{"id":3396,"date":"2023-12-01T18:57:20","date_gmt":"2023-12-01T17:57:20","guid":{"rendered":"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=3396"},"modified":"2026-04-27T12:45:27","modified_gmt":"2026-04-27T10:45:27","slug":"les-pavages-du-plan-2","status":"publish","type":"post","link":"https:\/\/math.univ-lyon1.fr\/dream\/?p=3396","title":{"rendered":"Les pavages du plan"},"content":{"rendered":"\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/\">Accueil<\/a> \u00bb <a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=3328\">Ressources p\u00e9dagogiques<\/a> \u00bb <a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=77\">Des exemples de SDRP<\/a> \u00bb Les pavages du plan<\/p>\r\n<h1>L&rsquo;\u00e9nonc\u00e9 de la situation<\/h1>\r\n<p><iframe loading=\"lazy\" title=\"Les Pavages du Plan\" src=\"https:\/\/tube-sciences-technologies.apps.education.fr\/videos\/embed\/c9d92b22-9c35-4bd6-82cd-a6901a025d69\" width=\"560\" height=\"315\" frameborder=\"0\" sandbox=\"allow-same-origin allow-scripts allow-popups allow-forms\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>Globalement il s&rsquo;agit de d\u00e9terminer l&rsquo;ensemble des pavages du plan utilisant des polygones r\u00e9guliers. Pour une recherche en classe il peut \u00eatre utile de se restreindre aux pavages stricts.<\/p>\r\n<p><strong>\u00c9nonc\u00e9 1<\/strong> : Comment paver le plan avec des polygones r\u00e9guliers ?<\/p>\r\n<p><strong>\u00c9nonc\u00e9 2 <\/strong>: Un polygone r\u00e9gulier est un polygone convexe dont tous les angles ont la m\u00eame mesure et tous les c\u00f4t\u00e9s la m\u00eame longueur.<\/p>\r\n<p>Un <strong>pavage archim\u00e9dien du plan<\/strong> est un recouvrement du plan par des polygones r\u00e9guliers, sans trou, ni superposition, et tel qu&rsquo;autour de chaque sommet, il y ait le m\u00eame assemblage de polygones.<\/p>\r\n<p>On exclut dans ce probl\u00e8me les pavages tels qu&rsquo;un sommet de polygones appartienne au c\u00f4t\u00e9 d&rsquo;un autre comme sur la figure ci-dessous :<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/clarolineconnect.univ-lyon1.fr\/file\/resource\/media\/3928668\" alt=\"Fauxpavage.png\" width=\"120\" height=\"123\" \/><\/p>\r\n<p>La recherche propos\u00e9e est la d\u00e9termination de tous les pavages archim\u00e9diens du plan.<\/p>\r\n<h1>Les math\u00e9matiques en jeu<\/h1>\r\n<ul>\r\n<li>Les polygones r\u00e9guliers : le triangle \u00e9quilat\u00e9ral, le carr\u00e9, l&rsquo;hexagone r\u00e9gulier &#8230; pentagones, heptagones, enn\u00e9agones, &#8230;<\/li>\r\n<li>Les polygones r\u00e9guliers constructibles \u00e0 la r\u00e8gle et au compas<\/li>\r\n<li>Les angles\u00a0 : l&rsquo;angle plein, l&rsquo;angle plat, l&rsquo;angle droit, les angles des polygones : il est \u00e0 noter que leur reconnaissance pourra \u00eatre contrari\u00e9e<\/li>\r\n<li>Les mesures des ces angles, avec ou sans unit\u00e9 de mesure, en portion de l&rsquo;angle plat, de l&rsquo;angle droit<\/li>\r\n<li>Les multiples de 30, 90, 120 ; les multiples des mesures des angles des polygones r\u00e9guliers ; les diviseurs de 360 &#8230; les valeurs enti\u00e8res, d\u00e9cimales non enti\u00e8res, rationnelles non d\u00e9cimales des angles ou de leurs mesures ; les d\u00e9compositions additives de 360.<\/li>\r\n<li>Les assemblages autour d&rsquo;un n\u0153ud<\/li>\r\n<li>Les fractions et les relations de la forme\u00a0 \u03a3 1\/n = k\/2 &#8211; 1<\/li>\r\n<li>Les pavages; les pavages r\u00e9guliers, les pavages archim\u00e9diens, les pavages moins r\u00e9guliers<\/li>\r\n<\/ul>\r\n<h4>Analyse didactique<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/04\/PavagesDuPlan_Analyse_didactique_DREAM.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Analyse math\u00e9matique<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/04\/PavagesDuPlan_Analyse_Maths_DREAM.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Exemples de mise en \u0153uvre<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/04\/PavagesDuPlan_Mise_en_oeuvre_DREAM.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Pour aller plus loin<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/04\/PavagesDuPlan_Aller_Plus_Loin_DREAM.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h2>Retours d&rsquo;exp\u00e9riences<\/h2>\r\n<p>Des retours d&rsquo;exp\u00e9riences partag\u00e9s par les enseignant.e.s du groupe d&rsquo;exp\u00e9rimentation qui ont mis en place ce probl\u00e8me dans le cadre d&rsquo;un enseignement fond\u00e9 sur les probl\u00e8mes<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>D\u00e9terminer tous les pavages archim\u00e9diens du 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CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?cat=106\" rel=\"category\">SDRP<\/a>","rttpg_excerpt":"D\u00e9terminer tous les pavages archim\u00e9diens du 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