge+Philosophy+Classics&rft.pages=16&rft.pub=Cambridge+University+Press&rft.date=2015&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3469698%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9781316286425&rft.isbn=978-1-107-53405-6&rft.aulast=Lakatos&rft.aufirst=Imre&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-sin-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-sin_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sin_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-sin_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrünbaum1994" class="citation cs2"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a> (1994), "Polyhedra with hollow faces", in Bisztriczky, Tibor; McMullen, Peter; Schneider, Rolf; Weiss, A. (eds.), <i>Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational</i>, Dordrecht: Kluwer Acad. Publ., pp. 43–70, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-011-0924-6_3">10.1007/978-94-011-0924-6_3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-010-4398-4" title="Special:BookSources/978-94-010-4398-4"><bdi>978-94-010-4398-4</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=1322057">1322057</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Polyhedra+with+hollow+faces&rft.btitle=Proceedings+of+the+NATO+Advanced+Study+Institute+on+Polytopes%3A+Abstract%2C+Convex+and+Computational&rft.place=Dordrecht&rft.pages=43-70&rft.pub=Kluwer+Acad.+Publ.&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1322057%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-94-011-0924-6_3&rft.isbn=978-94-010-4398-4&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>; for quote, see p. 43.</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="CITEREFLoeb2013" class="citation cs2"><a href="/wiki/Arthur_Lee_Loeb" title="Arthur Lee Loeb">Loeb, Arthur L.</a> (2013), "Polyhedra: Surfaces or solids?", in <a href="/wiki/Marjorie_Senechal" title="Marjorie Senechal">Senechal, Marjorie</a> (ed.), <i>Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination</i> (2nd ed.), Springer, pp. 65–75, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-92714-5_5">10.1007/978-0-387-92714-5_5</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-92713-8" title="Special:BookSources/978-0-387-92713-8"><bdi>978-0-387-92713-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Polyhedra%3A+Surfaces+or+solids%3F&rft.btitle=Shaping+Space%3A+Exploring+Polyhedra+in+Nature%2C+Art%2C+and+the+Geometrical+Imagination&rft.pages=65-75&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-0-387-92714-5_5&rft.isbn=978-0-387-92713-8&rft.aulast=Loeb&rft.aufirst=Arthur+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCormack1931" class="citation cs2">McCormack, Joseph P. (1931), <i>Solid Geometry</i>, D. Appleton-Century Company, p. 416</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+Geometry&rft.pages=416&rft.pub=D.+Appleton-Century+Company&rft.date=1931&rft.aulast=McCormack&rft.aufirst=Joseph+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" 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="CITEREFde_Bergvan_KreveldOvermarsSchwarzkopf2000" class="citation cs2"><a href="/wiki/Mark_de_Berg" title="Mark de Berg">de Berg, M.</a>; <a href="/wiki/Marc_van_Kreveld" title="Marc van Kreveld">van Kreveld, M.</a>; <a href="/wiki/Mark_Overmars" title="Mark Overmars">Overmars, M.</a>; <a href="/wiki/Otfried_Cheong" title="Otfried Cheong">Schwarzkopf, O.</a> (2000), <i>Computational Geometry: Algorithms and Applications</i> (2nd ed.), Springer, p. 64</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Geometry%3A+Algorithms+and+Applications&rft.pages=64&rft.edition=2nd&rft.pub=Springer&rft.date=2000&rft.aulast=de+Berg&rft.aufirst=M.&rft.au=van+Kreveld%2C+M.&rft.au=Overmars%2C+M.&rft.au=Schwarzkopf%2C+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatveev2001" class="citation cs2">Matveev, S.V. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Polyhedron,_abstract">"Polyhedron, abstract"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Polyhedron%2C+abstract&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Matveev&rft.aufirst=S.V.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPolyhedron%2C_abstract&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart1980" class="citation cs2"><a href="/wiki/Bonnie_Stewart" title="Bonnie Stewart">Stewart, B. M.</a> (1980), <a href="/wiki/Adventures_Among_the_Toroids" title="Adventures Among the Toroids"><i>Adventures Among the Toroids: A study of orientable polyhedra with regular faces</i></a> (2nd ed.), p. 6</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Adventures+Among+the+Toroids%3A+A+study+of+orientable+polyhedra+with+regular+faces&rft.pages=6&rft.edition=2nd&rft.date=1980&rft.aulast=Stewart&rft.aufirst=B.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-cromwell-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-cromwell_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-cromwell_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-cromwell_8-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCromwell1997" class="citation cs2">Cromwell, Peter R. (1997), <a href="/wiki/Polyhedra_(book)" title="Polyhedra (book)"><i>Polyhedra</i></a>, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55432-9" title="Special:BookSources/978-0-521-55432-9"><bdi>978-0-521-55432-9</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=1458063">1458063</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyhedra&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-55432-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1458063%23id-name%3DMR&rft.aulast=Cromwell&rft.aufirst=Peter+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>; for definitions of polyhedra, see pp. 206–209; for polyhedra with equal regular faces, see p. 86.</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="CITEREFO'Rourke1993" class="citation cs2"><a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (1993), <a rel="nofollow" class="external text" href="http://www.gbv.de/dms/goettingen/241632501.pdf">"Computational Geometry in C"</a> <span class="cs1-format">(PDF)</span>, <i>Computers in Physics</i>, <b>9</b> (1): 113–116, <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/1995ComPh...9...55O">1995ComPh...9...55O</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4823371">10.1063/1.4823371</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+in+Physics&rft.atitle=Computational+Geometry+in+C&rft.volume=9&rft.issue=1&rft.pages=113-116&rft.date=1993&rft_id=info%3Adoi%2F10.1063%2F1.4823371&rft_id=info%3Abibcode%2F1995ComPh...9...55O&rft.aulast=O%27Rourke&rft.aufirst=Joseph&rft_id=http%3A%2F%2Fwww.gbv.de%2Fdms%2Fgoettingen%2F241632501.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-acoptic-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-acoptic_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-acoptic_10-1"><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="CITEREFGrünbaum1999" class="citation cs2"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a> (1999), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210830211936/https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf">"Acoptic polyhedra"</a> <span class="cs1-format">(PDF)</span>, <i>Advances in discrete and computational geometry (South Hadley, MA, 1996)</i>, Contemporary Mathematics, vol. 223, Providence, Rhode Island: American Mathematical Society, pp. 163–199, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F223%2F03137">10.1090/conm/223/03137</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0674-6" title="Special:BookSources/978-0-8218-0674-6"><bdi>978-0-8218-0674-6</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=1661382">1661382</a>, archived from <a rel="nofollow" class="external text" href="https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2021-08-30<span class="reference-accessdate">, retrieved <span class="nowrap">2022-07-01</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Acoptic+polyhedra&rft.btitle=Advances+in+discrete+and+computational+geometry+%28South+Hadley%2C+MA%2C+1996%29&rft.place=Providence%2C+Rhode+Island&rft.series=Contemporary+Mathematics&rft.pages=163-199&rft.pub=American+Mathematical+Society&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1661382%23id-name%3DMR&rft_id=info%3Adoi%2F10.1090%2Fconm%2F223%2F03137&rft.isbn=978-0-8218-0674-6&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rft_id=https%3A%2F%2Fsites.math.washington.edu%2F~grunbaum%2FBG225.Acoptic%2520polyhedra.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" 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="CITEREFBokowskiGuedes_de_Oliveira2000" class="citation cs2">Bokowski, J.; 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(2000), "On the generation of oriented matroids", <i><a href="/wiki/Discrete_and_Computational_Geometry" class="mw-redirect" title="Discrete and Computational Geometry">Discrete and Computational Geometry</a></i>, <b>24</b> (2–3): 197–208, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs004540010027">10.1007/s004540010027</a></span>, <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=1756651">1756651</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+and+Computational+Geometry&rft.atitle=On+the+generation+of+oriented+matroids&rft.volume=24&rft.issue=2%E2%80%933&rft.pages=197-208&rft.date=2000&rft_id=info%3Adoi%2F10.1007%2Fs004540010027&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1756651%23id-name%3DMR&rft.aulast=Bokowski&rft.aufirst=J.&rft.au=Guedes+de+Oliveira%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-bursta-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-bursta_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bursta_12-1"><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="CITEREFBurgielStanton2000" class="citation cs2">Burgiel, H.; Stanton, D. (2000), "Realizations of regular abstract polyhedra of types {3,6} and {6,3}", <i><a href="/wiki/Discrete_and_Computational_Geometry" class="mw-redirect" title="Discrete and Computational Geometry">Discrete and Computational Geometry</a></i>, <b>24</b> (2–3): 241–255, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs004540010030">10.1007/s004540010030</a></span>, <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=1758047">1758047</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+and+Computational+Geometry&rft.atitle=Realizations+of+regular+abstract+polyhedra+of+types+%7B3%2C6%7D+and+%7B6%2C3%7D&rft.volume=24&rft.issue=2%E2%80%933&rft.pages=241-255&rft.date=2000&rft_id=info%3Adoi%2F10.1007%2Fs004540010030&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1758047%23id-name%3DMR&rft.aulast=Burgiel&rft.aufirst=H.&rft.au=Stanton%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-grunbaum-same-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-grunbaum-same_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-grunbaum-same_13-1"><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="CITEREFGrünbaum2003" class="citation cs2"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a> (2003), <a rel="nofollow" class="external text" href="https://faculty.washington.edu/moishe/branko/BG249.Your%20polyh-my%20polyh.pdf">"Are your polyhedra the same as my polyhedra?"</a> <span class="cs1-format">(PDF)</span>, in <a href="/wiki/Boris_Aronov" title="Boris Aronov">Aronov, Boris</a>; 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(1961), "3.2 Duality", <a href="/wiki/Mathematical_Models_(Cundy_and_Rollett)" title="Mathematical Models (Cundy and Rollett)"><i>Mathematical models</i></a> (2nd ed.), Oxford: Clarendon Press, pp. 78–79, <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=0124167">0124167</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.2+Duality&rft.btitle=Mathematical+models&rft.place=Oxford&rft.pages=78-79&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1961&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0124167%23id-name%3DMR&rft.aulast=Cundy&rft.aufirst=H.+Martyn&rft.au=Rollett%2C+A.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrünbaumShephard1969" class="citation cs2"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, B.</a>; <a href="/wiki/Geoffrey_Colin_Shephard" title="Geoffrey Colin Shephard">Shephard, G.C.</a> (1969), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170222114014/http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf">"Convex polytopes"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Bulletin_of_the_London_Mathematical_Society" class="mw-redirect" title="Bulletin of the London Mathematical Society">Bulletin of the London Mathematical Society</a></i>, <b>1</b> (3): 257–300, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fblms%2F1.3.257">10.1112/blms/1.3.257</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=0250188">0250188</a>, archived from <a rel="nofollow" class="external text" href="http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-02-22<span class="reference-accessdate">, retrieved <span class="nowrap">2017-02-21</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+London+Mathematical+Society&rft.atitle=Convex+polytopes&rft.volume=1&rft.issue=3&rft.pages=257-300&rft.date=1969&rft_id=info%3Adoi%2F10.1112%2Fblms%2F1.3.257&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0250188%23id-name%3DMR&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=B.&rft.au=Shephard%2C+G.C.&rft_id=http%3A%2F%2Fwww.wias-berlin.de%2Fpeople%2Fsi%2Fcourse%2Ffiles%2Fconvex_polytopes-survey-Gruenbaum.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>. See in particular the bottom of page 260.</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1947" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1947), <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)"><i>Regular Polytopes</i></a>, Methuen, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16">16</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Polytopes&rft.pages=16&rft.pub=Methuen&rft.date=1947&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" 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 id="CITEREFBarnette1973" class="citation cs2">Barnette, David (1973), <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.pjm/1102946311">"A proof of the lower bound conjecture for convex polytopes"</a>, <i>Pacific Journal of Mathematics</i>, <b>46</b> (2): 349–354, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1973.46.349">10.2140/pjm.1973.46.349</a></span>, <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=0328773">0328773</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pacific+Journal+of+Mathematics&rft.atitle=A+proof+of+the+lower+bound+conjecture+for+convex+polytopes&rft.volume=46&rft.issue=2&rft.pages=349-354&rft.date=1973&rft_id=info%3Adoi%2F10.2140%2Fpjm.1973.46.349&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D328773%23id-name%3DMR&rft.aulast=Barnette&rft.aufirst=David&rft_id=https%3A%2F%2Fprojecteuclid.org%2Feuclid.pjm%2F1102946311&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></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="CITEREFLuotoniemi2017" class="citation cs2">Luotoniemi, Taneli (2017), <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2017/bridges2017-17.html">"Crooked houses: Visualizing the polychora with hyperbolic patchwork"</a>, in Swart, David; Séquin, Carlo H.; Fenyvesi, Kristóf (eds.), <i>Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture</i>, Phoenix, Arizona: Tessellations Publishing, pp. 17–24, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-938664-22-9" title="Special:BookSources/978-1-938664-22-9"><bdi>978-1-938664-22-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Crooked+houses%3A+Visualizing+the+polychora+with+hyperbolic+patchwork&rft.btitle=Proceedings+of+Bridges+2017%3A+Mathematics%2C+Art%2C+Music%2C+Architecture%2C+Education%2C+Culture&rft.place=Phoenix%2C+Arizona&rft.pages=17-24&rft.pub=Tessellations+Publishing&rft.date=2017&rft.isbn=978-1-938664-22-9&rft.aulast=Luotoniemi&rft.aufirst=Taneli&rft_id=https%3A%2F%2Farchive.bridgesmathart.org%2F2017%2Fbridges2017-17.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1930" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (January 1930), "The polytopes with regular-prismatic vertex figures", <i>Philosophical Transactions of the Royal Society of London, Series A</i>, <b>229</b> (670–680), The Royal Society: 329–425, <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/1930RSPTA.229..329C">1930RSPTA.229..329C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1930.0009">10.1098/rsta.1930.0009</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London%2C+Series+A&rft.atitle=The+polytopes+with+regular-prismatic+vertex+figures&rft.volume=229&rft.issue=670%E2%80%93680&rft.pages=329-425&rft.date=1930-01&rft_id=info%3Adoi%2F10.1098%2Frsta.1930.0009&rft_id=info%3Abibcode%2F1930RSPTA.229..329C&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne2000" class="citation cs2"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (2000), "Example 44.2.3, the "punched-in icosahedron"<span class="cs1-kern-right"></span>", <i>Geometry: Euclid and beyond</i>, Undergraduate Texts in Mathematics, Springer-Verlag, New York, p. 442, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-22676-7">10.1007/978-0-387-22676-7</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98650-2" title="Special:BookSources/0-387-98650-2"><bdi>0-387-98650-2</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=1761093">1761093</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Example+44.2.3%2C+the+%22punched-in+icosahedron%22&rft.btitle=Geometry%3A+Euclid+and+beyond&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=442&rft.pub=Springer-Verlag%2C+New+York&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1761093%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-387-22676-7&rft.isbn=0-387-98650-2&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldman1991" class="citation cs2"><a href="/wiki/Ron_Goldman_(mathematician)" title="Ron Goldman (mathematician)">Goldman, Ronald N.</a> (1991), "Chapter IV.1: Area of planar polygons and volume of polyhedra", in Arvo, James (ed.), <i>Graphic Gems Package: Graphics Gems II</i>, Academic Press, pp. 170–171</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+IV.1%3A+Area+of+planar+polygons+and+volume+of+polyhedra&rft.btitle=Graphic+Gems+Package%3A+Graphics+Gems+II&rft.pages=170-171&rft.pub=Academic+Press&rft.date=1991&rft.aulast=Goldman&rft.aufirst=Ronald+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBüelerEngeFukuda2000" class="citation cs2">Büeler, B.; Enge, A.; Fukuda, K. (2000), "Exact Volume Computation for Polytopes: A Practical Study", <i>Polytopes — Combinatorics and Computation</i>, pp. 131–154, <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=10.1.1.39.7700">10.1.1.39.7700</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/10.1007%2F978-3-0348-8438-9_6">10.1007/978-3-0348-8438-9_6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-6351-2" title="Special:BookSources/978-3-7643-6351-2"><bdi>978-3-7643-6351-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Exact+Volume+Computation+for+Polytopes%3A+A+Practical+Study&rft.btitle=Polytopes+%E2%80%94+Combinatorics+and+Computation&rft.pages=131-154&rft.date=2000&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.39.7700%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2F978-3-0348-8438-9_6&rft.isbn=978-3-7643-6351-2&rft.aulast=B%C3%BCeler&rft.aufirst=B.&rft.au=Enge%2C+A.&rft.au=Fukuda%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSydler1965" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Jean-Pierre_Sydler" title="Jean-Pierre Sydler">Sydler, J.-P.</a> (1965), <a rel="nofollow" class="external text" href="https://eudml.org/doc/139296">"Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions"</a>, <i><a href="/wiki/Commentarii_Mathematici_Helvetici" title="Commentarii Mathematici Helvetici">Comment. Math. Helv.</a></i> (in French), <b>40</b>: 43–80, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02564364">10.1007/bf02564364</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=0192407">0192407</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:123317371">123317371</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comment.+Math.+Helv.&rft.atitle=Conditions+n%C3%A9cessaires+et+suffisantes+pour+l%27%C3%A9quivalence+des+poly%C3%A8dres+de+l%27espace+euclidien+%C3%A0+trois+dimensions&rft.volume=40&rft.pages=43-80&rft.date=1965&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0192407%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123317371%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fbf02564364&rft.aulast=Sydler&rft.aufirst=J.-P.&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F139296&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkel2001" class="citation cs2">Hazewinkel, M. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Dehn_invariant">"Dehn invariant"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Dehn+invariant&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Hazewinkel&rft.aufirst=M.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDehn_invariant&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDebrunner1980" class="citation cs2 cs1-prop-foreign-lang-source">Debrunner, Hans E. (1980), "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln", <i><a href="/wiki/Archiv_der_Mathematik" title="Archiv der Mathematik">Archiv der Mathematik</a></i> (in German), <b>35</b> (6): 583–587, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01235384">10.1007/BF01235384</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=0604258">0604258</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:121301319">121301319</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+der+Mathematik&rft.atitle=%C3%9Cber+Zerlegungsgleichheit+von+Pflasterpolyedern+mit+W%C3%BCrfeln&rft.volume=35&rft.issue=6&rft.pages=583-587&rft.date=1980&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D604258%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121301319%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01235384&rft.aulast=Debrunner&rft.aufirst=Hans+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexandrov2010" class="citation cs2">Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", <i>Journal of Geometry</i>, <b>99</b> (1–2): 1–13, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0901.2989">0901.2989</a></span>, <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=10.1.1.243.7674">10.1.1.243.7674</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/10.1007%2Fs00022-011-0061-7">10.1007/s00022-011-0061-7</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=2823098">2823098</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:17515249">17515249</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geometry&rft.atitle=The+Dehn+invariants+of+the+Bricard+octahedra&rft.volume=99&rft.issue=1%E2%80%932&rft.pages=1-13&rft.date=2010&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17515249%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00022-011-0061-7&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.243.7674%23id-name%3DCiteSeerX&rft_id=info%3Aarxiv%2F0901.2989&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2823098%23id-name%3DMR&rft.aulast=Alexandrov&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1992" class="citation cs2"><a href="/wiki/Jean_Taylor" title="Jean Taylor">Taylor, Jean E.</a> (1992), "Zonohedra and generalized zonohedra", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>99</b> (2): 108–111, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2324178">10.2307/2324178</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2324178">2324178</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=1144350">1144350</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Zonohedra+and+generalized+zonohedra&rft.volume=99&rft.issue=2&rft.pages=108-111&rft.date=1992&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1144350%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2324178%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2324178&rft.aulast=Taylor&rft.aufirst=Jean+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley1997" class="citation cs2"><a href="/wiki/Richard_P._Stanley" title="Richard P. Stanley">Stanley, Richard P.</a> (1997), <i>Enumerative Combinatorics, Volume I</i> (1 ed.), Cambridge University Press, pp. 235–239, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-66351-9" title="Special:BookSources/978-0-521-66351-9"><bdi>978-0-521-66351-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enumerative+Combinatorics%2C+Volume+I&rft.pages=235-239&rft.edition=1&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-66351-9&rft.aulast=Stanley&rft.aufirst=Richard+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCox2011" class="citation cs2">Cox, David A. (2011), <i>Toric varieties</i>, John B. Little, Henry K. Schenck, Providence, R.I.: American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4819-7" title="Special:BookSources/978-0-8218-4819-7"><bdi>978-0-8218-4819-7</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/698027255">698027255</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Toric+varieties&rft.place=Providence%2C+R.I.&rft.pub=American+Mathematical+Society&rft.date=2011&rft_id=info%3Aoclcnum%2F698027255&rft.isbn=978-0-8218-4819-7&rft.aulast=Cox&rft.aufirst=David+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley1996" class="citation cs2">Stanley, Richard P. (1996), <i>Combinatorics and commutative algebra</i> (2nd ed.), Boston: Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3836-9" title="Special:BookSources/0-8176-3836-9"><bdi>0-8176-3836-9</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/33080168">33080168</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorics+and+commutative+algebra&rft.place=Boston&rft.edition=2nd&rft.pub=Birkh%C3%A4user&rft.date=1996&rft_id=info%3Aoclcnum%2F33080168&rft.isbn=0-8176-3836-9&rft.aulast=Stanley&rft.aufirst=Richard+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemaineO'Rourke2007" class="citation cs2"><a href="/wiki/Erik_Demaine" title="Erik Demaine">Demaine, Erik D.</a>; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (2007), "23.2 Flexible polyhedra", <a href="/wiki/Geometric_Folding_Algorithms" title="Geometric Folding Algorithms"><i>Geometric Folding Algorithms: Linkages, origami, polyhedra</i></a>, Cambridge University Press, Cambridge, pp. 345–348, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511735172">10.1017/CBO9780511735172</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-85757-4" title="Special:BookSources/978-0-521-85757-4"><bdi>978-0-521-85757-4</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=2354878">2354878</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=23.2+Flexible+polyhedra&rft.btitle=Geometric+Folding+Algorithms%3A+Linkages%2C+origami%2C+polyhedra&rft.pages=345-348&rft.pub=Cambridge+University+Press%2C+Cambridge&rft.date=2007&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2354878%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9780511735172&rft.isbn=978-0-521-85757-4&rft.aulast=Demaine&rft.aufirst=Erik+D.&rft.au=O%27Rourke%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Rourke2008" class="citation cs2"><a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (2008), "Unfolding orthogonal polyhedra", <i>Surveys on discrete and computational geometry</i>, Contemp. Math., vol. 453, Amer. Math. Soc., Providence, RI, pp. 307–317, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F453%2F08805">10.1090/conm/453/08805</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4239-3" title="Special:BookSources/978-0-8218-4239-3"><bdi>978-0-8218-4239-3</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=2405687">2405687</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Unfolding+orthogonal+polyhedra&rft.btitle=Surveys+on+discrete+and+computational+geometry&rft.series=Contemp.+Math.&rft.pages=307-317&rft.pub=Amer.+Math.+Soc.%2C+Providence%2C+RI&rft.date=2008&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2405687%23id-name%3DMR&rft_id=info%3Adoi%2F10.1090%2Fconm%2F453%2F08805&rft.isbn=978-0-8218-4239-3&rft.aulast=O%27Rourke&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1966" class="citation cs2"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (November 1966), "Mathematical Games: Is it possible to visualize a four-dimensional figure?", <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>, <b>215</b> (5): 138–143, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican1166-138">10.1038/scientificamerican1166-138</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24931332">24931332</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Mathematical+Games%3A+Is+it+possible+to+visualize+a+four-dimensional+figure%3F&rft.volume=215&rft.issue=5&rft.pages=138-143&rft.date=1966-11&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1166-138&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24931332%23id-name%3DJSTOR&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1974" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1974), <i>Regular Complex Polytopes</i>, Cambridge: Cambridge University Press, <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=0370328">0370328</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Complex+Polytopes&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1974&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0370328%23id-name%3DMR&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2017)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopko2012" class="citation cs2">Popko, Edward S. (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA463"><i>Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere</i></a>, CRC Press, p. 463, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-0430-1" title="Special:BookSources/978-1-4665-0430-1"><bdi>978-1-4665-0430-1</bdi></a>, <q>A hosohedron is only possible on a sphere</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Divided+Spheres%3A+Geodesics+and+the+Orderly+Subdivision+of+the+Sphere&rft.pages=463&rft.pub=CRC+Press&rft.date=2012&rft.isbn=978-1-4665-0430-1&rft.aulast=Popko&rft.aufirst=Edward+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHjTSBQAAQBAJ%26pg%3DPA463&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKraynikReinelt2007" class="citation cs2">Kraynik, A.M.; Reinelt, D.A. (2007), "Foams, Microrheology of", in Mortensen, Andreas (ed.), <i>Concise Encyclopedia of Composite Materials</i> (2nd ed.), Elsevier, pp. 402–407</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foams%2C+Microrheology+of&rft.btitle=Concise+Encyclopedia+of+Composite+Materials&rft.pages=402-407&rft.edition=2nd&rft.pub=Elsevier&rft.date=2007&rft.aulast=Kraynik&rft.aufirst=A.M.&rft.au=Reinelt%2C+D.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>. See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zs_lGeGsuaAC&pg=PA403">p. 403</a>: "foams consist of polyhedral gas bubbles ... each face on a polyhedron is a minimal surface with uniform mean curvature ... no face can be a flat polygon with straight edges".</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPearce1978" class="citation cs2">Pearce, P. (1978), "14 Saddle polyhedra and continuous surfaces as environmental structures", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sfc2OEuE8oQC&pg=PA224"><i>Structure in nature is a strategy for design</i></a>, MIT Press, p. 224, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-66045-7" title="Special:BookSources/978-0-262-66045-7"><bdi>978-0-262-66045-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=14+Saddle+polyhedra+and+continuous+surfaces+as+environmental+structures&rft.btitle=Structure+in+nature+is+a+strategy+for+design&rft.pages=224&rft.pub=MIT+Press&rft.date=1978&rft.isbn=978-0-262-66045-7&rft.aulast=Pearce&rft.aufirst=P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dsfc2OEuE8oQC%26pg%3DPA224&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span>.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1985" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1985), "A special book review: M.C. Escher: His life and complete graphic work", <i>The Mathematical Intelligencer</i>, <b>7</b> (1): 59–69, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03023010">10.1007/BF03023010</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:189887063">189887063</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=A+special+book+review%3A+M.C.+Escher%3A+His+life+and+complete+graphic+work&rft.volume=7&rft.issue=1&rft.pages=59-69&rft.date=1985&rft_id=info%3Adoi%2F10.1007%2FBF03023010&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189887063%23id-name%3DS2CID&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span> Coxeter's analysis of <i>Stars</i> is on pp. 61–62.</span> </li> <li id="cite_note-:0-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_60-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrötschelLovászSchrijver1993" class="citation cs2"><a href="/wiki/Martin_Gr%C3%B6tschel" title="Martin Grötschel">Grötschel, Martin</a>; <a href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">Lovász, László</a>; <a href="/wiki/Alexander_Schrijver" title="Alexander Schrijver">Schrijver, Alexander</a> (1993), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hWvmCAAAQBAJ&pg=PA281"><i>Geometric algorithms and combinatorial optimization</i></a>, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-78240-4">10.1007/978-3-642-78240-4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-78242-8" title="Special:BookSources/978-3-642-78242-8"><bdi>978-3-642-78242-8</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=1261419">1261419</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+algorithms+and+combinatorial+optimization&rft.series=Algorithms+and+Combinatorics&rft.edition=2nd&rft.pub=Springer-Verlag%2C+Berlin&rft.date=1993&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1261419%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-78240-4&rft.isbn=978-3-642-78242-8&rft.aulast=Gr%C3%B6tschel&rft.aufirst=Martin&rft.au=Lov%C3%A1sz%2C+L%C3%A1szl%C3%B3&rft.au=Schrijver%2C+Alexander&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhWvmCAAAQBAJ%26pg%3DPA281&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">N.W. 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Historical Studies, vol. 59, Birkhäuser, p. 71, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-72487-4">10.1007/978-3-319-72487-4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-72486-7" title="Special:BookSources/978-3-319-72486-7"><bdi>978-3-319-72486-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Folding+in+Mathematics%3A+Mathematizing+the+Margins&rft.series=Science+Networks.+Historical+Studies&rft.pages=71&rft.pub=Birkh%C3%A4user&rft.date=2018&rft_id=info%3Adoi%2F10.1007%2F978-3-319-72487-4&rft.isbn=978-3-319-72486-7&rft.aulast=Friedman&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFederico1982" class="citation cs2">Federico, Pasquale Joseph (1982), <i>Descartes on Polyhedra: A Study of the "De solidorum elementis"</i>, Sources in the History of Mathematics and Physical Sciences, vol. 4, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90760-2" title="Special:BookSources/0-387-90760-2"><bdi>0-387-90760-2</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=0680214">0680214</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Descartes+on+Polyhedra%3A+A+Study+of+the+%22De+solidorum+elementis%22&rft.series=Sources+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=0-387-90760-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D680214%23id-name%3DMR&rft.aulast=Federico&rft.aufirst=Pasquale+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-83">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranceseRicheson2007" class="citation cs2">Francese, Christopher; Richeson, David (2007), "The flaw in Euler's proof of his polyhedral formula", <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>114</b> (4): 286–296, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2007.11920417">10.1080/00029890.2007.11920417</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=2281926">2281926</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:10023787">10023787</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+flaw+in+Euler%27s+proof+of+his+polyhedral+formula&rft.volume=114&rft.issue=4&rft.pages=286-296&rft.date=2007&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2281926%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10023787%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F00029890.2007.11920417&rft.aulast=Francese&rft.aufirst=Christopher&rft.au=Richeson%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-84">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexanderson2006" class="citation cs2">Alexanderson, Gerald L. (2006), "About the cover: Euler and Königsberg's bridges: a historical view", <i>American Mathematical Society</i>, New Series, <b>43</b> (4): 567–573, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-06-01130-X">10.1090/S0273-0979-06-01130-X</a></span>, <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=2247921">2247921</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Society&rft.atitle=About+the+cover%3A+Euler+and+K%C3%B6nigsberg%27s+bridges%3A+a+historical+view&rft.volume=43&rft.issue=4&rft.pages=567-573&rft.date=2006&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-06-01130-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2247921%23id-name%3DMR&rft.aulast=Alexanderson&rft.aufirst=Gerald+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-85">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEckmann2006" class="citation cs2">Eckmann, Beno (2006), "The Euler characteristic – a few highlights in its long history", <i>Mathematical Survey Lectures 1943–2004</i>, Springer, pp. 177–188, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-33791-1_15">10.1007/978-3-540-33791-1_15</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33791-1" title="Special:BookSources/978-3-540-33791-1"><bdi>978-3-540-33791-1</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=2269092">2269092</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Euler+characteristic+%E2%80%93+a+few+highlights+in+its+long+history&rft.btitle=Mathematical+Survey+Lectures+1943%E2%80%932004&rft.pages=177-188&rft.pub=Springer&rft.date=2006&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2269092%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-540-33791-1_15&rft.isbn=978-3-540-33791-1&rft.aulast=Eckmann&rft.aufirst=Beno&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-86">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrünbaum1994" class="citation cs2">Grünbaum, Branko (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZptYDwAAQBAJpg">"Regular polyhedra"</a>, in Grattan-Guinness, I. 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Teller">Teller, Seth J.</a>; <a href="/wiki/Pat_Hanrahan" title="Pat Hanrahan">Hanrahan, Pat</a> (1993), "Global visibility algorithms for illumination computations", in <a href="/wiki/Mary_Whitton" title="Mary Whitton">Whitton, Mary C.</a> (ed.), <i>Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1993, Anaheim, CA, USA, August 2–6, 1993</i>, Association for Computing Machinery, pp. 239–246, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F166117.166148">10.1145/166117.166148</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89791-601-8" title="Special:BookSources/0-89791-601-8"><bdi>0-89791-601-8</bdi></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:7957200">7957200</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Global+visibility+algorithms+for+illumination+computations&rft.btitle=Proceedings+of+the+20th+Annual+Conference+on+Computer+Graphics+and+Interactive+Techniques%2C+SIGGRAPH+1993%2C+Anaheim%2C+CA%2C+USA%2C+August+2%E2%80%936%2C+1993&rft.pages=239-246&rft.pub=Association+for+Computing+Machinery&rft.date=1993&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7957200%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F166117.166148&rft.isbn=0-89791-601-8&rft.aulast=Teller&rft.aufirst=Seth+J.&rft.au=Hanrahan%2C+Pat&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-98">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCherchi2019" class="citation cs2">Cherchi, Gianmarco (February 2019), <i>Polycube Optimization and Applications: From the Digital World to Manufacturing</i> (Doctoral dissertation), University of Cagliari, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11584%2F261570">11584/261570</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polycube+Optimization+and+Applications%3A+From+the+Digital+World+to+Manufacturing&rft.pub=University+of+Cagliari&rft.date=2019-02&rft_id=info%3Ahdl%2F11584%2F261570&rft.aulast=Cherchi&rft.aufirst=Gianmarco&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-99">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRote2011" class="citation cs2">Rote, Günter (2011), "Realizing planar graphs as convex polytopes", in van Kreveld, Marc J.; Speckmann, Bettina (eds.), <i>Graph Drawing – 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21–23, 2011, Revised Selected Papers</i>, Lecture Notes in Computer Science, vol. 7034, Springer, pp. 238–241, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-25878-7_23">10.1007/978-3-642-25878-7_23</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-25877-0" title="Special:BookSources/978-3-642-25877-0"><bdi>978-3-642-25877-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Realizing+planar+graphs+as+convex+polytopes&rft.btitle=Graph+Drawing+%E2%80%93+19th+International+Symposium%2C+GD+2011%2C+Eindhoven%2C+The+Netherlands%2C+September+21%E2%80%9323%2C+2011%2C+Revised+Selected+Papers&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=238-241&rft.pub=Springer&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2F978-3-642-25878-7_23&rft.isbn=978-3-642-25877-0&rft.aulast=Rote&rft.aufirst=G%C3%BCnter&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> <li id="cite_note-100"><span class="mw-cite-backlink"><b><a href="#cite_ref-100">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemaineO'Rourke2007" class="citation cs2"><a href="/wiki/Erik_Demaine" title="Erik Demaine">Demaine, Erik</a>; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (2007), <a href="/wiki/Geometric_Folding_Algorithms" title="Geometric Folding Algorithms"><i>Geometric Folding Algorithms: Linkages, Origami, Polyhedra</i></a>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Folding+Algorithms%3A+Linkages%2C+Origami%2C+Polyhedra&rft.pub=Cambridge+University+Press&rft.date=2007&rft.aulast=Demaine&rft.aufirst=Erik&rft.au=O%27Rourke%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span> </li> </ol></div></div> <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=Polyhedron&action=edit&section=48" 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"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/polyhedron" class="extiw" title="wiktionary:polyhedron">polyhedron</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/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" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Polyhedra" class="extiw" title="commons:Polyhedra"><span style="font-style:italic; font-weight:bold;">Polyhedra</span></a>.</div></div> </div> <div class="mw-heading mw-heading3"><h3 id="General_theory">General theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=49" title="Edit section: General theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Polyhedron"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Polyhedron.html">"Polyhedron"</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=Polyhedron&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPolyhedron.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyhedron" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.steelpillow.com/polyhedra/">Polyhedra Pages</a></li> <li><a rel="nofollow" class="external text" href="https://www.math.technion.ac.il/S/rl/docs/uniform.pdf">Uniform Solution for Uniform Polyhedra by Dr. Zvi Har'El</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151127053535/http://www.math.technion.ac.il/S/rl/docs/uniform.pdf">Archived</a> 2015-11-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170224151555/http://www.uwgb.edu/dutchs/symmetry/symmetry.htm">Symmetry, Crystals and Polyhedra</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Lists_and_databases_of_polyhedra">Lists and databases of polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=50" title="Edit section: Lists and databases of polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> – The Encyclopedia of Polyhedra.</li> <li><a rel="nofollow" class="external text" href="http://www.eg-models.de/index.html">Electronic Geometry Models</a> – Contains a peer reviewed selection of polyhedra with unusual properties.</li> <li><a rel="nofollow" class="external text" href="http://www.orchidpalms.com/polyhedra/index.html">Polyhedron Models</a> – Virtual polyhedra.</li> <li><a rel="nofollow" class="external text" href="http://www.polyedergarten.de/">Paper Models of Uniform (and other) Polyhedra</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Free_software">Free software</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=51" title="Edit section: Free software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170616023727/http://www.uff.br/cdme/pdp/pdp-html/pdp-en.html">A Plethora of Polyhedra</a> – An interactive and free collection of polyhedra in Java. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.</li> <li><a rel="nofollow" class="external text" href="http://dogfeathers.com/java/hyperstar.html">Hyperspace Star Polytope Slicer</a> – Explorer java applet, includes a variety of 3d viewer options.</li> <li><a rel="nofollow" class="external text" href="http://www.openscad.org/">openSCAD</a> – Free cross-platform software for programmers. Polyhedra are just one of the things you can model. The <a href="https://en.wikibooks.org/wiki/OpenSCAD_User_Manual" class="extiw" title="b:OpenSCAD User Manual">openSCAD User Manual</a> is also available.</li> <li><a rel="nofollow" class="external text" href="http://www.openvolumemesh.org/">OpenVolumeMesh</a> – An open source cross-platform C++ library for handling polyhedral meshes. Developed by the Aachen Computer Graphics Group, RWTH Aachen University.</li> <li><a rel="nofollow" class="external text" href="https://levskaya.github.com/polyhedronisme/">Polyhedronisme</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120425234840/http://levskaya.github.com/polyhedronisme/">Archived</a> 2012-04-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – Web-based tool for generating polyhedra models using <a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">Conway Polyhedron Notation</a>. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Resources_for_making_physical_models">Resources for making physical models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyhedron&action=edit&section=52" title="Edit section: Resources for making physical models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.korthalsaltes.com/">Paper Models of Polyhedra</a> Free nets of polyhedra.</li> <li><a rel="nofollow" class="external text" href="http://ldlewis.com/How-to-Build-Polyhedra/">Simple instructions for building over 30 paper polyhedra</a></li> <li><a rel="nofollow" class="external text" href="http://hbmeyer.de/flechten/indexeng.htm">Polyhedra plaited with paper strips</a> – Polyhedra models constructed without use of glue.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Polyhedra" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Polyhedra" title="Template:Polyhedra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polyhedra" title="Template talk:Polyhedra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polyhedra" title="Special:EditPage/Template:Polyhedra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Polyhedra" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Polyhedra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Listed by number of faces and type</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">1–10 faces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monohedron" class="mw-redirect" title="Monohedron">Monohedron</a></li> <li><a href="/wiki/Dihedron" title="Dihedron">Dihedron</a></li> <li><a href="/wiki/Hosohedron" title="Hosohedron">Trihedron</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li> <li><a href="/wiki/Pentahedron" title="Pentahedron">Pentahedron</a></li> <li><a href="/wiki/Hexahedron" title="Hexahedron">Hexahedron</a></li> <li><a href="/wiki/Heptahedron" title="Heptahedron">Heptahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Enneahedron" title="Enneahedron">Enneahedron</a></li> <li><a href="/wiki/Decahedron" title="Decahedron">Decahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 faces</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecahedron" title="Hendecahedron">Hendecahedron</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Tridecahedron" title="Tridecahedron">Tridecahedron</a></li> <li><a href="/wiki/Tetradecahedron" title="Tetradecahedron">Tetradecahedron</a></li> <li><a href="/wiki/Pentadecahedron" title="Pentadecahedron">Pentadecahedron</a></li> <li><a href="/wiki/Hexadecahedron" title="Hexadecahedron">Hexadecahedron</a></li> <li><a href="/wiki/Heptadecahedron" title="Heptadecahedron">Heptadecahedron</a></li> <li><a href="/wiki/Octadecahedron" title="Octadecahedron">Octadecahedron</a></li> <li><a href="/wiki/Enneadecahedron" title="Enneadecahedron">Enneadecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 faces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositetrahedron" title="Icositetrahedron">Icositetrahedron</a> (24)</li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">Triacontahedron</a> (30)</li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a> (32)</li> <li><a href="/wiki/Hexoctahedron" class="mw-redirect" title="Hexoctahedron">Hexoctahedron</a> (48)</li> <li><a href="/wiki/Hexecontahedron" title="Hexecontahedron">Hexecontahedron</a> (60)</li> <li><a href="/wiki/Rhombic_enneacontahedron" title="Rhombic enneacontahedron">Enneacontahedron</a> (90)</li> <li><a href="/wiki/Rhombic_hectotriadiohedron" title="Rhombic hectotriadiohedron">Hectotriadiohedron</a> (132)</li> <li><a href="/wiki/Skew_apeirohedron" title="Skew apeirohedron">Apeirohedron</a> (∞)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">elemental things</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Face_(geometry)" title="Face (geometry)">face</a></li> <li><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edge</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a></li> <li><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform polyhedron</a> (two infinite groups and 75) <ul><li><a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular polyhedron</a> (9)</li> <li><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">quasiregular polyhedron</a> (16)</li> <li><a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular polyhedron</a> (two infinite groups and 50)</li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">convex polyhedron</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a> (5)</li> <li><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solid</a> (13)</li> <li><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solid</a> (13)</li> <li><a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solid</a> (92)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">non-convex polyhedron</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedron</a> (4)</li> <li><a href="/wiki/Star_polyhedron" title="Star polyhedron">Star polyhedron</a> (infinite)</li> <li><a href="/wiki/Uniform_star_polyhedron" title="Uniform star polyhedron">Uniform star polyhedron</a> (57)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a>‌s</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prism</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprism</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Wedge_(geometry)" title="Wedge (geometry)">wedge</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a></li> <li><a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Convex_polyhedra" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Convex_polyhedron_navigator" title="Template:Convex polyhedron navigator"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Convex_polyhedron_navigator" title="Template talk:Convex polyhedron navigator"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Convex_polyhedron_navigator" title="Special:EditPage/Template:Convex polyhedron navigator"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Convex_polyhedra" style="font-size:114%;margin:0 4em">Convex <a class="mw-selflink selflink">polyhedra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> <span class="nobold">(<a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedron#Regular_tetrahedron" title="Tetrahedron">tetrahedron</a></li> <li><a href="/wiki/Cube" title="Cube">cube</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">octahedron</a></li> <li><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a></li> <li><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a><br /><span class="nobold">(<a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li> <li><a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li> <li><a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a></li> <li><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li> <li><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li> <li><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li> <li><a href="/wiki/Snub_cube" title="Snub cube">snub cube</a></li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li> <li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li> <li><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li> <li><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a><br /><span class="nobold">(duals of Archimedean)</span></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li> <li><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li> <li><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li> <li><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li> <li><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li> <li><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li> <li><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li> <li><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Dihedron" title="Dihedron">dihedron</a></i></li> <li><i><a href="/wiki/Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a></li> <li><a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral others</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></li> <li><a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedra</a></li> <li><a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramid</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Bicupola_(geometry)" class="mw-redirect" title="Bicupola (geometry)">bicupola</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Bifrustum" title="Bifrustum">bifrustum</a></li> <li><a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotunda</a></li> <li><a href="/wiki/Birotunda" title="Birotunda">birotunda</a></li> <li><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a></li> <li><a href="/wiki/Scutoid" title="Scutoid">scutoid</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Degenerate polyhedra are in <i>italics</i>.</div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" 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52.261 3 Template:Fix"," 3.83% 49.676 3 Template:Navbox"," 3.31% 42.912 1 Template:Polyhedra"," 3.18% 41.230 1 Template:Cn"]},"scribunto":{"limitreport-timeusage":{"value":"0.811","limit":"10.000"},"limitreport-memusage":{"value":10663749,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAlexanderson2006\"] = 1,\n [\"CITEREFAlexandrov2010\"] = 1,\n [\"CITEREFAndrews2022\"] = 1,\n [\"CITEREFAustin2013\"] = 1,\n [\"CITEREFBarnette1973\"] = 1,\n [\"CITEREFBerman1971\"] = 1,\n [\"CITEREFBoissonnatYvinec1989\"] = 1,\n [\"CITEREFBokowskiGuedes_de_Oliveira2000\"] = 1,\n [\"CITEREFBridge1974\"] = 1,\n [\"CITEREFBrunsGubeladze2009\"] = 1,\n [\"CITEREFBréardCook2019\"] = 1,\n [\"CITEREFBurgielStanton2000\"] = 1,\n [\"CITEREFBüelerEngeFukuda2000\"] = 1,\n [\"CITEREFCalvo-LópezAlonso-Rodríguez2010\"] = 1,\n [\"CITEREFCherchi2019\"] = 1,\n [\"CITEREFCox2011\"] = 1,\n [\"CITEREFCoxeter1930\"] = 1,\n [\"CITEREFCoxeter1947\"] = 1,\n [\"CITEREFCoxeter1974\"] = 1,\n [\"CITEREFCoxeter1985\"] = 1,\n [\"CITEREFCoxeterDu_ValFlatherPetrie1999\"] = 1,\n [\"CITEREFCromwell1997\"] = 1,\n [\"CITEREFCundy1952\"] = 1,\n [\"CITEREFCundyRollett1961\"] = 1,\n [\"CITEREFDebrunner1980\"] = 1,\n [\"CITEREFDemaineO\u0026#039;Rourke2007\"] = 2,\n [\"CITEREFDiudea2018\"] = 1,\n [\"CITEREFEckmann2006\"] = 1,\n [\"CITEREFEves1969\"] = 1,\n [\"CITEREFFederico1982\"] = 1,\n [\"CITEREFField1979\"] = 1,\n [\"CITEREFField1997\"] = 2,\n [\"CITEREFFranceseRicheson2007\"] = 1,\n [\"CITEREFFriberg2000\"] = 1,\n [\"CITEREFFriedman2018\"] = 1,\n [\"CITEREFGamba2012\"] = 1,\n [\"CITEREFGardner1966\"] = 1,\n [\"CITEREFGhomi2018\"] = 1,\n [\"CITEREFGoldman1991\"] = 1,\n [\"CITEREFGrünbaum1994\"] = 2,\n [\"CITEREFGrünbaum1999\"] = 1,\n [\"CITEREFGrünbaum2003\"] = 1,\n [\"CITEREFGrünbaum2007\"] = 1,\n [\"CITEREFGrünbaum2009\"] = 1,\n [\"CITEREFGrünbaumShephard1969\"] = 1,\n [\"CITEREFGunnPeet1929\"] = 1,\n [\"CITEREFHartshorne2000\"] = 1,\n [\"CITEREFHisarligilHisarligil2017\"] = 1,\n [\"CITEREFInchbald2006\"] = 1,\n [\"CITEREFKernBland1938\"] = 1,\n [\"CITEREFKitchen1991\"] = 1,\n [\"CITEREFKnorr1983\"] = 1,\n [\"CITEREFKraynikReinelt2007\"] = 1,\n [\"CITEREFLakatos2015\"] = 1,\n [\"CITEREFLimHaron2012\"] = 1,\n [\"CITEREFLitchenberg1988\"] = 1,\n [\"CITEREFLoeb2013\"] = 1,\n [\"CITEREFLuotoniemi2017\"] = 1,\n [\"CITEREFMalkevitch2018\"] = 1,\n [\"CITEREFMcCormack1931\"] = 1,\n [\"CITEREFMcMullenSchulte2002\"] = 1,\n [\"CITEREFMitchellMountPapadimitriou1987\"] = 1,\n [\"CITEREFMontebelli2015\"] = 1,\n [\"CITEREFO\u0026#039;KeefeHyde2020\"] = 1,\n [\"CITEREFO\u0026#039;Rourke1993\"] = 1,\n [\"CITEREFO\u0026#039;Rourke2008\"] = 1,\n [\"CITEREFPearce1978\"] = 1,\n [\"CITEREFPeirce1976\"] = 1,\n [\"CITEREFPitici2011\"] = 1,\n [\"CITEREFPopko2012\"] = 1,\n [\"CITEREFRashed2009\"] = 1,\n [\"CITEREFRicheson2008\"] = 1,\n [\"CITEREFRingel1974\"] = 1,\n [\"CITEREFRoberts2009\"] = 1,\n [\"CITEREFRote2011\"] = 1,\n [\"CITEREFSaffaro1992\"] = 1,\n [\"CITEREFSchramm1992\"] = 1,\n [\"CITEREFSparavigna2012\"] = 1,\n [\"CITEREFStanley1996\"] = 1,\n [\"CITEREFStanley1997\"] = 1,\n [\"CITEREFStewart1980\"] = 2,\n [\"CITEREFSydler1965\"] = 1,\n [\"CITEREFTaylor1992\"] = 1,\n [\"CITEREFTellerHanrahan1993\"] = 1,\n [\"CITEREFTimofeenko2010\"] = 1,\n [\"CITEREFZeeman2002\"] = 1,\n [\"CITEREFde_Bergvan_KreveldOvermarsSchwarzkopf2000\"] = 1,\n [\"CITEREFvan_der_Waerden1983\"] = 1,\n [\"Grünbaum-Convex-Polytopes\"] = 1,\n}\ntemplate_list = table#1 {\n [\"-\"] = 1,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 1,\n [\"Citation\"] = 92,\n [\"Citation needed\"] = 1,\n [\"Cite Geometric Algorithms and Combinatorial Optimization\"] = 1,\n [\"Cite conference\"] = 1,\n [\"Cn\"] = 1,\n [\"Commons\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Efn\"] = 1,\n [\"Ety\"] = 1,\n [\"Harvtxt\"] = 1,\n [\"ISBN\"] = 1,\n [\"Infobox\"] = 1,\n [\"Main\"] = 18,\n [\"Math\"] = 2,\n [\"Mathworld\"] = 1,\n [\"Multiple image\"] = 4,\n [\"Mvar\"] = 3,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 2,\n [\"Other uses\"] = 1,\n [\"Page needed\"] = 1,\n [\"Plural form\"] = 1,\n [\"Polyhedra\"] = 1,\n [\"Polyhedron navigator\"] = 1,\n [\"Redirect-distinguish\"] = 1,\n [\"Reflist\"] = 1,\n [\"Rp\"] = 1,\n [\"Sfnp\"] = 3,\n [\"Short description\"] = 1,\n [\"Slink\"] = 2,\n [\"SpringerEOM\"] = 2,\n [\"TOC limit\"] = 1,\n [\"Vanchor\"] = 1,\n [\"Webarchive\"] = 2,\n [\"Wiktionary\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-2x7bg","timestamp":"20241125134519","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Polyhedron","url":"https:\/\/en.wikipedia.org\/wiki\/Polyhedron","sameAs":"http:\/\/www.wikidata.org\/entity\/Q172937","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q172937","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-09-09T06:05:12Z","dateModified":"2024-11-06T17:46:24Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/83\/Tetrahedron.jpg","headline":"solid in three dimensions with flat faces"}</script> </body> </html>