Golden mean is all abt this..

1. JUST HAVE A
LOOK AT THESE
PHOTOS

6. LETS
UNDERSTAND
WHY THIS
LOOKS SO
BEAUTIFUL

7. PRESENTED BY :
THE GOLDEN MEAN AND
ITS APPLICATIONS

8. GOLDEN MEAN

9. • Phi (Golden Ratio) as a mysterious number has
been discovered in many places, such as art,
architectures, humans, and plants.
• According to the history of mathematics, Phi
was first understood and used by the ancient
mathematicians in Egypt, two to three
thousand years ago, due to its frequent
appearance in Geometry.
THE HISTORY OF THE GOLDEN RATIO

10. CONTD..
• The name "Golden Ratio" appears in the form
sectio aurea (Golden Section in Greek) by
Leonardo da Vinci who used this the Golden
ratio in many of his masterpieces, such as The
Last Supper and Mona Lisa.
• In 1900s, an American mathematician named
Mark Barr, represented the Golden Ratio by
using a greek symbol Φ.

11. Why are the objects that
contain the
GOLDEN RATIO
so pleasing?

12. THE SECRET OF THE GOLDEN RATIO
• Objects that meet the requirements of the Golden
Ratio are attractive and pleasing to the human eye.
• Throughout history, many experts have tried to find
reasonable explanations for this question. Two
thousand years ago, ancient Greeks discovered the
magic of this ratio from the Golden Rectangle.
• The Golden Rectangle is a rectangle that meets the
Golden Ratio requirements and contains an infinite
number of proportional Golden Rectangles within
itself.
• The ancient Greeks were attracted by this special
number and its unique characteristics.

13. MATHEMATICAL PROPERTIES OF THE
GOLDEN RATIO
The Golden Ratio - Φ is an irrational number
that has the following unique properties:
1. Taking the reciprocal of Φ and adding one yields Φ.
phi=(1/phi)+1, or Φ = (1/Φ) + 1.
2. Φ squared equals itself plus one.
Φ^2=Φ+1. it is the only number in the world that has such
properties.
3. If we convert the equation from 2 into the equation Φ^2-
Φ-1=0. Doing this we get x = (1 ± √5)/2. Together, these
two solutions are known as Phi (1.618033989) and phi
(0.618033989). Phi and phi are reciprocals.

14. Φ vs. Π
Φ and Π (pi) have this in
common: where Π is the ratio of
the circumference to its diameter,
Φ is the ratio of the length to the
width of a perfect rectangle.

15. WHAT IS THE GOLDEN RATIO?
The Golden Ratio is a unique number, approximately
1.618033989. It is also known as the Divine Ratio, the Golden
Mean, the Golden Number, and the Golden Section.
WHAT IS THE FIBONACCI SEQUENCE OF NUMBERS?
The Fibonacci numbers are a unique sequence of integers,
starting with 1, where each element is the sum of the two
previous numbers. For example: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, etc.

16. RELATIONSHIP BETWEEN THE FIBONACCI
SEQUENCE AND THE GOLDEN RATIO
• The Fibonacci Sequence is an infinite sequence,
which means it goes on for ever, and as it
develops, the ratio of the consecutive terms
converges (becomes closer) to the Golden Ratio,
~1.618.
• For example, to find the ratio of any two
successive numbers, take the latter number and
divide by the former. So, we will have: 1/1=1,
2/1=2, 3/2=1.5, 5/3=1.66, 8/5=1.6, 13/8=1.625,
21/13=1.615.

17. As we can see, the ratio approaches the Golden Ratio, which is
also known as the Golden Section and Golden Mean. Even
though we know it approaches this one particular constant,
we can see from the graph that it will never reach this exact
value.

18. • The Golden Ratio was originally derived from the
golden rectangle. The following is one method to
construct a golden rectangle:
CONSTRUCTING A GOLDEN RECTANGLE
Given: a square ABCD
1.Find midpoint on DC.
2.Connect MB.
3.Draw a circle with the center of M, radius of MB.
4.Expand the DC until it meets with the circle. The
intersection is one vertex of the rectangle.
5.Complete the rectangle.
During the whole process, we have made the two exactly proportional rectangles,
each with the side ratio Φ, the Golden Ratio.

19. CONTD…
• The Golden Rectangle is said to be one of the most
visually satisfying of all geometric forms; whose
length: width ratio is equal to Phi.

20. THE GOLDEN SPIRAL
• When a Golden Rectangle is progressively
subdivided into smaller and smaller Golden
Rectangles the pattern below is obtained.
From this, a spiral can be drawn which grows
logarithmically, where the radius of the
spiral, at any given point, is the length of the
corresponding square to a Golden Rectangle.
This is called the Golden Spiral.

22. The Human Face
• Dr. Stephen Marquardt a former plastic
surgeon, used the golden section and
some of its relatives to make a mask
that he claims that is the most beautiful
shape a human face can ever have, it
used decagons and pentagons as its
function that embodies phi in all their
dimensions. Dr. Marquardt has been
studying on human's facial beauty for
many years.
THE GOLDEN RATIO AND BEAUTY
IN HUMANS

23. THE HUMAN SMILE
A perfect smile: the front two teeth form a golden rectangle
(which is said to be one of the most visually satisfying of forms,
as it is formed with sides of 1 and 1.618). There is also a Golden
Ratio in the height to width of the center two teeth. And the
ratio of the width of the two center teeth to those next to
them is phi. And, the ratio of the width of the smile to the third
tooth from the center is also phi.

24. THE HUMAN LUNGS
The windpipe divides into two main bronchi, one long (the
left) and the other short (the right). This asymmetrical
division continues into the subsequent subdivisions of the
bronchi. It was determined that in all these divisions the
proportion of the short bronchus to the long was always
1:1.618.

25. WHAT KIND OF
RELATIONSHIP DO
FIBONACCI NUMBERS
AND PLANTS HAVE?

26. The process of the growing plant follows
the Fibonacci numbers, from the first shoot,
to the two shoots, three shoots, and five
shoots, and eight shoots, and on and on.
The branching rates in plants occur in the
Fibonacci pattern, where the first level has
one "branching" (the trunk), the second has
two branches, than 3, 5, 8, 13 and so on.
Also, the spacing of leaves around each
branch or stalk spirals with respect to the
Golden Ratio.
THE GOLDEN RATIO AND BEAUTY IN NATURE
Plant Growth

28. HAVE YOU EVER
WONDERED WHY
MONA LISA PAINTING IS
SO BEAUTIFUL?

29. • The Golden Ratio has a
great impact on art,
influencing artists'
perspectives of a pleasant
art piece. 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.
THE GOLDEN RATIO AND BEAUTY IN ART

30. • The Golden Ratio has appeared in ancient
architecture. The examples are many, such as
the Great Pyramid in Giza, Egypt which is
considered one of the Seven World Wonders
of the ancient world, and the Greek
Parthenon that was constructed between 447
and 472BC.
THE GOLDEN RATIO AND BEAUTY IN ARCHITECTURE

31. The Great Pyramid at Giza
• Half of the base, the
slant height, and the
height from the
vertex to the center
create a right
triangle. When that
half of the base
equal to one, the
slant height would
equal to the value of
Phi and the height
would equal to the
square root of Phi.

32. Fibonacci and phi are
used in the design of
violins and even in
the design of high
quality speaker wire.
GOLDEN RATIO IN MUSIC

33. MUSICAL SCALES ARE BASED ON
FIBONACCI NUMBERS
• There are 13 notes in the span of any note
through its octave.
Note too how the piano keyboard of C to C
above of 13 keys has 8 white keys and 5
black keys, split into groups of 3 and 2

34. CONCLUSION:
• An interesting result has occurred due to our
research, we now see the examples of the
Golden Ratio everywhere. It is as if our eyes
have been opened to something that existed
all around us but to see. For that we are
thankful.