Fibonacci’s Fountain

Fibonacci’s name is known to almost every high-school kid today. However, he was not very well known in his day. In the 15th century A.D., when he lived, he was regarded as a lunatic who found strange relationships between numbers and created series that were of no use to the people of his time. Though it would be harsh to call him a lunatic, that was how most people of his time saw him.

Over the past 500 years, our views of mathematics have changed and mathematicians have discovered the value of Fibonacci’s works. Fibonacci is best known for the series named after him. The Fibonacci series is a semi-infinite series, beginning with two 1’s, in which each subsequent term is the sum of the two terms preceding it. Thus, the series runs as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,………..

Although Fibonacci created many such number series and has other mathematical feats to his credit, this particular series is much more popular than most of the others. This popularity is not without reason; read on to discover….

At a glance, the above series looks like a few numbers written without any specific order. On closer inspection, we discover that from the third term onwards, each term is the sum of the two preceding terms. This is not the only feature of the series, there is a lot more than meets the eye. The Fibonacci series turns up in places where one would least expect it, in mathematical models which have nothing to do with the series in question, popular art and even in Nature!

Lets start with a simple mathematical example. Consider a cute little animal whose name we don’t know. Let us call it X. Now, suppose there are two types of the animal X. They are A and B. The only difference between A and B is that on reproduction, A gives B, while B gives one A and one B. In the next step, the reproduction occurs according to the same rules, and so on generation after generation. The set of tuples below illustrate the process described above.

{A}>{B}>{AB}>>{BAB}>>{ABBAB}>>{BABABBAB}>>{ABBABBABABBAB}>>{BABABBABABBABBABABBAB}….

If you look carefully, you can see that the population of each generation increases according to the corresponding terms of the Fibonacci series. If this process is followed indefinitely, it will continue to give the terms faithfully. This mathematical model was devised to simulate the reproduction behaviour of two different kinds of rabbit, and the researchers were stunned at this queer result, which was closely matched even in the real rabbits they were experimenting with!

The numbers that appear in the Fibonacci series have also been observed in Nature. In the sunflower, the structures within the dark centre of the flower are arranged in opposing spirals. It has been found that on following any spiral the number of structures encountered is always a number that appears in the Fibonacci series. It is usually 21 in the smallest flowers, 34 in the larger ones and 55 in the largest examples of the species.

There is yet another profound manifestation of this series. This is seen in both Nature and popular art. Mathematicians investigating the mysterious properties of the Fibonacci series, found that as we move on towards a very large number of terms, each term of the series bears a ratio to the preceding term that converges to a specific value as the number of terms approaches infinity. This value, known as the ‘Golden Ratio’, is a rational number and is approximately equal to 1.618. This ratio pops up in all kinds of strange places, which have puzzled mathematicians and scientists. It has been seen that in ancient architecture, most of the rectangular rooms had the ratio of their length to breadth as 1.618. Even today, in several art galleries it is seen that unintentionally, solely for aesthetic appeal an artist has drawn figures with the Golden Ratio prevailing between the dimensions of the figures.

Even Mother Nature has a strange attraction towards the Golden Ratio. Studying the spirals on a mollusc shell, marine biologists found that on any shell and at any position if a straight line were drawn from the centre of the shell to its periphery, the distance traversed between two successive intersections with the spiral always maintain the ratio of 1.618 irrespective of the species of the mollusc and its size.

There are many more instances of this series in several unrelated areas and it would fill a great volume to write about them all. One wonders if Fibonacci himself was aware of these facts surrounding his series. Even if he did, his secret shall remain with him. Now one can only try to unravel the peculiar relationship between Fibonacci’s series, the Golden Ratio, and Mother Nature.

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