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Originally Posted by Rekna
One more oddity I just found out Phi (1.61803399) has a multiplicative inverse known as phi(.61803399). The strange thing is these are the only 2 numbers whose multiplicative inverses vary by exactly one.
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Considering the geometric determination involves dividing a length Z into two sections X and Y, such that Y=X+n, n=1 and the ratio of X:Y::Y:Z. This "strange thing" you're pointing out is that X and X+n differ by exactly one when n=1. Oooooooh.
Furthermore, the Fibonacci sequence is basically a integral math implementation of the geometric proof for proportionally larger values of Z and n (I.E. when you divide the system by n, thereby returning to n=1, you get a progressively closer approximation of the golden ratio for larger values of Z). The math is kinda cool, but there's nothing mystical about it.