It's not just pleasing to the eye, there is a mathematical base to the golden ratio.
Take a line of unit length, and divide it into two non-equal parts. The larger is called the major (we'll call it x here), and the smaller is called the minor(y, or 1-x).
Now, the point of the golden ratio is that if you divide your line with exactly the golden ratio (GR), then the ratio of x:y is the same as the whole distance (1) to x. Or put mathematically:
1/x = x/y
1/x = x/(1-x)
this can be transformed into
using the standard formula for quadratic equations
x=(sqrt(5)-1)/2 [yes, there's a second solution, but it's a negative length in this case, so we can disregard it]
and the golden ration is 1/x, or x/y, which is both ~1.618