The Fibonacci Sequence in Barcelona

Photography by Marc Adillon

Photography by Núria Casals

The picture shows the Fibonacci’s sequence monument in Port Vell in Barcelona. We can see the numbers which are part of the sequence built on the pavement and separated by a proportional distance related on the ratio between them. The Fibonacci sequence was born in a mathematical problem related to the rabbit reproduction proposed by Leonardo of Pisa in his very famous Liber abaci (1202).

Leonardo Pisano, also known as Fibonacci, was born around 1170 in the city-state of Pisa. Leonardo’s father, Guglielmo Bonacci, was a Pisan engaged in business in northern Africa, in Bugia (now Béjaïa, Algeria). This Algerian city was a very important intellectual focus during the eleventh and twelfth centuries. Therefore, it is not surprising that a businessman like Mr. Bonacci realize the possibilities that had the contact of the two cultures.

Guglielmo Bonacci put his son in a calculus positional Hindu course as well, Leonardo started to be interested about mathematics. He took profit of the frequent business trips of his father to know mathematicians of the countries they visited —Egypt, Syria, Provence, Sicily, Greece— and to make a deep study about Euclid’s Elements, that always had as a logical model of rigor and style. Therefore, it was natural that Fibonacci should have been steeped in Arabic algebraic methods, including, fortunately the Hindu-Arabic numerals and, unfortunately, the rhetorical form of expression. Fibonacci decided to write about all the knowledge that he had been collecting and that impressed him so much. He wrote it in a serie of books, the first of which was Liber Abaci (1202, reprinted 1228), Practica geometricae (1223), Liber quadratorum (1225), Flos (1225) and Epistola ad Magistrum Theodorum (1225). In Liber abaci Fibonacci explains the Arabic positional numbering system and how to read numbers, add them, multiply them… and to solve all the kind of problems which could need any trader in the Medieval Ages. The famous Fibonacci sequence is one of the problems of the chapter twelve:

How many pairs of Rabbits Are Created by One Pair in One Year

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; [p284] to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.

You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months

Fibonacci solved the problem with the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and 377 and here one of the most famous mathematical set of numbers was introduced for the first time.

This post has been written by Marc Adillon and Núria Casals in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Pla de Miquel Taradell in Barcelona (map)

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