Appearances of the Golden Number

Date: 06/03/98 at 23:45:00
From: quinn matias
Subject: Why is phi such an ubiquitous irrational number?

I have been doing some research on the phenomenon of phyllotaxes (the 
study of the disposition of leaves around a stem, or any other pattern 
of the sort). I have seen the irrational number phi (1+SQR5)/2 crop up 
in most of the patterns under analysis. For example, the ratio between 
the number of parastichia formed in the sunflower capitulum or pine 
cone is the same as that which can be obtained by analysing the 
intrisic proportions present in the five regular solids or polar 
equotations of the growth of the shell of a snail or Nautilus, namely 
phi, also called the golden section. 

I have read many classic books on the topic of Fibonacci numbers and 
its relation to continuous fractions and phi, but I have never come 
across a proper mathematical reason which would explain why phi is so 
ubiquitous in the natural and non-biotic world. I am looking for some 
kind of scientific explanation. 

Thanks!

Date: 11/17/98 at 22:17:33
From: Doctor Ron
Subject: Re: Why is phi such an ubiquitous irrational number?

Hi Quinn:

This is an interesting question that people have pondered about for a 
while. It was suspected last century that plants use phi as an optimal 
solution to a problem - but what problem? Only a few years ago (1993) 
two French mathematicians, Stephane Douady and Yves Couder, found a 
mathematical explanation. It has to do with the way plants grow: from a 
meristem (a tiny tip of the growing point of plants) where new cells 
are formed. The principle nature uses is that of spiral growth and it 
produces new cells at a constant rate (or rather a constant amount of 
turn) for each new "point." The points may be leaves on a twig, or 
branches from a trunk, or seeds on a seed head, or petals round the 
edge. As the cells are then fixed (and the meristem grows upward and 
turns again before producing a new "cell") they then grow only outward 
and develop. So what is the "best" angle to use for this simple growing 
process? It turns out to be the "simplest" irrational number of 
"points" per turn and this is phi = 0.618034 which is also 
1.618034 = Phi = 1/phi "points" per whole turn.

It's explained in more detail on this web page at a large Fibonacci 
site:
 
 http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html   

As for "the non-biotic world" such as puzzles, the mathematical 
formulation of a problem will sometimes lead to a quadratic equation. 
The golden section numbers Phi and phi occur as the solutions (roots) 
to the quadratic equation x^2 + x - 1 = 0 and x^2 - x - 1 = 0. 
Sometimes a problem is solved by a "recursive" solution and, if it 
involves the sum of the (number of) solutions for the same probem but 
of sizes one less and two less, then it involves the Fibonacci numbers 
and related series. The Fibonacci numbers are f(n) = f(n-1) + f(n-2) 
(the sum of the previous two numbers), and it can be proved that no 
matter which two numbers we start from (the Fibonacci series starts 
from 0 and 1 or else 1 and 1 or perhaps 1 and 2), the ratio of two 
successive numbers is always phi or Phi, if we go far enough.  

I hope this helps!

- Doctor Ron, The Math Forum
  http://mathforum.org/dr.math