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The Golden Ratio

The Golden Ratio. Volkan UYGUN. In mathematics and the arts, 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.

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The Golden Ratio

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  1. The Golden Ratio Volkan UYGUN

  2. In mathematics and the arts, 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. • 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). .

  3. Also known as: • Extreme and mean ratio, • Medial section, • Divine proportion, • Divine section (Latin: sectio divina), • Golden proportion, • Golden cut, • Mean of Phidias

  4. Construction of the Golden Section • Firstly, divide a square such that it makes two precisely equal rectangles.

  5. 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.

  6. When we complete the shape to a rectangle, we will realize that the rectangle fits the optimum ratio of golden. • The base lenght of the rectangle (C) divided by the base lenght of the square (A) equals the golden ratio. • C / A =A / B = 1.6180339 = The Golden Ratio

  7. Each time we substract a square from the golden rectangle, what we will get is a golden rectangle again.

  8. The Golden Spiral • After doing the substraction infinitely many times, if we draw a spiral starting from the square of the smallest rectangle, (Sidelenght of 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.

  9. 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

  10. Golden Ratio In Arts and Architecture

  11. 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. 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.

  13. 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.

  14. The baselenght of Egyptian pyramids divided by the height of them gives the golden ratio..

  15. Golden Ratio In Human and Nature

  16. Golden Ratio in Human Hand and Arm The length of different parts in your arm also fits the golden ratio. 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

  17. Golden Ratio in the Human Face • The dividence of every long line to the short line equals the golden ratio. • Lenght of the face / Wideness of the face Lenght between the lips and eyebrows / Lenght of the nose, Lenght of the face / Lenght between the jaw and eyebrows Lenght of the mouth / Wideness of the nose, Wideness of the nose / Distance between the holes of the nose, Length between the pupils / Length between teh eyebrows. All contain the golden ratio.

  18. The Golden Spiral can be seen in the arrangement of seeds on flower heads.

  19. 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..

  20. Golden Ratio In the Snowflakes • The ratio of the braches of a snowflake results in the golden ratio.

  21. 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

  22. 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

  23. Thank You For Listening

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