Fibonacci Dizisi

Fibonacci serisi sayıları: 0,1,1,2,3,5,8,13,21,34,55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, … vb. şeklinde devam eder. Her sayı kendisinden önce gelen iki sayının toplamıdır. Bu durumda genel olarak n'inci Fibonacci sayısı F(n) şu şekilde ifade edilir:



Fibonacci sayılarının ilginç özellikleri vardır. Mesela n sayısı büyüdükçe iki ardışık Fibonacci sayısının oranı Altın oran'a yani 1.618... e yakınsar.
 

Altın Üçgen/Golden Triangle

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72

-->˚-36 -->˚-72 -->˚ üçgeni "altın üçgen" olarak adlandırılır.İkiz kenarlardan birinin uzunluğunun, taban uzunluğuna oranı altın oranı vermektedir.

Günlük Hayatımızda Altın Oran

Günlük hayatta, altın orana bir çok yerde rastlarız:





Bilgisayar yardımıyla, altın oranın virgülden sonra 1 milyonuncu basamağına kadar ulaşılmış...Buraya hepsi sığmaz ama birkaç bin basamağı yazıyorum:
1.6180339887498948482045868343656381177203091798057628621354486227052604628189
02449707207204189391137484754088075386891752126633862223536931793180060766726354
43338908659593958290563832266131992829026788067520876689250171169620703222104321
62695486262963136144381497587012203408058879544547492461856953648644492410443207
71344947049565846788509874339442212544877066478091588460749988712400765217057517
97883416625624940758906970400028121042762177111777805315317141011704666599146697
98731761356006708748071013179523689427521948435305678300228785699782977834784587
82289110976250030269615617002504643382437764861028383126833037242926752631165339
24731671112115881863851331620384005222165791286675294654906811317159934323597349
49850904094762132229810172610705961164562990981629055520852479035240602017279974
71753427775927786256194320827505131218156285512224809394712341451702237358057727
86160086883829523045926478780178899219902707769038953219681986151437803149974110
69260886742962267575605231727775203536139362107673893764556060605921658946675955
19004005559089502295309423124823552122124154440064703405657347976639723949499465
84578873039623090375033993856210242369025138680414577995698122445747178034173126
45322041639723213404444948730231541767689375210306873788034417009395440962795589
86787232095124268935573097045095956844017555198819218020640529055189349475926007
34852282101088194644544222318891319294689622002301443770269923007803085261180754
51928877050210968424936271359251876077788466583615023891349333312231053392321362
43192637289106705033992822652635562090297986424727597725655086154875435748264718
14145127000602389016207773224499435308899909501680328112194320481964387675863314
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40587461033781054443909436835835813811311689938555769754841491445341509129540700
50194775486163075422641729394680367319805861833918328599130396072014455950449779
21207612478564591616083705949878600697018940988640076443617093341727091914336501
37157660114803814306262380514321173481510055901345610118007905063814215270930858
80928757034505078081454588199063361298279814117453392731208092897279222132980642
94687824274874017450554067787570832373109759151177629784432847479081765180977872
68416117632503861211291436834376702350371116330725869883258710336322238109809012
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93902111569103927683845386333321565829659773103436032322545743637204124406408882
67375843395367959312322134373209957498894699565647360072959998391288103197426312
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00090862180297756378925999164994642819302229355234667475932695165421402109136301
81947227078901220872873617073486499981562554728113734798716569527489008144384053
27483781378246691744422963491470815700735254570708977267546934382261954686153312
09533579238014609273510210119190218360675097308957528957746814229543394385493155
33963038072916917584610146099505506480367930414723657203986007355076090231731250
16132048435836481770484818109916024425232716721901893345963786087875287017393593
03013359011237102391712659047026349402830766876743638651327106280323174069317334
48234356453185058135310854973335075996677871244905836367541328908624063245639535
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05132693448375299164998093602461784426757277679001919190703805220461232482391326
10432719168451230602362789354543246176997575368904176365025478513824631465833638
33760235778992672988632161858395903639981838458276449124598093704305555961379734
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18394918323709085748502665680398974406621053603064002608171126659954199368731609
45722888109207788227720363668448153256172841176909792666655223846883113718529919
21631905201568631222820715599876468423552059285371757807656050367731309751912239
73887224682580571597445740484298780735221598426676625780770620194304005425501583
12503017534094117191019298903844725033298802450143679684416947959545304591031381
16218704567997866366174605957000344597011352518134600656553520347888117414994127
48264152135567763940390710387088182338068033500380468001748082205910968442026446
40218770534010031802881664415309139394815640319282278548241451050318882518997007
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99391357478698957986439498023047116967157362283912018127312916589952759919220318
37235682727938563733126547998591246327503006059256745497943508811929505685493259

The Golden Ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is an irrational mathematical constant, approximately 1.6180339887...

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

The golden ratio is often denoted by the Greek letter ϕ (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:



This equation has as its unique positive solution the algebraic irrational number: