{"id":5845,"date":"2021-02-27T18:11:45","date_gmt":"2021-02-27T17:11:45","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=5845"},"modified":"2021-02-28T10:53:23","modified_gmt":"2021-02-28T09:53:23","slug":"spirale-et-carres-tikz-et-python","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/27\/spirale-et-carres-tikz-et-python\/","title":{"rendered":"Spirale et carr\u00e9s, TiKZ et Python"},"content":{"rendered":"\r\n<p>Spirale et carr\u00e9s, TiKZ et Python : il y a peu de temps, j&#8217;ai eu \u00e0 construire une spirale inscrite dans une suite de carr\u00e9s, avec TiKZ et \u00e0 l&#8217;aide de Python :<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image size-large is-resized fancybox\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale-carres-tikz-python-1024x681.png\" alt=\"Spirale et carr\u00e9s, TiKZ et Python\" class=\"wp-image-5846\" width=\"512\" height=\"341\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale-carres-tikz-python-1024x681.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale-carres-tikz-python-600x399.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale-carres-tikz-python-300x199.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale-carres-tikz-python-768x510.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale-carres-tikz-python.png 1115w\" sizes=\"auto, (max-width: 512px) 100vw, 512px\" \/><\/figure>\r\n\r\n\r\n\r\n<p>Ceci \u00e9tait pour illustrer la somme des termes d&#8217;une suite g\u00e9om\u00e9trique.<\/p>\r\n\r\n\r\n\r\n<!--more-->\r\n\r\n\r\n\r\n<p>Bien entendu, on peut la construire &#8220;\u00e0 la main&#8221; car finalement, il n&#8217;y a que 8 carr\u00e9s qui sont bien visibles. Mais le challenge informatique \u00e9tait bien trop all\u00e9chant pour que je n&#8217;utilise pas <a href=\"https:\/\/www.mathweb.fr\/euclide\/pythontex\/\" target=\"_blank\" rel=\"noreferrer noopener\">Pythontex<\/a>, et donc le langage Python coupl\u00e9 \u00e0 \\(\\LaTeX\\).<\/p>\r\n\r\n\r\n\r\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/27\/spirale-et-carres-tikz-et-python\/#Spirale_et_carres_TiKZ_et_Python_les_bases_du_programme\" >Spirale et carr\u00e9s, TiKZ et Python: les bases du programme<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/27\/spirale-et-carres-tikz-et-python\/#Spirale_et_carres_TiKZ_et_Python_le_programme_LaTeX\" >Spirale et carr\u00e9s, TiKZ et Python: le programme \\(\\LaTeX\\)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/27\/spirale-et-carres-tikz-et-python\/#Spirale_de_Fibonacci\" >Spirale de Fibonacci<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Spirale_et_carres_TiKZ_et_Python_les_bases_du_programme\"><\/span>Spirale et carr\u00e9s, TiKZ et Python: les bases du programme<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>L&#8217;id\u00e9e ici est de calculer les coordonn\u00e9es du coin inf\u00e9rieur gauche de chaque carr\u00e9s, puis la longueur du c\u00f4t\u00e9 de ce carr\u00e9, sachant qu&#8217;elle est \u00e9gale \u00e0 la moiti\u00e9 de celle du c\u00f4t\u00e9 du carr\u00e9 pr\u00e9c\u00e9dent.<\/p>\r\n\r\n\r\n\r\n<p>Ensuite, il faut dessiner le quart de cercles dans le carr\u00e9.<\/p>\r\n\r\n\r\n\r\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Spirale_et_carres_TiKZ_et_Python_le_programme_LaTeX\"><\/span>Spirale et carr\u00e9s, TiKZ et Python: le programme \\(\\LaTeX\\)<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{standalone}\r\n\\usepackage{tikz}\r\n\\usepackage{pythontex}\r\n\\begin{document}\r\n\\begin{tikzpicture}\r\n\\begin{pycode}\r\nx, y, cx, cy, a = 0, 0, 100, 100, 180\r\nfor i in range(10):\r\n    print( '\\\\draw({},{}) rectangle +({},{});\\n\\\\draw[red] ({},{}) arc({}:{}:{});\\n'.format(x,y,cx,cy,x,y,a,a-90,abs(cx)) );\r\n    x += cx\r\n    y += cy\r\n    a -= 90\r\n    if i % 2 == 0:\r\n        cx = cx \/ 2\r\n        cy = -cy \/ 2\r\n    else:\r\n        cx = -cx \/ 2\r\n        cy = cy \/ 2\r\n\\end{pycode}\r\n\\end{tikzpicture}\r\n\\end{document}<\/pre>\r\n\r\n\r\n\r\n<p>Pour celles et ceux qui ne seraient pas \u00e0 l&#8217;aise avec PythonTex, il existe aussi une solution en Python pur :<\/p>\r\n\r\n\r\n\r\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from os import system\r\nfrom os.path import isfile\r\n\r\n# construction du code LaTeX\r\n\r\nlatex = '\\\\documentclass{standalone}\\n\\\\usepackage{tikz}\\n\\\\begin{document}\\n\\\\begin{tikzpicture}'\r\n\r\nx, y, cx, cy, a = 0, 0, 100, 100, 180\r\nfor i in range(10):\r\n    latex +=  '\\\\draw({},{}) rectangle +({},{});\\n\\\\draw[red] ({},{}) arc({}:{}:{});\\n'.format(x,y,cx,cy,x,y,a,a-90,abs(cx))\r\n    x += cx\r\n    y += cy\r\n    a -= 90\r\n    if i % 2 == 0:\r\n        cx = cx \/ 2\r\n        cy = -cy \/ 2\r\n    else:\r\n        cx = -cx \/ 2\r\n        cy = cy \/ 2\r\n\r\nlatex += '\\\\end{tikzpicture}\\n\\\\end{document}'\r\n\r\n# g\u00e9n\u00e9ration du fichier\r\n\r\nif isfile('spirale_python.tex'):\r\n    system('del spirale_python.tex')\r\n\r\nfichier = open('spirale_python.tex',\"x\")\r\nfichier.write(latex)\r\nfichier.close()\r\nsystem('pdflatex  --shell-escape -synctex=1 -interaction=nonstopmode spirale_python.tex')\r\nsystem('start  spirale_python.pdf')<\/pre>\r\n\r\n\r\n\r\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Spirale_de_Fibonacci\"><\/span>Spirale de Fibonacci<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>Tant que nous sommes lanc\u00e9s sur les spirales inscrites dans des carr\u00e9s, autant \u00e9largir \u00e0 la spirale de Fibonacci.<\/p>\r\n\r\n\r\n\r\n<p>Mais l\u00e0, c&#8217;est tout de m\u00eame un peu plus cors\u00e9&#8230; <\/p>\r\n\r\n\r\n\r\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from os import system\r\nfrom os.path import isfile\r\n\r\n# spirale_fibonnacci\r\n\r\ndef spirale_fibonacci(n):\r\n    F = [1,1]\r\n    for k in range(2,n+1):\r\n        F.append(F[k-1]+F[k-2])\r\n    \r\n    latex = '\\\\documentclass{standalone}\\n'\r\n    latex += '\\\\usepackage{tikz}\\n'\r\n    latex += '\\\\begin{document}\\n'\r\n    latex += '\\\\begin{tikzpicture}\\n'\r\n    latex += '\\\\draw (0,0) rectangle +(1,1);\\n'\r\n    latex += '\\\\draw[red] (1,0) arc(0:90:1);\\n'\r\n    latex += '\\\\draw (-1,0) rectangle +(1,1);\\n'\r\n    latex += '\\\\draw[red] (0,1) arc(90:180:1);\\n'\r\n    \r\n    x = -1\r\n    y = 0\r\n    start_angle = 90\r\n    \r\n    for k in range(2,n+1):\r\n        start_angle += 90\r\n        if k % 4 == 2:\r\n            y -= F[k]\r\n            start_angle_x = x\r\n            start_angle_y = y + F[k]\r\n        elif k % 4 == 3:\r\n            x += F[k-1]\r\n            start_angle_x = x\r\n            start_angle_y = y\r\n        elif k % 4 == 0:\r\n            x -= F[k-2]\r\n            y += F[k-1]\r\n            start_angle_x = x + F[k]\r\n            start_angle_y = y\r\n        else:\r\n            x -= F[k]\r\n            y -= F[k-2]\r\n            start_angle_x = x + F[k]\r\n            start_angle_y = y + F[k]\r\n            \r\n        latex += '\\\\draw ({},{}) rectangle +({},{});' . format(x,y,F[k],F[k])\r\n        latex += '\\\\draw[red] ({},{}) arc({}:{}:{});\\n' . format(start_angle_x,start_angle_y,start_angle,start_angle+90,F[k])\r\n\r\n    latex += '\\\\end{tikzpicture}\\n\\\\end{document}'\r\n\r\n    # g\u00e9n\u00e9ration du fichier\r\n\r\n    if isfile('spirale_fibo.tex'):\r\n        system('del spirale_fibo.tex')\r\n\r\n    fichier = open('spirale_fibo.tex',\"x\")\r\n    fichier.write(latex)\r\n    fichier.close()\r\n    system('pdflatex  --shell-escape -synctex=1 -interaction=nonstopmode spirale_fibo.tex')\r\n    system('start  spirale_fibo.pdf')<\/pre>\r\n\r\n\r\n\r\n<p>J&#8217;ai d\u00e9fini une fonction <em>spirale_fibonacci(n)<\/em> qui construit en \\(\\LaTeX\\) la spirale d&#8217;or avec <em>n<\/em>+1 carr\u00e9s de Fibonacci. Par exemple,<\/p>\r\n\r\n\r\n\r\n<pre class=\"wp-block-preformatted\">&gt;&gt;&gt; spirale_fibonacci(9)<\/pre>\r\n\r\n\r\n\r\n<p>donnera:<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image size-large fancybox\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"632\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale_fibonacci_or-1024x632.png\" alt=\"spirale d'or (spirale de Fibonacci)\" class=\"wp-image-5849\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale_fibonacci_or-1024x632.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale_fibonacci_or-300x185.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale_fibonacci_or-600x371.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale_fibonacci_or-768x474.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/spirale_fibonacci_or.png 1203w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\r\n","protected":false},"excerpt":{"rendered":"<p>Spirale et carr\u00e9s, TiKZ et Python : il y a peu de temps, j&#8217;ai eu \u00e0 construire une spirale inscrite dans une suite de carr\u00e9s, avec TiKZ et \u00e0 l&#8217;aide de Python : Ceci \u00e9tait pour illustrer la somme des termes d&#8217;une suite g\u00e9om\u00e9trique.<\/p>\n","protected":false},"author":1,"featured_media":5846,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,5],"tags":[96,263,224],"class_list":["post-5845","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-python","tag-fibonacci","tag-spirale","tag-tikz"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Spirale et carr\u00e9s, TiKZ et Python - Mathweb.fr - Spirale de Fibonacci<\/title>\n<meta name=\"description\" content=\"Nous allons voir dans cet article comment construire en TiKZ (LaTeX) et Python une spirale inscrite dans des carr\u00e9s. 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