Fibonacci Sequence and Related

Numbers

• The famous Fibonacci sequence is the result of

a question posed by Leonardo de Fibonacci, a

mathematician during the Middle Ages.

If you begin with one pair of rabbits on the first

day of the year, how many pairs of rabbits will

you have on the first day of the next year?

It is assumed that each pair of rabbits produces a

new pair every month and each new pair begins to

produce two months after birth.

Who is Fibonacci?

The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD.

In Fibonacci's Liber Abaci book (1202), chapter 12, he introduces the following problem

Another Mathematical Contribution:

Introducing the Decimal Number system into Europe

The Fibonacci Problem

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.

Fibonacci cont’l

He died in the 1240's and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa.

Fig. 2-69, p. 113

Fig. 2-70, p. 114

Fibonacci Sequence, cont’d

• The solution to this question is shown in the

table below.

• The sequence that appears three times in the

table, 1, 1, 2, 3, 5, 8, 13, 21, … is called the

Fibonacci sequence.

Fibonacci Sequence, cont’d

• The Fibonacci sequence is the sequence

of numbers 1, 1, 2, 3, 5, 8, 13, 21, …

• The Fibonacci sequence is found in many

places in nature.

• Any number in the sequence is called a

Fibonacci number.

• The sequence is usually written

F1, F2, F3, …, Fn, …

Recursion

• Recursion, in a sequence, indicates that

each number in the sequence is found

using previous numbers in the sequence.

• Some sequences, such as the Fibonacci

sequence, are generated by a recursion

rule along with starting values for the first

two, or more, numbers in the sequence.

Fibonacci Sequence, cont’d

• For the Fibonacci sequence, the starting values

are F1 = 1 and F2 = 1.

• The recursion rule for the Fibonacci sequence

is:

• Example: Find the third number in the

sequence using the formula.

• Let n = 3.

1 2n n nF F F

3 3 1 3 2 2 1 1 1 2F F F F F

Another Version

Suppose a tree starts from one shoot that grows

for two months and then sprouts a second

branch. If each established branch begins to

sprout a new branch after one month’s growth,

and if every new branch begins to sprout its own

first new branch after two month’s growth, how

many branches does the tree have at the end of

the year?

• Solution: The number of branches each month

in the first year is given in the table and drawn

in the figure below.

Fibonacci Numbers In Nature

• The Fibonacci numbers are found many

places in the natural world, including:

• The number of flower petals.

• The branching behavior of plants.

• The growth patterns of sunflowers and

pinecones, ……

• It is believed that the spiral nature of plant

growth accounts for this phenomenon.

Fibonacci Numbers In Nature, cont’d

• The number of petals on a flower are

often Fibonacci numbers.

Fibonacci Numbers In Nature, cont’d

• Plants grow in a spiral pattern. The ratio of the

number of spirals to the number of branches is

called the phyllotactic ratio.

• The numbers in the phyllotactic ratio are usually

Fibonacci numbers.

Fibonacci Numbers In Nature, cont’d

• Example: The

branch at right has

a phyllotactic ratio

of 3/8.

• Both 3 and 8 are

Fibonacci numbers.

Fibonacci Numbers In Nature, cont’d

• Mature sunflowers have one set of spirals

going clockwise and another set going

counterclockwise.

• The numbers of spirals in each set are

usually a pair of adjacent Fibonacci

numbers.

• The most common number of spirals is 34

and 55.

The generating function for the

Fibonacci numbers is

Pascal Triangle

The Golden Ratio

• Consider the ratios of pairs of consecutive Fibonacci numbers.

• Some of the ratios are calculated in the table shown on the following slide.

The Golden Ratio, cont’d

The Golden Ratio, cont’d

• The ratios of pairs of consecutive Fibonacci

numbers are also represented in the graph

below.

• The ratios approach the dashed line which

represents a number around 1.618.

The Golden Ratio, cont’d

• The irrational number, approximately

1.618, is called the golden ratio.

• Other names for the golden ratio include

the golden section, the golden mean, and

the divine proportion.

• The golden ratio is represented by the

Greek letter φ, which is pronounced “fe”

or “fi”.

The Golden Ratio, cont’d

• The golden ratio has an exact value of

• The golden ratio has been used in

mathematics, art, and architecture for

more than 2000 years.

1 5

2

Golden Rectangles

• A golden rectangle has a ratio of the

longer side to the shorter side that is the

golden ratio.

• Golden rectangles are used in

architecture, art, and packaging.

Golden Rectangles, cont’d

• The rectangle enclosing the diagram of the

Parthenon is an example of a golden rectangle.

Temple of Athena, Athens, Greece, the 5th central B. C.

Golden Ratio

Golden Ratio

Da Vinci Code