The Fibonacci number sequence describes how things grow, and also how they decay. It can predict the increase in the population of a colony of rabbits, and of a colony of bacteria. It also predicts the way that a colony that overwhelms its resources will decline. More controversially, Fibonacci numbers are used to predict the behavior of financial assets, such as stock market indexes.
What it is
The Fibonacci number sequence is simple to generate. Simply add the last two numbers in the sequence together. Thus the series runs 0,1,1,2,3,5,8,13,21… One plus zero is one, one plus one is two, two plus one is three, and so on. Anyone can generate this curious sequence at home in their spare time, which is one source of its fascination. Anyone can play with it.
How it was discovered
Italian mathematician Leonardo of Pisa, also known as Leonardo Pisando Fibonacci, born sometime in the twelfth century, discovered the properties of the sequence, perhaps while investigating a sort of logic problem concerning the growth of a population of rabbits.
His work was informed by the knowledge of advanced Arab mathematicians he lived among in North Africa, and he wrote important books about math when he later returned to Italy. The series was only a part of his contribution to mathematics; he also concerned himself with practical applications like accounting. He himself never called his sequence Fibonacci numbers; it was named in his honor.
How it is used
Give a single bacterium good growing conditions and it population will increase predictably, according to a Fibonacci sequence. So will a population of rabbits or any other population. Therefore, the mathematical formula that describes the sequence is used in generating population estimates.
The sequence also describes growth within living things, for example the growth of a Nautilus shell. So the formula for the sequence can be used to predict growth in living things.
When things go wrong, populations also decline according to Fibonacci sequences. The sequence has many other uses in mathematics and science, as well.
In asset pricing, some students of the markets believe that prices move, at least in part, according to Fibonacci numbers. It is hard to see why this would be so, at least for those of us who are not familiar with the higher maths. It is tempting to believe that prices move according to Fibonacci rules because traders expect them to, and therefore buy and sell in anticipation of Fibonacci effects. In any case, many traders behave according to Fibonacci numbers.
Abstract mathematics is no doubt beautiful to its practitioners, but its utility can be hard for the rest of us to see at times. This series, though invented in the twelfth or thirteenth century, is plainly still useful and still presenting avenues for sophisticated research. The significance of the Fibonacci number sequence is that it describes the behavior of many natural systems with an elegant simplicity that opens to a world of complex abstract thought.