3. Leonardo Fibonacci
An Italian mathematician, born around 1170 in Pisa.
He contributed to the revival of classical exact sciences
after their decline in the early Middle ages.
Fibonacci travelled in the Mediterranean region in order
to get educated by the leading mathematicians of the
time.
4. Works
Fibonacci remains in the history of science mainly with
his
„Geometric practice“ deals with geometry and
trigonometry and “Book about squares” – with
algebra.
The book „Flos“ contains solutions of problems
described earlier by Joan from Palermo.
Today we also know about a few other works of his,
which haven’t been preserved.
„Book of calculation”.
5. Contribution to mathematics
Fibonacci is famous for introducing Arabic numbers in
Europe, which he used in his „Book of calculation“.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
During his travels in the Mediterranean region he
discovered a problem about a number sequence
with interesting properties, which was later named
after him by the French mathematician Luka.
6. The problem about a population of
rabbits
Fibonacci presents the sequence through a problem
involving reproduction of rabbits.
They put a pair of rabbits in a place, surrounded with
walls, so as to see how many pairs of rabbits will
be born in a year.
According to the nature of rabbits a pair of rabbits will
reproduce another pair in one month and rabbits
will be able to bear other rabbits from the second
month after they were born.
7. The problem about a population of
rabbits
І month
one pair of newborn
ІІ month
one pair
(they don’t reproduce)
ІІІ month
1 + 1 = 2 pairs
ІV month
2 + 1= 3 pairs
(only one pair reproduces)
V month
3 + 2 = 5 pairs
8. Fibonacci sequence
In the sequence of numbers each number is the sum
of the previous two numbers:
F1 = 1, F2 = 1,
F3 = F1+ F2 = 1 + 1 = 2,
F4 = F2 + F3 = 1 + 2 = 3,
F5 = 5, F6 = 8, F7 = 13, F8 = 21 ,etc:
Fn=Fn-1+Fn-2, for all n>2 .
9. Characteristics of the sequence
Each number in the sequence divided by the previous
one gives approximately 1.618. This number is
called Fibonacci constant and its symbol is the
Greek letter φ.
There is a difference only for the first few members of
the sequence.
An interesting fact is that the quotient of each number in
the sequence and the number that follows it is
presented by this formula:
11618.1618.01
−=−==−
ϕ
i
i
F
F
618.1
1
==
−i
i
F
F
ϕ
10. The number φ
The number φ was used in ancient Egypt and it was
known under various names: Golden ratio, Golden
number, Heavenly proportion and many others.
50 years ago, the mathematician Mark Barr
suggested that this ratio be noted with the Greek
letter φ.
It is the first letter of the name of the great ancient
Greek sculptor Phidias, who according to the
legend often used the Golden ratio in his statues.
11. Golden ratio
The number φ = 1.618 known also as “Golden ratio”
is a relationship of parts in which the larger part
relates to the smaller one, as the sum of the parts
to the larger part and each relationship equals φ.
a : b = φ (a + b) : а = φ
a : b = (a + b) : а
a b
a + b
12. Illustration of the sequence
We can illustrate the Fibonacci sequence with a row
of rectangles:
The first two members of this row are squares with a
side 1.
Then there is a rectangle with sides 1 and 2, made up
of two squares.
We add a new square with a side 2 (which is equal to
the larger side of the rectangle) and we make a
rectangle with sides 2 and 3, etc.
13.
14. In the ancient times
The Golden ratio was known in the remote past.
The sequence 3, 5, 8, 13, 18, 21 is represented in the
sunbeams in a drawing in the Magura cave in the
Northwestern Bulgaria (10000 B.C.)
The Golden ratio was encoded in small standards of
measures which have been found in the necropolis in
Varna, Bulgaria (5000 B.C.).
15. Harmony
In the past, architects, painters and theorists often
considered the Golden ratio to be an ideal
expression of beauty.
We find it in lots of buildings and sculptures: The
Egyptian pyramids, the Parthenon in Athens, the
Gothic cathedrals in Western Europe.
16. During the Renaissance
The book “Divine Proportion” by the monk Luca
Paccoli was published in 1509. The illustrations in
it are supposed to have been done by Leonardo
da Vinci.
Scientists’ and artists’ interest towards the Golden
ratio increased and it found application in
geometry, art and architecture.
The symbol of Renaissance is Leonardo’s Vitruvian
Man. The picture and the text are often called the
Canon of proportions.
17. The Golden ratio in nature
In 1754 Charles Bonnet discovered that if the places
where the leaves “attach” to the branch are
connected mentally, the result will be a few spirals,
the so called
The spaces between the cycles of the leaves are
proportional to the Fibonacci numbers.
a : b = b : c ≈ 1.6
genetic screw.
18.
19.
20. The Golden ratio in nature
Sunflowers, with their spirals of
seeds, have the ratio 1.618
between the diametres of
each rotation.
This coefficient is also observed in the relations
between various nature components.
21. Harmony
Apart from the human body,
the Golden ratio is a sign
of harmony in human
faces.
Psychologists have
determined that the faces
with more Golden ratios
are subconsciously
perceived as more
beautiful.
22.
23. „„Academic Ivan Tsenov”Academic Ivan Tsenov”
Secondary School of MathematicsSecondary School of Mathematics
and Natural Sciencesand Natural Sciences
Bulgaria, VratsaBulgaria, Vratsa
