The Golden Ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer cheated on it.
Definition
The most capacious definition of the golden ratio says that the smaller part refers to the larger, as large to the whole. Its approximate value is 1.6180339887. In a rounded percentage, the proportions of parts of a whole will relate as 62% to 38%. This relationship operates in the forms of space and time.
The ancients saw in the golden ratio a reflection of the cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as "asymmetric symmetry", calling it in a broad sense a universal rule reflecting the structure and order of our world order.
History
The ancient Egyptians had an idea of the golden proportions, they knew about them in Russia, but for the first time the golden ratio was explained by the monk Luca Pacioli in the book "Divine Proportion" (1509), the illustrations of which were supposedly made by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the Son, the large one - the Father, and the whole - the Holy Spirit.
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The name of the Italian mathematician Leonardo Fibonacci is directly related to the rule of the golden section. As a result of solving one of the problems, the scientist came up with a sequence of numbers, now known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relation of this sequence to the golden ratio: “It is arranged in such a way that the two lowest terms of this endless proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains indefinitely . Now the Fibonacci series is an arithmetic basis for calculating the proportions of the golden section in all its manifestations.
Leonardo da Vinci also devoted a lot of time to studying the features of the golden ratio, most likely it was he who owned the term itself. His drawings of a stereometric solid, formed by regular pentagons, prove that each of the rectangles obtained when cutting gives aspect ratios in gold division.
Over time, the rule of the golden ratio turned into an academic routine, and only the philosopher Adolph Zeising gave him a second life in 1855. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his "mathematical aesthetics" drew much criticism.
Nature
Without even going into the calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of the lizard, the distance between the leaves on the branch, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.
The Belarusian scientist Eduard Soroko, who studied the forms of gold divisions in nature, noted that everything that grows and strives to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting forms is spiral twisting.
Even Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Later, Goethe noted the gravitation of nature to spiral forms, calling the spiral "the curve of life." Modern scientists have found that such manifestations of spiral forms in nature, such as the snail shell, the arrangement of sunflower seeds, cobweb patterns, hurricane movement, DNA structure and even the structure of galaxies, contain the Fibonacci series.
Person
Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is a universal form for testing the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.
In the diary of Leonardo da Vinci there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo tried in a similar way to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo's Vitruvian Man, created his own scale of "harmonic proportions" that influenced the aesthetics of 20th century architecture.
Adolf Zeising, investigating the proportionality of man, did a tremendous job. He measured about two thousand human bodies, as well as many antique statues, and deduced that the golden ratio expresses the average law. In man, almost all parts of the body are subordinate to him, but the main indicator of the golden ratio is the division of the body by the navel point.
As a result of measurements, the researcher found that the proportions of the male body of 13: 8 are closer to the golden ratio than the proportions of the female body - 8: 5.
The art of spatial forms
The artist Vasily Surikov said, "that there is an immutable law in the composition, when nothing can be removed or added in a picture, even an extra point cannot be put, this is real mathematics." For a long time, artists have been following this law intuitively, but after Leonardo da Vinci, the process of creating a painting can no longer do without solving geometric problems. For example, Albrecht Durer used a proportional compass invented by him to determine the points of the golden section.
The art critic FV Kovalev, having examined in detail the painting by Nikolai Ge "Alexander Sergeevich Pushkin in the village of Mikhailovskoye", notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.
Researchers of the Golden Ratio tirelessly study and measure the masterpieces of architecture, claiming that they have become such because they were created according to the golden canons: in their list are the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling for the viewer.
Word, sound and filmstrip
Temporary art forms in their own way demonstrate to us the principle of the golden division. Literary critics, for example, noticed that the most popular number of lines in poems of the late period of Pushkin's work corresponds to the Fibonacci series - 5, 8, 13, 21, 34.
The rule of the golden section also applies in individual works of the Russian classic. So the climax of "The Queen of Spades" is the dramatic scene of Hermann and the Countess, ending with the death of the latter. There are 853 lines in the story, and the culmination is on line 535 (853: 535 = 1.6) - this is the point of the golden section.
The Soviet musicologist E. K. Rosenov notes the amazing accuracy of the golden ratio in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical decision usually falls on the golden section.
Film director Sergei Eisenstein deliberately coordinated the script of his film "Battleship Potemkin" with the rule of the golden section, dividing the tape into five parts. In the first three sections, the action takes place on the ship, and in the last two - in Odessa. The transition to the scenes in the city is the golden mean of the film.