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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.
yes, with fibonacci serie it is an approximation. but you now discovered the rule:
The golden ratio is defined by x=1/x + 1
that is:
1/1.618 = 0.618 and 1/0.618 = 1.618
you could write that formula as a 2nd grade equation:
1/x = x - 1
x*(1/x) = (x - 1)*x // multiply by x 1 = x^2 - x
then write as:
x^2 - x - 1 = 0
so, applying the formula for solving 2nd g eq:
x[1,2] = (1 ± sqrt(5))/2
you got *THE* magic ratio:
x1 = 1.618033989... x2 = -0.618033989...
(and the approximation depends on your calculator... hehe ;)
1.61803398874989484820458683436563811772030917980576286213544862270526 is enough?
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.