847 lines
110 KiB
Plaintext
847 lines
110 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h1 style=\"color: indigo\">  Tracer des fractales de Julia :  </h1>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Cette fois il s'agit de tracer des fractales, issues **des mathématiques et non de la nature** comme pour les Lsystem. \n",
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"Gaston Maurice JULIA est un mathématicien du début du XX eme siècle. Il a travaillé sur les itérations de fonctions complexes. \n",
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"Remarque: Il ne disposait pas alors d'ordinateur avec écran graphique pour représenter son travail ;-) \n",
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"Dans les années 1970, Benoît MANDELBROT c'est intéressé aux travaux de JULIA et a inventé le terme de fractale. \n",
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"Voir: https://fr.wikipedia.org/wiki/Ensemble_de_Mandelbrot\n",
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"\n",
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""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\"> 1/ Rappel: les équations du second degré : </h3>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"La théorie développée par Julia est basée sur les nombres complexes. Ces derniers trouvent leur justification dans la résolution des équations du second degré : \n",
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"$a.x² + b.x + c = 0$ \n",
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"La résolution d'une telle équation se fait par le calcul de son discriminant : \n",
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"$\\Delta = b² - 4.a.c$ \n",
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"Le signe de ce dernier nous renseigne sur les solutions de l'équation: \n",
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"$\\Delta > 0 \\rightarrow$ l'équation admet deux racines qui sont distinctes: $x_1 = (-b +\\sqrt \\Delta)/2.a$ et $x_2 = (-b -\\sqrt \\Delta)/2.a$ \n",
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"$\\Delta = 0 \\rightarrow$ l'équation a pour solution une racine double: $x_1 = -b/2.a$ \n",
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"$\\Delta < 0 \\rightarrow$ l'équation n'a pas de solution dans l'ensemble des réels: $\\mathbb{R}$ ... $\\Rightarrow$ mais peut-être dans un autre ensemble, tel que:$\\mathbb{C}$ "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"En résumé: \n",
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"\n",
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" \n",
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"$\\Rightarrow$ Écrire un script permettant à partir d'une équation donnée par l'utilisateur, de préciser si cette équation a des solutions et leurs valeurs."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"L'ensemble des solutions est {2.6666666666666665, -5.0}.\n"
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]
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}
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],
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"source": [
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"'''Résolution de l'équation du 2° degré '''\n",
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"\n",
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"from math import sqrt\n",
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"\n",
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"def solve(a, b, c):\n",
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" d = b**2 - 4*a*c\n",
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" if d<0:\n",
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" return set()\n",
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" return {(-b-sqrt(d))/(2*a), (-b+sqrt(d))/(2*a)}\n",
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"\n",
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"a,b,c = (float(x.strip()) for x in input(\"aX² + bX + c = 0. Entrer a, b, c [format: 'a,b,c']\").split(','))\n",
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"print(f\"L'ensemble des solutions est {solve(a,b,c)}.\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\">2/ Au secours Valentin : c'est quoi un nombre complexe ? De quoi on parle, tu nous expliques ?</h3> \n",
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"\n",
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"Introduction à l'ensemble des complexes: $\\mathbb{C}$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"'''Les nombres complexes avec Python'''\n",
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"\n",
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"a = int(input('Saisir la partie réelle du nombre complexe z: '))\n",
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"b = int(input('Saisir la partie imaginaire du nombre complexe z: '))\n",
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"z = complex(a,b)\n",
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"print(f'z = {z} \\n')\n",
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"print(f'La partie réelle de z est : {z.real}')\n",
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"print(f'La partie imaginaire de z est : {z.imag}i \\n')\n",
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"print(f'Le module de z est : {abs(z)}')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\"> Ce qu'il faut retenir : </h3> \n",
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"\n",
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"$\\Rightarrow$ Il existe un ensemble $\\mathbb{C}$ appelé ensemble des complexes, dans lequel l'ensemble des réels $\\mathbb{R}$ est inclus \n",
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"Un complexe est définit par deux nombres réels : \n",
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"le premier représente la partie réel du complexe, le deuxième sa partie imaginaire à laquelle est associée à la lettre i (ou j) : $z = -2 + 3i$ ou encore $z = -2 + 3j$ \n",
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"\n",
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"**Propriété de l'imaginaire i** : $~i^2 = -1$ (ou encore : $~j^2 = -1$). \n",
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"C'est cette dernière propriété qui permet de résoudre le cas $\\Delta < 0$ pour les équations du 2° degré : \n",
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"\n",
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"\n",
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"Puisque d'un nombre complexe est constitué de deux parties (partie réelle et partie imaginaire), il est tentant de lui associé un point du plan de tel sorte que : \n",
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"- sa partie réelle est portée par l'axe des abscisses (Ox) \n",
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"- sa partie imaginaire est portée par l'axe des ordonnées (Oy). \n",
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"\n",
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"Ainsi au nombre complexe $z = a + bi$ correspond le point M de coordonnées a et b : $M(a, b)$. \n",
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"On dit que : **z est l'affixe du point M**. \n",
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"\n",
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"Application: Utiliser le logiciel ```Geogebra``` (sélectionner le mode **Tableur** => voir menu *Affichage*) pour représenter dans **le plan complexe** les points d'affixe : \n",
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"$z_1 = 2 + 3i; ~ z_2 = 1; ~ z_3 = -1i = -i$ (ne pas oublier de cocher dans Géogebra : *Afficher l'objet*).\n",
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"\n",
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"On constate que la distance du point M (affixe de z) à l'origine est donnée par : $\\sqrt{a^2 + b^2}$. \n",
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"Cette distance représente aussi le module du nombre complexe $~z = a + ib$ : \n",
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"$|z| = \\sqrt{a^2 + b^2}$\n",
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"\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\">3/ Compléter votre algorithme sur la résolution d'une équation du 2° degré :</br>\n",
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" => Ajouter le cas où le discriminant est négatif </h3>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Le point maths, par Valentin\n",
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"\n",
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"## Théorème d'Alembert-Gauss\n",
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"\n",
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"Tout polynôme non constant à coefficients complexes admet une racine dans $\\mathbb{C}$.\n",
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"\n",
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"$\\boxed{\\forall P \\in \\mathbb{C}[X], \\deg P \\geqslant 1, \\exists z \\in \\mathbb{C} \\quad P(z) = 0}$\n",
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"\n",
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"On peut aussi dire que tout polynôme de $\\mathbb{C}[X]$ peut être factorisé, de la forme :\n",
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"\n",
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"$\\displaystyle P(X) = a_n\\prod_{k=1}^n (X - a_k)$ (où $n$ est le degré de $P$).\n",
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"\n",
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"Cela entraîne que la somme des multiplicités des racines distinctes d'un polynôme vaut le degré de ce polynôme.\n",
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"\n",
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"## Un cas particulier : $\\deg P = 2$\n",
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"\n",
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"Soit $P = aX^2 + bX + c$ et soit $\\Delta = b^2 - 4ac$.\n",
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"\n",
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"Soient $\\delta_1$ et $\\delta_2$ les deux racines carrées de $\\Delta$.\n",
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"\n",
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"Alors $z_1 = \\dfrac{-b-\\delta}{2a}$ et $z_2 = \\dfrac{-b+\\delta}{2a}$ vérifient $P(z_1) = 0$ et $P(z_2) = 0$\n",
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"\n",
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"Dans le cas où $z_1 = z_2$, on dit que $z_1$ est de multiplicité 2.\n",
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"\n",
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"Remarquons que $\\delta_1 + \\delta_2 = 0$.\n",
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"\n",
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"La fonction python sqrt du module cmath associe à un complexe $z$ une de ses deux racines carrées. Notons $\\delta$ une telle racine. Alors $\\{\\delta, -\\delta\\}$ est l'image de $z$ par la multifonction (i.e correspondance) qui à un complexe associe ses racines carrées. C'est aussi la coupe du graphe $\\{(x, y) \\in \\mathbb{C}^2 \\mid y^2 = x \\}$ suivant $\\{z\\}$. L'application $f: z \\mapsto z^2$ n'étant pas injective, la correspondance $f^{-1}$ n'est pas univoque."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{(-0.6180339887498949+0j), (1.618033988749895+0j)}"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"'''Résolution d'une équation du second degré dans l'ensemble des complexes'''\n",
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"\n",
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"from cmath import sqrt\n",
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"\n",
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"def solve(a, b, c):\n",
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" d = b**2 - 4*a*c\n",
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" return {(-b-sqrt(d))/(2*a), (-b+sqrt(d))/(2*a)}\n",
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"\n",
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"solve(1, -1, -1)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\"> 4/ Les suites de Julia :</h3> \n",
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"\n",
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"Une **suite d'éléments de $E$** est une application de $\\mathbb{N}$ dans $E$. Ces suites sont soit définies explicitement, soit définies par récurrence.\n",
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"Exemple: $u_{n+1} = u_n * 2 + 3$. On suppose ici que le terme initial est: $u_0 = 1$ :\n",
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"- $u_0 = 1$\n",
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"- $u_1 = 5$\n",
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"- $u_2 = 13$\n",
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"- ... \n",
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"\n",
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"Le calcul d'un terme n d'une suite, avec n très grand, peut-être long et fastidieux. \n",
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"$\\Rightarrow$ La technologie numérique et les ordinateurs s'avèrent d'une grande aide pour le calcul des suites. \n",
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"- Exemple 1 : La suite de Fibonacci, déjà rencontrée en exercice, pour laquelle l'élément $u_n$ dépend de $u_{n-1}$ et $u_{n-2}$ \n",
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"$\\Rightarrow ~f_0 = 0; ~ f_1 = 1 ~~ \\Rightarrow ~ f_n = f_{n-1} + f_{n-2}$ \n",
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"\n",
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"- Exemple 2 : avec le programme du calcul du Lsystem : \n",
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"$\\Rightarrow$ Voir le code ci-dessous : **mettre en évidence** la ligne permettant le ***calcul de la suite*** des termes du Lsystem: à l'aide d'un commentaire."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"ordre 0 => A\n",
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"ordre 1 => AB\n",
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"ordre 2 => ABA\n",
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"ordre 3 => ABAAB\n",
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"ordre 4 => ABAABABA\n"
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]
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}
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],
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"source": [
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"def codage_lSystem(chaine: str, ordre: int) -> str:\n",
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" \"\"\"\n",
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" Prends en entrée une chaîne initiale (l'axiome) et l'ordre souhaité\n",
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" Retourne la chaîne obtenue après n applications du L-system\n",
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" \"\"\"\n",
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" for n in range(ordre): # Itérer le lSystem jusqu'à l'ordre n\n",
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" chaine = lSystem_convert(chaine)\n",
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" return chaine\n",
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"\n",
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"def lSystem_convert(chaine: str) -> str:\n",
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" \"\"\"\n",
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" Prends en entrée une chaîne de caractères\n",
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" Retourne la chaîne obtenue après application des règles du L-System\n",
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" \"\"\"\n",
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" rules = {'A': 'AB', 'B': 'A'} # A compléter : règles\n",
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" return ''.join([rules[car] for car in chaine ]) \n",
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"\n",
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"axiome = 'A'\n",
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"for k in range (5): # Pour affichage des générations jusqu'à l'ordre n\n",
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" print(f'ordre {k} => {codage_lSystem(axiome, k)}')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\">5/ Les ensembles de Julia : </h3>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Un ensemble de Julia $J(c)$ est défini à partir d'une suite ```complexe```, et consiste à voir si la suite $u_n$ \"s'échappe\" d'un domaine donné, lorsque n augmente. \n",
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"Exemple : on considère la suite de Julia telle que: \n",
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"- $u_0 = z$\n",
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"- $u_{n+1} = u_n ^2 + c$ \n",
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"\n",
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"Dans laquelle $~z~$ et $~c~$ sont des nombres complexes : $~~z = a + bi~~$ et $~~c = p + qi$ , avec : \n",
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"- $~z~\\Rightarrow$ qui représente la valeur initiale de la suite $u_0$ ;\n",
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"- $~c~\\Rightarrow$ un paramètre (sous forme *complexe*) de la suite de Julia.\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"u0 = 0.0 + 0.0 i\n",
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"(-0.5+0.6j)\n",
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"(-0.61+0j)\n",
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"(-0.1279+0.6j)\n",
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"(-0.84364159+0.44652j)\n",
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"(0.012351021977728305-0.1534056855336j)\n",
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"(-0.5233807566101395+0.596210566012932j)\n",
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"(-0.5815396226356586-0.024090274277615786j)\n",
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"(-0.1623920086195465+0.6280188980251884j)\n",
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"(-0.8680365718132812+0.396029499397311j)\n",
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"(0.09664812561246894-0.08753617798754354j)\n",
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"(-0.49832172227226335+0.5830795849484489j)\n",
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"(-0.5916572634952607+0.018877554013384867j)\n",
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"(-0.1502980445988279+0.5776619161019134j)\n",
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"(-0.8111037871043028+0.42635708714134046j)\n",
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"(-0.02389101230070656-0.091639696078201j)\n",
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"(-0.5078270534285525+0.6043787302124746j)\n",
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"(-0.6073853333393171-0.013839739437382148j)\n",
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"(-0.13127459523198132+0.6168121095030072j)\n",
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"(-0.8632241590762293+0.43805647998161634j)\n",
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"(0.05326246915897892-0.1562818731200477j)\n"
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]
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}
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],
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"source": [
|
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"def suiteJulia(z: complex, c: complex) -> complex :\n",
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" \"\"\"\n",
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" Calcul de la suite de Julia: \n",
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" En entrée: z et c sont des complexes.\n",
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" Retourne un complexe.\n",
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" \"\"\"\n",
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" u = z\n",
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" print(f'u0 = {(u.real)} + {(u.imag)} i')\n",
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" for i in range (20):\n",
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" u = u **2 + c\n",
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" print(u)\n",
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"\n",
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"z = complex(0, 0)\n",
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"suiteJulia(z, complex(-0.5, 0.6))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<h3 style=\"color: DarkBlue\">6/ Analyse et exploitation des résultats : </h3>\n",
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"<h4 style=\"color: SeaGreen\" class=\"fa fa-book\">61/ Mise en place des points dans le plan complexe : <mark style=\"color: DarkBlue\"> => avec Geogebra : </mark></h4>\n",
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"\n",
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"Examiner les valeurs obtenues en retour à l’exécution du code précédent. \n",
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"Utiliser le logiciel ```Géogebra``` pour placer directement dans le *plan complexe*, les points obtenus : => tester tout d'abord pour un point, saisie en mode tableur et sans oublier de cocher \"Afficher l'objet\". \n",
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"***Remarque*** : *Le mode tableur de ```Géogebra``` accepte de charger des points à partir d'un fichier, .dat par exemple (clic droit).* \n",
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"**Cependant cette fonctionnalité dépend de la version de ```Géogebra``` à votre disposition:** *si votre version ne propose pas cette fonctionnalité, vous pouvez réaliser cette partie avec : Matplotlib*. \n",
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"**Remarque** : Matplolib est moins adapté, puisqu'il vous faudra vous même mettre en place les points dans le plan complexe, à l'aide de leur partie réelle et partie imaginaire. \n",
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"\n",
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"Afin de pouvoir charger les points à partir d'un fichier, il faut adapter le code ci-dessus afin d'enregistrer les points dans un fichier ```Julia.dat``` et transformer le type ```complex``` de Python pour l'adapter au type reconnu par ```Géogebra``` : \n",
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"$~a+bj \\Rightarrow a+bi~$. \n",
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"A voir : l'exercice sur le tracé d'une spirale dans les activités de révision et aussi [ici](https://python.doctor/page-lire-ecrire-creer-fichier-python \"python.doctor: lire ecrire dans un fichier\")."
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]
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},
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{
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||
"cell_type": "code",
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||
"execution_count": 17,
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||
"metadata": {},
|
||
"outputs": [
|
||
{
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||
"data": {
|
||
"text/plain": [
|
||
"0j"
|
||
]
|
||
},
|
||
"execution_count": 17,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"\" Saisir ici le code précédent, adapté pour générer un fichier : Julia.dat => et l'importer dans Géogebra \"\n",
|
||
"\n",
|
||
"def suiteJuliaFichier(z: complex, c: complex) -> complex :\n",
|
||
" \"\"\"\n",
|
||
" Calcul de la suite de Julia: \n",
|
||
" En entrée: z et c sont des complexes.\n",
|
||
" Retourne un complexe.\n",
|
||
" \"\"\"\n",
|
||
" with open('Julia.dat', 'w') as file:\n",
|
||
" u = z\n",
|
||
" print(f'u_0 = {(u.real)} + {(u.imag)} i', file=file)\n",
|
||
" for i in range (20):\n",
|
||
" u = u **2 + c\n",
|
||
" print(f'u_{i+1} = {(u.real)} + {(u.imag)} i', file=file)\n",
|
||
" return z\n",
|
||
"z = complex(0, 0)\n",
|
||
"suiteJuliaFichier(z, complex(-0.5, 0.6))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h4 ><mark style= \"color: DarkBlue\"> => avec Matplotlib : </mark></h4>\n",
|
||
"\n",
|
||
"Pour représenter les points avec Matplotlib, il vous faut extraire la partie réel et la partie imaginaire des résultats renvoyé par la fonction ```suiteJulia()```. \n",
|
||
"**=> Compléter le code suivant :**"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 22,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"u0 = 0.0 + 0.0 i\n"
|
||
]
|
||
},
|
||
{
|
||
"data": {
|
||
"image/png": 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",
|
||
"text/plain": [
|
||
"<Figure size 1000x1000 with 1 Axes>"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"\"Représentation des point dans le plan complexe, avec Matplotlib :\"\n",
|
||
"\n",
|
||
"import matplotlib.pyplot as plt\n",
|
||
"\n",
|
||
"def suiteJulia(u: complex, c: complex) -> 'list[complex]' :\n",
|
||
" \"\"\"\n",
|
||
" Calcul de la suite de Julia: \n",
|
||
" En entrée: z et c sont des complexes.\n",
|
||
" Retourne une liste de complexes.\n",
|
||
" \"\"\"\n",
|
||
" u = z\n",
|
||
" s = [u]\n",
|
||
" print(f'u0 = {(u.real)} + {(u.imag)} i')\n",
|
||
" for i in range (20):\n",
|
||
" u = u **2 + c\n",
|
||
" s.append(u)\n",
|
||
" return s\n",
|
||
"\n",
|
||
"def tracer_les_points(listePoints):\n",
|
||
" \"Pour tracer les points :\"\n",
|
||
" plt.figure(figsize=(10,10))\n",
|
||
" plt.xlim([-2, 2])\n",
|
||
" plt.ylim([-2, 2])\n",
|
||
" plt.scatter([z.real for z in listePoints], [z.imag for z in listePoints])\n",
|
||
" plt.show()\n",
|
||
" plt.close()\n",
|
||
"\n",
|
||
"u = complex(0, 0)\n",
|
||
"tracer_les_points(suiteJulia(u, complex(-0.5, 0.6)))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h4 style=\"color: SeaGreen\" class=\"fa fa-book\">62/ Le point d'affixe z = 0 + 0i appartient-il à l'ensemble de Julia J(c) ? </h4>\n",
|
||
"<h4><mark style=\"color: DarkBlue\"> => avec Geogebra : </mark></h4>\n",
|
||
"\n",
|
||
"Pour le savoir : tracer dans ```géogebra``` un cercle de centre 0 et de rayon 2. Si les points de la suite sont dans ce cercle, c'est donc que le point z appartient à l'ensemble de Julia (du moins en ce qui concerne les vingts premiers termes de la suite). \n",
|
||
"\n",
|
||
"Essayer maintenant la suite, pour laquelle: $~z = 0.239 + 0.2i~$ et $~c = -0.5 + 0.6i$ \n",
|
||
"Cette fois vous devriez constater que les dernières valeurs de la suite \"s'échappent\" du cercle. \n",
|
||
"$\\Rightarrow$ On en déduit que le point M d'affixe $~z = 0.239 + 0.2i~$ n'appartient pas à l'ensemble J(c). \n",
|
||
"\n",
|
||
"<h4 ><mark style= \"color: DarkBlue\"> => avec Matplotlib : </mark></h4>\n",
|
||
"\n",
|
||
"Reprendre les étapes décrites ci-dessus en les adaptant à Matplotlib **=> compléter le code ci-dessous:**"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"u0 = 0.0 + 0.0 i\n",
|
||
"u0 = 0.0 + 0.0 i\n"
|
||
]
|
||
},
|
||
{
|
||
"data": {
|
||
"image/png": 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",
|
||
"text/plain": [
|
||
"<Figure size 1000x1000 with 1 Axes>"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"\"Savoir si un point z appartient à l'ensemble de Julia J(c) :\"\n",
|
||
"\n",
|
||
"import matplotlib.pyplot as plt\n",
|
||
"\n",
|
||
"def suiteJulia(u: complex, c: complex) -> 'list[complex]' :\n",
|
||
" \"\"\"\n",
|
||
" Calcul de la suite de Julia: \n",
|
||
" En entrée: z et c sont des complexes.\n",
|
||
" Retourne une liste de complexes.\n",
|
||
" \"\"\"\n",
|
||
" u = z\n",
|
||
" s = [u]\n",
|
||
" print(f'u0 = {(u.real)} + {(u.imag)} i')\n",
|
||
" for i in range (20):\n",
|
||
" u = u **2 + c\n",
|
||
" s.append(u)\n",
|
||
" return s\n",
|
||
"\n",
|
||
"def tracer_les_points(listePoints):\n",
|
||
" \"Pour tracer les points :\"\n",
|
||
" plt.figure(figsize=(10,10))\n",
|
||
" plt.xlim([-2, 2])\n",
|
||
" plt.ylim([-2, 2])\n",
|
||
" plt.scatter([z.real for z in listePoints], [z.imag for z in listePoints])\n",
|
||
" cercle = plt.Circle((0, 0), 2,fill=False)\n",
|
||
" plt.gcf().gca().add_artist(cercle)\n",
|
||
" plt.show()\n",
|
||
" plt.close()\n",
|
||
"\"\"\"\n",
|
||
"def cercle_limite():\n",
|
||
" cercle = plt.Circle((0, 0), 2,fill=False)\n",
|
||
" plt.gcf().gca().add_artist(cercle) #gca = Get Current Axis\n",
|
||
" plt.show()\n",
|
||
" plt.close()\n",
|
||
"\"\"\"\n",
|
||
"u = complex(0, 0)\n",
|
||
"suiteJulia(u, complex(-0.5, 0.6))\n",
|
||
"tracer_les_points(suiteJulia(u, complex(-0.5, 0.6)))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h4 style=\"color: SeaGreen\" class=\"fa fa-book\">63/ Savoir si un point appartient à l'ensemble de Julia : </h4>\n",
|
||
"\n",
|
||
"Pour savoir si un point $M$ d'affixe $z$ appartient à l'ensemble de Julia J(c), on peut se limiter à examiner ***le module $u_n$*** de la suite de Julia : \n",
|
||
"D'après la définition du module d'un nombre complexe (distance du point $M$ d'affixe $z = a + jb$, à l'origine $O$ du plan complexe), regarder si un point de la suite \"s'échappe\" du domaine de Julia, limité par le cercle de centre $0$ et de rayon 2, revient à examiner si **le module du complexe $~u_n~$ calculé est supérieur à 2**. \n",
|
||
"On peut d'ailleurs démontrer que la suite tend vers l'infini (n'appartient donc pas à J(c)) lorsque justement ses termes $u_n$ sortent du cercle de rayon 2, centré en $O$. \n",
|
||
"\n",
|
||
"Appliquer cette propriété pour examiner et conclure sur l'évolution de $|u_n|$ pour les valeurs de $~z~$ et du paramètre $~c~$ suivantes :"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 29,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"u0 = 0.0 + 0.0 i\n",
|
||
"0.7810249675906654\n",
|
||
"0.61\n",
|
||
"0.6134805701894722\n",
|
||
"0.9545214731883868\n",
|
||
"0.15390208607399739\n",
|
||
"0.7933438443797638\n",
|
||
"0.5820383784682888\n",
|
||
"0.6486747264540702\n",
|
||
"0.9541105042909015\n",
|
||
"0.13039724936159686\n",
|
||
"0.7670113045268956\n",
|
||
"0.5919583427000827\n",
|
||
"0.5968942884001868\n",
|
||
"0.9163349383334626\n",
|
||
"0.0947027685237214\n",
|
||
"0.789406084171619\n",
|
||
"0.6075429874036964\n",
|
||
"0.6306268292602768\n",
|
||
"0.9680131344495009\n",
|
||
"0.16510879590991473\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"'''Modifier le code de la fonction suite de Julia afin d'afficher 20 itérations du module de u => abs(u)\n",
|
||
"Ceci pour les valeurs suivantes de la suite :\n",
|
||
"z = 0 + 0i et c = -0.5 + 0.6i\n",
|
||
"z = 0.239 + 0.2i et c = -0.5 + 0.6i\n",
|
||
"z = 0.25 + 0.2i et c = -0.5 + 0.6i\n",
|
||
"'''\n",
|
||
"\n",
|
||
"def suiteJulia(z: complex, c: complex) -> complex :\n",
|
||
" \"\"\"\n",
|
||
" Calcul de la suite de Julia: \n",
|
||
" En entrée: z et c sont des complexes.\n",
|
||
" Retourne un complexe.\n",
|
||
" \"\"\"\n",
|
||
" u = z\n",
|
||
" print(f'u0 = {(u.real)} + {(u.imag)} i')\n",
|
||
" for i in range (20):\n",
|
||
" u = u **2 + c\n",
|
||
" print(abs(u))\n",
|
||
"\n",
|
||
"z = complex(0, 0)\n",
|
||
"suiteJulia(z, complex(-0.5, 0.6))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h3 style=\"color: DarkBlue\">7/ Construction des fractales de Julia : </h3>\n",
|
||
"<h4 style=\"color: SeaGreen\">71/ Calculer l'ensemble de Julia : </h4>\n",
|
||
"\n",
|
||
"**Écrire une fonction ```ensembleJulia```** qui devra itérer 100 fois (ou plus) le calcul de la suite de Julia à partir du point $~M~$ d'affixe $~z~$ et du paramètre $~c~$. \n",
|
||
"Cette fonction doit renvoyer : \n",
|
||
"- -1 si le point $M$ appartient à J(c) => dans ce cas toutes les itérations donnent un module inférieur à 2;\n",
|
||
"- l'indice i du tour de boucle qui correspond au premier terme de la suite supérieur à 2. "
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 30,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"def ensembleJulia (z: complex, c: complex) -> int :\n",
|
||
" '''Prend une suite de Julia et un point d'affixe z :\n",
|
||
" Retourne => -1 si le point est dans l'ensemble J(c): le module de tous les termes est < 2\n",
|
||
" Retourne l'indice i du premier terme ayant un module supérieur à 2\n",
|
||
" '''\n",
|
||
"\n",
|
||
" u = z\n",
|
||
" print(f'u0 = {(u.real)} + {(u.imag)} i')\n",
|
||
" for i in range (20):\n",
|
||
" u = u **2 + c\n",
|
||
" if abs(u) > 2:\n",
|
||
" return i\n",
|
||
" return -1"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h4 style = 'color: SeaGreen'> 72/ Changement de repère :</h4>\n",
|
||
"\n",
|
||
"Puisqu'il s'agit de tracer une image à l'écran, nous allons faire en sorte que les points $M$ (de la suite de Julia) appartiennent à l'écran de l'ordinateur. Donc le plan complexe sera représenté par l'écran de l'ordinateur. \n",
|
||
"L'écran est constitué de pixels ayant $x$ et $y$ pour coordonnées. Les dimensions de l'image de la fractale seront comprises entre 0 et 400 pixels pour $x$ et pour $y$. \n",
|
||
"Les ensembles de Julia intéressant sont pour $z = a + ib$ ayant une partie réelle $a$ et une partie imaginaire $b$ comprises entre -1.25 et +1.25. \n",
|
||
"Un changement de repère est donc nécessaire pour faire correspondre la partie du plan complexe pertinente (-1.25 -> +1.25) avec les pixels de notre image fractale (0 -> 400). \n",
|
||
"\n",
|
||
"\n",
|
||
"\n",
|
||
"\n",
|
||
"**Établir les équations** pour passer de l'image en pixel, à la valeur de $z$, et **écrire** la fonction ```convert``` correspondante :\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"def convert(x:int, y:int) ->complex :\n",
|
||
" \"Convertit un point du plan (de l'image) de coordonnées x, y en un nombre complexe z = re + img*j\"\n",
|
||
" return complex(x/160 - 1.25, y/160 - 1.25)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h4 style = 'color: SeaGreen'> 73/ Gestion des couleurs :</h4>\n",
|
||
"\n",
|
||
"A chaque point de notre fractale correspond un affixe $z$ utilisé dans la fonction ```ensembleJulia``` pour savoir si ce point appartient ou pas à J(c). \n",
|
||
"Nous allons affecter une couleur à ce point, en fonction de sont appartenance ou pas à J(c). Dans ce dernier cas, sa couleur sera fonction du rang du terme pour lequel il s'échappe de J(c). \n",
|
||
"*Rappel: la couleur d'un pixel est définie par un triplet de nombres, chacun compris entre 0 et 255.* \n",
|
||
"Principe: si n représente le nombre renvoyé par la fonction ```ensembleJulia```:\n",
|
||
"- si n=-1 le pixel correspondant est noir.\n",
|
||
"- et pour 0< n <100 : \n",
|
||
"=> la couleur (n, n , n) donnera un dégradé allant de noir à gris moyen (100, 100, 100) \n",
|
||
"=> la couleur (2n, 0 , 0) donnera un dégradé allant de noir au rouge vif (200, 0, 0) \n",
|
||
"=> la couleur (0, 0, 3n%256) donnera un premier dégradé allant de noir au bleu vif (0, 0, 255), puis lorsque 3n dépasse 255 on reviend à zéro, jusqu'à atteindre la couleur (0, 0, 44) pour n=100. \n",
|
||
"\n",
|
||
"Pour commencer nous allons faire un \"mix\" de tout cela, soit par exemple : (4n%256, 2n, 6n%256). \n",
|
||
"Puis tester aussi la formule suivante: (255-log10(n)x127, 255-log10(n)x127, log10(n)x127). \n",
|
||
"Et pourquoi pas tester aussi une variante personnelle. \n",
|
||
"\n",
|
||
"**Écrire une fonction ```colorise```** permettant de fixer une couleur (d'après les formules ci-dessus) à chaque pixel d'un rectangle, en fonction d'une valeur de $n$, avec $-1<n<100$ : "
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 37,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from PIL import Image\n",
|
||
"from math import log10, sqrt\n",
|
||
"\n",
|
||
"from random import randint\n",
|
||
"\n",
|
||
"def colorise(img, taille):\n",
|
||
" \"Colorise un carré de pixel, avec des bandes, selon les formules proposées ci-dessus :\"\n",
|
||
" \n",
|
||
" for x in range(400):\n",
|
||
" for y in range(400):\n",
|
||
" n = int(sqrt((x-200)**2 + (y-200)**2)) # essayer de faire un truc joli\n",
|
||
" if n == -1:\n",
|
||
" img.putpixel((x, y), (0,0,0))\n",
|
||
" else:\n",
|
||
" img.putpixel((x, y), (4*n%256, 2*n, 6*n%256))\n",
|
||
"\n",
|
||
" # Faire appel aux méthodes :\n",
|
||
" # putpixel(); save() et show()\n",
|
||
" # Voir : #https://info.blaisepascal.fr/pillow\n",
|
||
"\n",
|
||
"taille = 400\n",
|
||
"img = Image.new('RGB',(taille,taille),(255,255,255))\n",
|
||
"colorise(img, taille)\n",
|
||
"img.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h4 style = 'color: SeaGreen'> 74/ Association des fonctions précédentes :</h4>\n",
|
||
"\n",
|
||
"Associer les fonctions précédentes pour obtenir le tracé des fractales de Julia. \n",
|
||
"La fonction ```dessineFractale``` est une évolution de la fonction ```colorise``` qui gère le tracé (position et couleur de chaque pixel) de la fractale de Julia, à l'aide des fonctions : ```convert``` et ```ensembleJulia```. \n",
|
||
"Tester les fractales pour les valeurs du paramètre $c$ successivement égal à : \n",
|
||
"- $-0.5+0.6i$\n",
|
||
"- $-0.8-0.18i$\n",
|
||
"- $0-0.8i$\n",
|
||
"- $0.285+0.013i$\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 53,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from PIL import Image\n",
|
||
"from math import log10\n",
|
||
"\n",
|
||
"#***Déclaration des constantes :***\n",
|
||
"taille = 1000\n",
|
||
"reMax = -0.5\n",
|
||
"reMin = 0.5\n",
|
||
"imgMax = -0.5\n",
|
||
"imgMin = 0.5\n",
|
||
"\n",
|
||
"#***Déclaration des fonction locales :***\n",
|
||
"def ensembleJulia (z: complex, c: complex) -> int :\n",
|
||
" '''Prend une suite de Julia et un point d'affixe z :\n",
|
||
" Retourne => -1 si le point est dans l'ensemble J(c): le module de tous les termes est < 2\n",
|
||
" Retourne l'indice i du premier terme ayant un module supérieur à 2\n",
|
||
" '''\n",
|
||
" u = z\n",
|
||
" for i in range (20):\n",
|
||
" u = u **2 + c\n",
|
||
" if abs(u) > 2:\n",
|
||
" return i\n",
|
||
" return -1\n",
|
||
"\n",
|
||
"def convert(x:int, y:int, taille:int) ->complex :\n",
|
||
" \"Convertit un point du plan (de l'image) de coordonnées x, y en un nombre complexe z = re + img*j\"\n",
|
||
" return complex(x*(reMax-imgMin)/taille - (reMax-reMin)/2, y*(imgMax-imgMin)/taille - (imgMax-imgMin)/2)\n",
|
||
"\n",
|
||
"def dessineFractale(img, taille, c):\n",
|
||
" \"Construit la fractale à partir d'une carré de pixels à colorer en fonction de la valeur de n :\"\n",
|
||
" for x in range(taille):\n",
|
||
" for y in range(taille):\n",
|
||
" n = ensembleJulia(convert(x, y, taille), c)\n",
|
||
" if n == -1:\n",
|
||
" img.putpixel((x, y), (0,0,0))\n",
|
||
" else:\n",
|
||
" img.putpixel((x, y), (4*n%256, 2*n, 6*n%256))\n",
|
||
"\n",
|
||
"#*** Programme principal :***\n",
|
||
"\"Tester les fractales pour le paramètre c = -0.5+0.6j; -0.8-0.18j; 0-0.8j; 0.285+0.013j \"\n",
|
||
"img = Image.new('RGB',(taille,taille),(255,255,255))\n",
|
||
"dessineFractale(img, taille, 0-0.8j)\n",
|
||
"img.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h3 style=\"color: DarkBlue\">8/ Mise en évidence de la structure de la fractale : </h3>\n",
|
||
"\n",
|
||
"Une figure fractale est un objet mathématique qui présente une structure similaire à toutes les échelles. \n",
|
||
"A partir d'une fractale de niveau 0, vérifier qu'il est possible d'obtenir une fractale de niveau 1, puis 2 et ainsi de suite : \n",
|
||
"=> Pour cela adapter les constantes du script (taille etc...) pour zoomer de plus en plus.\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<h3 style=\"color: DarkBlue\">9/ Ensemble de Mandelbrot : </h3>\n",
|
||
"\n",
|
||
"Prolongement possible aux ensembles de Julia : $\\Rightarrow$ les ensembles de Mandelbrot. \n",
|
||
"A voir :\n",
|
||
"- https://fr.wikipedia.org/wiki/Ensemble_de_Mandelbrot\n",
|
||
"- le logiciel Xaos."
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"kernelspec": {
|
||
"display_name": "Python 3.8.10 64-bit",
|
||
"language": "python",
|
||
"name": "python3"
|
||
},
|
||
"language_info": {
|
||
"codemirror_mode": {
|
||
"name": "ipython",
|
||
"version": 3
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython3",
|
||
"version": "3.8.10"
|
||
},
|
||
"orig_nbformat": 4,
|
||
"vscode": {
|
||
"interpreter": {
|
||
"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
||
}
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 2
|
||
}
|