On Fri, 30 Jun 2006, Chuckk Hubbard wrote:
It is an approximation, and gets closer and closer, and it works if you start with ANY 2 positive numbers and make a Fibonacci-esque series from them. You could start with 2 and 417, then 419, 836, 1255, etc, and gradually the ratios of consecutive members of the series start to approach the golden ratio.
This is because defining Newfib(1)-Newfib(0)!=1 just multiplies Fibonacci by a constant, which cancels out when taking ratios; and redefining Newfib(0) != 0 only shifts it like Newfib(n) = Fib(n+c), though c might be fractional (you normally have to use the formula i gave in the previous mail to get Fibonacci values on the positive real domain)
The golden ratio is defined by x=1/x + 1
More usually like x^2 - x - 1 = 0, which is really the same thing. Note how it looks like Fib(n+2) - Fib(n+1) - Fib(n) = 0 ! You may consider Fib(n+c) = x^c for extra fun.
when you take, for instance, 13/8, that is 1 + 5/8... or 21/13 is 1 + 8/13... 34/21 is 1 + 13/21, because of the way you obtain each number. Always 1 + the reciprocal of the previous ratio. So you will never quite reach the golden ratio, but you alternately go slightly higher and slightly lower, getting gradually closer.
BTW, Euclid's way of simplifying fractions (greatest common divisor) is slowest with consecutive Fibonacci numbers. Its average and worst running times are multiples of log((a+1/a)*n)/log(a), where a is the golden ratio.
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