# Fibonacci Sequence and Related Numbers

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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

   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:
Fn  Fn1  Fn2
• Example: Find the third number in the
sequence using the formula.
• Let n = 3.
F3  F31  F32  F2  F1  1  1  2
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

1 5

2
• The golden ratio has been used in
mathematics, art, and architecture for
more than 2000 years.
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

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