# Powers of the Golden Mean Sheikh Lotfollah Mosque  is one of the masterpieces of Iranian architecture that was built during the Safavid Empire, standing on the eastern side of Naqsh-i Jahan Square, Esfahan, Iran. Construction of the mosque started in 1603 and was finished in 1619.
for Wikipedia by Phillip Maiwald

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. Phi is a unique singularity in number. While irrational, shown here to only 6 figures, it is its infinite fractional part which is responsible for Phi’s special properties.

The Fibonacci series: Found in sacred buildings (above), it is also present in the way living forms develop. Many other series of initial number pairs tend towards generating better and better approximations to Phi. This was most famously the Fibonacci series of 0 1 1 2 3 5 8 13 21 34 55 89 … (each right hand result is the simple sum of the two preceding numbers (0+0 = 1, 1+1=2, etc.

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, Phi= 2 + 3 * Phi