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

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.

- Rational approximations of irrational numbers.
- Where do the "best" rational approximations come from?
- The most irrational number.
- Irrational numbers and the growth of
plants.

--*Tony Phillips*

© copyright 1999, American Mathematical Society.