Phi
The Golden Mean

The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.
The decimal representation of phi is 1.6180339887499... .

Pure Math
Proportion is the relationship of the size of two things.
Arithmetic proportion exists when a quantity is changed by adding some amount. Geometric proportion exists when a quantity is changed by multiplying by some amount. Phi possesses both qualities.
If you study the Fibonacci series and the Golden Rectangle carefully, you will eventually realize that phi + 1 = phi * phi.
The resulting proportion is both arithmetic and geometric. It is thus perfect proportion; you could think of it as the place on some imaginary graph where the curved line of multiplication crosses the straight line of addition.

Nature
In pure mathematics, an increase in size can be any imaginable number, even one like e or pi. But in the world of nature, things always grow by adding some unit, even if the unit is as small as a molecule. So it's not surprising that phi turns out to be an ideal rate of growth for things which grow by adding some quantity.
Some examples:
The Nautilus shell (Nautilus pompilius) grows larger on each spiral by phi.
The sunflower has 55 (see number list) clockwise spirals overlaid on either 34 or 89 (see number list) counterclockwise spirals, a phi proportion.

A really good book on this subject is Sacred Geometry - Philosophy and Practice by Robert Lawlor, published by Crossroad; Library of Congress number 81-67703.