Pourquoi les abeilles construisent des hexagones \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 --miel: #b8720a;\n --miel-clair: #f5c842;\n --miel-pale: #fdf0c0;\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: #2a1a00;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u2b21 \u2b21 \u2b21 \u2b21\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 5rem;\n opacity: 0.07;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; 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top: -10px;\n font-size: 3.5rem;\n color: var(--miel);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">G\u00e9om\u00e9trie & Nature \u00b7 Article #5<\/div>\n <h1>Pourquoi les abeilles<br>construisent des <em>hexagones<\/em><\/h1>\n <p class=\"hero-intro\">Les abeilles n’ont pas de dipl\u00f4me en math\u00e9matiques. Pourtant, elles ont r\u00e9solu l’un des probl\u00e8mes d’optimisation les plus \u00e9l\u00e9gants de toute la g\u00e9om\u00e9trie \u2014 des millions d’ann\u00e9es avant que les humains ne le formalisent.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 10 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Dans les villages du B\u00e9nin, on trouve souvent des ruches sauvages accroch\u00e9es aux fromagers ou aux palmiers \u00e0 huile. Si tu t’approches et observes attentivement la structure de la ruche, tu remarques quelque chose de frappant : chaque cellule est un <strong>hexagone parfait<\/strong>. Pas un carr\u00e9. Pas un triangle. Pas un cercle. Un hexagone.\n <\/p>\n\n <p>\n Pourquoi ? Est-ce un hasard ? Est-ce instinctif ? Ou est-ce que les abeilles ont, d’une fa\u00e7on ou d’une autre, d\u00e9couvert la <strong>solution optimale<\/strong> \u00e0 un probl\u00e8me math\u00e9matique profond ?\n <\/p>\n\n <p>\n La r\u00e9ponse est : les trois \u00e0 la fois. Et c’est magnifique.\n <\/p>\n\n <h2>Le probl\u00e8me : remplir un espace sans gaspiller<\/h2>\n\n <p>\n Les abeilles ont un objectif tr\u00e8s pr\u00e9cis : construire le maximum de cellules pour stocker le miel, en utilisant le minimum de cire. La cire est pr\u00e9cieuse \u2014 une abeille doit consommer environ 8 kg de miel pour produire 1 kg de cire. Chaque milligramme compte.\n <\/p>\n\n <p>\n Le probl\u00e8me math\u00e9matique est donc le suivant : <strong>quelle forme g\u00e9om\u00e9trique permet de remplir un plan entier, sans laisser d’espace vide, avec le p\u00e9rim\u00e8tre le plus petit possible pour une aire donn\u00e9e ?<\/strong>\n <\/p>\n\n <div class=\"definition\">\n <strong>Pavage du plan :<\/strong> Une fa\u00e7on de recouvrir enti\u00e8rement une surface plane avec des figures g\u00e9om\u00e9triques identiques, sans chevauchement ni espace vide. On appelle aussi cela une <em>tessellation<\/em>.\n <\/div>\n\n <h2>Quelles formes peuvent paver le plan ?<\/h2>\n\n <p>\n Parmi les polygones r\u00e9guliers \u2014 ceux dont tous les c\u00f4t\u00e9s et tous les angles sont \u00e9gaux \u2014 seulement trois peuvent paver le plan parfaitement :\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 Les trois pavages r\u00e9guliers du plan<\/div>\n <svg width=\"100%\" viewBox=\"0 0 560 200\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"560\" height=\"200\" fill=\"white\"\/>\n\n \n <g transform=\"translate(30,20)\">\n <polygon points=\"20,80 40,45 60,80\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"40,45 80,45 60,80\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,80 80,45 100,80\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"80,45 120,45 100,80\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"40,45 60,10 80,45\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"20,80 60,80 40,115\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,80 40,115 80,115\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,80 100,80 80,115\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <text x=\"55\" y=\"145\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\">Triangle<\/text>\n <text x=\"55\" y=\"160\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">6 triangles par sommet<\/text>\n <\/g>\n\n \n <g transform=\"translate(210,20)\">\n <rect x=\"20\" y=\"45\" width=\"35\" height=\"35\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"55\" y=\"45\" width=\"35\" height=\"35\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"90\" y=\"45\" width=\"35\" height=\"35\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"20\" y=\"80\" width=\"35\" height=\"35\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"55\" y=\"80\" width=\"35\" height=\"35\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"90\" y=\"80\" width=\"35\" height=\"35\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <text x=\"75\" y=\"145\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\">Carr\u00e9<\/text>\n <text x=\"75\" y=\"160\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">4 carr\u00e9s par sommet<\/text>\n <\/g>\n\n \n <g transform=\"translate(380,10)\">\n <polygon points=\"60,30 85,44 85,72 60,86 35,72 35,44\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"85,44 110,30 110,58 85,72\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"110,30 135,44 135,72 110,58\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,86 85,72 85,100 60,114 35,100 35,72\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"85,100 110,86 110,114 85,128 60,114\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <text x=\"85\" y=\"155\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\" font-weight=\"bold\">Hexagone \u2713<\/text>\n <text x=\"85\" y=\"170\" font-size=\"10\" fill=\"#b8720a\" text-anchor=\"middle\" font-weight=\"bold\">Optimal !<\/text>\n <\/g>\n <\/svg>\n <\/div>\n\n <p>\n Ces trois formes remplissent parfaitement le plan. Mais laquelle est la plus efficace ? C’est l\u00e0 que les math\u00e9matiques entrent en jeu.\n <\/p>\n\n <h2>Le calcul : qui gaspille le moins de cire ?<\/h2>\n\n <p>\n Pour comparer \u00e9quitablement, donnons \u00e0 chaque forme la <strong>m\u00eame aire<\/strong> \u2014 disons 1 cm\u00b2 \u2014 et calculons le p\u00e9rim\u00e8tre n\u00e9cessaire. Moins le p\u00e9rim\u00e8tre est grand, moins de cire l’abeille utilise.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Forme<\/th>\n <th>Aire = 1 cm\u00b2<\/th>\n <th>P\u00e9rim\u00e8tre (cm)<\/th>\n <th>Efficacit\u00e9<\/th>\n <\/tr>\n <tr>\n <td>Triangle \u00e9quilat\u00e9ral<\/td>\n <td>1 cm\u00b2<\/td>\n <td>4,56 cm<\/td>\n <td>\u274c Le moins efficace<\/td>\n <\/tr>\n <tr>\n <td>Carr\u00e9<\/td>\n <td>1 cm\u00b2<\/td>\n <td>4,00 cm<\/td>\n <td>\u26a0\ufe0f Interm\u00e9diaire<\/td>\n <\/tr>\n <tr>\n <td><strong>Hexagone r\u00e9gulier<\/strong><\/td>\n <td>1 cm\u00b2<\/td>\n <td><strong>3,72 cm<\/strong><\/td>\n <td>\u2705 <strong>Le plus efficace !<\/strong><\/td>\n <\/tr>\n <tr>\n <td>Cercle (r\u00e9f\u00e9rence)<\/td>\n <td>1 cm\u00b2<\/td>\n <td>3,54 cm<\/td>\n <td>\u2b55 Optimal mais ne pave pas<\/td>\n <\/tr>\n <\/table>\n <\/div>\n\n <p>\n L’hexagone a le p\u00e9rim\u00e8tre le plus petit parmi les formes qui peuvent paver le plan. C’est <strong>13% plus efficace<\/strong> que le carr\u00e9 et <strong>18% plus efficace<\/strong> que le triangle. Pour une ruche de 100 000 cellules, l’\u00e9conomie de cire est consid\u00e9rable.\n <\/p>\n\n <h2>Pourquoi l’hexagone est-il si efficace ? L’intuition g\u00e9om\u00e9trique<\/h2>\n\n <p>\n Voici l’intuition cl\u00e9 : parmi toutes les figures g\u00e9om\u00e9triques, c’est le <strong>cercle<\/strong> qui a le rapport aire\/p\u00e9rim\u00e8tre le plus favorable. Plus une forme ressemble \u00e0 un cercle, plus elle est efficace.\n <\/p>\n\n <p>\n Maintenant, parmi les polygones r\u00e9guliers qui peuvent paver le plan (triangle, carr\u00e9, hexagone), l’hexagone est celui qui ressemble le plus \u00e0 un cercle \u2014 il a 6 c\u00f4t\u00e9s, ses angles sont de 120\u00b0, et il est presque rond.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udd35 Du cercle \u00e0 l’hexagone \u2014 plus on a de c\u00f4t\u00e9s, plus on se rapproche du cercle<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 180\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"180\" fill=\"white\"\/>\n\n \n <polygon points=\"65,140 95,88 125,140\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <circle cx=\"95\" cy=\"120\" r=\"28\" fill=\"none\" stroke=\"#e8e2d8\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"95\" y=\"165\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">3 c\u00f4t\u00e9s<\/text>\n\n \n <rect x=\"175\" y=\"88\" width=\"55\" height=\"55\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <circle cx=\"202\" cy=\"115\" r=\"27\" fill=\"none\" stroke=\"#e8e2d8\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"202\" y=\"165\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">4 c\u00f4t\u00e9s<\/text>\n\n \n <polygon points=\"310,88 338,108 328,140 292,140 282,108\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <circle cx=\"310\" cy=\"118\" r=\"27\" fill=\"none\" stroke=\"#e8e2d8\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"310\" y=\"165\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">5 c\u00f4t\u00e9s<\/text>\n\n \n <polygon points=\"415,88 440,102 440,130 415,144 390,130 390,102\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2.5\"\/>\n <circle cx=\"415\" cy=\"116\" r=\"27\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"415\" y=\"165\" font-size=\"10\" fill=\"#b8720a\" text-anchor=\"middle\" font-weight=\"bold\">6 c\u00f4t\u00e9s \u2713<\/text>\n\n \n <path d=\"M 140,115 L 170,115\" stroke=\"#b8720a\" stroke-width=\"1.5\" marker-end=\"url(#arrow)\"\/>\n <path d=\"M 240,115 L 270,115\" stroke=\"#b8720a\" stroke-width=\"1.5\"\/>\n <path d=\"M 345,115 L 375,115\" stroke=\"#b8720a\" stroke-width=\"1.5\"\/>\n <text x=\"260\" y=\"40\" font-size=\"11\" fill=\"#6b6560\" text-anchor=\"middle\">De plus en plus proche du cercle \u2192<\/text>\n <\/svg>\n <\/div>\n\n <h2>La conjecture du nid d’abeille \u2014 2 000 ans d’histoire<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Histoire math\u00e9matique<\/span>\n <p>\n D\u00e8s l’Antiquit\u00e9, les math\u00e9maticiens grecs avaient remarqu\u00e9 l’efficacit\u00e9 de la structure hexagonale des ruches. Le math\u00e9maticien grec <strong>Pappus d’Alexandrie<\/strong> (vers 290-350 ap. J.-C.) avait m\u00eame sugg\u00e9r\u00e9 que les abeilles poss\u00e9daient une certaine forme de sagesse g\u00e9om\u00e9trique.\n <\/p>\n <p>\n Mais d\u00e9montrer rigoureusement que l’hexagone est <em>la<\/em> solution optimale \u2014 pas seulement parmi trois formes, mais parmi toutes les partitions possibles du plan \u2014 s’est r\u00e9v\u00e9l\u00e9 bien plus difficile.\n <\/p>\n <p>\n Cette conjecture, appel\u00e9e la <strong>\u00ab\u00a0conjecture du nid d’abeille\u00a0\u00bb<\/strong>, est rest\u00e9e sans preuve rigoureuse pendant plus de <strong>2 000 ans<\/strong>. Ce n’est qu’en <strong>1999<\/strong> que le math\u00e9maticien am\u00e9ricain Thomas Hales a finalement publi\u00e9 une d\u00e9monstration compl\u00e8te et formelle. Deux mill\u00e9naires pour prouver ce que les abeilles savaient depuis toujours.\n <\/p>\n <\/div>\n\n <h2>Les angles : pourquoi 120\u00b0 exactement ?<\/h2>\n\n <p>\n Un autre aspect remarquable : aux jonctions des cellules, les parois se rejoignent toujours \u00e0 <strong>120\u00b0<\/strong> exactement. Ce n’est pas un hasard \u2014 c’est la condition d’\u00e9quilibre des forces.\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\u2699\ufe0f L’\u00e9quilibre des forces<\/span>\n <p>Imagine trois forces \u00e9gales qui tirent depuis un point central. Pour que le syst\u00e8me soit en \u00e9quilibre parfait, les trois forces doivent \u00eatre espac\u00e9es de 360\u00b0 \/ 3 = <strong>120\u00b0<\/strong>.<\/p>\n <p>C’est exactement ce qui se passe dans une ruche : chaque paroi subit des tensions \u00e9gales de ses voisines. L’angle de 120\u00b0 est l’angle d’\u00e9quilibre naturel \u2014 la structure la plus <strong>stable m\u00e9caniquement<\/strong>.<\/p>\n <p>On retrouve ce m\u00eame angle dans les structures cristallines, les mousses de savon, et m\u00eame certains mat\u00e9riaux utilis\u00e9s dans l’a\u00e9ronautique.<\/p>\n <\/div>\n\n <h2>Les hexagones en Afrique et dans la nature<\/h2>\n\n <p>\n Ce n’est pas seulement dans les ruches qu’on trouve des hexagones. La nature les produit partout o\u00f9 un probl\u00e8me d’optimisation similaire se pose :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Hexagones partout autour de nous<\/span>\n <p><strong>Les carapaces de tortues<\/strong> que l’on voit souvent au B\u00e9nin pr\u00e9sentent des motifs proches de l’hexagonal \u2014 une structure qui maximise la r\u00e9sistance pour un minimum de mati\u00e8re.<\/p>\n <p><strong>La peau des girafes<\/strong> et les \u00e9cailles de certains poissons du lac Nokou\u00e9 pr\u00e9sentent des motifs hexagonaux similaires.<\/p>\n <p><strong>Les colonnes basaltiques<\/strong> \u2014 des formations rocheuses volcaniques \u2014 se fracturent naturellement en hexagones lorsque la lave refroidit et se contracte.<\/p>\n <p><strong>Les cristaux de neige<\/strong> ont toujours une sym\u00e9trie \u00e0 6 branches \u2014 li\u00e9e \u00e0 la structure hexagonale des mol\u00e9cules d’eau cristallis\u00e9es.<\/p>\n <p><strong>Le graph\u00e8ne<\/strong>, le mat\u00e9riau le plus r\u00e9sistant connu (100 fois plus r\u00e9sistant que l’acier), est un r\u00e9seau d’atomes de carbone arrang\u00e9s en… hexagones.<\/p>\n <\/div>\n\n <h2>Le calcul de l’aire et du p\u00e9rim\u00e8tre d’un hexagone<\/h2>\n\n <p>Pour un hexagone r\u00e9gulier de c\u00f4t\u00e9 <strong>a<\/strong> :<\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> P\u00e9rim\u00e8tre = 6a (6 c\u00f4t\u00e9s \u00e9gaux)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> Aire = (3\u221a3 \/ 2) \u00d7 a\u00b2 \u2248 2,598 \u00d7 a\u00b2<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> Un hexagone de c\u00f4t\u00e9 1 cm a une aire \u2248 2,598 cm\u00b2 et un p\u00e9rim\u00e8tre = 6 cm<\/div>\n\n <p>\n On peut aussi voir l’hexagone r\u00e9gulier comme compos\u00e9 de <strong>6 triangles \u00e9quilat\u00e9raux<\/strong> identiques, tous de c\u00f4t\u00e9 <em>a<\/em>. C’est d’ailleurs comme \u00e7a que les abeilles le construisent \u2014 en ajoutant cellule apr\u00e8s cellule, chacune partageant une paroi avec ses voisines.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 L’hexagone = 6 triangles \u00e9quilat\u00e9raux<\/div>\n <svg width=\"100%\" viewBox=\"0 0 300 260\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"300\" height=\"260\" fill=\"white\"\/>\n \n <polygon points=\"150,40 215,77 215,153 150,190 85,153 85,77\"\n fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2.5\"\/>\n \n <line x1=\"150\" y1=\"115\" x2=\"150\" y2=\"40\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"215\" y2=\"77\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"215\" y2=\"153\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"150\" y2=\"190\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"85\" y2=\"153\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"85\" y2=\"77\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n \n <text x=\"150\" y=\"75\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">1<\/text>\n <text x=\"193\" y=\"100\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">2<\/text>\n <text x=\"193\" y=\"140\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">3<\/text>\n <text x=\"150\" y=\"165\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">4<\/text>\n <text x=\"107\" y=\"140\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">5<\/text>\n <text x=\"107\" y=\"100\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">6<\/text>\n \n <circle cx=\"150\" cy=\"115\" r=\"3\" fill=\"#b8720a\"\/>\n \n <text x=\"155\" y=\"95\" font-size=\"10\" fill=\"#6b6560\">120\u00b0<\/text>\n \n <text x=\"150\" y=\"225\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\">6 triangles \u00e9quilat\u00e9raux identiques<\/text>\n <text x=\"150\" y=\"242\" font-size=\"11\" fill=\"#6b6560\" text-anchor=\"middle\">Chaque angle int\u00e9rieur = 120\u00b0<\/text>\n <\/svg>\n <\/div>\n\n <blockquote>\n La nature est le plus grand math\u00e9maticien qui soit. Elle r\u00e9sout des probl\u00e8mes d’optimisation que nous mettons des si\u00e8cles \u00e0 comprendre.\n <cite>\u2014 Inspiration des math\u00e9matiques naturelles<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Dans une ruche observ\u00e9e dans un village du Zou au B\u00e9nin, chaque cellule hexagonale a un c\u00f4t\u00e9 de <strong>3 mm<\/strong>.\n <\/p>\n <p><strong>a)<\/strong> Calcule le p\u00e9rim\u00e8tre d’une cellule.<\/p>\n <p><strong>b)<\/strong> Calcule l’aire d’une cellule (utilise la formule : Aire = (3\u221a3\/2) \u00d7 a\u00b2).<\/p>\n <p><strong>c)<\/strong> Si la ruche contient 80 000 cellules, quelle est la surface totale de stockage de miel (en cm\u00b2) ?<\/p>\n <p><strong>d)<\/strong> Quelle aurait \u00e9t\u00e9 la surface totale si les abeilles avaient utilis\u00e9 des carr\u00e9s de m\u00eame p\u00e9rim\u00e8tre ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> P\u00e9rim\u00e8tre = 6 \u00d7 3 = <strong>18 mm<\/strong><\/p>\n <p><strong>b)<\/strong> Aire = (3\u221a3\/2) \u00d7 3\u00b2 = (3 \u00d7 1,732 \/ 2) \u00d7 9 = 2,598 \u00d7 9 \u2248 <strong>23,38 mm\u00b2<\/strong><\/p>\n <p><strong>c)<\/strong> Surface totale = 80 000 \u00d7 23,38 = <strong>1 870 400 mm\u00b2<\/strong> = <strong>187,04 cm\u00b2<\/strong><\/p>\n <p><strong>d)<\/strong> Un carr\u00e9 de p\u00e9rim\u00e8tre 18 mm a un c\u00f4t\u00e9 de 18\/4 = 4,5 mm, donc une aire de 4,5\u00b2 = 20,25 mm\u00b2<\/p>\n <p>Surface totale avec carr\u00e9s = 80 000 \u00d7 20,25 = 1 620 000 mm\u00b2 = <strong>162 cm\u00b2<\/strong><\/p>\n <p>\u2705 L’hexagone offre <strong>187 cm\u00b2<\/strong> contre <strong>162 cm\u00b2<\/strong> pour le carr\u00e9 \u2014 soit <strong>15,4% de surface en plus<\/strong> pour la m\u00eame quantit\u00e9 de cire !<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n Les abeilles ne font pas de math\u00e9matiques conscientes. Mais l’\u00e9volution, sur des millions d’ann\u00e9es, a s\u00e9lectionn\u00e9 le comportement le plus efficace \u2014 celui qui correspond exactement \u00e0 la solution optimale du probl\u00e8me g\u00e9om\u00e9trique. La nature et les math\u00e9matiques parlent la m\u00eame langue.\n <\/p>\n <p>\n La prochaine fois que tu verras une ruche accroch\u00e9e \u00e0 un palmier au bord d’une route b\u00e9ninoise, souviens-toi : tu contemples 2 000 ans de math\u00e9matiques, r\u00e9solues sans crayon ni tableau noir.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans le monde du <strong>hasard et des probabilit\u00e9s<\/strong> \u2014 en nous demandant si le hasard existe vraiment, ou s’il n’est que de l’ignorance d\u00e9guis\u00e9e en myst\u00e8re.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Pourquoi les abeilles construisent des hexagones \u2014 MathsVivantes G\u00e9om\u00e9trie & Nature \u00b7 Article #5 Pourquoi les abeillesconstruisent des hexagones Les […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","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":""},"categories":[9],"tags":[],"class_list":["post-83","post","type-post","status-publish","format-standard","hentry","category-geometrie"],"yoast_head":"\n<title>Pourquoi les abeilles construisent des hexagones - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Pourquoi les abeilles construisent des hexagones ? 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{"id":83,"date":"2026-04-19T23:27:37","date_gmt":"2026-04-19T22:27:37","guid":{"rendered":"https:\/\/vmi3217629.contaboserver.net\/?p=83"},"modified":"2026-05-02T08:21:55","modified_gmt":"2026-05-02T07:21:55","slug":"abeilles-hexagones-geometrie","status":"publish","type":"post","link":"https:\/\/vmi3217629.contaboserver.net\/?p=83","title":{"rendered":"Pourquoi les abeilles construisent des hexagones"},"content":{"rendered":"\n\n\n\n\n\nPourquoi les abeilles construisent des hexagones \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 --miel: #b8720a;\n --miel-clair: #f5c842;\n --miel-pale: #fdf0c0;\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: #2a1a00;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u2b21 \u2b21 \u2b21 \u2b21\";\n position: absolute;\n top: 20px; 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letter-spacing: 0.18em; text-transform: uppercase; color: var(--miel-clair); margin-bottom: 0.8em; font-weight: 600; }\n\n blockquote {\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--miel);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">G\u00e9om\u00e9trie & Nature \u00b7 Article #5<\/div>\n <h1>Pourquoi les abeilles<br>construisent des <em>hexagones<\/em><\/h1>\n <p class=\"hero-intro\">Les abeilles n’ont pas de dipl\u00f4me en math\u00e9matiques. Pourtant, elles ont r\u00e9solu l’un des probl\u00e8mes d’optimisation les plus \u00e9l\u00e9gants de toute la g\u00e9om\u00e9trie \u2014 des millions d’ann\u00e9es avant que les humains ne le formalisent.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 10 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Dans les villages du B\u00e9nin, on trouve souvent des ruches sauvages accroch\u00e9es aux fromagers ou aux palmiers \u00e0 huile. Si tu t’approches et observes attentivement la structure de la ruche, tu remarques quelque chose de frappant : chaque cellule est un <strong>hexagone parfait<\/strong>. Pas un carr\u00e9. Pas un triangle. Pas un cercle. Un hexagone.\n <\/p>\n\n <p>\n Pourquoi ? Est-ce un hasard ? Est-ce instinctif ? Ou est-ce que les abeilles ont, d’une fa\u00e7on ou d’une autre, d\u00e9couvert la <strong>solution optimale<\/strong> \u00e0 un probl\u00e8me math\u00e9matique profond ?\n <\/p>\n\n <p>\n La r\u00e9ponse est : les trois \u00e0 la fois. Et c’est magnifique.\n <\/p>\n\n <h2>Le probl\u00e8me : remplir un espace sans gaspiller<\/h2>\n\n <p>\n Les abeilles ont un objectif tr\u00e8s pr\u00e9cis : construire le maximum de cellules pour stocker le miel, en utilisant le minimum de cire. La cire est pr\u00e9cieuse \u2014 une abeille doit consommer environ 8 kg de miel pour produire 1 kg de cire. Chaque milligramme compte.\n <\/p>\n\n <p>\n Le probl\u00e8me math\u00e9matique est donc le suivant : <strong>quelle forme g\u00e9om\u00e9trique permet de remplir un plan entier, sans laisser d’espace vide, avec le p\u00e9rim\u00e8tre le plus petit possible pour une aire donn\u00e9e ?<\/strong>\n <\/p>\n\n <div class=\"definition\">\n <strong>Pavage du plan :<\/strong> Une fa\u00e7on de recouvrir enti\u00e8rement une surface plane avec des figures g\u00e9om\u00e9triques identiques, sans chevauchement ni espace vide. On appelle aussi cela une <em>tessellation<\/em>.\n <\/div>\n\n <h2>Quelles formes peuvent paver le plan ?<\/h2>\n\n <p>\n Parmi les polygones r\u00e9guliers \u2014 ceux dont tous les c\u00f4t\u00e9s et tous les angles sont \u00e9gaux \u2014 seulement trois peuvent paver le plan parfaitement :\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 Les trois pavages r\u00e9guliers du plan<\/div>\n <svg width=\"100%\" viewBox=\"0 0 560 200\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"560\" height=\"200\" fill=\"white\"\/>\n\n \n <g transform=\"translate(30,20)\">\n <polygon points=\"20,80 40,45 60,80\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"40,45 80,45 60,80\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,80 80,45 100,80\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"80,45 120,45 100,80\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"40,45 60,10 80,45\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"20,80 60,80 40,115\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,80 40,115 80,115\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,80 100,80 80,115\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <text x=\"55\" y=\"145\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\">Triangle<\/text>\n <text x=\"55\" y=\"160\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">6 triangles par sommet<\/text>\n <\/g>\n\n \n <g transform=\"translate(210,20)\">\n <rect x=\"20\" y=\"45\" width=\"35\" height=\"35\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"55\" y=\"45\" width=\"35\" height=\"35\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"90\" y=\"45\" width=\"35\" height=\"35\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"20\" y=\"80\" width=\"35\" height=\"35\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"55\" y=\"80\" width=\"35\" height=\"35\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <rect x=\"90\" y=\"80\" width=\"35\" height=\"35\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <text x=\"75\" y=\"145\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\">Carr\u00e9<\/text>\n <text x=\"75\" y=\"160\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">4 carr\u00e9s par sommet<\/text>\n <\/g>\n\n \n <g transform=\"translate(380,10)\">\n <polygon points=\"60,30 85,44 85,72 60,86 35,72 35,44\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"85,44 110,30 110,58 85,72\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"110,30 135,44 135,72 110,58\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"60,86 85,72 85,100 60,114 35,100 35,72\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <polygon points=\"85,100 110,86 110,114 85,128 60,114\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <text x=\"85\" y=\"155\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\" font-weight=\"bold\">Hexagone \u2713<\/text>\n <text x=\"85\" y=\"170\" font-size=\"10\" fill=\"#b8720a\" text-anchor=\"middle\" font-weight=\"bold\">Optimal !<\/text>\n <\/g>\n <\/svg>\n <\/div>\n\n <p>\n Ces trois formes remplissent parfaitement le plan. Mais laquelle est la plus efficace ? C’est l\u00e0 que les math\u00e9matiques entrent en jeu.\n <\/p>\n\n <h2>Le calcul : qui gaspille le moins de cire ?<\/h2>\n\n <p>\n Pour comparer \u00e9quitablement, donnons \u00e0 chaque forme la <strong>m\u00eame aire<\/strong> \u2014 disons 1 cm\u00b2 \u2014 et calculons le p\u00e9rim\u00e8tre n\u00e9cessaire. Moins le p\u00e9rim\u00e8tre est grand, moins de cire l’abeille utilise.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Forme<\/th>\n <th>Aire = 1 cm\u00b2<\/th>\n <th>P\u00e9rim\u00e8tre (cm)<\/th>\n <th>Efficacit\u00e9<\/th>\n <\/tr>\n <tr>\n <td>Triangle \u00e9quilat\u00e9ral<\/td>\n <td>1 cm\u00b2<\/td>\n <td>4,56 cm<\/td>\n <td>\u274c Le moins efficace<\/td>\n <\/tr>\n <tr>\n <td>Carr\u00e9<\/td>\n <td>1 cm\u00b2<\/td>\n <td>4,00 cm<\/td>\n <td>\u26a0\ufe0f Interm\u00e9diaire<\/td>\n <\/tr>\n <tr>\n <td><strong>Hexagone r\u00e9gulier<\/strong><\/td>\n <td>1 cm\u00b2<\/td>\n <td><strong>3,72 cm<\/strong><\/td>\n <td>\u2705 <strong>Le plus efficace !<\/strong><\/td>\n <\/tr>\n <tr>\n <td>Cercle (r\u00e9f\u00e9rence)<\/td>\n <td>1 cm\u00b2<\/td>\n <td>3,54 cm<\/td>\n <td>\u2b55 Optimal mais ne pave pas<\/td>\n <\/tr>\n <\/table>\n <\/div>\n\n <p>\n L’hexagone a le p\u00e9rim\u00e8tre le plus petit parmi les formes qui peuvent paver le plan. C’est <strong>13% plus efficace<\/strong> que le carr\u00e9 et <strong>18% plus efficace<\/strong> que le triangle. Pour une ruche de 100 000 cellules, l’\u00e9conomie de cire est consid\u00e9rable.\n <\/p>\n\n <h2>Pourquoi l’hexagone est-il si efficace ? L’intuition g\u00e9om\u00e9trique<\/h2>\n\n <p>\n Voici l’intuition cl\u00e9 : parmi toutes les figures g\u00e9om\u00e9triques, c’est le <strong>cercle<\/strong> qui a le rapport aire\/p\u00e9rim\u00e8tre le plus favorable. Plus une forme ressemble \u00e0 un cercle, plus elle est efficace.\n <\/p>\n\n <p>\n Maintenant, parmi les polygones r\u00e9guliers qui peuvent paver le plan (triangle, carr\u00e9, hexagone), l’hexagone est celui qui ressemble le plus \u00e0 un cercle \u2014 il a 6 c\u00f4t\u00e9s, ses angles sont de 120\u00b0, et il est presque rond.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udd35 Du cercle \u00e0 l’hexagone \u2014 plus on a de c\u00f4t\u00e9s, plus on se rapproche du cercle<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 180\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"180\" fill=\"white\"\/>\n\n \n <polygon points=\"65,140 95,88 125,140\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <circle cx=\"95\" cy=\"120\" r=\"28\" fill=\"none\" stroke=\"#e8e2d8\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"95\" y=\"165\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">3 c\u00f4t\u00e9s<\/text>\n\n \n <rect x=\"175\" y=\"88\" width=\"55\" height=\"55\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <circle cx=\"202\" cy=\"115\" r=\"27\" fill=\"none\" stroke=\"#e8e2d8\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"202\" y=\"165\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">4 c\u00f4t\u00e9s<\/text>\n\n \n <polygon points=\"310,88 338,108 328,140 292,140 282,108\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2\"\/>\n <circle cx=\"310\" cy=\"118\" r=\"27\" fill=\"none\" stroke=\"#e8e2d8\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"310\" y=\"165\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">5 c\u00f4t\u00e9s<\/text>\n\n \n <polygon points=\"415,88 440,102 440,130 415,144 390,130 390,102\" fill=\"#fdf0c0\" stroke=\"#b8720a\" stroke-width=\"2.5\"\/>\n <circle cx=\"415\" cy=\"116\" r=\"27\" fill=\"none\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n <text x=\"415\" y=\"165\" font-size=\"10\" fill=\"#b8720a\" text-anchor=\"middle\" font-weight=\"bold\">6 c\u00f4t\u00e9s \u2713<\/text>\n\n \n <path d=\"M 140,115 L 170,115\" stroke=\"#b8720a\" stroke-width=\"1.5\" marker-end=\"url(#arrow)\"\/>\n <path d=\"M 240,115 L 270,115\" stroke=\"#b8720a\" stroke-width=\"1.5\"\/>\n <path d=\"M 345,115 L 375,115\" stroke=\"#b8720a\" stroke-width=\"1.5\"\/>\n <text x=\"260\" y=\"40\" font-size=\"11\" fill=\"#6b6560\" text-anchor=\"middle\">De plus en plus proche du cercle \u2192<\/text>\n <\/svg>\n <\/div>\n\n <h2>La conjecture du nid d’abeille \u2014 2 000 ans d’histoire<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Histoire math\u00e9matique<\/span>\n <p>\n D\u00e8s l’Antiquit\u00e9, les math\u00e9maticiens grecs avaient remarqu\u00e9 l’efficacit\u00e9 de la structure hexagonale des ruches. Le math\u00e9maticien grec <strong>Pappus d’Alexandrie<\/strong> (vers 290-350 ap. J.-C.) avait m\u00eame sugg\u00e9r\u00e9 que les abeilles poss\u00e9daient une certaine forme de sagesse g\u00e9om\u00e9trique.\n <\/p>\n <p>\n Mais d\u00e9montrer rigoureusement que l’hexagone est <em>la<\/em> solution optimale \u2014 pas seulement parmi trois formes, mais parmi toutes les partitions possibles du plan \u2014 s’est r\u00e9v\u00e9l\u00e9 bien plus difficile.\n <\/p>\n <p>\n Cette conjecture, appel\u00e9e la <strong>\u00ab\u00a0conjecture du nid d’abeille\u00a0\u00bb<\/strong>, est rest\u00e9e sans preuve rigoureuse pendant plus de <strong>2 000 ans<\/strong>. Ce n’est qu’en <strong>1999<\/strong> que le math\u00e9maticien am\u00e9ricain Thomas Hales a finalement publi\u00e9 une d\u00e9monstration compl\u00e8te et formelle. Deux mill\u00e9naires pour prouver ce que les abeilles savaient depuis toujours.\n <\/p>\n <\/div>\n\n <h2>Les angles : pourquoi 120\u00b0 exactement ?<\/h2>\n\n <p>\n Un autre aspect remarquable : aux jonctions des cellules, les parois se rejoignent toujours \u00e0 <strong>120\u00b0<\/strong> exactement. Ce n’est pas un hasard \u2014 c’est la condition d’\u00e9quilibre des forces.\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\u2699\ufe0f L’\u00e9quilibre des forces<\/span>\n <p>Imagine trois forces \u00e9gales qui tirent depuis un point central. Pour que le syst\u00e8me soit en \u00e9quilibre parfait, les trois forces doivent \u00eatre espac\u00e9es de 360\u00b0 \/ 3 = <strong>120\u00b0<\/strong>.<\/p>\n <p>C’est exactement ce qui se passe dans une ruche : chaque paroi subit des tensions \u00e9gales de ses voisines. L’angle de 120\u00b0 est l’angle d’\u00e9quilibre naturel \u2014 la structure la plus <strong>stable m\u00e9caniquement<\/strong>.<\/p>\n <p>On retrouve ce m\u00eame angle dans les structures cristallines, les mousses de savon, et m\u00eame certains mat\u00e9riaux utilis\u00e9s dans l’a\u00e9ronautique.<\/p>\n <\/div>\n\n <h2>Les hexagones en Afrique et dans la nature<\/h2>\n\n <p>\n Ce n’est pas seulement dans les ruches qu’on trouve des hexagones. La nature les produit partout o\u00f9 un probl\u00e8me d’optimisation similaire se pose :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Hexagones partout autour de nous<\/span>\n <p><strong>Les carapaces de tortues<\/strong> que l’on voit souvent au B\u00e9nin pr\u00e9sentent des motifs proches de l’hexagonal \u2014 une structure qui maximise la r\u00e9sistance pour un minimum de mati\u00e8re.<\/p>\n <p><strong>La peau des girafes<\/strong> et les \u00e9cailles de certains poissons du lac Nokou\u00e9 pr\u00e9sentent des motifs hexagonaux similaires.<\/p>\n <p><strong>Les colonnes basaltiques<\/strong> \u2014 des formations rocheuses volcaniques \u2014 se fracturent naturellement en hexagones lorsque la lave refroidit et se contracte.<\/p>\n <p><strong>Les cristaux de neige<\/strong> ont toujours une sym\u00e9trie \u00e0 6 branches \u2014 li\u00e9e \u00e0 la structure hexagonale des mol\u00e9cules d’eau cristallis\u00e9es.<\/p>\n <p><strong>Le graph\u00e8ne<\/strong>, le mat\u00e9riau le plus r\u00e9sistant connu (100 fois plus r\u00e9sistant que l’acier), est un r\u00e9seau d’atomes de carbone arrang\u00e9s en… hexagones.<\/p>\n <\/div>\n\n <h2>Le calcul de l’aire et du p\u00e9rim\u00e8tre d’un hexagone<\/h2>\n\n <p>Pour un hexagone r\u00e9gulier de c\u00f4t\u00e9 <strong>a<\/strong> :<\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> P\u00e9rim\u00e8tre = 6a (6 c\u00f4t\u00e9s \u00e9gaux)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> Aire = (3\u221a3 \/ 2) \u00d7 a\u00b2 \u2248 2,598 \u00d7 a\u00b2<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> Un hexagone de c\u00f4t\u00e9 1 cm a une aire \u2248 2,598 cm\u00b2 et un p\u00e9rim\u00e8tre = 6 cm<\/div>\n\n <p>\n On peut aussi voir l’hexagone r\u00e9gulier comme compos\u00e9 de <strong>6 triangles \u00e9quilat\u00e9raux<\/strong> identiques, tous de c\u00f4t\u00e9 <em>a<\/em>. C’est d’ailleurs comme \u00e7a que les abeilles le construisent \u2014 en ajoutant cellule apr\u00e8s cellule, chacune partageant une paroi avec ses voisines.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 L’hexagone = 6 triangles \u00e9quilat\u00e9raux<\/div>\n <svg width=\"100%\" viewBox=\"0 0 300 260\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"300\" height=\"260\" fill=\"white\"\/>\n \n <polygon points=\"150,40 215,77 215,153 150,190 85,153 85,77\"\n fill=\"none\" stroke=\"#b8720a\" stroke-width=\"2.5\"\/>\n \n <line x1=\"150\" y1=\"115\" x2=\"150\" y2=\"40\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"215\" y2=\"77\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"215\" y2=\"153\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"150\" y2=\"190\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"85\" y2=\"153\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n <line x1=\"150\" y1=\"115\" x2=\"85\" y2=\"77\" stroke=\"#b8720a\" stroke-width=\"1.5\" stroke-dasharray=\"5,3\"\/>\n \n <text x=\"150\" y=\"75\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">1<\/text>\n <text x=\"193\" y=\"100\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">2<\/text>\n <text x=\"193\" y=\"140\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">3<\/text>\n <text x=\"150\" y=\"165\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">4<\/text>\n <text x=\"107\" y=\"140\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">5<\/text>\n <text x=\"107\" y=\"100\" font-size=\"13\" fill=\"#b8720a\" text-anchor=\"middle\">6<\/text>\n \n <circle cx=\"150\" cy=\"115\" r=\"3\" fill=\"#b8720a\"\/>\n \n <text x=\"155\" y=\"95\" font-size=\"10\" fill=\"#6b6560\">120\u00b0<\/text>\n \n <text x=\"150\" y=\"225\" font-size=\"12\" fill=\"#2a1a00\" text-anchor=\"middle\" font-family=\"serif\">6 triangles \u00e9quilat\u00e9raux identiques<\/text>\n <text x=\"150\" y=\"242\" font-size=\"11\" fill=\"#6b6560\" text-anchor=\"middle\">Chaque angle int\u00e9rieur = 120\u00b0<\/text>\n <\/svg>\n <\/div>\n\n <blockquote>\n La nature est le plus grand math\u00e9maticien qui soit. Elle r\u00e9sout des probl\u00e8mes d’optimisation que nous mettons des si\u00e8cles \u00e0 comprendre.\n <cite>\u2014 Inspiration des math\u00e9matiques naturelles<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Dans une ruche observ\u00e9e dans un village du Zou au B\u00e9nin, chaque cellule hexagonale a un c\u00f4t\u00e9 de <strong>3 mm<\/strong>.\n <\/p>\n <p><strong>a)<\/strong> Calcule le p\u00e9rim\u00e8tre d’une cellule.<\/p>\n <p><strong>b)<\/strong> Calcule l’aire d’une cellule (utilise la formule : Aire = (3\u221a3\/2) \u00d7 a\u00b2).<\/p>\n <p><strong>c)<\/strong> Si la ruche contient 80 000 cellules, quelle est la surface totale de stockage de miel (en cm\u00b2) ?<\/p>\n <p><strong>d)<\/strong> Quelle aurait \u00e9t\u00e9 la surface totale si les abeilles avaient utilis\u00e9 des carr\u00e9s de m\u00eame p\u00e9rim\u00e8tre ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> P\u00e9rim\u00e8tre = 6 \u00d7 3 = <strong>18 mm<\/strong><\/p>\n <p><strong>b)<\/strong> Aire = (3\u221a3\/2) \u00d7 3\u00b2 = (3 \u00d7 1,732 \/ 2) \u00d7 9 = 2,598 \u00d7 9 \u2248 <strong>23,38 mm\u00b2<\/strong><\/p>\n <p><strong>c)<\/strong> Surface totale = 80 000 \u00d7 23,38 = <strong>1 870 400 mm\u00b2<\/strong> = <strong>187,04 cm\u00b2<\/strong><\/p>\n <p><strong>d)<\/strong> Un carr\u00e9 de p\u00e9rim\u00e8tre 18 mm a un c\u00f4t\u00e9 de 18\/4 = 4,5 mm, donc une aire de 4,5\u00b2 = 20,25 mm\u00b2<\/p>\n <p>Surface totale avec carr\u00e9s = 80 000 \u00d7 20,25 = 1 620 000 mm\u00b2 = <strong>162 cm\u00b2<\/strong><\/p>\n <p>\u2705 L’hexagone offre <strong>187 cm\u00b2<\/strong> contre <strong>162 cm\u00b2<\/strong> pour le carr\u00e9 \u2014 soit <strong>15,4% de surface en plus<\/strong> pour la m\u00eame quantit\u00e9 de cire !<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n Les abeilles ne font pas de math\u00e9matiques conscientes. Mais l’\u00e9volution, sur des millions d’ann\u00e9es, a s\u00e9lectionn\u00e9 le comportement le plus efficace \u2014 celui qui correspond exactement \u00e0 la solution optimale du probl\u00e8me g\u00e9om\u00e9trique. La nature et les math\u00e9matiques parlent la m\u00eame langue.\n <\/p>\n <p>\n La prochaine fois que tu verras une ruche accroch\u00e9e \u00e0 un palmier au bord d’une route b\u00e9ninoise, souviens-toi : tu contemples 2 000 ans de math\u00e9matiques, r\u00e9solues sans crayon ni tableau noir.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans le monde du <strong>hasard et des probabilit\u00e9s<\/strong> \u2014 en nous demandant si le hasard existe vraiment, ou s’il n’est que de l’ignorance d\u00e9guis\u00e9e en myst\u00e8re.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Pourquoi les abeilles construisent des hexagones \u2014 MathsVivantes G\u00e9om\u00e9trie & Nature \u00b7 Article #5 Pourquoi les abeillesconstruisent des hexagones Les […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","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":""},"categories":[9],"tags":[],"class_list":["post-83","post","type-post","status-publish","format-standard","hentry","category-geometrie"],"yoast_head":"\n<title>Pourquoi les abeilles construisent des hexagones - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Pourquoi les abeilles construisent des hexagones ? 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