2. Who Was
Fibonacci??
• A Famous Mathematician.
• Fibonacci (1170-1250) is a
short for the Latin "filius
Bonacci" which means "the
son of Bonacci“ but his full
name was “Leonardo
Pisano”
• He introduced the Hindu-
Arabic number system into
Europe.
3. About the Origin of
Fibonacci Sequence
Fibonacci Sequence was discovered after an
investigation on the reproduction of rabbits.
We get the following sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34 ,55,89,144....
This sequence, in which each number is a sum of
two previous is called Fibonacci sequence.
so there is the
simple rule: add the last two to get the next
4. Fibonacci sequence
in Nature
Spirals seen in the
arrangement of seeds in the
head of this sunflower number
Spirals seen in the
arrangement of seeds in the
head of this sunflower number
34 in a counterclockwise
direction
5. and 55 in a clockwise direction.
Note that 34 and 55 are the
ninth and tenth Fibonacci
numbers respectively.
Also note that The flower
itself has 34 petals.
7. The number of petals
on a flower are often
fibonacci number
8. The Fibonacci numbers
can be found in
pineapples and bananas.
Bananas have 3 or 5 flat
sides and
Pineapple scales have
Fibonacci spirals in sets of
8, 13, 21
9.
10. The Golden
Ratio
The golden ratio is an irrational mathematical
constant, approximately equals to 1.6180339887
The golden ratio is often denoted by the Greek
letter φ (Phi). So φ = 1.6180339887
Also known as:
• Golden Ratio, • Golden Section,
• Golden cut, • Divine proportion,
• Divine section, • Mean of Phidias
• Extreme and mean ratio, • Medial section,
11. Two quantities are in the
golden ratio if the ratio
between the sum of those
quantities and the larger
one is the same as the
ratio between the larger
one and the smaller.
One interesting thing
about Phi is its reciprocal
1/φ = 1/1.618 = 0.618.
12. It is highly unusual for
the decimal integers of
a number and its
reciprocal to be exactly
the same.
A golden rectangle is a
rectangle where the
ratio of its length to
width is the golden ratio.
That is whose sides are
in the ratio 1:1.618.
The Golden
Rectangle
13. This smaller rectangle can similarly be subdivided in
to another set of smaller golden rectangle and smaller
square.
The golden rectangle has the property that it can be
further subdivided in to two portions a square and a
golden rectangle.
And this process can be done repeatedly to produce
smaller versions of squares and golden rectangles.
14. Golden Spiral
Start with the smallest one on the
right connect the lower right corner
to the upper right corner with an arc
that is one fourth of a circle. Then
continue your line in to the second
square on the with an arc that is one
fourth of a circle , we will continue
this process until each square has an
arc inside of it, with all of them
connected as a continues line. The
line should look like a spiral when
we are done .
15. Golden
Triangle
The Golden triangle is a
special isosceles triangle.
The top angle is 360
while the bottom two
angles are 720 each
17. If we continue to look
at the ratios as the
numbers in the
sequence get larger
and larger the ratio
will eventually become
the same number, and
that number is the
Golden Ratio!
19. Golden ratio
in Art
Many artists who lived
after Phidias have used
this proportion. Leonardo
Da Vinci called it the
"divine proportion" and
featured it in many of his
paintings.
Mona Lisa's face is a
perfect golden rectangle,
according to the ratio of
the width of her forehead
compared to the length
from the top of her head
to her chin.
20. Golden Ratio in
the Human Body
Golden Ratio in Fingers
Golden Ratio in Hands
21. Golden ratio in
the Face
The blue line defines a
perfect square of the
pupils and outside corners
of the mouth. The golden
section of these four blue
lines defines the nose, the
tip of the nose, the inside
of the nostrils, the two
rises of the upper lip and
the inner points of the ear.
The blue line also defines
the distance from the
upper lip to the bottom of
the chin
22. The yellow line, a golden section of the blue line,
defines the width of the nose, the distance between
the eyes and eye brows and the distance from the
pupils to the tip of the nose.
The green line, a golden section of the yellow line
defines the width of the eye, the distance at the pupil
from the eye lash to the eye brow and the distance
between the nostrils.
The magenta line, a golden section of the green line,
defines the distance from the upper lip to the bottom
of the nose and several dimensions of the eye.
23. The front two incisor teeth form a golden rectangle,
with a phi ratio in the heighth to the width.The ratio
of the width of the first tooth to the second tooth
from the center is also phi.
The ratio of the width of the smile to the third tooth
from the center is phi as well.
