phi vs. Phi - a Coincidence?

Ancient and modern architecture reflect the 'golden ratio' (1.618
length to width) and this number is remarkably close to phi (.618...)
seen in nature for leaf dispersions, etc. Is this just a coincidence?


It's more than just coincidence: the golden ratio (as you define it)
is phi's twin, "Phi," where

Phi = (sqrt(5) + 1)/2 = 1.618...

phi = (sqrt(5) - 1)/2 = 0.618...

Phi = 1/phi

Phi = 1 + phi

The latter facts together give the definition of the golden ratio:

x = 1/x + 1

This equation (equivalent to x^2 - x - 1 = 0) is satisfied by both Phi
and -phi, which therefore can be called the _golden ratios_. Since
they are reciprocals, either could just as well be given that name.

Together, these are used in the formula for the Fibonacci sequence:

F[n] = (Phi^n - (-phi)^n) / sqrt(5)

*thank you amy for this research.

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