From the Gold Division to Fibonacci Numbers

From the Gold Division to Fibonacci Numbers

In 1202, Italian mathematician Leonardo Fibonacci developed a series of numbers called the Fibonacci sequence. By definition, the first two numbers in the Fibonacci sequence are 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two:  0, 1, 1, 2, 3, 5, 8, 13, 21, etc. Therefore, value of each number is calculated by using what you already have calculated so far, without having to memorize all the previous values ​​- because the last two are suffice. Fibonacci sequence is the first the known of this kind sequences.

As a result of dividing each of the numbers of its predecessor in the sequence is obtained quotient which oscillating around 1.618 (Golden Ratio). With increasing numbers decrease 

deviation from this value. The inverse of 1.618 is 0.618. In connection with this, the factor of each

 number divided by the next number oscillate around 0.618. Between any two numbers separated by

a single number occurs ratio of 2.618 and its inverse, 0382. The same procedure can be repeated for a number of more distant from each other. For example, numbers separated by three positions have coefficients 4.236 and 0.236; number four positions away have proportions expressed by factors 6.853 and 0.146.

Applications include computer algorithms such as the Fibonacci search technique and the

Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting 

parallel and distributed systems. They also appear in biological settings, such as 

branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone.

Scientists have concluded that the phenomenon, whose structure is based on the Fibonacci 

sequence, give pleasure to the senses of sight and hearing beings human. The proportions 

gold ratio based on the Fibonacci sequence were used by Leonardo da Vinci in his paintings as well as by Botticelli.Gold proportions were also used during the climb the pyramid of Cheops at Giza and the Parthenon 

in Greece.

Reference:

Abrohms, A. Literature-Based Math Activities, 1992, New York: Scholastic Inc.