The golden ratio - the harmony of the universe

Definition

The most capacious definition of the golden ratio states that the lesser part refers to the greater part as the greater part refers to the whole. Its approximate value is 1.6180339887. In rounded percentages, the proportions of the parts of the whole would relate as 62% to 38%. This ratio is valid in the forms of space and time.

The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science sees 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 about golden ratio, the Russians also knew about them, but the first scientific explanation of the golden ratio was given by a monk Luca Pacioli in the book "The divine proportion" (1509), the illustrations to which supposedly made Leonardo da Vinci. Pacioli saw the golden ratio as a divine trinity: the smallest section represented the Son, the largest section the Father, and the whole the Holy Spirit.

Directly related to the golden ratio rule is the name of Italian mathematician Leonardo Fibonacci. As a result of solving one of his problems, the scientist found a sequence of numbers, now known as the Fibonacci numbers: 0, 1, 1, 2, 3... etc. Kepler noticed the attitude of this sequence to the golden proportion: "It is made in such a way that two least members of this infinite proportion, added together, give a third term, and any two last members, if added together, give the next one, and the same proportion is kept till infinity". Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations.

Leonardo da Vinci also spent a lot of time studying the features of the golden ratio, most likely he owns the term itself. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained in the section gives the ratio of the sides in the golden ratio.

Over time, the rule of the golden ratio turned into an academic routine, and only the philosopher Adolf Zeising gave it a second life back in 1855. He brought the proportions of the golden ratio to the absolute, making them universal for all phenomena of the world. His "mathematical aesthetics," however, provoked much criticism.

The nature of

Even without going into the calculations, the golden ratio can easily be found in nature. For example, it fits the ratio of the tail and body of a lizard, the distance between the leaves on a branch, there is a golden ratio in the shape of an egg, if the conventional line is drawn through its widest part.

Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything that grows and strives to take its place in space is endowed with the proportions of the golden ratio. In his opinion, one of the most interesting forms is the spiral curl.

Even Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in engineering. Goethe later noted nature's attraction to spiral forms, calling the spiral "the curve of life." Modern scientists have found that such manifestations of spiral forms in nature as the snail shell, the arrangement of sunflower seeds, the patterns of spider webs, the movement of a hurricane, the structure of DNA and even the structure of galaxies comprise the Fibonacci series.

Man

Fashion designers and designers of clothing 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 Leonardo da Vinci's diary, there is a drawing of a nude man inscribed in a circle in two overlapping positions. Relying on the research of the Roman architect Vitruvius, Leonardo similarly tried 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", which influenced the aesthetics of 20th century architecture Adolph Zeising did a tremendous amount of work investigating human proportionality. He measured about two thousand human bodies, as well as many antique statues, and deduced that the golden ratio expresses the average law. In humans, it obeyed almost all parts of the body, but the main indicator of the golden ratio is the division of the body point of the belly button. As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions of the female body 8:5.

The art of spatial forms Artist Vasily Surikov said "that there is an immutable law in composition, when you can not remove or add anything in the picture, not even an extra point can be put, it's real mathematics. For a long time, artists followed this law intuitively, but after Leonardo da Vinci the process of creating a painting is no longer without the solution of geometric problems. For example, Albrecht Dürer used his invention of the proportional compass to determine the points of the golden ratio.

After examining in detail the painting "Alexander Pushkin in the Village of Mikhailovskoye" by Nikolai Ge, the art historian Fyodor Kovalev notes that every detail on the canvas, be it the fireplace, the bookstand, the chair or the poet himself, is strictly inscribed in golden proportions.

Researchers of the golden ratio relentlessly study and measure architectural masterpieces, arguing that they became such because they were created according to golden canons: their list includes the Great Pyramids of Giza, Notre Dame de Paris, St. Basil's Cathedral and the Parthenon.

And today in any art of spatial forms try to follow the proportions of the golden ratio, as they, according to art historians, facilitate the perception of the work and form the viewer's aesthetic sense.

Word, Sound, and Film.

Forms of temporal art demonstrate to us, in their own way, the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in the poems of Pushkin's late period corresponds to the Fibonacci number - 5, 8, 13, 21, 34.

The rule of golden ratio also works in individual works of the Russian classics. So the climax of "The Queen of Spades" is a dramatic scene of Herman and the Countess, which ends with the death of the latter. The story has 853 lines, and the climax is on line 535 (853:535=1,6) - this is the point of the golden ratio.

Soviet musicologist E. K. Rosenow notes the striking accuracy of the ratio of the golden ratio in strict and free forms of Johann Sebastian Bach, which corresponds to a thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the point of the golden ratio is usually the most striking or unexpected musical solution.

Film director Sergei Eisenstein deliberately conformed to the Golden Section rule for his film "Battleship Potemkin" by dividing the film into five parts. In the first three sections the action unfolds on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden mean of the film.