{"id":464,"date":"2018-09-01T12:11:37","date_gmt":"2018-09-01T10:11:37","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=464"},"modified":"2018-09-24T10:31:34","modified_gmt":"2018-09-24T08:31:34","slug":"le-tipi-optimal","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/09\/01\/le-tipi-optimal\/","title":{"rendered":"Le tipi optimal"},"content":{"rendered":"

Penchons-nous sur la forme des tipis des indiens d\u2019Am\u00e9rique. Il s\u2019agit d\u2019un c\u00f4ne dont la hauteur vaut environ 75\u00a0% du diam\u00e8tre de la base. Des calculs montrent que cette forme minimise la toile \u00e0 utiliser pour un volume donn\u00e9, comme les abeilles \u00e9conomisent la cire pour cr\u00e9er leurs alv\u00e9oles. Est-ce un hasard\u00a0? Difficile de r\u00e9pondre \u00e0 la question car d\u2019autres param\u00e8tres comme la solidit\u00e9 de l\u2019ensemble entrent en jeu. Peu importe, ces probl\u00e8mes d\u2019optimisation se retrouvent souvent dans la nature comme dans la vie pratique.<\/span><\/span><\/p>\n

\"\"
Un tipi.<\/figcaption><\/figure>\n

Analyse math\u00e9matique<\/h2>\n

Analysons celui-ci math\u00e9matiquement. Un tipi est une tente conique caract\u00e9ris\u00e9e par le rayon de sa base, R, et par sa hauteur, que nous notons proportionnellement \u00e0 R, k<\/i>\u00a0R, car le probl\u00e8me tient essentiellement \u00e0 ce rapport k<\/i>. La capacit\u00e9 du tipi est \u00e9gale \u00e0 son volume et la surface de toile, \u00e0 son aire lat\u00e9rale.<\/span><\/span><\/p>\n

\"\"
Le tipi est un c\u00f4ne caract\u00e9ris\u00e9 par le rayon de sa base R et par sa hauteur k R.<\/figcaption><\/figure>\n

Le volume est \u00e9gal \u00e0 Pi \/\u00a03 multipli\u00e9 par le carr\u00e9 du rayon R et par la hauteur k<\/em>\u00a0R. Imposer un volume de 10 m\u00e8tres cube (par exemple) lie le rapport k<\/em> au rayon R. L\u2019aire lat\u00e9rale d\u00e9pend alors uniquement de ce rapport. Cette d\u00e9pendance se traduit par une courbe en forme de J \u00e0 l\u2019envers. Nous y constatons un minimum de l\u2019aire pour une valeur de k<\/em> de l\u2019ordre de 1,4, autrement dit pour une hauteur 40\u00a0% sup\u00e9rieure au rayon de la base. De fa\u00e7on plus pr\u00e9cise, le calcul diff\u00e9rentiel montre que ce minimum est atteint pour k<\/em> \u00e9gal \u00e0 la racine carr\u00e9e de 2, ce qui fait 1,414 \u00e0 0,001 pr\u00e8s.<\/p>\n

\"\"
Variation de l\u2019aire lat\u00e9rale en fonction du rapport entre la hauteur et le rayon. Le calcul montre que le minimum est atteint quand k est \u00e9gal \u00e0 la racine de 2, soit 1,414 \u00e0 0,001 pr\u00e8s.<\/figcaption><\/figure>\n

\u00c0 volume \u00e9gal, l\u2019aire lat\u00e9rale du tipi est donc minimale pour un rapport proche de 1,4. La courbe montre de plus que la variation de l\u2019aire lat\u00e9rale est faible autour de ce rapport, ce qui explique que, dans la pratique, il oscille autour de 1,4.<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Penchons-nous sur la forme des tipis des indiens d\u2019Am\u00e9rique. Il s\u2019agit d\u2019un c\u00f4ne dont la hauteur vaut environ 75\u00a0% du diam\u00e8tre de la base. Des calculs montrent que cette forme minimise la toile \u00e0 utiliser pour un volume donn\u00e9, comme les abeilles \u00e9conomisent la cire pour cr\u00e9er leurs alv\u00e9oles. Est-ce un hasard\u00a0? Difficile de r\u00e9pondre … Continuer la lecture de Le tipi optimal<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":469,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[29,4,98,10],"tags":[176,178,177,175,180,179,174],"class_list":["post-464","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths-2","category-regard-sur-le-monde","category-viequotidienne","category-voyages","tag-amerindiens","tag-amerique","tag-indiens","tag-optimiser","tag-racine-de-deux","tag-sioux","tag-tipi"],"yoast_head":"\nLe tipi optimal, 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\/2018\/09\/01\/le-tipi-optimal\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le tipi optimal, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Penchons-nous sur la forme des tipis des indiens d\u2019Am\u00e9rique. Il s\u2019agit d\u2019un c\u00f4ne dont la hauteur vaut environ 75\u00a0% du diam\u00e8tre de la base. Des calculs montrent que cette forme minimise la toile \u00e0 utiliser pour un volume donn\u00e9, comme les abeilles \u00e9conomisent la cire pour cr\u00e9er leurs alv\u00e9oles. Est-ce un hasard\u00a0? 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