Powers of the Golden Mean

The Golden Mean (1.618034) or Phi (Greek letter) is renowned for the behavior of it's reciprocal and square which are 0.618034 and 2.618034 respectively; that is the fractional part stays the same. This is a singularity in number that makes the Phi unique.It is irrational but shown here to 6 figures in the all important fractional part.

Whilst found in sacred buildings it is also present in living forms, partly due to the fact that many series of added number pairs tend towards Phi between one result and the next, most famously the Fibonacci series of 0 1 1 2 3 5 8 13 21 34 55 89 ... where each right hand result is the simple sum of the two preceeding numbers.

In looking at my notes of 31st Jan 1994 I came across a generalisation of the powers of Phi that I had worked out algebraically, using the fact that Phi2 can always be reduced to (1 + Phi). This allows any power of Phi to be reduced to a number plus another number, times Phi.

Phi1  = 0 + 1 * Phi

Phi2  = 1 + 1 * Phi

Phi3  = 1 + 2 * Phi

Phi4  = 2 + 3 * Phi (see below)

Phi5  = 3 + 5 * Phi

Phi6  = 5 + 8 * Phi

Phi7  = 8 + 13 * Phi

Phi8  = 13 +21 * Phi

Which should be enough to get the gist, that the Fibonacci series is turning up as the two numbers.

The general form is PhiN = FibN + FibN+1 * Phi

This happens because the Fibonacci arises through decomposition of powers.

The method of reducing powers works like this:

Phi4

= Phi2 * Phi2

= (1+Phi) * (1+ Phi)

= Phi2 + 2 * Phi +1

= (1 + Phi) + 2 * Phi +1

therefore, Phi4 = 2 + 3 * Phi