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The Golden Ratio

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Diego Martín

on 24 May 2013

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Transcript of The Golden Ratio

THE GOLDEN RATIO The golden ratio is also called the golden section or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, and golden number. In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. It is an irrational algebraic number that holds many interesting properties, which was discovered in ancient times, not as "unity" but as a ratio or proportion between two segments of a straight line. This proportion is found in some geometrical figures and nature. Ancient Greeks said that “The whole is to the part what the part is for the rest.” In addition, an aesthetic character is attributed to objects whose actions saved the golden ratio, such as credit cards, identity cards, or books. If the points A, B and C are aligned then those triangles keep the golden ratio. In this video we can find some characteristic about the golden rate. Some authors suggest that the Golden number is as a proportion in Babylon and Assyria over 2000 a. C. However, there is no historical documentation that shows that the golden ratio was used consciously by them. When measuring a complex structure, it is easy to get curious results if you have many sizes available. In addition, so we can say that the Golden number is present, measures should be taken from significant points of the object, but this is not the case of many hypotheses that defend the presence of the golden ratio. For all these reasons, Mario Livio concludes that it is highly unlikely that the Babylonians have discovered the Golden number. HISTORY Euclid (300-265 BC) was the first person to make a formal study of the Golden ratio. Euclid also showed that this number can not be described as the reason for two whole numbers, so, is an irrational number. MATHEMATICAL REPRESENTATION Here we can find several ways of representing it that we have studied along the year OTHER RELATIONS WITH MATHEMATICS Respect Fibonacci numbers (0, 1, 1, 2, 3, 5, 8...), we discovered that, as the numbers increase, this reason is alternately lower and higher than the Golden ratio. We can also notice that the continued fraction that describes the golden ratio always produces Fibonacci numbers as it increases the number of ones in the fraction. FIBONACCI NUMBERS The Golden ratio and the golden section are present in all the geometric objects in which exist semiregulares or regular pentagonal symmetry, which are pentagons or appear somehow containing the square root of five. As for example the relations between parts of the Pentagon. GOLDEN RATIO IN GEOMETRY THE GOLDEN RECTANGLE OF EUCLID Euclid obtains aureus AEFD rectangle from the square ABCD. The rectangle BEFC is also Golden. VITRUVIAN MAN Leonardo da Vinci drew his famous Vitruvian Man in 1492 in one of his notebooks. Inspired by texts of ancient Rome, Marco Vitruvius Pollio, architect Leonardo proposed a canon of beauty to the human proportions that marked the Renaissance. THE MODULOR The Modulor is a system of measures applicable standard in the functional design and architecture based on human proportions and the Golden ratio.
Measures are based on the man with raised hand and its half, at the level of the navel.
From the first step, multiplying and dividing successively in the same way by the gold ratio we get the blue series.
From the second measurement proceed in the same way, we get the red series. THE NATURE The arrangement of the petals of the flowers.
The distribution of leaves on a stem.
The relationship between the veins of the leaves of the trees.
The relationship between the thickness of the main branches and the trunk.
The relationship between the distance between the coils of any spiral coiled inside. THAT WAS ALL THANK FOR YOUR
ATTENTION!!
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