phi vs. Phi - a Coincidence?

Date: 10/16/2001 at 14:10:08
From: S. Cotton
Subject: phi versus "golden ratio"

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?

Thanks for your reply.

S. Cotton

Date: 10/16/2001 at 17:00:15
From: Doctor Peterson
Subject: Re: phi versus "golden ratio"

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)

See Ron Knott's page for details:

   The Golden section ratio: Phi
   http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/