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). .

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Also known as:

Extreme and mean ratio,

Medial section,

Divine proportion,

Divine section (Latin: sectio divina),

Golden proportion,

Golden cut,

Mean of Phidias

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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

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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

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Golden Ratio In

Arts and

Architecture

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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.

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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.

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The baselength of Egyptian pyramids divided by

the height of them gives the golden ratio.

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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.

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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.

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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.

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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/Fibonacci/fibnat.html

http://www.matematikce.net/maltinoran.html

Youtube

Goldenratio.net

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Thank You For

Listening

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