{"id":5213,"date":"2022-11-11T19:23:09","date_gmt":"2022-11-11T15:23:09","guid":{"rendered":"http:\/\/curvica974.re\/?p=5213"},"modified":"2025-12-23T20:44:30","modified_gmt":"2025-12-23T16:44:30","slug":"pavages-non-reguliers-orthogonaux","status":"publish","type":"post","link":"https:\/\/curvica974.re\/?p=5213","title":{"rendered":"Pavages non r\u00e9guliers orthogonaux"},"content":{"rendered":"\n<p>Apr\u00e8s avoir abord\u00e9 les classiques pavages r\u00e9guliers hyperboliques dans un article consacr\u00e9 <a rel=\"noreferrer noopener\" href=\"http:\/\/curvica974.re\/?p=5132\" data-type=\"URL\" data-id=\"http:\/\/curvica974.re\/?p=5132\" target=\"_blank\">aux cercles de pavage<\/a>, nous poursuivons l&rsquo;illustration des pavages hyperboliques par des objets moins r\u00e9guliers, et dans un premier temps, par la construction la plus simple qu&rsquo;il soit, celle de polygones orthogonaux  c&rsquo;est-\u00e0-dire des polygones n&rsquo;ayant que des angles droits.<\/p>\n\n\n\n<p>Ils sont \u00e9l\u00e9mentaires \u00e0 construire g\u00e9om\u00e9triquement pour peu que l&rsquo;on puisse construire la perpendiculaire communes \u00e0 deux droites. Voici la construction du polygone de base.<\/p>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/15ALGUFPdWVR45vMJ5Xe7sc0e7FvQVmu1\/view?usp=drive_link\" style=\"width:720px;height:500px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\"><em>Pour chaque changement de type de polygone, d\u00e9placer un des points \\(A,B,C\\) ou s&rsquo;il existent \\(D, E\\) ou \\(F\\).<\/em><\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1gJ9jIYvCOeGdoErNvutGLR9CmfRd8DMp\/view?usp=drive_link\" data-type=\"URL\" data-id=\"https:\/\/www.dgpad.net\/index.php?url=http:\/\/curvica974.re\/FigSite\/PavagesNonReg\/Demo_Ortho.dgp\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet<\/p>\n\n\n\n<p>Ensuite les constructions se font par sym\u00e9trie orthogonale, ce qui, dans le mod\u00e8le <strong>DP<\/strong>, est une inversion et ne prend qu&rsquo;un objet pour un point. Donc on obtient des figures tr\u00e8s l\u00e9g\u00e8res ce qui permet d&rsquo;envisager la r\u00e9alisation de la \u00ab\u00a0g\u00e9n\u00e9ration 2\u00a0\u00bb du pavage, y compris en remplissant les polygones. Voici un aper\u00e7u de ce que l&rsquo;on peut r\u00e9aliser. <\/p>\n\n\n\n<p class=\"has-small-font-size\"><em>Dans cette page, deux figures &#8211; de g\u00e9n\u00e9ration 2 &#8211; sont plus longues \u00e0 charger. Il peut \u00eatre n\u00e9cessaire de les relancer, avec l&rsquo;icone de l&rsquo;iframe, en g\u00e9n\u00e9ral il faut cliquer deux fois.<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Pavages PnR(5,4)<\/h2>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/1H2gqkwGk7iVayEm6e_nUz05vpkgn8oCQ\/view?usp=drive_link\" style=\"width:600px;height:600px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\">D<em>\u00e9placer les points \\(A,B,C\\). Il faut que \\(D\\) et \\(E\\)<\/em> <em>existent<\/em>.<\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1BEjr0lMswyD3GK7L8HO29GmbQM4wooRm\/view?usp=drive_link\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet<\/p>\n\n\n\n<p>Et on peut ensuite aller un peu plus loin, c&rsquo;est encore tr\u00e8s fluide, mais un peu long \u00e0 charger peut-\u00eatre<\/p>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/13V476D-3xP47kWHoZrAXNcaMHWuNxon9\/view?usp=drive_link\" style=\"width:600px;height:600px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\"><em>M\u00eames remarques que la figure pr\u00e9c\u00e9dente<\/em><\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1P2wF24k3B_Cytex2wvA7vrwiQZm39sZb\/view?usp=drive_link\" data-type=\"URL\" data-id=\"https:\/\/www.dgpad.net\/index.php?url=http:\/\/curvica974.re\/FigSite\/PavagesNonReg\/PnR54_Gene2net.dgp\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Pavages PnR(6,4)<\/h2>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/19XO6VpraaLQTMMjDxHOWIS2L3C3M9keV\/view?usp=drive_link\" style=\"width:600px;height:600px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\"><em>L\u00e0 encore d\u00e9placer les points bleus \u00e9pais de telle fa\u00e7on que les deux points roses existent.<\/em><br><em>On rappelle que tous ces hexagones ont pour aire \u03c0.<\/em><\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1VrCY_jlokwOTDh9NZX5t0xdgQ4YV63I3\/view?usp=drive_link\" data-type=\"URL\" data-id=\"https:\/\/www.dgpad.net\/index.php?url=http:\/\/curvica974.re\/FigSite\/PavagesNonReg\/PnR64_Gene1.dgp\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet<\/p>\n\n\n\n<p>On peut encore faire la g\u00e9n\u00e9ration 2, la figure est  plus longue \u00e0 ouvrir, mais ne prend que quelques secondes. Il y a en effet 85 hexagones qui sont remplis par des \u00ab\u00a0listes segments\u00a0\u00bb dont on a d\u00e9j\u00e0 parl\u00e9 dans les articles sur la <strong>PSH<\/strong> par exemple.<\/p>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/1NqZA9R0_VnTykwp4gxf6PBvAHazkA-Ee\/view?usp=drive_link\" style=\"width:620px;height:620px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\"><em>M\u00eames consignes &#8230;<\/em><\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1OPC5sHtE8WraH9dntS2PdvAwlJqZTZE7\/view?usp=drive_link\" data-type=\"URL\" data-id=\"https:\/\/www.dgpad.net\/index.php?url=http:\/\/curvica974.re\/FigSite\/PavagesNonReg\/PnR64_Gene2.dgp\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Pavages PnR(7,4) et PnR(8,4)<\/h2>\n\n\n\n<p>On termine par les pavages pour les deux valeurs de \\(n\\) suivantes, mais seulement pour la g\u00e9n\u00e9ration 1.<\/p>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/1fZjBxUlMdjev9tD4yN4TuEk5n0buSu2-\/view?usp=drive_link\np\" style=\"width:600px;height:600px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\"><em>G\u00e9n\u00e9ration 1 de pavages d&rsquo;heptagones orthogonaux<\/em><\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1wioMLJSo2JgB8RM3hbCnUtM02hqvPJ10\/view?usp=drive_link\" data-type=\"URL\" data-id=\"https:\/\/www.dgpad.net\/index.php?url=http:\/\/curvica974.re\/FigSite\/PavagesNonReg\/PnR74_Gene1.dgp\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet<\/p>\n\n\n\n<p>Et enfin, avec l&rsquo;octogone<\/p>\n\n\n<p><center><iframe src=\"https:\/\/www.dgpad.net\/responsive.php?url=https:\/\/drive.google.com\/file\/d\/1Amiowvw5lNngOBNKOBwVQOTZMqd7S_4H\/view?usp=drive_link\" style=\"width:600px;height:600px;border-style:solid;border-width:1px;box-shadow: 6px 6px 3px #888888;\"><\/iframe><\/center><\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\"><em>G\u00e9n\u00e9ration 1 de pavages d&rsquo;octogones orthogonaux<\/em><\/p>\n\n\n\n<p>Pr\u00e9f\u00e9rer <a href=\"https:\/\/www.dgpad.net\/index.php?url=https:\/\/drive.google.com\/file\/d\/1Gpph0Chk3y3sBKKQVPLXag5eLm-k708r\/view?usp=drive_link\" data-type=\"URL\" data-id=\"https:\/\/www.dgpad.net\/index.php?url=http:\/\/curvica974.re\/FigSite\/PavagesNonReg\/PnR84_Gene1.dgp\" target=\"_blank\" rel=\"noreferrer noopener\">ouvrir cette figure<\/a> dans un nouvel onglet<\/p>\n\n\n\n<p>On aura bien compris que r\u00e9aliser ces figures ne posent aucune difficult\u00e9. On se propose, dans de prochains articles, d&rsquo;aborder les pavages \\(PnR(5,5)\\) ou \\(PnR(6,6)\\) qui, l\u00e0, vont demander un s\u00e9rieux d\u00e9veloppement trigonom\u00e9trique si on veut r\u00e9aliser des figures dynamiques.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Apr\u00e8s avoir abord\u00e9 les classiques pavages r\u00e9guliers hyperboliques dans un article consacr\u00e9 aux cercles de pavage, nous poursuivons l&rsquo;illustration des pavages hyperboliques par des objets moins r\u00e9guliers, et dans un premier temps, par la construction la plus simple qu&rsquo;il soit, celle de polygones orthogonaux c&rsquo;est-\u00e0-dire des polygones n&rsquo;ayant que des [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[10,27,18],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/curvica974.re\/index.php?rest_route=\/wp\/v2\/posts\/5213"}],"collection":[{"href":"https:\/\/curvica974.re\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/curvica974.re\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/curvica974.re\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/curvica974.re\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5213"}],"version-history":[{"count":12,"href":"https:\/\/curvica974.re\/index.php?rest_route=\/wp\/v2\/posts\/5213\/revisions"}],"predecessor-version":[{"id":8442,"href":"https:\/\/curvica974.re\/index.php?rest_route=\/wp\/v2\/posts\/5213\/revisions\/8442"}],"wp:attachment":[{"href":"https:\/\/curvica974.re\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5213"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/curvica974.re\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5213"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/curvica974.re\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}