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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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    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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    $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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    you're welcome :)2012-12-30