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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](http://www.youtube.com/watch?v=ahXIMUkSXX0) like this ratio2012-12-30
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    @experimentX. WOW!!! so wonderful is it?? so it is everywhere in the nature?2012-12-30
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    yep!! also check the other two of the videos in the series.2012-12-30
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    Woow, man!! can you please write something as an answer, i will check it, you gave something which is more than an answer..2012-12-30
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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. http://math.stackexchange.com/questions/261359/how-to-show-that-closed-form-of-fibonacci-number-is-roots-ratio-difference-of-n2012-12-30
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    okay, thanks a lot. another great insight into the world of numbers!2012-12-30
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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.2012-12-30
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    [here's a link](http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/least-squares-determinants-and-eigenvalues/diagonalization-and-powers-of-a/MIT18_06SCF11_Ses2.9sum.pdf) to the closed form of Fibonacci number2012-12-30
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    great, thanks dude2012-12-30

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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! +2012-12-30
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    Great, Thanks Adi, very nice2012-12-30
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    i watched this video: https://www.khanacademy.org/math/algebra/rational-expressions/ratios_algebra/v/the-golden-ratio now it is clear like anything :D2012-12-30
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    @AdiDani how -1 in numerator it should be +1 ???2017-01-30
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    $1+\frac{-1+\sqrt 5}{2}=\frac{1+\sqrt 5}{2}$2017-02-04
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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: $\Delta =(-1)^2-4\cdot 1 (-1)=5$ and then $x_{positive}=\frac{1+\sqrt{5}}{2}=1,618$ 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 [ $\epsilon$ ] ;)2012-12-30
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    you're welcome :)2012-12-30