{"id":506,"date":"2026-10-28T09:49:39","date_gmt":"2026-10-28T07:49:39","guid":{"rendered":"https:\/\/liceubolintin.ro\/eduhub\/?p=506"},"modified":"2026-10-28T09:57:57","modified_gmt":"2026-10-28T07:57:57","slug":"filotaxia-si-sirul-lui-fibonacci-prof-adriana-militaru","status":"publish","type":"post","link":"https:\/\/liceubolintin.ro\/eduhub\/2026\/10\/28\/filotaxia-si-sirul-lui-fibonacci-prof-adriana-militaru\/","title":{"rendered":"Filotaxia si Sirul lui Fibonacci – Prof. Adriana Militaru"},"content":{"rendered":"

Matematic\u0103\u00a0 \u00b7\u00a0 2026\u20132027<\/em>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 Fi\u0219\u0103 de lucru \u00a0\u00a0<\/strong><\/p>\n

 <\/p>\n

Spiralele din natur\u0103 \u0219i num\u0103rul de aur<\/em><\/p>\n

 <\/p>\n\n\n\n\n\n
Elevi<\/strong><\/td>\nBalta Claudiu Mihai<\/p>\n

Toma Alexandru Gabriel<\/p>\n

Constantinescu Denis Mihai<\/p>\n

Coca Alin<\/td>\n<\/tr>\n

Clasa<\/strong><\/td>\na XII-a A<\/td>\n<\/tr>\n
Profesor coordonator<\/strong><\/td>\nMilitaru Adriana<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

 <\/p>\n

\u00a0\u00a0\u00a0\u0218irul lui Fibonacci<\/h1>\n

\u0218irul lui Fibonacci a fost f\u0103cut cunoscut \u00een Europa de Leonardo din Pisa (Fibonacci), \u00een anul 1202. Primii doi termeni sunt 1 \u0219i 1, iar fiecare termen urm\u0103tor este suma celor doi termeni dinaintea lui:<\/p>\n

Fn<\/sub> = Fn\u22121<\/sub> + Fn\u22122<\/sub><\/p>\n\n\n\n
\u00a0\u00a0Primii termeni ai \u0219irului<\/em><\/p>\n

1\u00a0\u00a0\u00a0 1\u00a0\u00a0\u00a0 2\u00a0\u00a0\u00a0 3\u00a0\u00a0\u00a0 5\u00a0\u00a0\u00a0 8\u00a0\u00a0\u00a0 13\u00a0\u00a0\u00a0 21\u00a0\u00a0\u00a0 34\u00a0\u00a0\u00a0 55\u00a0\u00a0\u00a0 89\u00a0\u00a0\u00a0 144\u00a0\u00a0 \u2026<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Raportul dintre doi termeni consecutivi se apropie tot mai mult de num\u0103rul de aur, \u03a6 \u2248 1,618.<\/p>\n

\u00a0\u00a0\u00a0Filotaxia<\/h1>\n\n\n\n
\u00a0\u00a0Din fi\u0219a de lucru<\/em><\/p>\n

Filotaxia studiaz\u0103 modul \u00een care organele plantelor (frunze, ramuri, flori, solzi) sunt aranjate pe tulpin\u0103 sau pe un ax central. Ca o consecin\u021b\u0103 a matematicii, structurile \u00een spiral\u0103 vizibile pe conurile de conifere, pe ananas sau la floarea-soarelui urmeaz\u0103 numerele din \u0219irul lui Fibonacci.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Solzii, semin\u021bele sau frunzele apar rotite cu aproximativ 137,5\u00b0 unele fa\u021b\u0103 de altele, a\u0219a c\u0103 ochiul vede dou\u0103 familii de spirale: \u00een sens orar (dextrogire) \u0219i \u00een sens antiorar (levogire). Pe un con de pin \u0219i pe un ananas sunt de obicei 8 \u0219i 13 spirale, iar la floarea-soarelui 34 \u0219i 55.<\/p>\n

Figura 1. 8 spirale \u00een sens orar (ro\u0219u) \u0219i 13 spirale \u00een sens antiorar (albastru).<\/p>\n

\u00a0\u00a0\u00a0Obiective \u0219i materiale<\/h1>\n\n\n\n
\u00a0\u00a0Obiectivele din fi\u0219a de lucru<\/em><\/p>\n

\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Eviden\u021bierea structurilor spiralate la conuri \/ ananas.<\/p>\n

\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Determinarea numeric\u0103 a spiralelor dextrogire (sens orar) \u0219i levogire (sens antiorar).<\/p>\n

\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Verificarea ipotezei conform c\u0103reia sistemele biologice folosesc numere Fibonacci consecutive.<\/p>\n

 <\/p>\n

\u00a0\u00a0Materialele din fi\u0219a de lucru<\/em><\/p>\n

\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 conuri de pin (alte conifere), fruct ananas proasp\u0103t;<\/p>\n

\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 markere de dou\u0103 culori diferite.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u00a0\u00a0Mod de lucru<\/h2>\n
    \n
  1. Alege\u021bi un solz de pornire pe con (sau pe ananas).<\/li>\n
  2. Cu markerul ro\u0219u marca\u021bi spiralele \u00een sensul acelor de ceasornic \u0219i num\u0103ra\u021bi-le.<\/li>\n
  3. Cu markerul albastru marca\u021bi \u0219i num\u0103ra\u021bi spiralele \u00een sens invers.<\/li>\n
  4. Trece\u021bi numerele \u00een Tabelul 2 \u0219i calcula\u021bi raportul: valoarea mare \/ valoarea mic\u0103.<\/li>\n<\/ol>\n

    \u00a0\u00a0\u00a0Num\u0103rul de aur<\/h1>\n

    Tabelul 1. Rapoartele dintre termeni consecutivi<\/p>\n\n\n\n\n\n\n\n\n
    Raport<\/td>\nValoare<\/td>\nRaport<\/td>\nValoare<\/td>\n<\/tr>\n<\/thead>\n
    2 \/ 1<\/td>\n2<\/td>\n13 \/ 8<\/td>\n1,625<\/td>\n<\/tr>\n
    3 \/ 2<\/td>\n1,5<\/td>\n21 \/ 13<\/td>\n1,615<\/td>\n<\/tr>\n
    5 \/ 3<\/td>\n1,667<\/td>\n34 \/ 21<\/td>\n1,619<\/td>\n<\/tr>\n
    8 \/ 5<\/td>\n1,6<\/td>\n55 \/ 34<\/td>\n1,618<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Rapoartele se apropie de \u03a6 = (1 + \u221a5) \/ 2 \u2248 1,618, numit num\u0103rul de aur sau raportul de aur. P\u0103tratele cu laturile 1, 1, 2, 3, 5, 8 \u0219i 13 formeaz\u0103 un dreptunghi de aur, iar arcele desenate \u00een ele dau spirala de aur.<\/p>\n

    Figura 2. P\u0103tratele Fibonacci \u0219i spirala de aur.<\/p>\n

    \u00a0\u00a0\u00a0\u00cenregistrarea \u0219i prelucrarea datelor<\/h1>\n

    Tabelul 2. Num\u0103rul de spirale \u0219i raportul lor (se completeaz\u0103 \u00een clas\u0103)<\/p>\n\n\n\n\n\n\n\n
    E\u0219antion analizat<\/td>\nNr. spirale orar (ro\u0219u)<\/td>\nNr. spirale antiorar (albastru)<\/td>\nRaportul (valoare mare \/ valoare mic\u0103)<\/td>\n<\/tr>\n<\/thead>\n
    E\u0219antion 1<\/td>\n <\/td>\n <\/td>\n <\/td>\n<\/tr>\n
    E\u0219antion 2<\/td>\n <\/td>\n <\/td>\n <\/td>\n<\/tr>\n
    Con de pin \/ ananas standard<\/td>\n8<\/td>\n13<\/td>\n13 \/ 8 = 1,625<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Calcul pentru e\u0219antionul standard: <\/strong>13 \/ 8 = 1,625; diferen\u021ba fa\u021b\u0103 de \u03a6 este 0,007 (aproximativ 0,4%).<\/p>\n

    \u00a0\u00a0\u00a0Evaluare \u0219i concluzii<\/h1>\n\n\n\n
    \u00a0\u00a0Din fi\u0219a de lucru<\/em><\/p>\n

    Numerele ob\u021binute \u00een urma num\u0103r\u0103rii spiralelor fac parte din \u0219irul lui Fibonacci.<\/p>\n

     <\/p>\n

    \u00a0\u00a0\u00centrebarea din fi\u0219a de lucru<\/em><\/p>\n

    C\u00e2t de aproape este raportul calculat \u00een tabel de valoarea teoretic\u0103 a num\u0103rului de aur, \u03a6 = 1,618?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

     <\/p>\n

    Pentru e\u0219antionul standard, 13 \/ 8 = 1,625 difer\u0103 de \u03a6 cu doar 0,007. Cu c\u00e2t numerele de spirale sunt mai mari, cu at\u00e2t raportul este mai aproape de \u03a6 (de exemplu 55 \/ 34 \u2248 1,618). \u03a6 \u2013 num\u0103rul de aur, adic\u0103 raportul de aur \u2013 este o propor\u021bie aproape perfect\u0103 \u00een natur\u0103.<\/p>\n

    Concluziile noastre dup\u0103 test: ________________________________________________________________<\/p>\n

    _____________________________________________________________________________________________<\/p>\n","protected":false},"excerpt":{"rendered":"

    Matematic\u0103\u00a0 \u00b7\u00a0 2026\u20132027\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 Fi\u0219\u0103 de lucru \u00a0\u00a0   Spiralele din natur\u0103 \u0219i num\u0103rul de aur   Elevi Balta Claudiu Mihai Toma Alexandru Gabriel Constantinescu Denis Mihai Coca Alin Clasa a XII-a…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-506","post","type-post","status-publish","format-standard","hentry","category-resources"],"_links":{"self":[{"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/posts\/506","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/comments?post=506"}],"version-history":[{"count":3,"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/posts\/506\/revisions"}],"predecessor-version":[{"id":510,"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/posts\/506\/revisions\/510"}],"wp:attachment":[{"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/media?parent=506"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/categories?post=506"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/liceubolintin.ro\/eduhub\/wp-json\/wp\/v2\/tags?post=506"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}