The "golden ratio" (also called the "golden mean") is the ratio between the sides of a rectangle considered to be "perfect" by some of the mathematical greeks. This rectangle has the interesting property that if you cut a square off the end, the remaining rectangle is similar (same ration of sides) to the original.
This means if s = short side and l = long side:
l / s = phi (golden ratio)
s / (l-s) = phi
s / (l-s) = l / s
s^2 = l(l-s)
l^2 - ls - s^2 = 0
Solving this quadratic, we get l = s*(sqrt(5) + 1)/2
So phi, the golden ratio = (sqrt(5) + 1) / 2 = 1.618...
This is also the limit of the ratios of two adjacent elements of the Fibonacci series! In other words, in the series:
1, 1, 2, 3, 5, 8, 13, ...
1 / 1 = 1.0
2 / 1 = 2.0
3 / 2 = 1.5
8 / 5 = 1.6
13 / 8 = 1.625
So if you construct a figure by taking two unit squares, putting a 2x2 square on the side (making a 3x2 rectangle), then adding a 3x3 square, a 5x5 square, and so on, the resulting rectangles approach this "golden ratio". And if you draw a curve through successive vertices of the squares, you get a smooth logarithmic curve, like a snail shell.