Golden section

Article

October 28, 2021

The golden ratio or gold ratio is a proportionality that often appears in both nature and art, creating a natural balance between symmetry and asymmetry. Gold engraving proportions can be found in many ancient buildings, medieval and renaissance works of art. The ancient Pythagoreans (Pythagoras and his followers), who argued that reality rests on mathematical foundations, believed in discovering one of the fundamental laws of existence in the golden ratio, as this ratio can also be recognized in nature (e.g., the human body or snail vase). Forms containing the proportions of the golden ratio still have great aesthetic value, and are used in many fields (such as typography or photography). The irrational number Φ ≈ 1,618 (Greek large fí), which numerically expresses the gold ratio, has several interesting mathematical properties. Mathematical definition Two parts (a and b, a> b) are proportioned to each other according to the golden ratio, if the whole (a + b) is proportional to the larger part (a) as the larger part (a) to the smaller part (b): the + b the the b {\ displaystyle {a + b \ over a} {a \ over b}} .That is, the larger part is equal to the geometric mean of the sum and the smaller part: the 2 ( the + b ) ⋅ b {\ displaystyle a ^ {2} (a + b) \ cdot b} .The wording equivalent to the above is that the major part is proportional to the smaller part as the smaller part is the difference between the two parts: the b b the - b , {\ displaystyle {a \ over b} {b \ over a-b},} say: b 2 the ⋅ ( the - b ) {\ displaystyle b ^ {2} a \ cdot (a-b)} .

History

Due to its frequent appearance in geometry, the golden ratio has already been studied by ancient mathematicians. It is arguably that this rule was also understood and used in ancient Egypt. The i. e. The proportions of the golden section can also be discovered in the proportions of the Great Pyramid of Giza, built around 2600. Half of the base edge of the pyramid (average 115.18 m) and the height of its sides (approximately 186.42 m) are related to each other according to the golden section (with a deviation of 0.03%, which can be considered within the margin of error). The ancient Greeks also knew this ratio. Pythagoras, Theodorus, and Euclid also dealt with it. The notation of the golden ratio, Φ (Greek capital letter), comes from the name of the Greek sculptor Pheidias, who often used it in his work. (Additional notations below.) In his work Aus experimenteller Ästhetik (From Experimental Aesthetics), Adolf Zeising (1810–1876) writes about his measurements on a large number of people. He placed the first division point of the well-developed human figure on the navel and found that the points of fit of the torso and major members of the body were also disproportionate to the golden ratio. Undoubtedly, the proportions of earlier sculptures, especially Greek sculptures, are also in line with Zeising's theory: if the height of the body is 1000, the lower part of the body is 618 from the navel, the upper part of the body is 382 from the navel, and the length of the head is 146. These are all according to the golden ratio rule relate to each other. Zeising also attempted antiquity and

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