{"id":791,"date":"2019-05-29T17:45:41","date_gmt":"2019-05-29T15:45:41","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=791"},"modified":"2019-06-04T10:15:41","modified_gmt":"2019-06-04T08:15:41","slug":"cercle-qui-roule-se-quarre","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/","title":{"rendered":"Cercle qui roule se quarre"},"content":{"rendered":"

La quadrature du cercle consiste \u00e0 construire un carr\u00e9 de m\u00eame aire qu’un cercle donn\u00e9. Si le cercle a pour rayon R, il s’agit donc de construire un carr\u00e9 de c\u00f4t\u00e9 R multipli\u00e9 par la racine carr\u00e9e du nombre Pi. On peut donc r\u00e9aliser la quadrature du cercle avec une r\u00e8gle gradu\u00e9e \u00e0 la pr\u00e9cision que l’on veut.<\/p>\n

Des r\u00e8gles qui changent tout<\/h2>\n

Quand le probl\u00e8me est apparu dans l’Antiquit\u00e9, il n’\u00e9tait pas question d’approximations, la r\u00e8gle \u00e9tait que la construction devait \u00eatre exacte. Il en existe plusieurs. L’une d’entre elle demande de faire rouler un cercle sur une droite. La voici sous forme de tableau :<\/p>\n

\"\"
Le cercle part de la position \u00e0 gauche (en rouge) pour arriver \u00e0 celle de droite (en jaune) ce qui permet de d\u00e9finir le carr\u00e9 de droite, de m\u00eame aire que le cercle initial.<\/figcaption><\/figure>\n

En utilisant uniquement le th\u00e9or\u00e8me de Pythagore, on d\u00e9montre que le carr\u00e9 est de c\u00f4t\u00e9 racine de Pi, ce qui prouve que le carr\u00e9 et le cercle ont m\u00eame aire (voir \u00e0 la fin pour une d\u00e9monstration).<\/p>\n

Cette utilisation d’un proc\u00e9d\u00e9 m\u00e9canique (faire rouler le cercle) ne convenait pas aux anciens, il fallait construire le carr\u00e9 \u00e0 la r\u00e8gle (non gradu\u00e9e) et au compas. Dans ces conditions, le probl\u00e8me devient impossible, ce qui n’a \u00e9t\u00e9 prouv\u00e9 qu’au XIX-i\u00e8me si\u00e8cle en d\u00e9montrant que le nombre Pi est transcendant c’est-\u00e0-dire qu’il n’est pas solution d’une \u00e9quation alg\u00e9brique \u00e0 coefficients entiers.<\/p>\n

De fa\u00e7on \u00e9tonnante, un probl\u00e8me purement g\u00e9om\u00e9trique et tr\u00e8s conditionn\u00e9 par des visions antiques a eu des cons\u00e9quences importantes en alg\u00e8bre et en analyse.<\/p>\n

Un peu de g\u00e9om\u00e9trie<\/h2>\n

La figure essentielle est la suivante :<\/p>\n

\"\"
Il s’agit de montrer que HC a pour longueur la racine carr\u00e9e de Pi.<\/figcaption><\/figure>\n

En appliquant le th\u00e9or\u00e8me de Pythagore dans les trois triangles rectangles HBC, HC et ABC, on obtient :<\/p>\n

HC\u00b2 + 1 = BC\u00b2, HC\u00b2+ Pi\u00b2 = AC\u00b2 et BC\u00b2 + AC\u00b2 = (Pi + 1)\u00b2<\/p>\n

En faisant la somme des deux premi\u00e8res expressions puis un peu d’alg\u00e8bre, il vient que HC\u00b2 est \u00e9gal \u00e0 Pi ce qu’il fallait d\u00e9montrer.<\/p>\n

Une lampe pour une id\u00e9e lumineuse<\/h2>\n

Cette quadrature du cercle, dont l’auteur initial nous est inconnu, m’a inspir\u00e9 la lampe suivant :<\/p>\n

\"\"
Cercle qui roule se quarre par Herv\u00e9 Lehning.<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"

La quadrature du cercle consiste \u00e0 construire un carr\u00e9 de m\u00eame aire qu’un cercle donn\u00e9. Si le cercle a pour rayon R, il s’agit donc de construire un carr\u00e9 de c\u00f4t\u00e9 R multipli\u00e9 par la racine carr\u00e9e du nombre Pi. On peut donc r\u00e9aliser la quadrature du cercle avec une r\u00e8gle gradu\u00e9e \u00e0 la pr\u00e9cision … Continuer la lecture de Cercle qui roule se quarre<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":792,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[5,12,63,112,29],"tags":[314,312,313],"class_list":["post-791","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-arts","category-graphismes","category-histoire","category-humour","category-maths-2","tag-lampe-formule","tag-quadrature-du-cercle","tag-regle-et-compas"],"yoast_head":"\nCercle qui roule se quarre, par Herv\u00e9 Lehning<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cercle qui roule se quarre, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"La quadrature du cercle consiste \u00e0 construire un carr\u00e9 de m\u00eame aire qu’un cercle donn\u00e9. Si le cercle a pour rayon R, il s’agit donc de construire un carr\u00e9 de c\u00f4t\u00e9 R multipli\u00e9 par la racine carr\u00e9e du nombre Pi. On peut donc r\u00e9aliser la quadrature du cercle avec une r\u00e8gle gradu\u00e9e \u00e0 la pr\u00e9cision … Continuer la lecture de Cercle qui roule se quarre →\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/\" \/>\n<meta property=\"og:site_name\" content=\"MATH'MONDE, le blog d'Herv\u00e9 LEHNING, agr\u00e9g\u00e9 de math\u00e9matiques\" \/>\n<meta property=\"article:published_time\" content=\"2019-05-29T15:45:41+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-04T08:15:41+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/05\/LampeQuadrature112007.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1659\" \/>\n\t<meta property=\"og:image:height\" content=\"2212\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/\",\"name\":\"Cercle qui roule se quarre, par Herv\u00e9 Lehning\",\"isPartOf\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/05\/LampeQuadrature112007.jpg\",\"datePublished\":\"2019-05-29T15:45:41+00:00\",\"dateModified\":\"2019-06-04T08:15:41+00:00\",\"author\":{\"@id\":\"\"},\"breadcrumb\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/#primaryimage\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/05\/LampeQuadrature112007.jpg\",\"contentUrl\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/05\/LampeQuadrature112007.jpg\",\"width\":1659,\"height\":2212},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/29\/cercle-qui-roule-se-quarre\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\/\/blogs.futura-sciences.com\/lehning\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Cercle qui roule se quarre\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/#website\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/\",\"name\":\"MATH'MONDE, le blog d'Herv\u00e9 LEHNING, agr\u00e9g\u00e9 de math\u00e9matiques\",\"description\":\"Comment comprendre le monde moderne sans culture math\u00e9matique ? 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