{"id":945,"date":"2015-05-05T13:59:26","date_gmt":"2015-05-05T12:59:26","guid":{"rendered":"http:\/\/www.ozone3d.net\/blogs\/lab\/?p=945"},"modified":"2015-10-16T10:53:44","modified_gmt":"2015-10-16T09:53:44","slug":"theorie-des-graphes-v-e-r-2-formule-deuler","status":"publish","type":"post","link":"https:\/\/www.ozone3d.net\/blogs\/lab\/20150505\/theorie-des-graphes-v-e-r-2-formule-deuler\/","title":{"rendered":"Th\u00e9orie des Graphes: V &#8211; E + R = 2 (Formule d&#8217;Euler)"},"content":{"rendered":"<p><!--more--><\/p>\n<p>La formule d&#8217;<b>Euler<\/b> pour les graphes connect\u00e9s vous connaissez? Je l&#8217;ai decouverte il n&#8217;y a pas longtemps en tombant sur <a href=\"http:\/\/jdh.hamkins.org\/math-for-eight-year-olds\/\">ce cours de math pour gamins de 8 ans<\/a>.<\/p>\n<p>Cette formule d\u00e9couverte par le <a href=\"http:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler\">suisse Euler<\/a> est toute simple:<\/p>\n<p>\n<font size=\"4\"><b>V &#8211; E + R = 2<\/b><\/font><\/p>\n<p>Cette \u00e9galit\u00e9, valable pour les <a href=\"http:\/\/en.wikipedia.org\/wiki\/Planar_graph\">graphes connect\u00e9s planaires<\/a>, nous dit que le nombre de sommets (V pour vertices) moins le nombre de cot\u00e9s (E pour edges) additionn\u00e9 du nombre de faces (ou regions R) est toujours egal \u00e0 2.<\/p>\n<p>Tant que les segments qui repr\u00e9sentent les cot\u00e9s ne se croisent pas (pas d&#8217;intersection), cette formule est vraie pour n&#8217;importe quel graphe:<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"http:\/\/www.ozone3d.net\/public\/jegx\/201505\/planar_graph_euler_formula_01.jpg\" alt=\"planar graph - Euler formula\"\/><br \/>\n<\/center><br \/>\nLe triangle pr\u00e9c\u00e9dant poss\u00e8de 3 <b>V<\/b>ertices, 3 <b>E<\/b>dges et d\u00e9fini 2 <b>R<\/b>egions (ou espaces: \u00e0 l&#8217;int\u00e9rieur du triangle et \u00e0 l&#8217;exterieur du triangle).<\/p>\n<p>\n<center><br \/>\n<img decoding=\"async\" src=\"http:\/\/www.ozone3d.net\/public\/jegx\/201505\/planar_graph_euler_formula_02.jpg\" alt=\"planar graph - Euler formula\"\/><br \/>\n<br \/>\n<img decoding=\"async\" src=\"http:\/\/www.ozone3d.net\/public\/jegx\/201505\/planar_graph_euler_formula_03.jpg\" alt=\"planar graph - Euler formula\"\/><br \/>\n<\/center><\/p>\n<p>\nEt la formulke d&#8217;Euler est aussi applicable aux <b>objets 3D<\/b>. Dans le cas des solides 3D, les r\u00e9gions sont les <b>faces<\/b>. Le cube par example, a 8 <b>V<\/b>ertices, 12 <b>E<\/b>dges et 6 <b>R<\/b>egions:<\/p>\n<p>\n<center><br \/>\n<img decoding=\"async\" src=\"http:\/\/www.ozone3d.net\/public\/jegx\/201505\/planar_graph_euler_formula_04.jpg\" alt=\"planar graph - Euler formula\"\/><br \/>\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":1079,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[27],"tags":[611,612,642,613,609,121,610,600],"class_list":["post-945","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-news","tag-connecte","tag-euler","tag-feature_post","tag-formule","tag-graphe","tag-math","tag-planaire","tag-visualization"],"aioseo_notices":[],"views":1093,"_links":{"self":[{"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/posts\/945","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/comments?post=945"}],"version-history":[{"count":0,"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/posts\/945\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/media\/1079"}],"wp:attachment":[{"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/media?parent=945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/categories?post=945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ozone3d.net\/blogs\/lab\/wp-json\/wp\/v2\/tags?post=945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}