Fibonacci, his numbers, his geometry

But let us first consider the principles according to which the sunflower arranges its seeds. It is a matter of two laws which we come across in nature, art, architecture, music and many other areas in innumerable places: the Fibonacci sequence and the golden section.

Leonardo of Pisa, better known by the name Leonardo Fibonacci, lived roughly from 1170 to after 1240 and is considered to be the first important European mathematician. On journeys to Africa, Byzantium and Syria he came into contact with Arabic mathematics, which in the Christian occident was largely unknown. In his work Liber abaci (The Book of the Abacus), which was published in 1202, he combined this knowledge with his own reflections. The book long remained unexcelled in the history of occidental mathematics and contributed to, among other things, Europe adopting the Arabic system of numbers.

Liber abaci contains a thought experiment which Fibonacci himself probably regarded as pure curiosity and did not pursue further, but which later was to become famous as the “Fibonacci sequence.”

Fibonacci asked himself how many pairs of rabbits originated from a single pair in one year. He assumed that each pair of rabbits gave birth to another pair in one month, which, in turn, would be productive from the second month after their birth.

The problem can be clearly solved by means of an example. Let us assume that in May there are 8 pairs of rabbits, in April it was 5. This means that in May 3 were newly born and therefore not yet productive. Thus in June there will be 8 pairs from May plus 5 new pairs, namely the offspring from the fertile rabbits in April. Of the 13 pairs from the month of June 5 are still not productive, so that 8 new pairs are added in July.

To find out the number of pairs of rabbits, Fibonacci observed, all one has to do is to count up the sum of the two previous numbers.

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

In the first month there is one pair of rabbits, in the second month there are 2, in the third month 3, in the fourth month 5, in the fifth month 8 and so on – by the end of the year 233 pairs of rabbits have resulted from the first pair.

Although Fibonacci’s thought-experiment is based, of course, on unrealistic assumptions, it does describe the essential features of growth processes. While for Fibonacci his problem was thus solved, it was later discovered that the Fibonacci sequence also occurs in nature and in art – be it in the position of leaves of plants, in the spiral form of snail shells, in the structure of clouds in an area of low pressure and in paintings, the architecture of buildings and in music.

It is also possible to approach the Fibonacci numbers geometrically. Let us assume a square whose sides measure 1. Beside it we construct a second square of the same size. We attach a longer side to another square which has the length 2. To this is added a square with the length of the side 3, one with a length of the side 5, one with the length of the side 8 and so on. It is not difficult to recognise the numbers of the Fibonacci sequence.

Now we draw a quarter of a circle in each square. The resultant spiral is called the Fibonacci spiral. It can be clearly seen in certain mussels and snail shells.