Folha Column: Fibonacci Numbers are Everywhere
Reproduction of Marcelo Viana's column in Folha de S.Paulo.
I was a teenager when I first read about Fibonacci. It was in an illustrated book that explained that these numbers are related to rabbit breeding and, incredibly, claimed that they govern the growth of leaves around the stem of plants. Since then, I've learned much more about these magical numbers, but nothing has erased the fascination of that first reading.
Leonardo Pisano, who would be nicknamed Fibonacci long after his death, was born in 1170 in the prosperous city of Pisa (he was a contemporary of the beginning of the construction of the famous leaning tower). The son of a merchant, he became interested in the calculation methods of his countrymen.
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In 1202, he published "Liber Abaci" (Book of the Abacus), the most important mathematics book written in the West in a millennium. In it, he presented to Europe much of the knowledge acquired from Arab and Jewish mathematicians, with particular emphasis on the Hindu (decimal) numbering system, which we still use today.
But what made Fibonacci famous, more than anything else, was a small exercise he included in chapter XII of "Liber Abaci": "A man placed a pair of rabbits in an enclosed space. How many pairs will be produced in a year, if we assume that each pair produces another per month from its second month of life?"
Representing the number of rabbit pairs in the nth month by F <sub>n</sub> , we have F <sub>1</sub> = F <sub>2 </sub> = 1 (this is the initial pair, which has not yet become reproductive) and from there F <sub>n </sub> = F<sub> n-1</sub> + F <sub>n-2 </sub>: the pairs in the nth month are those that already existed in the previous month plus the offspring of those that are two or more months old. Thus, F <sub>3 </sub> = 2, F <sub>4</sub> = 3, F <sub>5</sub> = 5, F <sub>6</sub> = 8, F <sub>7</sub> = 13, etc.
It is extraordinary that this seemingly naive sequence is related to so many profound ideas. Because of this, it also emerges in areas where mathematics plays a discreet but sovereign role, such as music and painting.
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