Is it just the most structurally stable ratio or something? Why do so many things evolve to have golden proportions?

The spirals originate from the fibonacci series. This video (and the sequels) explains it all.

 

In short: it has to do with the mathematically optimal way for plants to arrange branches, leafs or scales.

There is a similar recent thread about this.

 

http://www.scienceforums.net/topic/103089-the-golden-ratio/

The 1.62 Golden Mean ratio indeed is ubiquitous in biological organisms. It is usually comprises of just bring a cross section from a larger segment, that is based on a Fibonacci Sequence Spiral. Two famous examples are Nautilus shells and pinecones.

 

Do the spiral ratio is the real magic here. I believe it is structurally optimal. Also it's excellent for space allocation. And it's aesthetically​ pleasing to prospective polinators. Just as symmetry is provenly attractive in humans!

 

The Golden Mean as a stand alone rectangle is seen more commonly in man-made things. Buildings, architecture. Paintings, artwork. But in nature I think it's almost always a part of the Fib Spiral.

 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55..........