The document discusses Fibonacci numbers, the golden ratio, and their prevalence in nature. It begins by introducing Fibonacci and describing his famous rabbit problem, which led to the discovery of the Fibonacci sequence. This sequence appears throughout nature, such as in the spirals of sunflowers and pinecones. The golden ratio of approximately 1.618 is also discussed, along with its relationship to the Fibonacci sequence and appearances in geometry, art, architecture, and the human body. Examples include the Mona Lisa painting and proportions of the human face, fingers, and full body.

1. Seena.V
Assistant Professor
Department of Mathematics

2. Fibonacci Numbers And
Golden Ratio

3. Who Was Fibonacci?

4. • A Famous Mathematician
• Fibonacci (1170-1250) is a
short for theLatin "filius
Bonacci" which means
"the son of Bonacci“ but his
full name was Leonardo
Pisano
• He introduced the Hindu-
Arabic number system into
Europe

5. About the
Origin of
Fibonacci Sequence

6. Fibonacci Sequence was
discovered after an investigation
on the reproduction of rabbits.

7. Fibonacci’s Rabbits
Problem:
Suppose a newly-born pair of rabbits (one male, one female)
are put in a field. Rabbits are able to mate at the age of
one month so that at the end of its second month, a female
can produce another pair of rabbits. Suppose that the rabbits
never die and that the female always produces one new pair
(one male, one female) every month from the second month
on. How many pairs will there be in one year?

8. Pairs
1 pair
At the end of the first month there is still only one pair

9. Pairs
1 pair
End first month… only one pair
1 pair
At the end of the second month the female produces a
new pair, so now there are 2 pairs of rabbits
2 pairs

10. Pairs
1 pair
1 pair
2 pairs
3 pairs
End second month… 2 pairs of rabbits
At the end of the
third month, the
original female
produces a second
pair, making 3 pairs
in all in the field.
End first month… only one pair

11. Pairs
1 pair
1 pair
2 pairs
3 pairs
End third month…
3 pairs
5 pairs
End first month… only one pair
End second month… 2 pairs of rabbits
At the end of the fourth month, the first pair produces yet another new pair, and the female
born two months ago produces her first pair of rabbits also, making 5 pairs.

12. Thus 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!

13. So 144 Pairs will be there
at the end of One Year….

14. Fibonacci sequence in
Nature

15. Spirals seen in the
arrangement of
seeds in the head of
this sunflower number

18. Spirals seen in the arrangement of seeds in the head of this sunflower
number 34 in a counterclockwise direction

19. and 55 in a clockwise direction

20. Note that 34 and 55 are the ninth and
tenth Fibonacci numbers respectively.

21. Also note that The flower itself has
34 petals.

22. Pinecones
clearly show
the Fibonacci
Spiral

26. Note that 8 and 13 are
Consecutive Fibonacci
numbers

27. The number of petals on a
flower are often
Fibonacci numbers.

30. The Fibonacci numbers can be found in
pineapples and bananas. Bananas have 3
or 5 flat sides, Pineapple scales have
Fibonacci spirals in sets of 8, 13, 21

32. The
Golden Ratio

33. The golden ratio is an irrational
mathematical constant,
approximately equals to
1.6180339887

34. The golden ratio is
often denoted by the
Greek
letter φ(Phi)
So φ= 1.6180339887

35. Also known as:
• Golden Ratio,
• Golden Section,
• Golden cut,
• Divine proportion,
• Divine section,
• Mean of Phidias
• Extreme and mean ratio,
• Medial section,

36. 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
.

37. a b
a+b
a+b
a
=
a
b
= φ

38. φ= 1+√5
2
= 1.618

39. One interesting thing about Phi is
its reciprocal
1/φ = 1/1.618 = 0.618.
It is highly unusual for the
decimal integers of a number and
its reciprocal to be exactly the
same.

40. 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

41. The golden rectangle has the property
that it can be further subdivided in to two
portions a square and a golden rectangle
This smaller rectangle can similarly be
subdivided in to another set of smaller
golden rectangle and smaller square.
And this process can be done repeatedly
to produce smaller versions of squares
and golden rectangles

42. Golden Rectangle

43. 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 .

44. Golden Triangle
The Golden
triangle is a special
isosceles triangle.
The top angle is
360 while the
bottom two angles
are 720 each

45. Relation between
Fibonacci
Sequence and
Golden ratio

46. Aha! Notice that as
we continue down
the sequence, the
ratios seem to be
converging upon one
number (from both
sides of the number)!
2/1 = 2.0 (bigger)
3/2 = 1.5 (smaller)
5/3 = 1.67(bigger)
8/5 = 1.6(smaller)
13/8 = 1.625 (bigger)
21/13 = 1.615 (smaller)
34/21 = 1.619 (bigger)
55/34 = 1.618(smaller)
89/55 = 1.618
The Fibonacci sequence is 1,1,2,3,5,8,13,21,34,55,….

47. 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!

48. 1
1
2
3 1.5000000000000000
5 1.6666666666666700
8 1.6000000000000000
13 1.6250000000000000
21 1.6153846153846200
34 1.6190476190476200
55 1.6176470588235300
89 1.6181818181818200
144 1.6179775280898900
233 1.6180555555555600
377 1.6180257510729600
610 1.6180371352785100
987 1.6180327868852500
1,597 1.6180344478216800
2,584 1.6180338134001300
4,181 1.6180340557275500
6,765 1.6180339631667100
10,946 1.6180339985218000
17,711 1.6180339850173600
28,657 1.6180339901756000
46,368 1.6180339882053200
75,025 1.6180339889579000

49. Golden ratio
in
Nature

50. Nautilus Shell

51. 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

52. 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.

53. Golden Ratio
in the
Human Body

54. Golden Ratio in Fingers

55. Golden Ratio in Hands

56. 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.
• 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.

57. • 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.

58. Golden Ratio in Human body
• The white line is the body's height.
• The blue line, a golden section of the white line,
defines the distance from the head to the finger
tips
• The yellow line, a golden section of the blue line,
defines the distance from the head to the navel
and the elbows.
• The green line, a golden section of the yellow line,
defines the distance from the head to the
pectorals and inside top of the arms, the width of
the shoulders, the length of the forearm and the
shin bone.
• The magenta line, a golden section of
the green line, defines the distance from the head
to the base of the skull and the width of the
abdomen. The sectioned portions of the magenta
line determine the position of the nose and the
hairline.
• Although not shown, the golden section of
the magenta line (also the short section of the
green line) defines the width of the head and half
the width of the chest and the hips.

59. Golden Mean Gauge

61. More Examples of Golden Sections