The Golden Ratio is, for our
intensive purposes, a mathematical constant with the purpose of
facilitating the calculating of Fibonacci numbers. The Golden Proportionobserved in Fibonacci series is the following:
1, 1, 2, 3, 5,8, 13, 21, 34, 55, 89, 144, 233...
Notice a pattern?
Well, 2 = 1+1 and 3 = I +2 so that F3 = F2 + F I or
Fibonacci numbers are extremely closely related to the Golden Ratio... Take the following Fibonacci Numbers and divide the first number into the next.










1 
1 
2 
3 
5 
8 
13 
21 
34 
55 

1/1=1 
1/2=.5 
2/3=.667 
3/5=.6 
5/8=.625 
8/13=.615 
13/21=.619 
21/34=.617 
34/55=.618 
55/89=.618 
=
[5^{(1/2)} +1] / 2 which is about 1.61803. For a big expanded form of Phi, click here)
Phi has many special properties one of which is Phi * Phi = Phi + 1 and another is that sqrt(Phi) = Phi  1
F(n) = [(^{n}  ()^{n })] / [5^{(1/2)}].
You see that! Isn't that amazing? Fibonacci numbers, in a way, explain why we attach so much importance to the Golden Ratio and at the same time, the Golden Ratio clarifies Fibonacci sequences.
Colin has made a program that will generate as many Fibonacci numbers as you like. It is for any Macintosh, and you can read the code here (It's written in pascal). Click here to download the program (24k).
This image, from Surrey University shows the relationship between Fibonacci numbers and their ratios. Notice the numbers are approaching Phi.