Golden ratio

A PRESENTATION ON
By
Nikhil R
12/24/2014 1

 In mathematics, two quantities are in the golden ratio if
their ratio is the same as the ratio of their sum to the larger of
the two quantities.
What is this ‘Golden Ratio’ ?
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 The golden ratio is a mathematical
constant approximately 1.6180339887.
 The golden ratio is also known as the most
aesthetic ratio between the two sides of a
rectangle.
 The golden ratio is often denoted by the
Greek letter (phi). .
12/24/2014 3

Also known as:
 Extreme and mean ratio,
 Medial section,
 Divine proportion,
 Divine section (Latin: sectio divina),
 Golden proportion,
 Golden cut,
 Mean of Phidias
12/24/2014 4

HOW IS GOLDEN RATIO CALCULATED?
2 quantities a and b are said to be in the golden ratio φ if
One method for finding the value of φ is to start with the left fraction.
Through simplifying the fraction and substituting in b/a = 1/φ,
Therefore,
Multiplying by φ gives
12/24/2014 5

which can be rearranged to,
Using the quadratic formula, two solutions are obtained:
and
Because φ is the ratio between positive quantities φ is necessarily
positive:
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Construction of the Golden Section
 Firstly, divide a square such that it makes two
precisely equal rectangles.
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 Take the diagonal of the rectangle as the radius to
contsruct a circle to touch the next side of the
square.
 Then, extend the base of the square so that it
touches the circle.
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 When we complete the shape to a rectangle, we will
realize that the rectangle fits the optimum ratio of golden.
 The base Lengthof the rectangle (C) divided by the base
Lengthof the square (A) equals the golden ratio.
 C / A =A / B = 1.6180339 = The Golden Ratio12/24/2014 10

 Each time we substract a square from the golden
rectangle, what we will get is a golden rectangle again.
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The Golden Spiral
 After doing the substraction infinitely many times, if we
draw a spiral starting from the square of the smallest
rectangle, (SideLengthof the square=Radius of the
spiral) we will get a Golden Spiral.
 The Golden spiral determines the structure and the
shape of many organic and inorganic assets.
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Relations with the Fibonacci Numbers
 Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
610, 987, 1597, ...
 Fibonacci numbers have an interesting attribute. When we divide one of
the fibonacci numbers to the previous one, we will get results that are so
close to each other. Moreover, after the 13th number in the serie, the
divison will be fixed at 1.618, namely the golden number.
Golden ratio= 1.618
233 / 144 = 1.618
377 / 233 = 1.618
610 / 377 = 1.618
987 / 610 = 1.618
1597 / 987 = 1.618
2584 / 1597 = 1.618
12/24/2014 13

Golden Ratio In
Arts and
Architecture
12/24/2014 14

Leonardo Da Vinci
 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, for
example in the famous
"Mona Lisa". Try drawing a
rectangle around her face.
You will realize that the
measurements are in a
golden proportion. You can
further explore this by
subdividing the rectangle
formed by using her eyes
as a horizontal divider.
12/24/2014 15

The “Vitruvian Man”
 Leonardo did an
entire exploration of
the human body and
the ratios of the
lengths of various
body parts. “Vitruvian
Man” illustrates that
the human body is
proportioned
according to the
Golden Ratio.
12/24/2014 16

The Parthenon
 “Phi“ was named
for the Greek
sculptor Phidias.
 The exterior
dimensions of
the Parthenon in
Athens, built in
about 440BC,
form a perfect
golden rectangle.
12/24/2014 17

 The baselength of Egyptian pyramids divided by
the height of them gives the golden ratio.
12/24/2014 18
EGYPTIAN PYRAMIDS ARE IN GOLDEN
RATIO TOO!!

Golden Ratio In
Human and
Nature
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Golden Ratio in Human Hand and Arm
Look at your own hand:
 You have ...
 2 hands each of which
has ...
 5 fingers, each of which
has...
 3 parts separated by ...
 2 knuckles
The length of different
parts in your arm also fits
the golden ratio.
12/24/2014 20

Golden Ratio in the Human Face
 The dividence of every long line to the short line equals the golden
ratio.
 Length of the face / Wideness of the face
Length between the lips and eyebrows / Lengthof the nose,
Length of the face / Lengthbetween the jaw and eyebrows
Length of the mouth / Wideness of the nose,
Wideness of the nose / Distance between the holes of the nose,
Length between the pupils / Length between the eyebrows.
All contain the golden ratio.12/24/2014 21

The Golden Spiral can be seen in the
arrangement of seeds on flower heads
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Golden Ratio In The Sea Shells
 The shape of the inner and outer surfaces
of the sea shells, and their curves fit the
golden ratio..
12/24/2014 24

Golden Ratio In the Snowflakes
 The ratio of the braches of a snowflake results in
the golden ratio.
12/24/2014 25

Disputed Sightings
 Some specific proportions in the bodies of many
animals (including humans) and parts of the shells of
mollusks and cephalopods are often claimed to be in
the golden ratio. There is actually a large variation in the
real measures of these elements in a specific individual
and the proportion in question is often significantly
different from the golden ratio.
 The proportions of different plant components (numbers
of leaves to branches, diameters of geometrical figures
inside flowers) are often claimed to show the golden
ratio proportion in several species. In practice, there are
significant variations between individuals, seasonal
variations, and age variations in these species. While
the golden ratio may be found in some proportions in
some individuals at particular times in their life cycles,
there is no consistent ratio in their proportions.
12/24/2014 26

Referances
 http://tr.wikipedia.org/wiki/Alt%C4%B1n_oran
 http://www.geom.uiuc.edu/~demo5337/s97b/art.htm
 http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fibnat.html
 http://www.matematikce.net/maltinoran.html
 Youtube
 Goldenratio.net
12/24/2014 27

Thank You For
Listening
12/24/2014 28

Golden ratio

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Golden ratio