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
.

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
.

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.

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.

**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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