{"id":1087,"date":"2019-03-19T13:34:03","date_gmt":"2019-03-19T12:34:03","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1087"},"modified":"2020-09-06T17:17:36","modified_gmt":"2020-09-06T15:17:36","slug":"python-turtle-et-un-arbre","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/19\/python-turtle-et-un-arbre\/","title":{"rendered":"Python, turtle et un arbre fractal"},"content":{"rendered":"\n

Utiliser Python, notamment module turtle<\/em>, pour construire un arbre fractal, c’est possible ! Je ne suis pas trop fan de ce module (car tr\u00e8s lent), mais il faut bien avouer que le r\u00e9sultats est sympatoche… comme disent les jeunes !<\/p>\n\n\n\n\n\n\n\n

Je me suis donc mis \u00e0 la recherche d’un code donnant un tel sch\u00e9ma pour tenter de me familiariser avec le module. J’avais une id\u00e9e bien pr\u00e9cise : celle d’un arbre fractal, que j’avais vu quelque part.<\/p>\n\n\n\n

Apr\u00e8s quelques recherches, je suis enfin tomb\u00e9 sur le code suivant (que je me suis permis de commenter):<\/p>\n\n\n\n

from turtle import *\n\nangle = 30\ncolor('#3f1905')\n\ndef arbre(n,longueur):\n    if n==0:\n        color('green')\n        forward(longueur) # avance\n        backward(longueur) # recule\n        color('#3f1905')\n    else:\n        width(n)\n        forward(longueur\/3) #avance\n        left(angle) # tourne vers la gauche de angle degr\u00e9s\n        arbre(n-1,longueur*2\/3)\n        right(2*angle) # tourne vers la droite de angle degr\u00e9s\n        arbre(n-1,longueur*2\/3)\n        left(angle) # tourne vers la gauche de angle degr\u00e9s\n        backward(longueur\/3) # recule\n\nhideturtle() # cache la tortue\nup() # l\u00e8ve le stylo\nright(90) # tourne de 90 degr\u00e9s vers la droite \nforward(300) # avance de 300 pixels\nleft(180) # fait un demi-tour\ndown() # pose le stylo\narbre(11,700) # ex\u00e9cute la macro\nshowturtle() # affiche la tortue\nmainloop()<\/pre>\n\n\n\n
qui donne ceci:<\/p>\n\n\n\n
\"python<\/a>
Arbre fractal r\u00e9alis\u00e9 avec Python avec un angle de 30\u00b0<\/figcaption><\/figure><\/div>\n\n\n\n
Mais attention… Ce magnifique dessin Python obtenu avec turtle<\/em> repr\u00e9sentant un arbre fractal met un certain temps avant d’\u00eatre fini (par exemple, sur mon ordinateur 16 Mo de RAM, processeur Intel Core i5, il met une vingtaine de minutes… arf !).<\/p>\n\n\n\n
Si on modifie l’angle (disons pour le mettre \u00e0 50\u00b0), on obtient ceci:<\/p>\n\n\n\n
\"python<\/a>
Arbre fractal r\u00e9alis\u00e9 avec Python avec un angle de 50\u00b0<\/figcaption><\/figure><\/div>\n\n\n\n
Et avec un angle de 110\u00b0 :<\/p>\n\n\n\n
\"python<\/a>
Arbre fractal r\u00e9alis\u00e9 avec Python avec un angle de 110\u00b0<\/figcaption><\/figure><\/div>\n\n\n\n
N’est-il pas choupinou ? \ud83d\ude42<\/p>\n\n\n\n
Vous pouvez remarquer dans le code la fonction r\u00e9cursive : c’est une notion que l’on verra en NSI (nouveau lyc\u00e9e)… mais pr\u00e9vue en cycle de maturit\u00e9 (en Terminale quoi !)… m\u00eame si je n’ai pas pu r\u00e9sister \u00e0 l’envie d’en mettre dans mon livre d’exercices corrig\u00e9s qui sortira en version papier pour la rentr\u00e9e 2019.<\/p>\n\n\n\n
Une fonction r\u00e9cursive est une fonction qui fait appel \u00e0 elle-m\u00eame dans sa d\u00e9finition. On retrouve ce genre de fonction par exemple pour calculer le pgcd de deux nombres:<\/p>\n\n\n\n
def pgcd(a,b):\n    if b==0:\n        return a\n    else:\n        r=a%b\n        return pgcd(b,r)\n \nprint(pgcd(56,42))<\/pre>\n\n\n\n
Ici, la r\u00e9cursivit\u00e9 s’appuie sur la propri\u00e9t\u00e9 du pgcd qui stipule que:$$\\text{pgcd}(a,b)=\\text{pgcd}(b,r)$$o\u00f9 \\(a = bq + r,\\ 0\\leq r < b\\).<\/p>\n\n\n\n
Les math\u00e9matiques sont donc tr\u00e8s importantes pour construire des algorithmes (et par suite des programmes) performants. En effet, on aurait tr\u00e8s bien pu calculer le pgcd de la mani\u00e8re suivante:<\/p>\n\n\n\n
def pgcd(a,b):\n    while b<>0:\n        r=a%b\n        a,b=b,r\n    return a\n \nprint(pgcd(56,42))<\/pre>\n\n\n\n
Pour comparer deux algorithmes, on parle souvent de complexit\u00e9<\/em>. Cet article ne parle pas de cette notion, mais sachez tout de m\u00eame que la complexit\u00e9 d’une fonction r\u00e9cursive est donn\u00e9e par une suite arithm\u00e9tico-g\u00e9om\u00e9trique. Dans la fonction pgcd, il y a 2 instructions \u00e9l\u00e9mentaires (le test sur b et l’affectation de r). La complexit\u00e9 \\(C_n\\) est donc \u00e9gale \u00e0 \\(2+C_{n-1}\\), o\u00f9 \\(C_{n-1}\\) est celle de la fonction pgcd(b,r). On d\u00e9montre alors que la complexit\u00e9 de la fonction r\u00e9cursive est de l’ordre de O<\/em>(n<\/em>) [complexit\u00e9 lin\u00e9aire<\/em>]. En fait, la complexit\u00e9 est linaire dans le pire des cas (quand a<\/em> et b<\/em> sont deux nombres successifs de la suite de Fibonacci), car dans les autres cas, la complexit\u00e9 est logarithmique [O<\/em>(ln(n<\/em>))].<\/p>\n\n\n\n
L’\u00e9valuation moyenne de la complexit\u00e9 de l’algorithme d’Euclide version r\u00e9cursive est assez compliqu\u00e9e. Le nombre d’appels moyen de la fonction pgcd est:$$\\frac{12\\ln(2\\ln n)}{\\pi^2}+1,47.$$<\/p>http:\/\/imss-www.upmf-grenoble.fr\/prevert\/Prog\/Complexite\/euclide.html<\/a><\/cite><\/blockquote>\n\n\n\n
La complexit\u00e9 de l’algorithme non r\u00e9cursif est quasi-identique \u00e0 sa version r\u00e9cursive.<\/p>\n\n\n\n
 
La version r\u00e9cursive de l’algorithme d’Euclide est peut-\u00eatre un peu plus facile \u00e0 \u00e9crire que la version it\u00e9rative. Les deux versions ont fondamentalement la m\u00eame complexit\u00e9, avec un petit avantage \u00e0 la version it\u00e9rative, car l’appel d’une fonction n’est pas gratuit. <\/p>https:\/\/www.labri.fr\/perso\/betrema\/deug\/poly\/euclide.html<\/a><\/cite><\/blockquote>\n\n\n\n
Mais l\u00e0, je m’\u00e9gare… Saint-Lazare !<\/p>\n","protected":false},"excerpt":{"rendered":"
Utiliser Python, notamment module turtle, pour construire un arbre fractal, c’est possible ! Je ne suis pas trop fan de ce module (car tr\u00e8s lent), mais il faut bien avouer que le r\u00e9sultats est sympatoche… comme disent les jeunes !<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,4,6,5],"tags":[93,92,91,90],"class_list":["post-1087","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-informatique","category-mathematiques","category-python","tag-complexite","tag-fractale","tag-recursivite","tag-turtle"],"yoast_head":"\nPython, turtle et un arbre fractal fractal - Mathweb.fr - Exemple de code<\/title>\n<meta name=\"description\" content=\"Utiliser Python, notamment module turtle, pour construire un arbre fractal, c'est possible ! 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