The ancient Greeks, beginning with Pythagoras and the school of Pythagoras, observed certain patterns and number relationships occurring in nature. This occurrance, they said, explained the harmony and beauty found in nature because it related to the science of numbers. Not only did the early mathematicians recognize the golden ratio, but so did Greek architects and sculptors. In fact, Phidias, a famous Greek sculptor, used it in his work; the symbol which is commonly used for the golden ratio is the first letter of his name, . The formula for phi is or 1.618033989 .

Fibonacci Numbers

We are all familiar with the series of numbers formed by assigning 1 to the first two elements and then the next follows as the sum of the two preceeding it, or 1, 1, 2, 3, 5, 8, ... --the so called Fibonnacci series. This series has special properties which yield some amazing results.

The series was first explored by Leonardo of Pisa ( Filius Bonacci, son of Bonacci, shortened to Fibonacci). He is thought to have recognized the series while examining the Chinese Triangle of the 1300s. Where are the connections and applications of the Fibonacci numbers? Just about anywhere you want to look. They are found in probability and statistics, such as the rise and fall of stocks on the Wall Street circuit; they are found in plants and their growth patterns; they are found in the breeding of rabbits; they are found in the genealogy of the bee; they are found in the behavior of light and atoms; they are found in spirals of a Nautilus seashell, galaxies, horns on mountain sheep (mouflons), elephant tusks, a growing fern plant, on the tail of a sea horse, an ocean wave pounding the shore, a hurricane or storm, etc. Endless examples can be found. What connection, though, to Phi, ?

Using Excel, a commonly known spreadsheet program, we now look at the following limit:


1. We will look at the Fibonacci series and the limit to which it converges. Notice how relatively fast the series converges to phi, at n = 39. As n increases, the size of the numbers in the left column do also; this causes the convergence to the best value for phi, or 1.618033988749895 .

Fibonacci series

2. We look at the Lucas sequence, which is formed in the same fashion, but with different seed numbers: first n = 1 and second n = 3. Using the same procedure as before, observe the convergence of this sequence; not much different from the first one; here n = 40.

Lucas sequence

3. We look at the "Banker" sequence, formed in the same fashion ( but very insignificant, indeed), where the convergence is faster, probably because of the spread on the seed numbers, but virtually the same; here n = 41, seed numbers are first n = 2 and n = 26.

Banker sequence

So, what conclusion can we draw from three different series with the same convergence?

Simply this: the value, phi, is not dependent on the numbers in the series, but how the numbers are formed. The general term of the series, according to Binet, is as follows:

With large values of n, the second term of Binet's formula may be neglected. Since the golden ratio has been shown to be


then phi is approximately equal to


which, of course, equals phi, the value of convergence of all three series/sequences.

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