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Leonardo Fibonacci 

     Fibonacci was born Leonardo of Pisa in Italy, 1175 AD. He chose the nickname Fibonacci which means "son of Bonacci" or "son of good fortune." As a young boy, Fibonacci moved to the coast of Africa where it is believed he was educated by Muslim school masters who taught him Hindu-Arabic Numerals. He recognized the superiority of these numerals over the Roman ones used at that time. At the age of 27, Fibonacci wrote a book on how to do arithmetic in the decimal system. The book was titled Liber abaci which meant book of the abacus, or book of calculating. This book was very influential in introducing Arabic numerals &emdash;0, 1, 2, 3, 4, 5, 6, 7, 8, and 9&emdash;into Western culture, namely Europe. The book also contained many solutions to many story problems, one of which made him famous. He became the first person to record permanently specific numbers which generate the divine proportion. He simply came upon the golden sequence in a hypothetical problem about rabbit breeding. The discovery of these golden numbers, as they were later termed, have had a profound affect on many others and their disciplines including art, biology and music. Fibonacci's contributions have kept his name alive for many centuries and continue to open the doors for many to follow. He is considered the greatest mathematician of the middle ages.

     Fibonacci was interested in the rate at which rabbits reproduced. He pondered the idea of how fast they could breed given ideal circumstances. In his book titled Liber Abaci Fibonacci gave a solution to this problem which, unbeknownst to him, would have implications in many other fields beyond mathematics.

     Problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if in one month each pair bears a new pair which becomes productive from the second month on?

Solution:

Notice the pattern emerging from the illustration can be represented by the Fibonacci sequence.

Month           J   F  M  A  M  J   J

# of Pairs        1   1   2   3  5   8  13

How is each term of the sequence found?

How many pairs of rabbits will there be in a year?

     Consider the Fibonacci sequence. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... We have seen that each term in the sequence is the sum of the two preceding terms. However, if we look closer, beyond the 15th term, we can see that the ratio of any consecutive pairs of Fibonacci numbers is �. In other words, the further along the sequence the ratios are, the closer the decimal equivalent comes to 0.618034, which is the golden ratio.

Let n be the position of the term in the Fibonacci sequence.

Let Fn be the Fibonacci number in the nth position.

The figure shows the first 20 numbers in the Fibonacci sequence.

n                        Fn                      Ratios

1                             1                     F4/F3= 3/2=1.5000

2                             1                       

3                             2

4                             3

5                              5                  F8/ F7 = 21/13 = 1.61538

6                             8                         

7                             13

8                             21

9                             34                 F11/F10 = 89/55 = 1.61818

10                            55                      

11                            89

12                            144

13                            233                F15/F14 = 610/377 =1.61804

14                            377     

15                            610

16                            987

17                            1597              F16/F15 = 987/610 = 1.61803

18                            2584     

19                           4181

20                           6765

Beyond F15 the ratio of any consecutive pair of Fibonacci numbers is � the golden ratio, correct to at least 5 decimal places.

Investigation : What happens if we take the ratios the other way around, for example 1/2, 1/2, 2/3, 3/5, etc.? Plot a graph of these ratios and find another fundamental property of the ratio.

Given Line Segment PQ

Choose a unit length.

Locate the midpoint of segment PQ (point A).

Mark off half the unit on either side of A (points B and C).

Mark off units in sequence in both directions from B and C.

With BC = 1, move to the right marking the points so that they correspond to lengths measuring 1,2,3, and 5 units, respectively (points D, E, F, G).

Repeat to the left to locate points H, I, J, and K.

Now line segment PQ has been divided into lengths corresponding to palindromic sequences of Fibonacci numbers.

     Since the ratio of two consecutive Fibonacci numbers rapidly approaches � as n increases, using a pair toward the beginning of the sequence offers a tool for constructing an approximation of the golden rectangle.

Construction of a golden rectangle using the Fibonacci numbers:

1. Begin with a 1-unit square.

2. Attach another 1-unit square.

3. Attach a 2-unit square to the left of the two 1-unit squares, so that it fits like a puzzle.

4. Attach a 3-unit square beneath the 2 and 1-unit squares.

5. Continue in this manner using squares of 5, 8, 13, 21, and 34 units (see figure below).

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