THE FIBONACCI SEQUENCE AND THE GOLDEN NUMBER

Leonardo Fibonacci (about 1170 – 1242) was one of the greatest mathematicians of the Middle Ages. His most famous works are:

Liber Abaci (1202) - a treatise on arithmetic and algebra with which, at the beginning of the 13th century, in Europe, Fibonacci introduced the Indo-Arabic decimal numerical system and the main methods of calculation associated with it. The series of numbers that is known as the "Fibonacci sequence" is part of this treatise.

Practica geometriae (1220) - an important treatise on the practice of geometry, in which algebra is applied for the solution of geometrical problems.

Liber quadratorum (1225) - an algebra treatise.

The Fibonacci sequence originates from the solution of a problem whose topic concerned the growth of a population of rabbits, starting from an initial parent pair. The text of the problem was this:

″A man put a pair of rabbits in a place completely surrounded by walls to find out how many pairs of rabbits descended from this one in a year (by nature, pairs of rabbits generate another pair every month and begin to procreate in the second month after birth )”.

The solution to this problem, found by Fibonacci, is schematized in the following table:

Months

Birth evolution

Number of pairs of rabbits

Function

0

At the beginning of the observation, the rabbit population is composed of the first not-fertile couple A.

A

1

F(0)=1

1

After a month, there will be still only one pair A, which has become fertile and can mate.

A

1

F(1)=1

2

The following month of observation there will be two couples, one of which is fertile A, ready to mate again and one not-fertile B, born from A.

A B

2

F(2)=2--->F(0)+F(1)=1+1=2

3

After three months, the fertile couple A generates another couple C (not-fertile), while the couple B will have become fertile and will be ready to mate again. In total there will be three pairs of rabbits.

A C B

3

F(3)=3--->F(2)+F(1)=2+1=3

4

The following month there will be a new couple D (not-fertile), born from the first fertile couple A, and there will be another new couple E (not-fertile), born from the second fertile couple B; while the third couple C will have become fertile. In total there will be five pairs of rabbits.

A D C B E

5

F(4)=5--->F(3)+F(2)=3+2=5

5

After the fifth month there will be a new couple F (not-fertile), generated by the fertile couple A; the couple D becomes fertile; the fertile couple C generates another couple G (not-fertile) and also the fertile couple B generates a new couple H (not-fertile), while the couple E becomes fertile. The total number of pairs of rabbits will be eight.

A F D C G B H E

8

F(5)=8--->F(4)+F(3)=5+3=8

Ultimately, for each of the months that have been examined, the pairs that make up the population of rabbits are equal in number to: 1, 1, 2, 3, 5, 8.

It is possible to proceed in this way indefinitely. It is immediately noted that each number of the series is equal to the sum of the two previous numbers. Therefore, the rabbit population will grow as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

this is the Fibonacci sequence.

The general rule concerning Fibonacci numbers is:

n = number of months.

The Fibonacci sequence is related to the golden number φ = 1.618033... since:

the ratio between two consecutive terms of the Fibonacci sequence tends to approximate the golden number φ better and better.

In effect:

The general formula that allows to obtain the generic term of the Fibonacci sequence, starting from the golden number φ, is: