Who was Fibonacci?
Born in Pisa in 1170 (probably) and died in 1250 (possibly), L Fibonacci was better known at the time under the name Leonardo Pisano (Leonard of Pisa) or Leonardo Bigollo (negative term meaning ' wandering good-for-nothing'). His modern-day name came from his father, Guilielmo Bonacci (hence Fibonacci: 'Son of Bonacci') who occupied a diplomatic post in North Africa during the boyhood of the future mathematician.
Until 1200, Fibonacci supposedly travelled around the Mediterranean acquiring his knowledge of mathematics in Egypt, Syria, Greece, in Sicily, and in Provence (as mentioned in the Liber Abacci of 1202).
The Abacci Liber (concerning abacuses, the ancestors of modern computers
and cousins of the Chinese counting frames) was the first of several
mathematical works :
Pratica Geometriae (1220), Flos (1225), Liber quadratorum ( 1225), and Di Minor Guisa, of which there remains no manuscript (it should be remembered that all that occured at least two centuries before Gutenberg !).
Fibonacci is best known for the number series which carries his name (1,1,2,3,5,8,13,21,34,55... see below), but it is worth knowing that it is to him that we owe the systematic introduction of the Hindu-Arabic decimal system to Europe; however, he was more famous in his time for the practical applications his methods offered merchants in the exercise of their trade.
From the 1230's until the end of his life, we have little information about Fibonacci; a text of 1240 states that a sum of money was granted to him by the state of Pisa, in return for services rendered to the town; we know nothing of the last years of this great visionary.
The Fibonacci Sequence
F(n+2) = F(n+1) + F(n) is the official definition of the Fibonacci sequence, as provided by the Fibonacci Association, formed of mathematicians from several American universities, and which publishes the Fibonacci Quarterly, a journal devoted entirely to the study of Fibonacci numbers.
This sequence arises from the resolution of one of the problems of Abacci Liber:
If one places a rabbit couple in an enclosed place, how many rabbits would one obtain after a certain time if it is presupposed that they reproduce once per month, and that those born can reproduce at the age of a month ?
One obtains 1,1,2,3,5,8,13,21,34,55... after each month, and one can check that the relation F(n+2) = F(n+1) + F(n) is correct, for example for n=8, one has : 55=34+21.
The presence of Fibonacci numbers in nature is quite astounding: the spirals found in sea shells and vegetables, the growth pattern of many plant species, and almost anything relating to reproduction, as mentioned.
In fact, the Fibonacci Sequence contains much more than the 'natural proportions' that artists have traditionally seen. Indeed, very recently, Robert Devaney, eminent professor at the University of Boston, has discovered the appearance of Fibonacci numbers in the Mandelbrot Group.
In visual arts and music, the use of Fibonacci numbers is generally related to proportion, particularly linked to the golden section; examples of this can be found in the work of Michel-Angelo, J.S Bach, Brahms, Scriabin, Bartok, and many artists in the twentieth century including Mondrian.
The Golden Section
If we take a stick and divide it into two parts, the smaller part having the same proportion to the larger part as the larger part has to the whole, we have divided the stick at its golden mean, we have found its golden section. Apart from a few exceptions, the relationship between every Fibonacci number and its neighbour in the series gives the same relationship.