La suite de Fibonacci est-elle vraiment partout dans la nature ? \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --jungle: #1a4a2e;\n --jungle-clair: #2ecc71;\n --jungle-pale: #eafaf1;\n --or: #c8a010;\n --or-clair: #f0d060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n .hero {\n background: #0a1f12;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"1 1 2 3 5 8\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 4rem;\n opacity: 0.06;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; 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}\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n .fib-num { width: 36px; height: 36px; font-size: 0.8rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Nature & Maths \u00b7 Article #10<\/div>\n <h1>La suite de Fibonacci<br>est-elle <em>vraiment<\/em><br>partout dans la nature ?<\/h1>\n <p class=\"hero-intro\">On nous dit que les tournesols, les coquillages, les ananas et m\u00eame la galaxie suivent la suite de Fibonacci. C’est vrai \u2014 en partie. Mais c’est aussi l’un des mythes math\u00e9matiques les plus exag\u00e9r\u00e9s de l’histoire. Aujourd’hui, on d\u00e9m\u00eale le vrai du faux.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 12 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Regarde un ananas au march\u00e9 de Dantokpa. Compte les rang\u00e9es de ses \u00e9cailles en spirale. Tu trouveras 8 rang\u00e9es dans un sens et 13 dans l’autre. Ou peut-\u00eatre 13 et 21. Regarde un tournesol s\u00e9ch\u00e9. Compte ses graines en spirale : 34 dans un sens, 55 dans l’autre. Ouvre une pomme de pin de la for\u00eat de la Lama : 8 spirales d’un c\u00f4t\u00e9, 13 de l’autre.\n <\/p>\n\n <p>\n 8, 13, 21, 34, 55… Ces nombres t’\u00e9voquent quelque chose ? Ils appartiennent tous \u00e0 la m\u00eame suite math\u00e9matique \u2014 la <strong>suite de Fibonacci<\/strong>. Et leur pr\u00e9sence dans la nature n’est pas un hasard. Mais elle n’est pas non plus le myst\u00e8re cosmique qu’on te fait parfois croire.\n <\/p>\n\n <h2>Qui \u00e9tait Fibonacci ?<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Leonardo de Pise, dit Fibonacci \u2014 XII<sup>e<\/sup> si\u00e8cle<\/span>\n <p>\n Leonardo Fibonacci est n\u00e9 \u00e0 Pise (Italie) vers 1170. Son p\u00e8re \u00e9tait commer\u00e7ant et voyageait beaucoup en Afrique du Nord \u2014 notamment \u00e0 B\u00e9ja\u00efa, en Alg\u00e9rie actuelle, o\u00f9 le jeune Leonardo apprit les math\u00e9matiques arabes et les chiffres indo-arabes que nous utilisons encore aujourd’hui (1, 2, 3…).\n <\/p>\n <p>\n En 1202, il publie le <em>Liber Abaci<\/em> \u2014 le livre du calcul \u2014 qui r\u00e9volutionne les math\u00e9matiques europ\u00e9ennes. C’est dans ce livre qu’appara\u00eet, \u00e0 titre d’exemple sur la reproduction des lapins, la suite qui portera son nom.\n <\/p>\n <p>\n Ironie de l’histoire : Fibonacci n’a pas d\u00e9couvert cette suite. Elle \u00e9tait connue des math\u00e9maticiens indiens depuis le VI<sup>e<\/sup> si\u00e8cle, notamment par Pingala qui l’utilisait dans l’\u00e9tude de la po\u00e9sie sanskrite. Fibonacci l’a simplement popularis\u00e9e en Europe.\n <\/p>\n <\/div>\n\n <h2>La suite : simple \u00e0 d\u00e9finir, infinie en richesse<\/h2>\n\n <p>\n La r\u00e8gle est d’une simplicit\u00e9 enfantine : <strong>chaque terme est la somme des deux pr\u00e9c\u00e9dents<\/strong>. On commence avec 0 et 1.\n <\/p>\n\n <div class=\"fib-sequence\">\n <div class=\"fib-num\">0<\/div>\n <div class=\"fib-plus\">+<\/div>\n <div class=\"fib-num\">1<\/div>\n <div class=\"fib-plus\">=<\/div>\n <div class=\"fib-num or\">1<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">2<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">3<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">5<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">8<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">13<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">21<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">34<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">55<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">…<\/div>\n <\/div>\n\n <div class=\"definition\">\n <strong>Suite de Fibonacci :<\/strong> F(0) = 0, F(1) = 1, et pour tout n \u2265 2 : F(n) = F(n\u22121) + F(n\u22122).\n <br><br>\n La suite compl\u00e8te : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…\n <\/div>\n\n <h2>Le nombre d’or : le grand secret de Fibonacci<\/h2>\n\n <p>\n Voici la propri\u00e9t\u00e9 la plus stup\u00e9fiante de cette suite. Divise chaque terme par le pr\u00e9c\u00e9dent et observe ce qui se passe :\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>F(n)<\/th>\n <th>F(n-1)<\/th>\n <th>Rapport F(n)\/F(n-1)<\/th>\n <\/tr>\n <tr><td>2<\/td><td>1<\/td><td>2,000 000<\/td><\/tr>\n <tr><td>3<\/td><td>2<\/td><td>1,500 000<\/td><\/tr>\n <tr><td>5<\/td><td>3<\/td><td>1,666 667<\/td><\/tr>\n <tr><td>8<\/td><td>5<\/td><td>1,600 000<\/td><\/tr>\n <tr><td>13<\/td><td>8<\/td><td>1,625 000<\/td><\/tr>\n <tr><td>21<\/td><td>13<\/td><td>1,615 385<\/td><\/tr>\n <tr><td>55<\/td><td>34<\/td><td>1,617 647<\/td><\/tr>\n <tr><td>144<\/td><td>89<\/td><td>1,617 978<\/td><\/tr>\n <tr><td>377<\/td><td>233<\/td><td><strong>1,618 026<\/strong><\/td><\/tr>\n <\/table>\n <\/div>\n\n <p>\n Les rapports convergent vers une valeur fixe : <strong>\u03c6 \u2248 1,618033…<\/strong> C’est le <strong>nombre d’or<\/strong> \u2014 aussi appel\u00e9 phi (\u03c6), l’un des nombres les plus c\u00e9l\u00e8bres et les plus myst\u00e9rieux des math\u00e9matiques.\n <\/p>\n\n <div class=\"formule\">\u03c6 = (1 + \u221a5) \/ 2 \u2248 1,618033988…<\/div>\n\n <p>\n Le nombre d’or a une propri\u00e9t\u00e9 unique : si tu lui enl\u00e8ves 1, tu obtiens son inverse. Autrement dit : \u03c6 \u2212 1 = 1\/\u03c6. Aucun autre nombre ne poss\u00e8de cette propri\u00e9t\u00e9.\n <\/p>\n\n <h2>Dans la nature : ce qui est vrai<\/h2>\n\n <h3>\u2705 Les phyllotaxies \u2014 vrai et expliqu\u00e9<\/h3>\n\n <p>\n La <strong>phyllotaxie<\/strong> est l’arrangement des feuilles, graines et p\u00e9tales sur les plantes. Et l\u00e0, les nombres de Fibonacci sont bien r\u00e9els \u2014 et ont une explication scientifique solide.\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf3f Exemples v\u00e9rifi\u00e9s en Afrique et ailleurs<\/span>\n <p><strong>L’ananas :<\/strong> 8 spirales dans un sens, 13 dans l’autre. Toujours des nombres de Fibonacci cons\u00e9cutifs.<\/p>\n <p><strong>Le tournesol :<\/strong> 34 spirales d’un c\u00f4t\u00e9, 55 de l’autre (ou 55 et 89 pour les grands). V\u00e9rifi\u00e9 scientifiquement.<\/p>\n <p><strong>La pomme de pin :<\/strong> 8 et 13 spirales. Ou 5 et 8 pour les petites.<\/p>\n <p><strong>Le bananier :<\/strong> les feuilles sortent selon un angle de 137,5\u00b0 \u2014 l’angle d’or, directement li\u00e9 \u00e0 \u03c6.<\/p>\n <p><strong>Le chou romanesco<\/strong> (vendu dans certains march\u00e9s b\u00e9ninois) : spirales fractales dont le nombre suit Fibonacci.<\/p>\n <\/div>\n\n <p>\n Pourquoi ? La r\u00e9ponse est math\u00e9matique et biologique \u00e0 la fois. Quand une plante produit de nouvelles graines ou feuilles, chaque nouvelle pousse se place \u00e0 environ <strong>137,5\u00b0<\/strong> de la pr\u00e9c\u00e9dente \u2014 c’est l’angle qui maximise l’espace disponible et la lumi\u00e8re re\u00e7ue. Et 137,5\u00b0 est directement li\u00e9 au nombre d’or : 360\u00b0 \/ \u03c6\u00b2 \u2248 137,5\u00b0.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83c\udf3b Spirales de Fibonacci dans un tournesol (sch\u00e9ma)<\/div>\n <svg width=\"100%\" viewBox=\"0 0 400 380\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"400\" height=\"380\" fill=\"white\"\/>\n \n <circle cx=\"200\" cy=\"190\" r=\"155\" fill=\"#fdf8e8\" stroke=\"#c8a010\" stroke-width=\"2\"\/>\n <circle cx=\"200\" cy=\"190\" r=\"10\" fill=\"#c8a010\"\/>\n\n \n <g id=\"graines\">\n \n \n <circle cx=\"200\" cy=\"160\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"220\" cy=\"175\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"215\" cy=\"200\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"195\" cy=\"215\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"175\" cy=\"205\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"170\" cy=\"182\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"183\" cy=\"163\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"207\" cy=\"155\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"228\" cy=\"165\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"237\" cy=\"188\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"228\" cy=\"211\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"207\" cy=\"225\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"183\" cy=\"225\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"162\" cy=\"213\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"152\" cy=\"192\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"158\" cy=\"170\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"173\" cy=\"152\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n \n <circle cx=\"200\" cy=\"135\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"230\" cy=\"145\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"250\" cy=\"168\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"255\" cy=\"195\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"243\" cy=\"220\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"222\" cy=\"238\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"197\" cy=\"243\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"172\" cy=\"237\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"152\" cy=\"220\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"140\" cy=\"197\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"143\" cy=\"172\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"157\" cy=\"150\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"178\" cy=\"136\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <\/g>\n\n \n <path d=\"M 200,190 Q 215,170 230,160 Q 250,152 265,165\"\n stroke=\"#c8a010\" stroke-width=\"1.5\" fill=\"none\" stroke-dasharray=\"4,3\" opacity=\"0.6\"\/>\n <path d=\"M 200,190 Q 185,170 172,162 Q 155,155 145,170\"\n stroke=\"#c8a010\" stroke-width=\"1.5\" fill=\"none\" stroke-dasharray=\"4,3\" opacity=\"0.6\"\/>\n\n \n <text x=\"200\" y=\"355\" font-size=\"11\" fill=\"#6b6560\" text-anchor=\"middle\">Arrangement selon l’angle d’or (137,5\u00b0)<\/text>\n <text x=\"200\" y=\"372\" font-size=\"11\" fill=\"#1a4a2e\" text-anchor=\"middle\">Spirales : 13 dans un sens, 21 dans l’autre<\/text>\n <\/svg>\n <\/div>\n\n <h3>\u2705 La reproduction des lapins \u2014 le probl\u00e8me original<\/h3>\n\n <p>\n C’est le probl\u00e8me que Fibonacci posait dans son livre : si on part d’une paire de lapins, et que chaque paire mature produit une nouvelle paire chaque mois, combien de paires aura-t-on apr\u00e8s n mois ?\n <\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">Mois 1<\/span> 1 paire (nouveau-n\u00e9e, pas encore mature)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 2<\/span> 1 paire (maintenant mature)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 3<\/span> 2 paires (la mature a produit une nouvelle)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 4<\/span> 3 paires (la mature + la nouvelle mature aussi)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 5<\/span> 5 paires \u2192 la suite de Fibonacci !<\/div>\n\n <h2>Ce qu’on exag\u00e8re : le mythe d\u00e9mystifi\u00e9<\/h2>\n\n <div class=\"alerte\">\n <span class=\"alerte-titre\">\u26a0\ufe0f Attention au mythe<\/span>\n <p><strong>Le corps humain :<\/strong> Certains affirment que le rapport entre ta hauteur totale et la hauteur de ton nombril est \u03c6. En r\u00e9alit\u00e9, ce rapport varie selon les individus et oscille entre 1,5 et 1,7 \u2014 pas toujours 1,618. C’est une g\u00e9n\u00e9ralisation abusive.<\/p>\n <p><strong>Le Parth\u00e9non et les pyramides :<\/strong> On pr\u00e9tend que leurs proportions suivent le nombre d’or. Les mesures pr\u00e9cises montrent des approximations tr\u00e8s variables \u2014 les b\u00e2tisseurs n’avaient pas \u03c6 en t\u00eate.<\/p>\n <p><strong>La galaxie spirale :<\/strong> Les bras de la Voie Lact\u00e9e ne suivent pas la spirale de Fibonacci \u2014 ils suivent une spirale logarithmique, ce qui est diff\u00e9rent.<\/p>\n <p><strong>Les visages \u00ab\u00a0beaux\u00a0\u00bb :<\/strong> Des \u00e9tudes s\u00e9rieuses n’ont trouv\u00e9 aucune corr\u00e9lation solide entre le nombre d’or et l’attractivit\u00e9 per\u00e7ue des visages humains.<\/p>\n <\/div>\n\n <p>\n La v\u00e9rit\u00e9 est plus nuanc\u00e9e et plus belle \u00e0 la fois : la suite de Fibonacci appara\u00eet <em>r\u00e9ellement<\/em> dans la phyllotaxie des plantes parce que c’est la solution optimale \u00e0 un probl\u00e8me de croissance. Mais elle n’est pas partout \u2014 et voir \u03c6 dans tout est un biais de confirmation : on cherche, on mesure approximativement, et on \u00ab\u00a0trouve\u00a0\u00bb.\n <\/p>\n\n <blockquote>\n Les math\u00e9matiques n’ont pas besoin d’\u00eatre mythifi\u00e9es pour \u00eatre belles. Leur vraie pr\u00e9sence dans la nature est d\u00e9j\u00e0 assez stup\u00e9fiante.\n <cite>\u2014 Le\u00e7on de rigueur math\u00e9matique<\/cite>\n <\/blockquote>\n\n <h2>La formule de Binet : calculer Fibonacci sans r\u00e9currence<\/h2>\n\n <p>\n Il existe une formule explicite pour calculer directement le n-i\u00e8me terme de Fibonacci, sans avoir \u00e0 calculer tous les pr\u00e9c\u00e9dents. Elle a \u00e9t\u00e9 d\u00e9couverte par le math\u00e9maticien fran\u00e7ais Jacques Philippe Marie Binet en 1843 :\n <\/p>\n\n <div class=\"formule\">F(n) = (\u03c6\u207f \u2212 \u03c8\u207f) \/ \u221a5<\/div>\n\n <p>\n O\u00f9 \u03c6 = (1+\u221a5)\/2 \u2248 1,618 et \u03c8 = (1\u2212\u221a5)\/2 \u2248 \u22120,618.\n <\/p>\n\n <p>\n Ce qui est remarquable : bien que F(n) soit toujours un entier, cette formule utilise des nombres irrationnels (\u221a5) et leur combinaison donne toujours un entier exact. C’est l’un de ces petits miracles math\u00e9matiques qui donnent envie de continuer \u00e0 explorer.\n <\/p>\n\n <h2>Le code Python<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># Suite de Fibonacci : r\u00e9currence, formule de Binet, visualisation<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># — M\u00e9thode 1 : R\u00e9currence classique —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">fibonacci_recursif<\/span>(n):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bbG\u00e9n\u00e8re les n premiers termes de Fibonacci\u00a0\u00bb\u00a0\u00bb\u00a0\u00bb<\/span>\n suite = [<span class=\"code-number\">0<\/span>, <span class=\"code-number\">1<\/span>]\n <span class=\"code-keyword\">for<\/span> i <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">2<\/span>, n):\n suite.<span class=\"code-function\">append<\/span>(suite[-<span class=\"code-number\">1<\/span>] + suite[-<span class=\"code-number\">2<\/span>])\n <span class=\"code-keyword\">return<\/span> suite\n\n<span class=\"code-comment\"># — M\u00e9thode 2 : Formule de Binet —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">fibonacci_binet<\/span>(n):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bbCalcule F(n) directement avec la formule de Binet\u00a0\u00bb\u00a0\u00bb\u00a0\u00bb<\/span>\n phi = (<span class=\"code-number\">1<\/span> + np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>)) \/ <span class=\"code-number\">2<\/span>\n psi = (<span class=\"code-number\">1<\/span> – np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>)) \/ <span class=\"code-number\">2<\/span>\n <span class=\"code-keyword\">return<\/span> <span class=\"code-function\">round<\/span>((phi**n – psi**n) \/ np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>))\n\n<span class=\"code-comment\"># Afficher et comparer les deux m\u00e9thodes<\/span>\nsuite = <span class=\"code-function\">fibonacci_recursif<\/span>(<span class=\"code-number\">15<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0Suite de Fibonacci (15 termes) :\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(suite)\n\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\\nComparaison r\u00e9currence vs Binet :\u00a0\u00bb<\/span>)\n<span class=\"code-keyword\">for<\/span> n <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">1<\/span>, <span class=\"code-number\">11<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbF({n}) = {suite[n]} | Binet = {fibonacci_binet(n)}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Convergence vers le nombre d’or —<\/span>\nrapports = [suite[i]\/suite[i-<span class=\"code-number\">1<\/span>] <span class=\"code-keyword\">for<\/span> i <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">2<\/span>, <span class=\"code-function\">len<\/span>(suite))]\nphi = (<span class=\"code-number\">1<\/span> + np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>)) \/ <span class=\"code-number\">2<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\\nNombre d’or \u03c6 = {phi:.10f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbDernier rapport = {rapports[-1]:.10f}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Graphique de convergence —<\/span>\nplt.<span class=\"code-function\">figure<\/span>(figsize=(<span class=\"code-number\">9<\/span>, <span class=\"code-number\">5<\/span>))\nplt.<span class=\"code-function\">plot<\/span>(<span class=\"code-function\">range<\/span>(<span class=\"code-number\">2<\/span>, <span class=\"code-function\">len<\/span>(suite)), rapports,\n <span class=\"code-string\">‘o-‘<\/span>, color=<span class=\"code-string\">‘#1a4a2e’<\/span>, linewidth=<span class=\"code-number\">2<\/span>, markersize=<span class=\"code-number\">6<\/span>)\nplt.<span class=\"code-function\">axhline<\/span>(y=phi, color=<span class=\"code-string\">‘#c8a010’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>,\n linewidth=<span class=\"code-number\">2<\/span>, label=<span class=\"code-string\">f’\u03c6 \u2248 {phi:.4f}’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Indice n’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘F(n) \/ F(n-1)’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘Convergence du rapport de Fibonacci vers le nombre d\\’or \u03c6’<\/span>)\nplt.<span class=\"code-function\">legend<\/span>()\nplt.<span class=\"code-function\">grid<\/span>(alpha=<span class=\"code-number\">0.3<\/span>)\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Un \u00e9l\u00e8ve de 3\u00e8me \u00e0 Abomey observe un ananas au march\u00e9. Il compte 8 spirales dans un sens et 13 dans l’autre.\n <\/p>\n <p><strong>a)<\/strong> V\u00e9rifie que 8 et 13 sont bien des nombres de Fibonacci cons\u00e9cutifs.<\/p>\n <p><strong>b)<\/strong> Calcule le rapport 13\/8. \u00c0 quel nombre c\u00e9l\u00e8bre se rapproche-t-il ?<\/p>\n <p><strong>c)<\/strong> Calcule F(10) avec la formule de r\u00e9currence (en partant de F(0)=0, F(1)=1).<\/p>\n <p><strong>d)<\/strong> La suite de Lucas est d\u00e9finie par L(0)=2, L(1)=1 et L(n)=L(n-1)+L(n-2). Calcule les 8 premiers termes. Vers quel nombre tendent les rapports L(n)\/L(n-1) ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> Suite de Fibonacci : 0, 1, 1, 2, 3, 5, <strong>8<\/strong>, <strong>13<\/strong>, 21…<\/p>\n <p>Oui, 8 = F(6) et 13 = F(7) \u2014 ce sont bien deux termes cons\u00e9cutifs \u2705<\/p>\n <p><strong>b)<\/strong> 13\/8 = <strong>1,625<\/strong><\/p>\n <p>Ce nombre se rapproche du nombre d’or \u03c6 \u2248 <strong>1,618<\/strong> \u2014 l’erreur est de moins de 0,5% !<\/p>\n <p><strong>c)<\/strong> F(0)=0, F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5, F(6)=8, F(7)=13, F(8)=21, F(9)=34<\/p>\n <p>\u2192 <strong>F(10) = F(9) + F(8) = 34 + 21 = 55<\/strong><\/p>\n <p><strong>d)<\/strong> L(0)=2, L(1)=1, L(2)=3, L(3)=4, L(4)=7, L(5)=11, L(6)=18, L(7)=29<\/p>\n <p>Rapports : 3\/2=1,5 | 4\/3=1,33 | 7\/4=1,75 | 11\/7=1,571 | 18\/11=1,636 | 29\/18=1,611…<\/p>\n <p>\u2192 Les rapports tendent \u00e9galement vers <strong>\u03c6 \u2248 1,618<\/strong> \u2014 fascinant ! \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La suite de Fibonacci est bien pr\u00e9sente dans la nature \u2014 mais de fa\u00e7on pr\u00e9cise et localis\u00e9e, pas universelle. Elle appara\u00eet l\u00e0 o\u00f9 la croissance optimale exige de maximiser l’espace ou la lumi\u00e8re, notamment dans la phyllotaxie des plantes. Ce n’est pas de la magie \u2014 c’est de l’optimisation math\u00e9matique fa\u00e7onn\u00e9e par des millions d’ann\u00e9es d’\u00e9volution.\n <\/p>\n <p>\n Et le nombre d’or \u03c6 qui en \u00e9merge est bien r\u00e9el, bien beau, et bien myst\u00e9rieux \u2014 sans avoir besoin d’\u00eatre sur-mythifi\u00e9. La prochaine fois que tu ach\u00e8teras un ananas au march\u00e9 de Dantokpa, prends le temps de compter ses spirales. Tu liras, dans ses \u00e9cailles brunes et dor\u00e9es, une \u00e9quation que la nature a r\u00e9solue bien avant nous.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans <strong>la loi normale et la courbe en cloche<\/strong> \u2014 cette forme qui gouverne les notes de tes \u00e9l\u00e8ves, la taille des B\u00e9ninois, et presque tout ce qui varie dans la nature.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>La suite de Fibonacci est-elle vraiment partout dans la nature ? \u2014 MathsVivantes Nature & Maths \u00b7 Article #10 La […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-122","page","type-page","status-publish","hentry"],"yoast_head":"\n<title>Suite de Fibonacci - AfrikIntelligent<\/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:\/\/vmi3217629.contaboserver.net\/?page_id=122\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Suite de Fibonacci - AfrikIntelligent\" \/>\n<meta property=\"og:description\" content=\"La suite de Fibonacci est-elle vraiment partout dans la nature ? 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}\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n .fib-num { width: 36px; height: 36px; font-size: 0.8rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Nature & Maths \u00b7 Article #10<\/div>\n <h1>La suite de Fibonacci<br>est-elle <em>vraiment<\/em><br>partout dans la nature ?<\/h1>\n <p class=\"hero-intro\">On nous dit que les tournesols, les coquillages, les ananas et m\u00eame la galaxie suivent la suite de Fibonacci. C’est vrai \u2014 en partie. Mais c’est aussi l’un des mythes math\u00e9matiques les plus exag\u00e9r\u00e9s de l’histoire. Aujourd’hui, on d\u00e9m\u00eale le vrai du faux.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 12 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Regarde un ananas au march\u00e9 de Dantokpa. Compte les rang\u00e9es de ses \u00e9cailles en spirale. Tu trouveras 8 rang\u00e9es dans un sens et 13 dans l’autre. Ou peut-\u00eatre 13 et 21. Regarde un tournesol s\u00e9ch\u00e9. Compte ses graines en spirale : 34 dans un sens, 55 dans l’autre. Ouvre une pomme de pin de la for\u00eat de la Lama : 8 spirales d’un c\u00f4t\u00e9, 13 de l’autre.\n <\/p>\n\n <p>\n 8, 13, 21, 34, 55… Ces nombres t’\u00e9voquent quelque chose ? Ils appartiennent tous \u00e0 la m\u00eame suite math\u00e9matique \u2014 la <strong>suite de Fibonacci<\/strong>. Et leur pr\u00e9sence dans la nature n’est pas un hasard. Mais elle n’est pas non plus le myst\u00e8re cosmique qu’on te fait parfois croire.\n <\/p>\n\n <h2>Qui \u00e9tait Fibonacci ?<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Leonardo de Pise, dit Fibonacci \u2014 XII<sup>e<\/sup> si\u00e8cle<\/span>\n <p>\n Leonardo Fibonacci est n\u00e9 \u00e0 Pise (Italie) vers 1170. Son p\u00e8re \u00e9tait commer\u00e7ant et voyageait beaucoup en Afrique du Nord \u2014 notamment \u00e0 B\u00e9ja\u00efa, en Alg\u00e9rie actuelle, o\u00f9 le jeune Leonardo apprit les math\u00e9matiques arabes et les chiffres indo-arabes que nous utilisons encore aujourd’hui (1, 2, 3…).\n <\/p>\n <p>\n En 1202, il publie le <em>Liber Abaci<\/em> \u2014 le livre du calcul \u2014 qui r\u00e9volutionne les math\u00e9matiques europ\u00e9ennes. C’est dans ce livre qu’appara\u00eet, \u00e0 titre d’exemple sur la reproduction des lapins, la suite qui portera son nom.\n <\/p>\n <p>\n Ironie de l’histoire : Fibonacci n’a pas d\u00e9couvert cette suite. Elle \u00e9tait connue des math\u00e9maticiens indiens depuis le VI<sup>e<\/sup> si\u00e8cle, notamment par Pingala qui l’utilisait dans l’\u00e9tude de la po\u00e9sie sanskrite. Fibonacci l’a simplement popularis\u00e9e en Europe.\n <\/p>\n <\/div>\n\n <h2>La suite : simple \u00e0 d\u00e9finir, infinie en richesse<\/h2>\n\n <p>\n La r\u00e8gle est d’une simplicit\u00e9 enfantine : <strong>chaque terme est la somme des deux pr\u00e9c\u00e9dents<\/strong>. On commence avec 0 et 1.\n <\/p>\n\n <div class=\"fib-sequence\">\n <div class=\"fib-num\">0<\/div>\n <div class=\"fib-plus\">+<\/div>\n <div class=\"fib-num\">1<\/div>\n <div class=\"fib-plus\">=<\/div>\n <div class=\"fib-num or\">1<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">2<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">3<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">5<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">8<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">13<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">21<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">34<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">55<\/div>\n <div class=\"fib-plus\">\u2192<\/div>\n <div class=\"fib-num\">…<\/div>\n <\/div>\n\n <div class=\"definition\">\n <strong>Suite de Fibonacci :<\/strong> F(0) = 0, F(1) = 1, et pour tout n \u2265 2 : F(n) = F(n\u22121) + F(n\u22122).\n <br><br>\n La suite compl\u00e8te : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…\n <\/div>\n\n <h2>Le nombre d’or : le grand secret de Fibonacci<\/h2>\n\n <p>\n Voici la propri\u00e9t\u00e9 la plus stup\u00e9fiante de cette suite. Divise chaque terme par le pr\u00e9c\u00e9dent et observe ce qui se passe :\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>F(n)<\/th>\n <th>F(n-1)<\/th>\n <th>Rapport F(n)\/F(n-1)<\/th>\n <\/tr>\n <tr><td>2<\/td><td>1<\/td><td>2,000 000<\/td><\/tr>\n <tr><td>3<\/td><td>2<\/td><td>1,500 000<\/td><\/tr>\n <tr><td>5<\/td><td>3<\/td><td>1,666 667<\/td><\/tr>\n <tr><td>8<\/td><td>5<\/td><td>1,600 000<\/td><\/tr>\n <tr><td>13<\/td><td>8<\/td><td>1,625 000<\/td><\/tr>\n <tr><td>21<\/td><td>13<\/td><td>1,615 385<\/td><\/tr>\n <tr><td>55<\/td><td>34<\/td><td>1,617 647<\/td><\/tr>\n <tr><td>144<\/td><td>89<\/td><td>1,617 978<\/td><\/tr>\n <tr><td>377<\/td><td>233<\/td><td><strong>1,618 026<\/strong><\/td><\/tr>\n <\/table>\n <\/div>\n\n <p>\n Les rapports convergent vers une valeur fixe : <strong>\u03c6 \u2248 1,618033…<\/strong> C’est le <strong>nombre d’or<\/strong> \u2014 aussi appel\u00e9 phi (\u03c6), l’un des nombres les plus c\u00e9l\u00e8bres et les plus myst\u00e9rieux des math\u00e9matiques.\n <\/p>\n\n <div class=\"formule\">\u03c6 = (1 + \u221a5) \/ 2 \u2248 1,618033988…<\/div>\n\n <p>\n Le nombre d’or a une propri\u00e9t\u00e9 unique : si tu lui enl\u00e8ves 1, tu obtiens son inverse. Autrement dit : \u03c6 \u2212 1 = 1\/\u03c6. Aucun autre nombre ne poss\u00e8de cette propri\u00e9t\u00e9.\n <\/p>\n\n <h2>Dans la nature : ce qui est vrai<\/h2>\n\n <h3>\u2705 Les phyllotaxies \u2014 vrai et expliqu\u00e9<\/h3>\n\n <p>\n La <strong>phyllotaxie<\/strong> est l’arrangement des feuilles, graines et p\u00e9tales sur les plantes. Et l\u00e0, les nombres de Fibonacci sont bien r\u00e9els \u2014 et ont une explication scientifique solide.\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf3f Exemples v\u00e9rifi\u00e9s en Afrique et ailleurs<\/span>\n <p><strong>L’ananas :<\/strong> 8 spirales dans un sens, 13 dans l’autre. Toujours des nombres de Fibonacci cons\u00e9cutifs.<\/p>\n <p><strong>Le tournesol :<\/strong> 34 spirales d’un c\u00f4t\u00e9, 55 de l’autre (ou 55 et 89 pour les grands). V\u00e9rifi\u00e9 scientifiquement.<\/p>\n <p><strong>La pomme de pin :<\/strong> 8 et 13 spirales. Ou 5 et 8 pour les petites.<\/p>\n <p><strong>Le bananier :<\/strong> les feuilles sortent selon un angle de 137,5\u00b0 \u2014 l’angle d’or, directement li\u00e9 \u00e0 \u03c6.<\/p>\n <p><strong>Le chou romanesco<\/strong> (vendu dans certains march\u00e9s b\u00e9ninois) : spirales fractales dont le nombre suit Fibonacci.<\/p>\n <\/div>\n\n <p>\n Pourquoi ? La r\u00e9ponse est math\u00e9matique et biologique \u00e0 la fois. Quand une plante produit de nouvelles graines ou feuilles, chaque nouvelle pousse se place \u00e0 environ <strong>137,5\u00b0<\/strong> de la pr\u00e9c\u00e9dente \u2014 c’est l’angle qui maximise l’espace disponible et la lumi\u00e8re re\u00e7ue. Et 137,5\u00b0 est directement li\u00e9 au nombre d’or : 360\u00b0 \/ \u03c6\u00b2 \u2248 137,5\u00b0.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83c\udf3b Spirales de Fibonacci dans un tournesol (sch\u00e9ma)<\/div>\n <svg width=\"100%\" viewBox=\"0 0 400 380\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"400\" height=\"380\" fill=\"white\"\/>\n \n <circle cx=\"200\" cy=\"190\" r=\"155\" fill=\"#fdf8e8\" stroke=\"#c8a010\" stroke-width=\"2\"\/>\n <circle cx=\"200\" cy=\"190\" r=\"10\" fill=\"#c8a010\"\/>\n\n \n <g id=\"graines\">\n \n \n <circle cx=\"200\" cy=\"160\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"220\" cy=\"175\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"215\" cy=\"200\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"195\" cy=\"215\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"175\" cy=\"205\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"170\" cy=\"182\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"183\" cy=\"163\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"207\" cy=\"155\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"228\" cy=\"165\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"237\" cy=\"188\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"228\" cy=\"211\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"207\" cy=\"225\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"183\" cy=\"225\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"162\" cy=\"213\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"152\" cy=\"192\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"158\" cy=\"170\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n <circle cx=\"173\" cy=\"152\" r=\"5\" fill=\"#1a4a2e\" opacity=\"0.8\"\/>\n \n <circle cx=\"200\" cy=\"135\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"230\" cy=\"145\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"250\" cy=\"168\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"255\" cy=\"195\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"243\" cy=\"220\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"222\" cy=\"238\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"197\" cy=\"243\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"172\" cy=\"237\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"152\" cy=\"220\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"140\" cy=\"197\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"143\" cy=\"172\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"157\" cy=\"150\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <circle cx=\"178\" cy=\"136\" r=\"5\" fill=\"#2ecc71\" opacity=\"0.9\"\/>\n <\/g>\n\n \n <path d=\"M 200,190 Q 215,170 230,160 Q 250,152 265,165\"\n stroke=\"#c8a010\" stroke-width=\"1.5\" fill=\"none\" stroke-dasharray=\"4,3\" opacity=\"0.6\"\/>\n <path d=\"M 200,190 Q 185,170 172,162 Q 155,155 145,170\"\n stroke=\"#c8a010\" stroke-width=\"1.5\" fill=\"none\" stroke-dasharray=\"4,3\" opacity=\"0.6\"\/>\n\n \n <text x=\"200\" y=\"355\" font-size=\"11\" fill=\"#6b6560\" text-anchor=\"middle\">Arrangement selon l’angle d’or (137,5\u00b0)<\/text>\n <text x=\"200\" y=\"372\" font-size=\"11\" fill=\"#1a4a2e\" text-anchor=\"middle\">Spirales : 13 dans un sens, 21 dans l’autre<\/text>\n <\/svg>\n <\/div>\n\n <h3>\u2705 La reproduction des lapins \u2014 le probl\u00e8me original<\/h3>\n\n <p>\n C’est le probl\u00e8me que Fibonacci posait dans son livre : si on part d’une paire de lapins, et que chaque paire mature produit une nouvelle paire chaque mois, combien de paires aura-t-on apr\u00e8s n mois ?\n <\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">Mois 1<\/span> 1 paire (nouveau-n\u00e9e, pas encore mature)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 2<\/span> 1 paire (maintenant mature)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 3<\/span> 2 paires (la mature a produit une nouvelle)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 4<\/span> 3 paires (la mature + la nouvelle mature aussi)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">Mois 5<\/span> 5 paires \u2192 la suite de Fibonacci !<\/div>\n\n <h2>Ce qu’on exag\u00e8re : le mythe d\u00e9mystifi\u00e9<\/h2>\n\n <div class=\"alerte\">\n <span class=\"alerte-titre\">\u26a0\ufe0f Attention au mythe<\/span>\n <p><strong>Le corps humain :<\/strong> Certains affirment que le rapport entre ta hauteur totale et la hauteur de ton nombril est \u03c6. En r\u00e9alit\u00e9, ce rapport varie selon les individus et oscille entre 1,5 et 1,7 \u2014 pas toujours 1,618. C’est une g\u00e9n\u00e9ralisation abusive.<\/p>\n <p><strong>Le Parth\u00e9non et les pyramides :<\/strong> On pr\u00e9tend que leurs proportions suivent le nombre d’or. Les mesures pr\u00e9cises montrent des approximations tr\u00e8s variables \u2014 les b\u00e2tisseurs n’avaient pas \u03c6 en t\u00eate.<\/p>\n <p><strong>La galaxie spirale :<\/strong> Les bras de la Voie Lact\u00e9e ne suivent pas la spirale de Fibonacci \u2014 ils suivent une spirale logarithmique, ce qui est diff\u00e9rent.<\/p>\n <p><strong>Les visages \u00ab\u00a0beaux\u00a0\u00bb :<\/strong> Des \u00e9tudes s\u00e9rieuses n’ont trouv\u00e9 aucune corr\u00e9lation solide entre le nombre d’or et l’attractivit\u00e9 per\u00e7ue des visages humains.<\/p>\n <\/div>\n\n <p>\n La v\u00e9rit\u00e9 est plus nuanc\u00e9e et plus belle \u00e0 la fois : la suite de Fibonacci appara\u00eet <em>r\u00e9ellement<\/em> dans la phyllotaxie des plantes parce que c’est la solution optimale \u00e0 un probl\u00e8me de croissance. Mais elle n’est pas partout \u2014 et voir \u03c6 dans tout est un biais de confirmation : on cherche, on mesure approximativement, et on \u00ab\u00a0trouve\u00a0\u00bb.\n <\/p>\n\n <blockquote>\n Les math\u00e9matiques n’ont pas besoin d’\u00eatre mythifi\u00e9es pour \u00eatre belles. Leur vraie pr\u00e9sence dans la nature est d\u00e9j\u00e0 assez stup\u00e9fiante.\n <cite>\u2014 Le\u00e7on de rigueur math\u00e9matique<\/cite>\n <\/blockquote>\n\n <h2>La formule de Binet : calculer Fibonacci sans r\u00e9currence<\/h2>\n\n <p>\n Il existe une formule explicite pour calculer directement le n-i\u00e8me terme de Fibonacci, sans avoir \u00e0 calculer tous les pr\u00e9c\u00e9dents. Elle a \u00e9t\u00e9 d\u00e9couverte par le math\u00e9maticien fran\u00e7ais Jacques Philippe Marie Binet en 1843 :\n <\/p>\n\n <div class=\"formule\">F(n) = (\u03c6\u207f \u2212 \u03c8\u207f) \/ \u221a5<\/div>\n\n <p>\n O\u00f9 \u03c6 = (1+\u221a5)\/2 \u2248 1,618 et \u03c8 = (1\u2212\u221a5)\/2 \u2248 \u22120,618.\n <\/p>\n\n <p>\n Ce qui est remarquable : bien que F(n) soit toujours un entier, cette formule utilise des nombres irrationnels (\u221a5) et leur combinaison donne toujours un entier exact. C’est l’un de ces petits miracles math\u00e9matiques qui donnent envie de continuer \u00e0 explorer.\n <\/p>\n\n <h2>Le code Python<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># Suite de Fibonacci : r\u00e9currence, formule de Binet, visualisation<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># — M\u00e9thode 1 : R\u00e9currence classique —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">fibonacci_recursif<\/span>(n):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bbG\u00e9n\u00e8re les n premiers termes de Fibonacci\u00a0\u00bb\u00a0\u00bb\u00a0\u00bb<\/span>\n suite = [<span class=\"code-number\">0<\/span>, <span class=\"code-number\">1<\/span>]\n <span class=\"code-keyword\">for<\/span> i <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">2<\/span>, n):\n suite.<span class=\"code-function\">append<\/span>(suite[-<span class=\"code-number\">1<\/span>] + suite[-<span class=\"code-number\">2<\/span>])\n <span class=\"code-keyword\">return<\/span> suite\n\n<span class=\"code-comment\"># — M\u00e9thode 2 : Formule de Binet —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">fibonacci_binet<\/span>(n):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bbCalcule F(n) directement avec la formule de Binet\u00a0\u00bb\u00a0\u00bb\u00a0\u00bb<\/span>\n phi = (<span class=\"code-number\">1<\/span> + np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>)) \/ <span class=\"code-number\">2<\/span>\n psi = (<span class=\"code-number\">1<\/span> – np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>)) \/ <span class=\"code-number\">2<\/span>\n <span class=\"code-keyword\">return<\/span> <span class=\"code-function\">round<\/span>((phi**n – psi**n) \/ np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>))\n\n<span class=\"code-comment\"># Afficher et comparer les deux m\u00e9thodes<\/span>\nsuite = <span class=\"code-function\">fibonacci_recursif<\/span>(<span class=\"code-number\">15<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0Suite de Fibonacci (15 termes) :\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(suite)\n\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\\nComparaison r\u00e9currence vs Binet :\u00a0\u00bb<\/span>)\n<span class=\"code-keyword\">for<\/span> n <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">1<\/span>, <span class=\"code-number\">11<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbF({n}) = {suite[n]} | Binet = {fibonacci_binet(n)}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Convergence vers le nombre d’or —<\/span>\nrapports = [suite[i]\/suite[i-<span class=\"code-number\">1<\/span>] <span class=\"code-keyword\">for<\/span> i <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">2<\/span>, <span class=\"code-function\">len<\/span>(suite))]\nphi = (<span class=\"code-number\">1<\/span> + np.<span class=\"code-function\">sqrt<\/span>(<span class=\"code-number\">5<\/span>)) \/ <span class=\"code-number\">2<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\\nNombre d’or \u03c6 = {phi:.10f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbDernier rapport = {rapports[-1]:.10f}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Graphique de convergence —<\/span>\nplt.<span class=\"code-function\">figure<\/span>(figsize=(<span class=\"code-number\">9<\/span>, <span class=\"code-number\">5<\/span>))\nplt.<span class=\"code-function\">plot<\/span>(<span class=\"code-function\">range<\/span>(<span class=\"code-number\">2<\/span>, <span class=\"code-function\">len<\/span>(suite)), rapports,\n <span class=\"code-string\">‘o-‘<\/span>, color=<span class=\"code-string\">‘#1a4a2e’<\/span>, linewidth=<span class=\"code-number\">2<\/span>, markersize=<span class=\"code-number\">6<\/span>)\nplt.<span class=\"code-function\">axhline<\/span>(y=phi, color=<span class=\"code-string\">‘#c8a010’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>,\n linewidth=<span class=\"code-number\">2<\/span>, label=<span class=\"code-string\">f’\u03c6 \u2248 {phi:.4f}’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Indice n’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘F(n) \/ F(n-1)’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘Convergence du rapport de Fibonacci vers le nombre d\\’or \u03c6’<\/span>)\nplt.<span class=\"code-function\">legend<\/span>()\nplt.<span class=\"code-function\">grid<\/span>(alpha=<span class=\"code-number\">0.3<\/span>)\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Un \u00e9l\u00e8ve de 3\u00e8me \u00e0 Abomey observe un ananas au march\u00e9. Il compte 8 spirales dans un sens et 13 dans l’autre.\n <\/p>\n <p><strong>a)<\/strong> V\u00e9rifie que 8 et 13 sont bien des nombres de Fibonacci cons\u00e9cutifs.<\/p>\n <p><strong>b)<\/strong> Calcule le rapport 13\/8. \u00c0 quel nombre c\u00e9l\u00e8bre se rapproche-t-il ?<\/p>\n <p><strong>c)<\/strong> Calcule F(10) avec la formule de r\u00e9currence (en partant de F(0)=0, F(1)=1).<\/p>\n <p><strong>d)<\/strong> La suite de Lucas est d\u00e9finie par L(0)=2, L(1)=1 et L(n)=L(n-1)+L(n-2). Calcule les 8 premiers termes. Vers quel nombre tendent les rapports L(n)\/L(n-1) ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> Suite de Fibonacci : 0, 1, 1, 2, 3, 5, <strong>8<\/strong>, <strong>13<\/strong>, 21…<\/p>\n <p>Oui, 8 = F(6) et 13 = F(7) \u2014 ce sont bien deux termes cons\u00e9cutifs \u2705<\/p>\n <p><strong>b)<\/strong> 13\/8 = <strong>1,625<\/strong><\/p>\n <p>Ce nombre se rapproche du nombre d’or \u03c6 \u2248 <strong>1,618<\/strong> \u2014 l’erreur est de moins de 0,5% !<\/p>\n <p><strong>c)<\/strong> F(0)=0, F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5, F(6)=8, F(7)=13, F(8)=21, F(9)=34<\/p>\n <p>\u2192 <strong>F(10) = F(9) + F(8) = 34 + 21 = 55<\/strong><\/p>\n <p><strong>d)<\/strong> L(0)=2, L(1)=1, L(2)=3, L(3)=4, L(4)=7, L(5)=11, L(6)=18, L(7)=29<\/p>\n <p>Rapports : 3\/2=1,5 | 4\/3=1,33 | 7\/4=1,75 | 11\/7=1,571 | 18\/11=1,636 | 29\/18=1,611…<\/p>\n <p>\u2192 Les rapports tendent \u00e9galement vers <strong>\u03c6 \u2248 1,618<\/strong> \u2014 fascinant ! \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La suite de Fibonacci est bien pr\u00e9sente dans la nature \u2014 mais de fa\u00e7on pr\u00e9cise et localis\u00e9e, pas universelle. Elle appara\u00eet l\u00e0 o\u00f9 la croissance optimale exige de maximiser l’espace ou la lumi\u00e8re, notamment dans la phyllotaxie des plantes. Ce n’est pas de la magie \u2014 c’est de l’optimisation math\u00e9matique fa\u00e7onn\u00e9e par des millions d’ann\u00e9es d’\u00e9volution.\n <\/p>\n <p>\n Et le nombre d’or \u03c6 qui en \u00e9merge est bien r\u00e9el, bien beau, et bien myst\u00e9rieux \u2014 sans avoir besoin d’\u00eatre sur-mythifi\u00e9. La prochaine fois que tu ach\u00e8teras un ananas au march\u00e9 de Dantokpa, prends le temps de compter ses spirales. Tu liras, dans ses \u00e9cailles brunes et dor\u00e9es, une \u00e9quation que la nature a r\u00e9solue bien avant nous.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans <strong>la loi normale et la courbe en cloche<\/strong> \u2014 cette forme qui gouverne les notes de tes \u00e9l\u00e8ves, la taille des B\u00e9ninois, et presque tout ce qui varie dans la nature.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>La suite de Fibonacci est-elle vraiment partout dans la nature ? \u2014 MathsVivantes Nature & Maths \u00b7 Article #10 La […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-122","page","type-page","status-publish","hentry"],"yoast_head":"\n<title>Suite de Fibonacci - AfrikIntelligent<\/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:\/\/vmi3217629.contaboserver.net\/?page_id=122\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Suite de Fibonacci - AfrikIntelligent\" \/>\n<meta property=\"og:description\" content=\"La suite de Fibonacci est-elle vraiment partout dans la nature ? 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