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http___fractalenlightenment.com_bs_15458_fractals_understanding-the-fibonacci-sequence-and-golden-ratio_1.xml
2.1 2.2 2.3 3.1 4.1 4.2 4.3 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6 7.1 7.2 7.3 8.1 9.1 10.1 10.2 10.3 10.4 10.5 11.1 11.2 12.1 12.2 12.3 13.1 15.1 16.1 17.1 17.2 17.3 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9
The Fibonacci sequence is possibly the most simple recurrence relation occurring in nature. Այն է, 0,1,1,2,3,5,8,13,21,34,55,89, 144… each number equals the sum of the two numbers before it, and the difference of the two numbers succeeding it. It is an infinite sequence which goes on forever as it develops. The Golden Ratio/Divine Ratio or Golden Mean - The quotient of any Fibonacci number and it’s predecessor approaches Phi, represented as ϕ (1.618), the Golden ratio. The Golden Ratio is best understood geometrically by the golden rectangle. A rectangle unevenly divided resulting into one square and one rectangle, the square’s sides would have the ratio of 1:1, and the new rectangle would be exactly proportionate to the original rectangle – 1:1.618. This iteration can continue both ways, infinitely. If you plot a quarter circle inside each of the squares as they reiterate, the golden spiral is formed. The golden spiral is possibly the most simple mathematic pattern that occurs in nature like shells of snails, sea shells, horns, Ծաղիկներ, plants. Numbers are only what we use to organize quantitative information. The Golden Ratio can be applied to any number of geometric forms including circles, triangles, pyramids, prisms, and polygons. The golden ratio is formed by thirds within thirds, sixths, the connection between two and three, including every even and odd number itself. The ratio itself represents the transcendence of numbers, understanding our world is not numbers, but what numbers represent. Through the spiral, the ratio illustrates how the numbers, all quantities, are quality. Ի վերջո,, all quality can be represented through quantity. Properties qualitative and quantitative are just labels of information, our gathered indisputable fact. If you graph any number system, eventually patterns appear. In mathematics, numbers and their patterns do not only continue infinitely linear, but in all directions. ??????,, considering infinite decimal expansion, even the shortest segments have an infinite amount of points. Our universe and the numbers not only go on infinitely linear, but even it’s short segments have infinite points. (A beautiful short film on Fibonacci sequence in Nature – ‘Nature by Numbers’) The golden ratio is not the only mathematical pattern that reaches infinity, there are many other patterns as well that reach infinity. Knowing this, ask yourself, how could infinity occur twice? If something were to happen infinitely, how could it happen twice? The answer is simple, infinity represents what is eternal, what is truly whole. ??????,, if infinity were to be used as a variable in mathematics like all other numbers, it would be denoted as 1∞, 2∞, 3∞, 4∞, եւ այլն. The oneness of everything factual is what you know, what you perceive, what you are aware of, is all the universe looking at itself. This is the universe, even you are the universe, us and everything we know is all the same thing. Since the numbers are everywhere, everything is a part of a pattern. Reflections of reflections, wheels within wheels. Life itself is a Fractal. (Another video on the Egyptian ‘Temple of Man’ is a symbolic representation of the Fibonacci series of balanced organic expansion and an expression of the underlying principles of life in the universe) Comments That’s GOD’s math! You forgot the “1” in front of .618…. It’s 1.618…… The formula is: (1+square root of 5) divided by 2. The real gem here is the Golden Proportion. The Fibonacci Sequence is the most well known sequence, but it’s beauty is merely a reflection of the Golden Proportion, also known as phi=.6180…. The Fibonacci Sequence starts with 0 ??? 1, but to illustrate this, you can start with any combinations of numbers. Give it a try. I’ll use my birthday day – 10?? – and my age, 50. We are nearly there already. I’ll call this the DavidA Sequence!
1.1 2.1 2.2 2.3 3.1 4.1 4.2 4.3 5.1 5.2 5.3 5.4 6.1 6.2 6.3 7.1 7.2 7.3 8.1 8.2 8.3 9.1 10.1 11.1 11.2 11.3 12.1 12.2 13.1 13.2 14.1 14.2 14.3 15.1 16.1
Razumijevanje Fibonacci Sequence i Golden Ratio Fibonacci Redoslijed je možda najjednostavniji ponavljanje vezi javljaju u prirodi. It is 0,1,1,2,3,5,8,13,21,34,55,89, 144… each number equals the sum of the two numbers before it, and the difference of the two numbers succeeding it. It is an infinite sequence which goes on forever as it develops. The Golden Ratio/Divine Ratio or Golden Mean – Količnik bilo koji broj Fibonacci i prethodnik približava Phi, predstavljeni kao φ (1.618), Zlatni omjer. The Golden Ratio najbolje shvatiti geometrijski od zlatnog pravokutnika. A pravougaonik neravnomjerno podijeljena je rezultiralo u jedan kvadrat i jedan pravougaonik, strane trga je imati omjer 1:1, a novi pravokutnik biti upravo proporcionalna originalni pravougaonik – 1:1.618. Takvo rješenje može nastaviti u oba smjera, beskonačno. Ako iscrtati četvrt kruga unutar svakog kvadrata jer su ponovili, se formira zlatnu spiralu. Zlatna spirala je možda najjednostavniji matematički obrazac koji javlja u prirodi kao što su ljušture puževa, školjke, rogove, cvijeće, plants. Brojevi su samo ono što mi koristimo organizirati kvantitativnih informacija. The Golden Ratio se može primijeniti na bilo koji broj geometrijskih oblika, uključujući krugovima, trouglova, piramida, prizme, i poligona. Zlatni omjer se formira trećine u roku od trećine, šestina, vezu između dva i tri, uključujući svake parne i neparne sam broj. Odnos sam po sebi predstavlja transcendencija brojeva, razumijevanje naš svijet nije brojevi, ali ono što brojevi predstavljaju. Kroz spiralu, odnos pokazuje kako brojevi, sve količine, su kvaliteta. Na kraju, sve kvalitete mogu biti predstavljeni kroz količina. Osobine kvalitativnih i kvantitativnih su samo etikete informacija, naše je okupio nespornu činjenicu. Ako grafikon bilo koji broj sistem, na kraju obrasce pojaviti. U matematici, Brojevi i njihove obrasce nastave ne samo beskonačno linearno, ali u svim pravcima. Na primjer, uzimajući u obzir beskonačni decimalni proširenja, čak i najkraće segmenti imaju beskonačno bodova. Naš svemir i brojevi ne samo ići na beskonačno linearno, Ali, čak i to je kratko segmenti imaju beskonačno poena. (Prekrasna kratki film o Fibonacci niz u Nature – "Nature by Numbers") Zlatni omjer nije jedini matematički obrazac koji doseže beskonačnost, postoje mnoge druge obrasce, kao i da je domet beskonačnosti. Znajući ovo, ask yourself, kako bi mogao doći do beskonačno dva puta? Ako se nešto dogodi beskonačno, Kako se to moglo dogoditi dva puta? Odgovor je jednostavan, infinity predstavlja ono što je vječan, ono što je zaista cijeli. Na primjer, ako je beskonačno su da se koristi kao varijabla u matematici kao i svi ostali brojevi, to će biti označen kao 1∞, 2∞, 3∞, 4∞, itd. Jedinstvo svega činjeničnog je ono što znate, ono što vide, ono što su svjesni, Sve je svemir gledajući sebi. Ovo je svemir, čak si svemir, nas i sve što znamo je sve isto. Budući da su brojevi su svuda, sve je dio obrasca. Refleksije refleksija, kotači unutar kotača. Život je Fractal. (Još jedan video na egipatskom "Temple of Man’ je simboličan prikaz Fibonacci niza uravnotežene organske ekspanzije i izraz osnovnih načela života u svemiru) Izvor Slika