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The Golden Ratio

The Golden Ratio, AKA The Divine Proportion

Rami Sharon

on 6 January 2013

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Transcript of The Golden Ratio

The Golden Ratio
The Divine Proportion Phi Drawing the Golden Rectangle The Golden Ratio
In nature The Golden Ratio
in Art The Golden Ratio
in Architecture The Golden Ratio
in Web Design Some Fun Facts Given a rectangle having sides in the ratio 1:x, Phi is defined as the unique number such that partitioning the original rectangle into a square and new rectangle as illustrated above results in a new rectangle which also has sides in the ratio 1:x From the definition Hence Phi is approximately 1.61803..... We can also define Phi as follows: ...Which brings us to: There is also special relationship between the golden ratio and the Fibonacci Sequence...
0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

I you take two successive Fibonacci numbers, their ratio is very close to the Golden Ratio, and it's getting closer as the numbers get higher... And not only when starting with 2 and 3,
but with any two numbers...
(192 and 16 in this case) Draw a square of size 1
Place a dot half way along one side
Draw a line from that point to an opposite corner (it will be √5/2 in length)
Turn that line so that it runs along the square's side The facial features of a koala bear show golden ratio proportions in the dimensions and positions of the eyes, nose and mouth in relation to the dimensions of the face. The eye, fins and tail all fall at golden sections of the length of a dolphin’s body. The dimensions of the dorsal fin are golden sections (yellow and green). The thickness of the dolphin’s tail section corresponds to same golden section of the line from head to tail. The eye-like markings of this moth fall at golden sections of the lines that mark its width and length. The spiral growth of sea shells provide a simple, but beautiful, example. In this world famous painting, we see each major section divided in accordance to the golden ratio. Since the ancient Greeks, people have known that the human body has symmetry. What is interesting is how the major portions of the human body follow the Golden Ratio. The bones making up your fingers are in the golden ratio. Golden Ratio in the Human Body: Height and length between naval point and foot, Length between shoulder line and length of the head, Length between finger tip to elbow and length between wrist and elbow and on and on... Statue of Athena: When viewed from the side the golden ratio can be seen. One golden ratio is the length from the front head to the opening of the ear compared to length of the forehead to the chin. The other ratio is the length of the nostril to the earlobe compared to the length from the nostril to the chin The Last Supper: contains a golden ratio in several places, appearing in both the ceiling and the position where the people sit. 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. Human health is affected by facial proportions. Biologically, people who have long faces have more chances of having breathing problem, suffering from sleep apnea. And People with shorter faces tend to have abnormal jaw development due to the excessive pressure on the jaw joint, suffering from headaches because their jaws are positioned in a manner that can restrict blood flow to the brain. The exterior dimensions of the Parthenon form a Golden Ratio in many of the proportions. 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. The UN Building: In the United Nations building, the width of the building compared with the height of every ten floors is a Golden Ratio. The Taj Mahal: In India, it was used in the construction, which was completed in 1648 The CN Tower in Toronto: the tallest tower and freestanding structure in the world, has contains the golden ratio in its design. The ratio of observation deck at 342 meters to the total height of 553.33 is 0.618 or phi, the reciprocal of Phi! the homepage for Life magazine, which won a Webby in 2011 for
“BEST VISUAL DESIGN – AESTHETIC” Thank You The value of phi, namely the golden number is approximately 1.6180339887...
This is an irrational number... Remember? The American mathematician, Mark Barr, has chosen Phi as the symbol for the golden ratio. Phi is named after Phidias. Phidias was a famous Greek sculptor who designed the Parthenon. The famous quadratic equation can be used to derive the golden ratio directly is x^2-x-1=0.
The 2 solutions to the quadratic equation x^2-x-1=0 are (1+5^0.5)/2 and (1-5^0.5)/2 respectively. The former is the golden ratio itself, namely 1.61803 while the latter gives a value of -0.61803.. The earliest written definition of the golden ratio was "the extreme and mean ratio", provided by a famous mathematician Euclid. Euclid wrote this in his book "Elements". The day do we celebrate phi day is 18 June. The value of phi is 1.618... Besides, its reciprocal is 0.618... And if you square phi, you will get 2.618... The 0.618 portion keeps repeating in all these 3 cases. Therefore, the date 18 June is chosen. The ancient Egyptians used the golden ratio in their pyramids. At that time, the golden ratio was known to them by another name, The sacred ratio. Besides pyramids, the Egyptians also used the golden ratio in many of their other architectures (temples) and arts. The golden ratio is claimed to appear in many fields, such as cosmology, theology, arts, architecture, botany and others. In fact, the golden ratio has captivated many mathematicians and there are some researches going on regarding the application of the golden ratio in stock market and foreign exchange!
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