{"id":1936,"date":"2024-09-30T17:19:56","date_gmt":"2024-09-30T16:19:56","guid":{"rendered":"https:\/\/blog.univ-angers.fr\/mathsinfo\/?p=1936"},"modified":"2024-09-30T17:27:37","modified_gmt":"2024-09-30T16:27:37","slug":"fractales","status":"publish","type":"post","link":"https:\/\/blog.univ-angers.fr\/mathsinfo\/2024\/09\/30\/fractales\/","title":{"rendered":"Fractales"},"content":{"rendered":"\n

Sur cette page de Wolfram<\/a>, on peut voir diff\u00e9rentes fractales :<\/p>\n\n\n

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\"\"<\/a><\/figure>\n<\/div>\n\n\n

Et en bas de cette m\u00eame page les algorithmes utilis\u00e9s pour transformer les cases :<\/p>\n\n\n

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\"\"<\/a><\/figure>\n<\/div>\n\n\n

Ci-dessous un programme Python qui permet d’appliquer le motif voulu, il suffit d’indiquer les coordonn\u00e9es des cases blanches. Par exemple pour le (a)<\/em>, la seule case blanche<\/strong> est au milieu (coordonn\u00e9es (1,1)<\/strong>)<\/p>\n\n\n

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\"\"<\/a>
La case blanche en (1,1)<\/figcaption><\/figure>\n<\/div>\n\n\n
from PIL import Image, ImageDraw\nimport numpy as np\n\n# Dimensions de l'image finale (\u00e0 modifier comme vous le voulez)\nsource = Image.new(\"RGB\", (600, 600), color=\"white\")\ndraw = ImageDraw.Draw(source)\n\n# Recherche s'il y a une case blanche\ndef has_common(str1, str2, pattern):\n    min_len = min(len(str1), len(str2))\n    for i in range(1, min_len + 1):\n        for p in pattern:\n            # Ne pas afficher si on trouve une case blanche\n            if str1[-i] == p[0] and str2[-i] == p[1]:\n                return False\n    # Sinon case noire\n    return True\n\n# Construction de la fractale\ndef fractal(pattern):\n    # On parcourt les colonnes\n    for c in range(600):\n        # Convertir 'c' en base 3 (ici \u00e0 l'aide de numpy)\n        t1 = np.base_repr(c, base=3)\n        # On parcourt les lignes\n        for l in range(600):\n            t2 = np.base_repr(l, base=3)\n            # Si pas de case blanche, afficher le point\n            if has_common(t1, t2, pattern):\n                draw.point((c, l), fill=(0, 0, 0))\n\n# Exemple avec le tapis de Sierpi\u0144ski\nfractal([\"11\"])\nsource.show()<\/code><\/pre>\n\n\n
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\"\"<\/a>
Motif (a) = case blanche en (1,1)<\/figcaption><\/figure>\n<\/div>\n\n\n
Quelques motifs<\/h2>\n\n\n
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\"\"<\/a>
(b) = fractal([’00’,’02’,’20’,’22’])<\/figcaption><\/figure>\n<\/div>\n\n
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\"\"<\/a>
(d) = fractal([’10’,’20’,’21’])<\/figcaption><\/figure>\n<\/div>\n\n
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\"\"<\/a>
(e) = fractal([’00’,’02’,’11’,’20’,’22’])<\/figcaption><\/figure>\n<\/div>\n\n
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\"\"<\/a>
(g) = fractal([’11’,’20’])<\/figcaption><\/figure>\n<\/div>\n\n
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\"\"<\/a>
(h) = fractal([’02’,’11’,’20’])<\/figcaption><\/figure>\n<\/div>\n\n\n
Et on peut en inventer de nouveaux :<\/p>\n\n\n
\n
\"\"<\/a>
fractal([’02’,’10’,’20’])<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"
Sur cette page de Wolfram, on peut voir diff\u00e9rentes fractales : Et en bas de cette m\u00eame page les algorithmes utilis\u00e9s pour transformer les cases : Ci-dessous un programme Python qui permet d’appliquer le motif voulu, il suffit d’indiquer les … Continuer la lecture →<\/span><\/a><\/p>\n","protected":false},"author":4913,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1936","post","type-post","status-publish","format-standard","hentry","category-non-classe"],"_links":{"self":[{"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/posts\/1936","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/users\/4913"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/comments?post=1936"}],"version-history":[{"count":6,"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/posts\/1936\/revisions"}],"predecessor-version":[{"id":1954,"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/posts\/1936\/revisions\/1954"}],"wp:attachment":[{"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/media?parent=1936"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/categories?post=1936"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.univ-angers.fr\/mathsinfo\/wp-json\/wp\/v2\/tags?post=1936"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}