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golden ratio
The “Golden Ratio”, or $\phi$, has the value
$1.61803398874989484820\ldots$ 
This number gets its rather illustrious name from the fact that the Greeks thought that a rectangle with ratio of side lengths equal to $\phi$ was the most pleasing to the eye, and much of classical Greek architecture is based on this premise. In addition, an aesthetically pleasing aspect of a rectangle with this ratio, from a mathematical viewpoint, is that if we embed and remove a $w\times w$ square in the below diagram, the remaining rectangle also has a widthtolength ratio of $\phi$.
l$\;\;\;\;\;\;\;\;l$
\psfragw$\!\!\!w$
Above: The golden rectangle; $l/w=\phi$.
$\phi$ has plenty of interesting mathematical properties, however. Its value is exactly
$\frac{1+\sqrt{5}}{2}$ 
The value
$\frac{1\sqrt{5}}{2}$ 
is often called $\phi^{{\prime}}$. $\phi$ and $\phi^{{\prime}}$ are the two roots of the recurrence relation given by the Fibonacci sequence. The following identities hold for $\phi$ and $\phi^{{\prime}}$ :

$\frac{1}{\phi}=\phi^{{\prime}}$

$1\phi=\phi^{{\prime}}$

$\frac{1}{\phi^{{\prime}}}=\phi$

$1\phi^{{\prime}}=\phi$
and so on. These give us
$\phi^{{1}}+\phi^{0}=\phi^{{1}}$ 
which implies
$\phi^{{n1}}+\phi^{n}=\phi^{{n+1}}$ 
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Comments
on your pic
ugh.. what illustrative.. hehehe a rectangle
sorry for sarcasm
I was thinking on something like

  
  
  
  

showing that cutting a square leaves a rectangle similar to the original
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: on your pic
sorry that was meant to be a correction, not a post
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: on your pic
I could do another pic below it showing that you get the "golden spiral" that way.
apk