Fibonacci Series

Fibonacci (1170 - 1240) was an Italian mathematician; his full name was Leonardo Fibonacci. He is also known as Leonardo of Pisa, hailing as he did from that great city. Fibonacci made several attempts to find mathematical formulae that would model the way in which animal and plant populations grew, basing his work on a study of rabbit populations. He was aware that such formulae would be dependent on earlier generations, so he looked at several ways of generating series of numbers, starting from any one or two numbers.

We now know that population growth is based on an exponential series, and that, where there are predator/prey relationships, the mathematical model is complex and cyclical. Fibonacci devised a series which generated numbers that grew rapidly, but he soon realised that it did not match the growth of animal populations. He did not pursue the matter any further.

A Fibonacci series is a series of numbers that grows from two 'seed' numbers, using the simple rule that the next member of the series is the sum of the previous two members. It has been used to describe such diverse phenomena as the stock market, plant growth, and musical themes. Here is an example:

2, 4, 6, 10, 16, 26 and so on...

In this case, 2 and 4 are the seed; they add up to 6 (the next number). 4 and 6 add to 10, and so on...

The simplest series starts with 1 and 1, so it runs:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on...

For any series of numbers, the first differential is the series you get by writing down the differences between adjacent members of the series, and the second differential is the series you get by writing down the differences between the differences. Try doing this for a Fibonacci series. Surprise! The first differential becomes the same as the series itself, and so does the second differential, and so do they all. If you stop and think about it, you'll figure out why; but Fibonacci series are almost the only series where all the differentials are the same as the original series. The only other family of series with the same differentials is the 'keep on doubling it' family.

Even more peculiar is the fact that, although Fibonacci series are composed entirely of integers, there is a formula for calculating the nth member of the series, which is based on the square root of 5. One marvels that a formula with so many irrational elements results in simple integers every time! The formula is given below:

F(n) = (Phi^n + Phi^-n)/sqrt(5) if n is odd
F(n) = (Phi^n - Phi^-n)/sqrt(5) if n is even

where Phi = (1 + sqrt(5))/2