{"id":538,"date":"2018-10-06T15:50:24","date_gmt":"2018-10-06T13:50:24","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=538"},"modified":"2019-03-21T14:32:14","modified_gmt":"2019-03-21T13:32:14","slug":"les-valeurs-de-%cf%80","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/10\/06\/les-valeurs-de-%cf%80\/","title":{"rendered":"Les valeurs de \u03c0"},"content":{"rendered":"

En 1897, une r\u00e9solution \u00e9tablissant que \u03c0<\/i>\u00a0=\u00a04 fut propos\u00e9e au vote des repr\u00e9sentants de l\u2019\u00e9tat de l\u2019Indiana (\u00c9tats-Unis d\u2019Am\u00e9rique). Avant de sourire, le math\u00e9maticien se posera une question\u00a0: pour quelle notion de distance\u00a0?<\/span><\/span><\/p>\n

Qu’est-ce que \u03c0<\/i> ?<\/h2>\n

Archim\u00e8de a r\u00e9pondu \u00e0 cette question voici fort longtemps. Il s\u2019agit du rapport entre la circonf\u00e9rence d\u2019un cercle et son diam\u00e8tre. Qu\u2019est-ce qu\u2019un cercle\u00a0? L\u2019ensemble des points du plan \u00e0 \u00e9gale distance d\u2019un point donn\u00e9. Qu\u2019est-ce que la distance\u00a0? Ici, nous ne pouvons que marquer une pause dans nos r\u00e9ponses toutes faites. Plusieurs distances sont envisageables\u00a0!<\/span> <\/span><\/p>\n

Distance \u00e0 vol d’oiseau<\/h2>\n

En math\u00e9matiques, la distance la plus utilis\u00e9e est qualifi\u00e9e d\u2019euclidienne. Dans la vie courante, on parle souvent de distance \u00e0 vol d\u2019oiseau. La distance d\u2019un point A \u00e0 un point B est la longueur du vecteur V qui m\u00e8ne de A \u00e0 B. En tenant compte du th\u00e9or\u00e8me de Pythagore, elle s\u2019exprime sous la forme\u00a0:<\/span><\/span><\/p>\n

|\u00a0V\u00a0|2<\/span><\/sup>\u00a0=\u00a0x<\/i>2<\/span><\/sup>\u00a0+\u00a0y<\/i>2<\/span><\/sup>.<\/span><\/span><\/p>\n

Les cercles associ\u00e9s \u00e0 cette distance ont la forme ronde usuelle. Le nombre \u03c0<\/i> a la valeur connue, 3,14 \u00e0 0,01 pr\u00e8s<\/span>.<\/span><\/span><\/p>\n

Distance Manhattan<\/h2>\n

M\u00eame pour un oiseau, la distance euclidienne correspond \u00e0 une certaine vision du monde, o\u00f9 le vol est possible dans toutes les directions. \u00c0 Manhattan, m\u00eame pour voler, mieux vaut suivre les avenues, qui forment un maillage rectangulaire. La longueur d\u2019un vecteur s\u2019y exprime sous la forme\u00a0:<\/span><\/span><\/p>\n

|\u00a0V\u00a0|\u00a0=\u00a0|\u00a0x<\/i>\u00a0|\u00a0+\u00a0|\u00a0y<\/i>\u00a0|.<\/span><\/span><\/p>\n

\"\"
La distance Manhattan correspond au plus court chemin, si l\u2019on marche le long des rues d\u2019une ville au plan rectangulaire (comme Manhattan)<\/figcaption><\/figure>\n

Le cercle unit\u00e9 a alors la forme d\u2019un losange, sa circonf\u00e9rence est \u00e9gale \u00e0 8 donc, pour cette distance, \u03c0<\/i>\u00a0=\u00a04.<\/span><\/span><\/p>\n

On retrouve la valeur 4 pour une autre distance <\/span><\/span>(appel\u00e9e distance infinie), celle donnant comme longueur \u00e0 V, la plus grande des valeurs absolues de ses coordonn\u00e9es. Les cercles ont alors une forme de carr\u00e9.<\/span><\/p>\n

\"\"
Les “cercles” de m\u00eame centre et de m\u00eame rayon pour les trois distances.<\/figcaption><\/figure>\n

Le d\u00e9cret de l’Indiana : humour ou sottise ?<\/h2>\n

Nous avons trouv\u00e9 deux fois 4 et une seule fois 3,14. On pourrait en conclure que \u03c0<\/i>\u00a0=\u00a04 est la valeur la plus raisonnable \u00e0 retenir. Quand les rues des villes se coupent \u00e0 angle droit, la distance Manhattan est la plus pertinente. Est-ce pour cette raison qu\u2019une loi visant \u00e0 adopter la valeur \u03c0<\/i>\u00a0=\u00a04 fut propos\u00e9e au vote de l\u2019assembl\u00e9e g\u00e9n\u00e9rale de l\u2019\u00e9tat de l\u2019Indiana\u00a0? Vous pouvez en juger vous-m\u00eame en allant lire le texte plein d\u2019humour de ce projet de loi sur l\u2019Internet (utilisez un moteur de recherche pour en trouver une copie). Nous laisserons de toutes fa\u00e7ons la question aux amateurs d\u2019histoire (s).<\/span><\/span><\/p>\n

Autres distances<\/h2>\n

Les trois distances utilis\u00e9es se g\u00e9n\u00e9ralisent en utilisant un nombre p<\/em>\u00a0\u00b3\u00a01 quelconque. Plus pr\u00e9cis\u00e9ment, on pose\u00a0:<\/p>\n

|\u00a0V\u00a0|p<\/sup><\/em>\u00a0=\u00a0|\u00a0x<\/em>\u00a0| p<\/sup><\/em>\u00a0+\u00a0|\u00a0y<\/em>\u00a0| p<\/sup><\/em>.<\/p>\n

La distance euclidienne correspond au cas\u00a0: p<\/em>\u00a0=\u00a02, la distance Manhattan au cas\u00a0: p<\/em>\u00a0=\u00a01. On d\u00e9montre, par un passage \u00e0 la limite, que la distance infinie correspond bien au cas\u00a0: p<\/em>\u00a0=\u00a0\u221e.<\/p>\n

Pour chacune de ces distances, nous obtenons une valeur de \u03c0<\/em>, que nous notons \u03c0p<\/sub><\/em>. Comment en calculer une valeur approch\u00e9e\u00a0? Tout simplement en proc\u00e9dant comme dans le cas de la distance euclidienne, c\u2019est-\u00e0-dire en rempla\u00e7ant le cercle par des polygones r\u00e9guliers ayant un grand nombre de c\u00f4t\u00e9s. Si nous effectuons ces calculs pour p<\/em> variant de 1 \u00e0 2 avec une pr\u00e9cision de 0,001, nous obtenons le tableau\u00a0:<\/p>\n

 <\/p>\n\n\n\n\n
p<\/em><\/td>\n1,0<\/td>\n1,1<\/td>\n1,2<\/td>\n1,3<\/td>\n1,4<\/td>\n1,5<\/td>\n1,6<\/td>\n1,7<\/td>\n1,8<\/td>\n1,9<\/td>\n2,0<\/td>\n<\/tr>\n
\u03c0p<\/sub><\/em><\/td>\n4,000<\/td>\n3,757<\/td>\n3,573<\/td>\n3,434<\/td>\n3,333<\/td>\n3,260<\/td>\n3,209<\/td>\n3,176<\/td>\n3,155<\/td>\n3,145<\/td>\n3,142<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Ainsi, \u03c0p<\/sub><\/em> semble d\u00e9croissant de 1 \u00e0 2. Le ph\u00e9nom\u00e8ne inverse se produit de 2 \u00e0 l\u2019infini. On est amen\u00e9 \u00e0 plusieurs conjectures\u00a0:<\/p>\n

1) \u03c0<\/em> est la valeur minimale des \u03c0p<\/sub><\/em>,<\/p>\n

2) \u03c0p<\/sub><\/em> prend toutes les valeurs entre \u03c0<\/em> et 4,<\/p>\n

3) \u03c0p<\/sub><\/em>\u00a0=\u00a0\u03c0q<\/sub><\/em> si 1\/p<\/em> + 1\/q<\/em> = 1.<\/p>\n

On d\u00e9montre que les trois sont exactes, en utilisant des raisonnements de calcul int\u00e9gral d\u00e9passant le cadre de cet article.<\/p>\n

Pour ces calculs, voir sur mon site.<\/a><\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

En 1897, une r\u00e9solution \u00e9tablissant que \u03c0\u00a0=\u00a04 fut propos\u00e9e au vote des repr\u00e9sentants de l\u2019\u00e9tat de l\u2019Indiana (\u00c9tats-Unis d\u2019Am\u00e9rique). Avant de sourire, le math\u00e9maticien se posera une question\u00a0: pour quelle notion de distance\u00a0? Qu’est-ce que \u03c0 ? Archim\u00e8de a r\u00e9pondu \u00e0 cette question voici fort longtemps. Il s\u2019agit du rapport entre la circonf\u00e9rence d\u2019un cercle … Continuer la lecture de Les valeurs de \u03c0<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":543,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[63,112,29],"tags":[208,209,207],"class_list":["post-538","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-histoire","category-humour","category-maths-2","tag-decret-de-lindiana","tag-distances","tag-pi"],"yoast_head":"\nLes valeurs de \u03c0, 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\/10\/06\/les-valeurs-de-\u03c0\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Les valeurs de \u03c0, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"En 1897, une r\u00e9solution \u00e9tablissant que \u03c0\u00a0=\u00a04 fut propos\u00e9e au vote des repr\u00e9sentants de l\u2019\u00e9tat de l\u2019Indiana (\u00c9tats-Unis d\u2019Am\u00e9rique). Avant de sourire, le math\u00e9maticien se posera une question\u00a0: pour quelle notion de distance\u00a0? Qu’est-ce que \u03c0 ? Archim\u00e8de a r\u00e9pondu \u00e0 cette question voici fort longtemps. 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