The Golden Ratio

The number $\phi = \frac{1+\sqrt{5}}{2}$, known as the Golden Ratio, turns up in a lot of places.  This one that I read about in National Geographic surprised me.  The following coordinates describe an icosahedron:
$$
(\phi,1,0), (-\phi,1,0), (-\phi,-1,0), (\phi,-1,0) \\ (0,\phi,1), (0,-\phi,1), (0,-\phi,-1), (0,\phi,-1) \\
(1,0,\phi), (1,0,-\phi), (-1,0,-\phi), (-1,0,\phi) \\
$$I thought I'd have a go at figuring out why, and the above picture is the result.  All you have to do is observe that the vertices of an icosahedron are the same as those of a set of three identical intersecting rectangles.  The proof then reduces to showing that the ratio of the long side to the short side is $\phi$ - i.e. that these are golden rectangles.  The proof above takes the short side to be of length 1 and the long side to be of length $x$.

So, why does $\phi$ turn up so often in maths and geometry?  I think the answer is the same as the answer to the question "why does $\frac{1}{2}$ turn up so often?", namely that both are roots of particularly simple polynomials with small integer coefficients.  In the case of the Golden Ratio it's $x^2 - x - 1$ and in the case of $\frac{1}{2}$ it's $2x-1$.  But for some reason, $\phi$ has acquired much more mystical significance.

POSTSCRIPT

The golden ratio $\phi$ also turns up in the pentagon and the pentangle.  In fact in the pentangle if you take the ratio of any line segment with the next shortest line segment you get $\phi$.  Here's a proof of one of the cases

 

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