There are many web resources on the golden mean and Fibonacci numbers. See in particular Ron Knott's Fibonacci Numbers and the Golden Section and Steven Finch's MathSoft page The Golden Mean which has many useful links. There is not much information on the Web about continued fractions*. The theorems I have quoted in typed format appear in C. D. Olds, Continued Fractions (Random House New Mathematical Library, New York, 1963) or in A. Rockett and P. Szüsz Continued Fractions (World Scientific, Singapore, 1992). There is a very nice page on Symmetry in Plants: Phyllotaxis at Smith College, with illustrative applets.
An irrational number by definition is one which cannot be written as the ratio of whole numbers. So it would seem that all irrational numbers are equally irrational. All pigs are equal, Orwell said, but some are more equal than others. And in fact there is a precise sense in which some irrational numbers are more irrational than others. This phenomenon has had important consequences in the organization of the natural world. In packing seeds around a core, many plants choose the strategy of placing each one at the most irrational angle possible to the one directly below it.
Rotation through the most irrational angle leads to seed packing patterns
with numbers of left- and right-diagonal rows given by consectutive Fibonacci numbers:
(3,5) and (5,8) for the two pine cones shown here. Larger image.
This column will present the ideas of rational approximation and the concept of some numbers being better approximated by rationals than others. We will look at the most irrational number and at how its use in plant growth leads to the occurrence of Fibonacci numbers in seed-packing patterns.
*: Update: There are now additional good web sources for continued fractions, including a page on Wikipedia.