JRG wrote:
> Aah, the golden ratio. I always thought of it just as
> the number to which the ratio between two consequential
> numbers of Fibonacci's series converges...

It all ties together.

How do you find the Fibonacci limit ratio?
You assume that the sequence approaches
	... ax^(n-2), ax^(n-1), ax^n ...
and the definition of the sequence gives
	ax^n = ax^(n-1) + ax^(n-2) ;
divide through by the last term:
	x^2 = x + 1
which is the same quadratic as before.


Or: given a continued fraction form
	a[0] + 1/( a[1] + 1/( a[2] + 1/( a[3] + ...)))
you want to find rational approximations.  Here's the procedure:
start with	num[-2] / den[-2] = 0/1
		num[-1] / den[-1] = 1/0
then		num[i] = a[i]*num[i-1] + num[i-2]
		den[i] = a[i]*den[i-1] + den[i-2]

For tau, 1 + 1/1+(1/1+(1/1+(1/1+(1/1+(...))))),
guess what that gives:
	0/1, 1/0, 1/1, 2/1, 3/2, 5/3, 8/5, 13/8 ...


The golden ratio turns up so often, that I was a bit startled one time
when a limit proved to be 6/pi^2 instead!


-- 
          --  Anton Sherwood  --  http://www.ogre.nu/  --
"If nobody said anything unless he knew what he was talking about, 
a ghastly hush would descend upon the earth."  ---Sir Alan Herbert

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