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i am very interested in Golden Ratio and its value. the Golden Ratio itself is not hard thing to visualize and understand in 5 minutes. But i am trying to reach the historical, logical reasons of origin of this ratio.

my first question is: why is the value of the ratio $\frac{a+b}{a}=\frac{a}{b}=1.618$ ? it is the positive root of $a^2-a-1=0$. can someone pls give me more clues,facts and properties of this phenomenon?

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it seems that plants like this ratio –  Santosh Linkha Dec 30 '12 at 10:41
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@experimentX. WOW!!! so wonderful is it?? so it is everywhere in the nature? –  doniyor Dec 30 '12 at 10:47
    
yep!! also check the other two of the videos in the series. –  Santosh Linkha Dec 30 '12 at 10:49
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sorry as a matter of fact, i know too little on this topic. As far as I know, as $n \to \infty $, the consecutive Fibonacci numbers tend to be in Golden ratio, and the roots of Golden ratio are used to express the n-th Fibonacci numbers in closed form. math.stackexchange.com/questions/261359/… –  Santosh Linkha Dec 30 '12 at 10:53
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also there is some scientific explanation why Plants love golden ratio so much, it's because when the leaves sprout up consecutively, the optimum angle is around the angle made my one of these numbers. –  Santosh Linkha Dec 30 '12 at 10:58
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2 Answers

up vote 5 down vote accepted

Suppose we have a stick $AB$ of length $1$ and we need to cut that at position $C$ and let be $AC>CB$ by golden cut then we have $${AC\over CB}={AB\over AC}$$If $AB=1,AC=x,CB=1-x$ we get $${x\over 1-x}={1\over x}$$ $$x^2=1-x$$ $$x^2+x-1=0$$ positive solution of this equation $$x=\frac{-1+\sqrt5}{2}=\phi$$ is golden ratio

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Very simple approach! Understandable one! + –  B. S. Dec 30 '12 at 11:23
    
Thank you Babak –  Adi Dani Dec 30 '12 at 11:25
    
Great, Thanks Adi, very nice –  doniyor Dec 30 '12 at 12:03
    
i watched this video: khanacademy.org/math/algebra/rational-expressions/… now it is clear like anything :D –  doniyor Dec 30 '12 at 19:03
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Regarding your question about the root...

$\frac{a+b}{a}=\frac{a}{b} \rightarrow \frac{a}{a}+\frac{b}{a}=\frac{a}{b}\rightarrow 1+\frac{1}{\frac{a}{b}}=\frac{a}{b}$. Let ,$\frac{a}{b}=x$ then $1+\frac{1}{x}=x$ and if you multiple both sides with $x$ you get the equation $x+1=x^2\rightarrow x^2-x-1=0$. Now the soloution of this equation: $\Delta =(-1)^2-4 1 (-1)=5$ and then $x_{1,2}=\frac{1+\sqrt{5}}{2}=1,618$ approximately.

and...

golden ratio

if :

  1. you a draw a circle with $radius=\frac{a}{2}$ which has AB as a tangent

  2. draw the line that crosses from K and A

  3. draw the circle with radius AE

you have the desired point in AB

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wow, that was it.. thanks, epsilon [ $\epsilon$ ] ;) –  doniyor Dec 30 '12 at 12:05
    
you're welcome :) –  epsilon Dec 30 '12 at 21:15
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