**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 MeanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which its square and reciprocal share the same fractional part {.618034}. It is associated with the synodic period of the planet Venus, which is 8/5 {1.6} of the practical year {365 days}, by approximation. It is a key proportion found in Greco-Roman and later "classical" architecture, and commonly encountered in the forms living bodies take.** (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 Phi^{2} can always be reduced to (1 + Phi). This allows any power of Phi to be reduced to a number plus another number, times Phi.

Phi^{1} = 0 + 1 * Phi

Phi^{2} = 1 + 1 * Phi

Phi^{3} = 1 + 2 * Phi

**Phi ^{4} = 2 + 3 * Phi (see below)**

Phi^{5} = 3 + 5 * Phi

Phi^{6} = 5 + 8 * Phi

Phi^{7} = 8 + 13 * Phi

Phi^{8} = 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 Phi
^{N}= Fib^{N}+ Fib^{N+1}* Phi - This happens because the Fibonacci arises through decomposition of powers.

The method of reducing powers works like this:

Phi^{4}

= Phi^{2} * Phi^{2}

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

= Phi^{2} + 2 * Phi +1

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

therefore,** Phi ^{4 }= 2 + 3 * Phi**

This article was first published February 16, 2009