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

(1-x)=x2

x2+x-1=0

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]

x=0.618...

y=1-x=0.381...

and the golden ration is 1/x, or x/y, which is both ~1.618