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Divine harmony: what is the golden ratio in simple words. Secrets of the universe in numbers

Divine harmony: what is the golden ratio in simple words. Secrets of the universe in numbers

27.04.2019

A special type of fine art of ancient Greece should be highlighted the manufacture and painting of all kinds of vessels. In an elegant form, the proportions of the golden section are easily guessed.

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In painting and sculpture of temples, on household items, the ancient Egyptians most often depicted gods and pharaohs. The canons of the image of a standing person walking, sitting, etc. were established. Artists were required to memorize individual forms and schemes of images from tables and samples. Ancient Greek artists made special trips to Egypt to learn how to use the canon.

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Before you is the canon of the image of a standing person, all the proportions of a person are connected by the “golden section” formula.

Turning to examples of the "golden section" in painting, one cannot but stop one's attention on the work of Leonardo da Vinci.

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Leonardo da Vinci

His identity is one of the mysteries of history. Leonardo da Vinci himself said: “Let no one who is not a mathematician dare to read my works.” The term itself "golden section" introduced by Leonardo da Vinci. He talked about the proportions of the human body.

“If we tie a human figure – the most perfect creation of the Universe – with a belt and then measure the distance from the belt to the feet, then this value will refer to the distance from the same belt to the top of the head, as the entire height of a person to the length from the belt to the feet.”

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In the most famous painting by Leonardo, the portrait of the Mona Lisa (the so-called "La Gioconda", circa 1503, Louvre), the image of a wealthy townswoman appears as a mysterious personification of nature as such, without losing a purely feminine cunning; The internal significance of the composition is given by the cosmically majestic and at the same time disturbingly alienated landscape, melting in a cold haze. Its composition is based on golden triangles, which are parts of a regular star pentagon.

There is no painting more poetic than the painting of Sandro Botticelli, and the great Sandro has no painting more famous than his “Venus”. For Botticelli, his Venus is the embodiment of the idea of universal harmony of the “golden section” that prevails in nature.

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The proportional analysis of Venus convinces us of this.

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Is it possible to talk about the "golden section" in music? You can, if you measure a piece of music by the time of its performance. In music, the golden ratio reflects the peculiarities of human perception of temporal proportions. The “golden section” point serves as a guideline for shaping. It often has a climax. It can also be the brightest moment, or the quietest, or the highest pitched place. (Listen to a piece of music.)

Thus, with the help of the "golden section" we saw the relationship between the arts: music and architecture, painting, mathematics and literature. (Message "The Tale of Igor's Campaign".)

A sensational discovery was made by the St. Petersburg poet and translator of "The Tale of Igor's Campaign" Andrey Chernov. He found that the construction of the verses of the mysterious ancient Russian monument obeys a mathematical law. The research allowed Chernov to conclude that the "Tale of Igor's Campaign", consisting of nine songs, was based on a circular composition.

And the reason for checking the harmony of the poem with algebra was an article about the life of the ancient Greek mathematician Pythagoras. Chernov's attention was attracted by arguments about the "golden section" and about the number, which go back to Pythagoras. An unexpected association arose: after all, in the compositional construction of a poem, it is also a circle and, therefore, there must be a “diameter” and a certain mathematical regularity.

Already the first calculations began to confirm the pattern, and what a pattern! If the number of verses in all three parts (there are 804 of them) is divided by the number of verses in the first and last part (256), we get 3.14, i.e. number up to three decimal places.

Chernov's discovery leads to a natural question: how did the ancient author of the Tale of Igor's Campaign, knowing nothing about numbers or other mathematical formulas, introduce an organizing mathematical principle into this text? Chernov suggests that the author used this intuitively, obeying the images of ancient Greek architectural monuments. In those days, the temple was a comprehensive, artistic ideal, and therefore influenced the rhythm of poetic self-expression.

We were convinced that there is still a connection between mathematics and literature, between architecture and music. And this is not accidental, because every art is characterized by the desire for harmony, proportion, harmony. Nature is perfect, and it has its own laws, expressed through mathematics and manifesting in all arts, regardless of whether it is literature or mathematics. These properties are not invented by people. They reflect the properties of nature itself.

MINESTERSVO OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

Federal State Budgetary Educational Institution

Higher professional education

"Far Eastern State University for the Humanities"

FACULTY OF FINE ARTS AND DESIGN

COURSE WORK

"Golden Ratio in Art"

2nd year students

P. A. Sorokina

Scientific director

FROM. Titov

Art. teacher

Khabarovsk 2012

Introduction

The history of the development of the golden section

Antiquity

Middle Ages

rebirth

The meaning of the golden ratio in art

Painting

Architecture

Literature

Conclusion

References

Application

Introduction

There are things that cannot be explained. So you come to an empty bench and sit on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely not one or the other. You will sit so that the ratio of one part of the bench to another, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive. Sitting down on a bench, you have produced a "golden ratio".

The objectives of the work are, first of all, to study the history of the golden section, to study the use of the "divine proportion" in art and to get acquainted with the modern use of the golden section.

A strange, mysterious, inexplicable thing: this divine proportion mystically accompanies all living things. Inanimate nature does not know what the "golden section" is. But you will certainly see this proportion in the curves of sea shells, and in the form of flowers, and in the form of beetles, and in a beautiful human body. Everything living and everything beautiful - everything obeys the divine law, the name of which is the "golden section".

So what is the "golden ratio"? What is this perfect, divine combination? Maybe it's the law of beauty? Or is it still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The "golden section" is both that, and another, and the third. Only not separately, but at the same time ... And this is his true mystery, his great secret.

Sometimes professional artists, having learned to draw and paint from nature, due to their own weak fundamental training, believe that knowledge of the laws of beauty (in particular, the law of the golden section) interferes with free intuitive creativity. This is a big and deep delusion of many artists who have not become true creators. The masters of Ancient Greece, who knew how to consciously use the golden ratio, which, in fact, is very simple, skillfully applied its harmonic values in all types of art and achieved such perfection in the structure of forms expressing their social ideals, which is rarely found in the practice of world art. All ancient culture passed under the sign of the golden ratio. This proportion was also known in ancient Egypt.

Knowledge of the laws of the golden section or continuous division, as some researchers of the doctrine of proportions call it, helps the artist to create consciously and freely. Using the laws of the golden section, you can explore the proportional structure of any work of art, even if it was created on the basis of creative intuition. This side of the matter is of no small importance in the study of the classical heritage and in the analysis of art criticism of works of all types of art.

Now we can say with confidence that the golden ratio is the basis of shaping, the use of which ensures the diversity of compositional forms in all types of art and gives rise to the creation of a scientific theory of composition and a unified theory of plastic arts.

The paper discusses the first mention of the golden section, the history of its development, its use in art and the modern vision of the golden section.

The history of the development of the golden section

Antiquity

The history of the Golden Section is the history of human knowledge of the world. The concept of the "Golden Section" has gone through all the stages of knowledge in its development. The first stage of knowledge was the discovery of the "golden section" by the ancient Pythagoreans. There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians.

Indeed, the proportions of the pyramid of Cheops, (1) temples, household items and decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. At the beginning of the 20th century in Saqqara (Egypt), archaeologists uncovered a crypt in which the remains of an ancient Egyptian architect named Khesi-Ra were buried. In literature, this name is often found as Khesira. It is assumed that Khesi-Ra was a contemporary of Imhotep, who lived during the reign of Pharaoh Djoser (27th century BC)

From the crypt, along with various material values, wooden boards-panels covered with magnificent carvings, which were made by the hand of an impeccable craftsman, were extracted. In total, 11 boards were placed in the crypt; of these, only five survived, and the rest of the panels were completely destroyed. For a long time, the purpose of the panels from the Khesi-Ra burial was unclear.(2) Initially, Egyptologists mistook these panels for false doors. However, starting from the 60s of the 20th century, the situation with panels began to clear up. In the early 60s, the Russian architect I. Shevelev drew attention to the fact that on one of the panels the wands that the architect holds in his hands correlate with each other as, that is, as a small side and a diagonal with an aspect ratio of 1: 2 ("double-adjacent square"). It was this observation that became the starting point for the research of the Russian architect I. Shmelev, who conducted a thorough geometric analysis of the "Hesi-Ra panels" and as a result came to a sensational discovery described in the brochure "The Phenomenon of Ancient Egypt" (1993).

“But now, after a comprehensive and reasoned analysis by the method of proportions, we get sufficient grounds to assert that the Hesi-Ra panels are a system of harmony rules encoded in the language of geometry...

So, we have in our hands concrete material evidence, "plain text" telling about the highest level of abstract thinking of intellectuals from Ancient Egypt. The author, who cut the boards, with amazing accuracy, jewelry elegance and virtuoso ingenuity, demonstrated the rule of the GS (golden section) in its widest range of variations. As a result, the GOLDEN SYMPHONY was born, represented by an ensemble of highly artistic works, not only testifying to the genius giftedness of their creator, but also convincingly confirming that the author was initiated into the magical mysteries of harmony. That genius was a Goldsmith named Hesi-Ra."

The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division.

All ancient Greek culture developed under the sign of the golden ratio. The idea of harmony, based on the golden ratio, could not help touching Greek art. Nature, taken in a broad sense, included the creative world of man, art, music, where the same laws of rhythm and harmony operate. To take the material and exclude everything superfluous - such is the aphoristically captured plan of the sculptor, who absorbed all the seriousness of the philosophical wisdom of the ancient thinker. And this is the main idea of Greek art, for which the "golden section" for the first time became some kind of aesthetic canon.

The basis of art is the theory of proportions. And, of course, questions of proportionality could not pass by Pythagoras. Of the philosophers of Greece, Pythagoras, perhaps for the first time, tries to mathematically analyze the essence of harmonic proportions. Pythagoras knew that the intervals of the octave can be expressed by numbers that correspond to the corresponding vibrations of the string, and these numerical relationships were put by Pythagoras as the basis of their musical harmony. Pythagoras is credited with knowledge of arithmetic, geometric and harmonic proportions, as well as the law of the golden section. Pythagoras attached special, outstanding significance to the latter, making the pentagram or star-shaped pentagon the hallmark of his "union".

Plato, borrowing the Pythagorean doctrine of harmony, uses five regular polyhedra ("Platonic solids") and emphasizes their "ideal" beauty.

Not only the philosophers of ancient Greece, but also many Greek artists and architects paid considerable attention to achieving proportionality. And this is confirmed by the analysis of the architectural structures of Greek architects. The Phrygian tombs and the ancient Parthenon, the "Canon" of Polykleitos and Praxiteles' Aphrodite of Cnidus, the most perfect Greek theater in Epidaurus and the most ancient theater of Dionysus in Athens that have come down to us - all these are vivid examples of sculpture and creativity, full of deep harmony based on the golden section.

The theater in Epidaurus was built by Polykleitos the Younger in the 40th Olympias. Designed for 15 thousand people. The theater (place for spectators) is divided into two tiers: the first has 34 rows of seats, the second - 21 (Fibonacci numbers!). The opening of the angle enclosing the space between the theatron and the skene (an extension for dressing the actors and storing props) divides the circumference of the base of the amphitheater in the ratio 137°.5: 222°.5 = 0.618 (golden proportion). This ratio is implemented in almost all ancient theaters. This proportion in Vitruvius in his schematic representations of such buildings is 5:8, that is, it is considered as the ratio of Fibonacci numbers.

Theater of Dionysus in Athens three-tiered. The first tier has 13 sectors, the second -21 (Fibonacci numbers!). The ratio of the openings of the angles dividing the circumference of the base into two parts is the same, that is, the golden ratio.

When building temples, a person was taken as a basis as a "measure of all things": he must enter the temple "with his head held high". His height was divided into 6 units (Greek feet), which were plotted on the ruler, and a scale was applied to it, rigidly connected with the sequence of six members of the Fibonacci series: 1, 2, 3, 5, 8, 13 (their sum is 32 = 25) . By adding or subtracting these reference segments, the necessary proportions of the structure were achieved. A sixfold increase in all the sizes laid down on the ruler retained the harmonic proportion. In accordance with this scale, temples, theaters or stadiums were built.

Plato also knew about the golden division. His dialogue "Timaeus" is devoted to the mathematical and aesthetic views of the school of Pythagoras and, in particular, to the questions of the golden division. In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division.

Thus, antiquity was completely subordinated to the proportions of the golden section. There was a proportional division in architecture, sculpture, painting and music. Harmony was inherent in all life.

Middle Ages

One of the most interesting personalities of the era of the Crusades, the harbinger of the Renaissance, was Emperor Friedrich Hohenstaufen, a student of the Sicilian Arabs and a fan of Arab culture. The greatest European mathematician of the Middle Ages Leonardo Pisano (nicknamed Fibonacci) lived and worked at his palace in Pisa.

Fibonacci wrote several mathematical works: "Liber abaci", "Liber quadratorum", "Practica geometriae". The most famous of these is "Liber abaci". This work was published during the life of Fibonacci in two editions in 1202 and 1228. The book consists of 15 sections. It should be noted that Fibonacci conceived his work as a manual for merchants, however, in terms of its significance, it went far beyond the limits of trading practice and, in essence, represented a kind of mathematical encyclopedia of the Middle Ages. From this point of view, the 12th section is of particular interest, in which Fibonacci (3) formulated and solved a number of mathematical problems that are of interest from the point of view of the general prospects for the development of mathematics.

The most famous of the problems formulated by Fibonacci is the "rabbit breeding problem" discussed above, which led to the discovery of the numerical sequence 1, 1, 2, 3, 5, 8, 13, ..., later called the "Fibonacci series".

Fibonacci was almost two centuries ahead of Western European mathematicians of his time. Like Pythagoras, who received his "scientific education" from the Egyptian and Babylonian priests and then contributed to the transfer of his knowledge to Greek science, Fibonacci received his mathematical education in Arabic educational institutions and much of the knowledge gained there, in particular, the Arabic-Hindu decimal number system. , he tried to "introduce" into Western European science. And like Pythagoras, the historical role of Fibonacci for the Western world was that with his mathematical books he contributed to the transfer of the mathematical knowledge of the Arabs to Western European science and thereby laid the foundation for the further development of Western European mathematics.

So, the Middle Ages learned about the golden ratio in a mathematical version (in the form of a sequence of Fibonacci numbers). The preservation of knowledge about the "divine proportion" served as the basis for the further development of art already in the Renaissance.

rebirth

Renaissance in the history of culture of the countries of Western and Central Europe is a transitional era from medieval culture to the culture of modern times. The most characteristic feature of this era is the humanistic worldview and the appeal to the ancient cultural heritage, as it were, the "revival" of ancient culture. The Renaissance was marked by major scientific advances in the field of natural science. A specific feature of the science of this era was a close connection with art, and this association was sometimes expressed in the work of one person. The most striking example of such a multifaceted personality is Leonardo da Vinci - an artist, scientist, engineer.

Together with other achievements of ancient culture, scientists and artists of the Renaissance with great enthusiasm perceived the Pythagorean idea of the harmony of the Universe and the golden ratio. And it is no coincidence that it was Leonardo da Vinci, who is one of the most prominent personalities of the Renaissance, who introduces the name "golden section", which immediately becomes the aesthetic canon of the Renaissance.

The idea of harmony turned out to be among those conceptual constructions of ancient culture, to which the church reacted with great interest. According to Christian doctrine, Velene was a creation of God and implicitly obeyed his will. And the Christian God in the creation of the world was guided by mathematical principles. This Catholic doctrine in the science and art of the Renaissance took the form of a search for a mathematical plan according to which God created the Universe.

The belief that nature was created according to a mathematical plan and that the Lord God is the creator of harmony was expressed at that time not only by scientists, but also by poets, as well as representatives of art.

According to the modern American historian of mathematics Maurice Kline, it was the close fusion of the religious doctrine of God as the creator of the Universe and the ancient idea of the numerical harmony of the Universe that became one of the most important reasons for the huge surge of culture in the Renaissance. The main goal of Renaissance science is most clearly stated in the following statement by Johannes Kepler:

"The main goal of all exploration of the external world should be the discovery of rational order and harmony, which God sent down to the world and revealed to us in the language of mathematics."

The same idea, the idea of the harmony of the world, the expression of its orderliness and perfection, turns into the main idea of the art of the Renaissance. In the works of Bramante, Leonardo da Vinci, Raphael, Giordano, Titian, Alberti, Donatello, Michelangelo, there is a strict proportion and harmony of the plot, subject to a verified proportion. The most convex law of harmony, the law of number, with which the beauty of the work was associated, was revealed in the works of art and scientific and methodological studies of Leonardo, Dürer, Alberti.

During the period of the Italian Renaissance, research continues in the field of the theory of proportionality of works of sculpture and architecture. During this period, the works of the famous Roman architect Vitruvius, which had a decisive influence on the works of Italian art theorists (Alberti), were republished in Italy. Originating in Florence, the classical style of the High Renaissance created its most monumental monuments in Rome, Venice and other cultural centers of Italy.

In addition to artists, architects and sculptors of this era, the entire musical culture was strongly influenced by ancient ideas about harmony. During this period, the famous philosopher, physicist and mathematician M. Mersenne introduces a 12-sound temperament system into music. In a number of his works - "Treatise on General Harmony", "General Harmony" Mersenne considers music as an integral part of mathematics and sees in it - in its consonant sound - one of the main ways of manifesting world harmony and beauty.

It was during this period that the first book devoted to the "golden section" appeared.

19th century

In the 19th century the nature of science is changing radically. The problem of the structural unity of the world, put forward in antiquity, is gradually being revived in its epistemological status, provided with the entire heritage of science. The idea of the structural unity of the world is confirmed by the evolutionary doctrine in biology (C. Darwin), which introduced the idea of development into natural science, the periodic law (D.I. Mendeleev), which made it possible to predict the properties of still unknown chemical elements, the law of conservation and transformation of energy (R. Mayer, J. Joule, G. Helmholtz), who put all the laws of physics and chemistry on a single basis, the cell theory (T. Schwann, M. Schleiden), which showed the uniform structure of all living organisms, and other outstanding scientific discoveries of science of the 19th century, which proved the existence internal connection between all known types of matter.

The thesis of the unity of man and nature, consistently carried out in antiquity, is revived again at the end of the 19th and mainly in the first half of the 20th century in a number of conceptual constructions, especially within the framework of the so-called "Russian cosmism" (V.I. Vernadsky, N.F. Fedorov, K. E. Tsiolkovsky, P. A. Florensky, A. L. Chizhevsky and others). The most important direction of research is the search for invariants of being - special stability, found in entire classes of outwardly different or heterogeneous phenomena, capable of revealing and expressing the general nature of the latter.

This direction of scientific research inevitably raised the question of knowing the objective laws of harmony, the need for an accurate calculation of harmonic relations. Against this background, interest in the harmonic proportion, the golden section, the Fibonacci numbers is awakening again.

In the 19th century, a great contribution to the development of the theory of proportionality was made by the German scientist A. Zeising, (4) whose book "Neue Lehre von den Prportionen des menschlichen Korpers" (1854) is still widely cited among the works devoted to the problem of proportionality.

Based on the position that proportionality is the ratio of two unequal parts to each other and to the whole in their most perfect combination, Zeising formulates the law of proportionality as follows:

"The division of the whole into unequal parts is proportional when the ratio of the parts of the whole to each other is the same as their ratio to the whole, i.e. the ratio that gives the golden ratio."

Trying to prove that the entire universe obeys this law, Zeising tries to trace it both in the organic and inorganic world.

In support of this, he cites data on the relationship of the mutual distances between the celestial bodies corresponding to the golden ratio, establishes the same relationship in the structure of the human figure, in the configuration of minerals, plants, in the sound chords of music in architectural works.

Having examined the statues of Apollo Belvedere and Venus Medicea, Zeising establishes that when dividing the total height in the indicated ratio, the dividing lines pass through the natural articulations of the body. The first section passes through the navel, the second through the middle of the neck, etc., that is, all sizes of individual parts of the body are obtained by dividing the whole according to the golden section.

Dwelling on the significance of the law of the golden section in music, Zeising points out that the ancient Greeks attributed the aesthetic impression of chords to the proportional division of the octave using the arithmetic mean and harmonic proportion. The first is the ratio of the fundamental tone to the fifth and to the octave - 6:9:12; the second is the ratio of the fundamental tone to the fourth and to the octave - 6:8:12. In the same way, the Greeks explained the harmony and other harmonies.

Based on the assumption that only those combinations of tones are beautiful, the intervals of which are proportional to each other and to the whole, and on the fact that the combination of only two tones does not give complete harmony, Zeising shows that the most pleasing to the ear consonances have such intervals. that the ratio of the frequencies included in the chord is closest to the golden ratio. For example, the combination of a small third with an octave of the main sound corresponds to a frequency ratio of 3:5, the connection of a major third with an octave of the main sound - 5:8 (3, 5, 8 - Fibonacci numbers!).

Zeising further concludes that since these two combinations of sounds between two-valued are the most pleasant to hear, this, apparently, explains the fact that musical periods end only with them. In the same way, he explains why the improvised folk melody and the simple music of two horns (or English horns) moves in sixths and their additions - thirds.

Zeising draws attention to another curious fact. As you know, major (male) and minor (female) modes are built on the basis of major and minor triads. A major triad built on the basis of a major third is an acoustically correct consonance. It creates the impression of balance, physical perfection, giving it the character of strength, light, vigor, united in life by the concept of "major".

A minor triad built on the basis of a minor third is an acoustically incorrect consonance. It creates the impression of a broken sound and has the character of gloom, sadness, weakness, united in life by the concept of "minor".

These conclusions of Zeising, with his interpretation of the reasons for the consonance of intervals, are confirmed by the studies of acousticians.

Turning to the meaning of the law of proportionality in architecture, Zeising points out that architecture in the field of arts occupies the same position as the organic world in nature, spiritualizing inert matter on the basis of world laws. At the same time, regularity, symmetry and proportionality are its indispensable attributes, which implies that the question of the laws of proportionality in architecture is much more acute than in sculpture or painting.

Thus, the science of the 19th century again returned to the search for an answer to those "eternal" questions that were posed by the ancient Greeks. The conviction has matured that the world is dominated by a "universal law" of number and rhythm, expressing its structural and functional aspects. In this regard, in the science of the 19th century, interest in the golden ratio is awakened again.

The meaning of the golden ratio in art

So, before you define the golden ratio, you need to familiarize yourself with the concept of proportion. In mathematics, proportion (Latin proportio) is an equality between two ratios of four quantities: a: b = c: d. Let's take a line segment as an example. The segment AB can be divided into two equal parts (/). This will be the ratio of equal values - AB: AC = AB: BC. The same straight line (5) can be divided into two unequal parts in any ratio. These parts do not form proportions. There is a ratio of a small segment to a large one or a smaller one to a larger one, but there is no ratio (proportion). And, finally, the line AB can be divided according to the golden section, when AB: AC, as AC: BC. This is the golden division or division in the extreme and average ratio. From the foregoing, it follows that the golden section is such a proportional harmonic division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller segment is related to the larger one as the larger one is to everything, i.e. a: b = b: c or c \ b = b: a. The definition - division in extreme and mean ratio - becomes clearer if we express it geometrically, namely, a: b as b: c.

We derive the golden ratio. (6) A perpendicular equal to half AB is restored from point B. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is plotted, ending with point D. The segment AD is transferred to the straight line AB. The resulting point f divides the segment AB in the ratio of the golden ratio. Arithmetically, the segments of the golden ratio are expressed as an infinite irrational fraction. AE \u003d 0.618 ..., if AB is taken as a unit, ff \u003d 0.382 .... In practice, rounding is used: 0.62 and 0.38. If the segment AB is taken as 100 parts, then the largest part of the segment is 62, and the smaller one is 38 parts.

Spirals are very common in nature. The concept of the golden ratio will be incomplete, if not to say about the spiral.(7)

The shape of the spirally curled shell attracted the attention of the ancient Greek scientist Archimedes. He studied it and deduced the equation of the spiral. A spiral drawn according to this equation is called an Archimedes spiral. The increase in her step is always uniform.

So where can we meet the golden ratio in art.

Painting

Very often in the same work of painting there is a combination of symmetrical division into equal parts along the vertical and division into unequal parts along the golden section along the horizontal. Consider examples.

In the famous portrait of Monna Lisa ("Gioconda") (8), which was completed by Leonardo da Vinci in 1503, an important element of the composition becomes a cosmically vast landscape, melting in a cold haze. The picture of the brilliant artist attracted the attention of researchers who discovered that the compositional construction of the picture is based on two "golden" triangles, which are parts of the "pentagram".

Leonardo da Vinci's painting "Madonna in the Grotto" (9) is not strictly symmetrical, but its construction is based on symmetry. The entire content of the picture is expressed in the figures that are located in its lower part. They fit into a square. But the artist was not content with this format. He completes the golden ratio rectangle above the square. As a result of this construction, the whole picture received the format of a golden rectangle placed vertically. With a radius equal to half the side of the square, he described a circle and received a semicircle of the upper part of the picture. At the bottom, the arc crossed the axis of symmetry and indicated the size of another golden ratio rectangle at the bottom of the picture. Then, with a radius equal to the side of the square, a new arc is described, which gave points on the vertical sides of the picture. These points helped build an equilateral triangle, which was the framework for building the entire group of figures. All proportions in the picture were derived from the height of the picture. They form a series of relationships of the golden section and serve as the basis for the harmony of forms and rhythm, which carry a hidden charge of emotional impact.

Raphael's painting "The Betrothal of Mary" is constructed in a similar way.

The widespread use of the "golden" spiral is characteristic of the works of art by Raphael, Michelangelo and other Italian artists.

The multi-figure composition "Massacre of the Innocents" (10), made in 1509-1510 by Raphael, is distinguished by the dynamism and drama of the plot. On the preparatory sketch of Raphael, a smooth line is drawn, covering the whole picture. The line starts at the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle, and then goes along the figure of a child, a woman holding him close to her, a warrior with a raised sword, and then along the figures of the same group on the right side of the sketch. If you naturally connect all these pieces of the curve with a dotted line, then you get a "golden" spiral with very high accuracy!

The figure of A. S. Pushkin in N. N. Ge’s painting “Alexander Sergeevich Pushkin in the village of Mikhailovsky” (11) was placed by the artist on the golden section line on the left side of the canvas. But all other values in width are not at all random: the width of the oven is 24 parts from the width of the picture, the whatnot is 14 parts, the distance from the whatnot to the oven is also 14 parts, etc.

If we turn to ancient Russian painting, icons of the 15th - 16th centuries, we will see the same methods of constructing an image. Vertical format icons are vertically symmetrical, and horizontal divisions are made according to the golden ratio. The icon "Descent into Hell" by Dionysius and the workshop is calculated with mathematical accuracy in the proportions of the golden section.

In the icon of the late XV century. "The Miracle of Flora and Lavra" carried out the triple ratio of the golden section. First, the master divided the height of the icon into two equal parts. He took the top one under the image of an angel and saints. He divided the lower part into two unequal segments in a ratio of 3: 2. As a result, he got the ratio of the three values of the golden section: a: b, like b: c. In numbers it will look like this: 100, 62, 38, and halved - 50, 31, 19.

A lot has been written about the symmetry of the "Trinity" (12) by Andrei Rublev. But no one paid attention to the fact that the principle of golden proportions is also implemented along the horizontal lines. The height of the middle angel is related to the height of the side angels, just as their height is related to the height of the entire icon. The line of the golden section crosses the axis of symmetry in the middle of the table and the bowl with the sacrificial calf. This is the compositional lock of the icon. The figure also shows smaller values of the golden section series. Along with the smoothness of the lines and color, the proportions of the icons play a significant role in creating the overall impression that the viewer experiences when viewing it.

The icon of Theophan the Greek "Assumption" appears to our eyes with a mighty chorale. Symmetry and the golden ratio in construction give this icon such power and harmony that we see and feel when we see Greek temples and listen to Bach's fugues. It is easy to see that the composition of Theophan the Greek's "Assumption" and Andrey Rublev's "Trinity" is one and the same. Researchers of the work of ancient Russian artists note that the merit of Theophan the Greek is not so much that he painted frescoes and icons for Russian cathedrals and churches, but that he taught Andrey Rublev the ancient wisdom.

Music

Music is a kind of art that reflects reality and influences a person through meaningful and specially organized sound sequences consisting of tones. While retaining some similarity to the sounds of real life, musical sounds fundamentally differ from the latter in their strict pitch and temporal (rhythmic) organization ("musical harmony"). Starting from the ancient period, elucidation of the laws of "musical harmony" has been one of the important areas of scientific research.

Pythagoras is credited with establishing two basic laws of harmony in music:

1) if the ratio of the oscillation frequencies of two sounds is described by small numbers, then they give a harmonic sound;

2) to get a harmonic triad, you need to add a third sound to a chord of two consonant sounds, the oscillation frequency of which is in harmonic proportional relationship with the first two. The importance of Pythagoras' works on the scientific explanation of the foundations of musical harmony can hardly be overestimated. It was the first scientifically substantiated theory of musical harmony.

Any piece of music has a temporal extension and is divided by some milestones ("aesthetic milestones") into separate parts that draw attention to themselves and facilitate the perception of the whole. These milestones can be dynamic and intonational culmination points of a musical work. Are there any regularities in the emergence of "aesthetic milestones" in a piece of music? An attempt to answer this question was made by the Russian composer L. Sabaneev. In a large article "Chopin's Etudes in the Light of the Golden Ratio" (1925), he shows that the individual time intervals of a piece of music, connected by a "climactic event", as a rule, are in the ratio of the golden ratio. Sabaneev writes:

"All such events, by the author's instinct, are timed to such points of the length of the whole that they themselves divide the time spans into separate parts that are in the relations of the "golden section." gold "relation is often carried out with great accuracy, which is all the more surprising that, in the absence of any knowledge of such things among poets and authors of music, this is all solely a consequence of an inner sense of harmony."

An analysis of a huge number of musical works allowed Sabaneev to conclude that the organization of a musical work is built in such a way that its cardinal parts, separated by milestones, form rows of the golden section. Such an organization of the work corresponds to the most economical perception of the mass of relations and therefore gives the impression of the highest "harmoniousness" of form. According to Sabaneev, the number and frequency of using the golden section in a musical composition depends on the "rank of the composer." The highest percentage of coincidences is noted among brilliant composers, that is, "the intuition of form and harmony, as one should expect, is the strongest among geniuses of the first class."

According to Sabaneev's observations, in the musical works of various composers, not one golden section is usually stated, associated with an "aesthetic event" taking place near it, but a whole series of such sections. Each such section reflects its own musical event, a qualitative leap in the development of a musical theme. In the 1770 works of 42 composers he studied, 3275 golden sections were observed; the number of works in which at least one golden section was observed was 1338. The largest number of works in which there is a golden section is by Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin (92%), Schubert (91%).

Much attention was paid to the study of the laws of musical harmony by the famous Russian art critic E.K. Rosenov. He argued that there are strict proportional relationships in musical works and poetry:

"Obvious features of" natural creativity "we must recognize in those cases when in the highly inspired creations of brilliant authors, generated by the powerful aspiration of the spirit for truth and beauty, we quite unexpectedly discover some mysterious regularity of numerical relations that is not amenable to direct consciousness."

E. Rosenov believed that the golden section should play an outstanding role in music as a means for bringing homogeneous phenomena into line, created by nature itself:

"The golden division could:

1) establish in a musical work an elegant, proportionate relationship between the whole and its parts;

2) to be a special place of prepared expectation, combined with culminating points (forces, masses, movements of sounds) and with all sorts of outstanding, from the point of view of the author, effects;

3) to direct the listener's attention to those thoughts of a musical work, to which the author attaches the most importance, which he wants to put in connection and correspondence with each other.

Rosenov selects for analysis a number of typical works of outstanding composers: Bach, Beethoven, Chopin, Wagner. For example, when studying Bach's Chromatic Fantasy and Fugue, the duration of a quarter was taken as a unit of measure in time. This work contains 330 such units of measure. The golden division of this interval falls on the 204th quarter from the beginning.

E. Rosenov analyzed in detail: the finale of Beethoven's cis-moll sonata, Chopin's Fantasia-Impromtu, the introduction to Wagner's "Tristan and Isolde". In all these works, the golden ratio is very common. The author pays special attention to Chopin's fantasy, which was created impromptu and was not subject to any editing, which means that there was no conscious application of the law of the golden section, which is present in this piece of music down to small musical formations.

So, we can recognize that the golden ratio is a criterion for the harmony of the composition of a musical work.

Architecture

In architecture, you can also observe the principle of the golden section. For example, the Church of the Intercession on the Nerl (1165) (13) is considered the most perfect creation of Vladimir architects.

Acquaintance with the Nerl temple creates an image of harmony, architectural beauty. And the question involuntarily arises: what "secrets" did the Russian architects, who created eight centuries ago, own?

Studying the architecture of the Church of the Intercession on the Nerl, the Russian architect I. Shevelev came to the conclusion that this masterpiece of architecture manifests a proportion, which is the ratio of the larger side to the diagonal of the "two-adjacent square", that is, a rectangle with an aspect ratio of 1: 2. Thus, the interconnected proportions of this architectural structure are based on the proportions of the "two-adjacent" square and its derivative - the golden ratio. The presence of these proportions determined the beauty of the temple. “The striking beauty and harmony of the architecture of the Church of the Intercession of the Virgin on the Nerl,” writes the theorist of architecture K.N. Afanasiev, “is shaped by a chain of interconnected relations of the “golden section”.

Another example is St. Basil's Cathedral on Red Square in Moscow. (14) The history of the creation of this temple is as follows. On October 2, 1552, Kazan fell, saving Russia forever from the Tatar invasion. To glorify the "Kazan capture", which went down in the history of Russia along with the Battle of Kulikovo, Tsar Ivan the Terrible decided to lay the Cathedral of the Intercession on the Red Square of Moscow; later this temple was nicknamed by the people "Basil the Blessed" in honor of the holy fool, who was buried near the walls of the temple in the 16th century.

The composition of the buildings of the cathedral is characterized by a harmonious combination of symmetrical and asymmetric proportions. The temple, symmetrical in its basis, contains many geometric "irregularities". Thus, the central volume of the tent is displaced by 3 m to the west of the geometric center of the entire composition. However, the inaccuracy makes the composition more picturesque, "alive" and it wins in general. The architectural decoration of the cathedral is characterized by the growth of decorative forms upwards; the forms grow out of one another, stretching upwards, rising now in large elements, now forming groups consisting of smaller decorative parts.

In accordance with this compositional idea, the proportions of the cathedral were also built. The researchers found in it a proportion based on the golden section series:

where j = 0.618. This articulation contains the main architectural idea of creating the cathedral, which is the same for all the domes, uniting them into one commensurate composition.

When considering St. Basil's Cathedral, the question involuntarily arises: is it by chance that the number of domes in it is 8 (around the central cathedral)? Were there any canons that determined the number of domes in the temple? Obviously they existed. The simplest Orthodox cathedrals of the early period were single-domed. After the reform of Patriarch Nikon in the middle of the 17th century, it was forbidden to build one-domed churches as they did not correspond to the five-domed rank of the Orthodox Church.

In addition to one- and two-domed Orthodox churches, many had 5 and 8 domes. However, the Novgorod St. Sophia Cathedral (10th century) was the 13th chapter, and the Church of the Transfiguration in Kizhi, carved out of wood 2.5 centuries ago, is crowned with 21 chapters. Is such an increase in the number of domes "according to Fibonacci" (1, 2, 3, 5, 8, 13, 21) random, reflecting the natural law of growth - from simple to complex?

The expression "architecture is frozen music" has become winged. It is not the result of a rigorous scientific analysis, but rather the result of a figurative, intuitive feeling of a certain connection between a harmonic architectural form and musical harmony. The musical melody is based on the alternation of sounds of different heights and durations, it is based on the temporal ordering of sounds. At the heart of the architectural composition is the spatial ordering of forms. It would seem that there is nothing in common between them. But in order to estimate the dimensions of the spatial structure of a geometric figure, we must trace this figure from beginning to end with our eyes, and the longer, for example, its length, the longer the perception will be. Obviously, here lies the organic connection between the spatial and temporal perception of objects by a person.

Literature

Of undoubted interest is the analysis of the novel "Eugene Onegin" made by N. Vasyutinskiy. This novel consists of 8 chapters, each with an average of about 50 verses. The most perfect, the most refined and emotionally rich is the eighth chapter. It has 51 verses. Together with Yevgeny's letter to Tatyana (60 lines), this exactly corresponds to the Fibonacci number 55!

N. Vasyutinskiy states:

"The culmination of the chapter is Eugene's explanation of his love for Tatyana - the line "Get pale and fade ... that's bliss!" This line divides the entire eighth chapter into two parts - in the first 477 lines, and in the second - 295 lines. Their ratio is 1.617 "The subtlest correspondence to the value of the golden ratio! This is a great miracle of harmony, accomplished by the genius of Pushkin!"

Much in the structure of poetic works makes this art form related to music. A clear rhythm, a regular alternation of stressed and unstressed syllables, an ordered dimensionality of poems, their emotional richness make poetry a sister of musical works. Each verse has its own musical form - its own rhythm and melody. It can be expected that in the structure of poems some features of musical works, patterns of musical harmony, and, consequently, the golden ratio, will appear. Lermontov's famous poem "Borodino" is divided into two parts: an introduction addressed to the narrator and occupying only one stanza ("Tell me, uncle, it's not without reason ..."), and the main part, representing an independent whole, which is divided into two equivalent parts. In the first of them, the expectation of the battle is described with increasing tension, in the second - with a gradual decrease in tension towards the end of the poem. The border between these parts is the climax of the work and falls exactly on the point of dividing it by the golden section.

The main part of the poem consists of 13 seven lines, that is, 91 lines. Dividing it by the golden ratio (91:1.618 = 56.238), we make sure that the division point is at the beginning of the 57th verse, where there is a short phrase: "Well, it was a day!". It is this phrase that represents the "culminating point of the excited expectation", which completes the first part of the poem (expectation of the battle) and opens its second part (the description of the battle).

Thus, the golden ratio plays a very meaningful role in poetry, highlighting the climax of the poem.

The application of the golden ratio in the modern world

In today's age of high technology, a person needs to contemplate harmony even in everyday things. Designers apply the principle of the golden ratio in almost everything from creating a logo to designing a car.

Design

In design, the Fibonacci series is most often used to calculate ideal proportions. But progress does not stand still, and today special extremely convenient programs have appeared that make it easy to calculate the golden ratio. You only need to give a number and get the corresponding value.

Perhaps you are a little surprised and cannot understand why the golden ratio is used in design? The answer can be illustrated as follows. The aspect ratio of the iPod Shuffle 1.59, the iPod Classic 1.67 and the iPhone4 1.7 have sales of more than 1,700,000 units in the first 4 days of trading. These sales results do not surprise fans of Apple products, of course, the device is evaluated by other characteristics. But it seems to me that Jonathan Ive did not accidentally stop at such proportions. It is no coincidence that Moleskine has been selling notebooks all over the world for 200 years. Matisse, Van Gogh, Hemingway and many others left notes and sketches in Moleskine books. This is the real history of mankind in books with proportions of 1.57

The golden ratio is found in the objective world both in direct reading, as a theme for stylization, and as a basic constructive principle, like the violin of the great master Stradivarius.

That is why in web design it is a powerful leverage on visitors. But not every designer can master this art.

In web design, the golden ratio helps to accomplish the following tasks:

1) Determine what size the picture and all the elements on the page should be.

2) By mastering the golden section method, a web designer can easily determine the centers of attention on the page - i.e. exactly those points where the eyes of all visitors are directed. It is enough to place the necessary illustration or text there - and it will fall into the field of view of potential customers.

Twitter during the 2011 redesign used the principle of the golden ratio in the new interface. (15) But it saves the ratio of site elements only in the standard, narrow version, if the window is larger, then the content is stretched.

The It "s Numbered site does not apply the golden ratio principle to the entire interface, but only to the content + image bundle. (16)
And the MmDesign site uses the golden ratio to display the main visual on the homepage.

Using the golden ratio does not guarantee that the design of the site will be good, there are a number of other equally important factors that contribute to the development of the right design. However, the golden ratio can help to bring balance and finish to the work, as well as ease of perception of the interface by users, which is often not very easy to achieve.

Using the golden section rule helps to find a balance and an optimal combination in the arrangement of various elements on the page.

Thus, the golden ratio is used in the creation of logos, in industrial design, in the creation of Internet resources.

Conclusion

golden ratio painting music

So, we conclude that among the countless variety of forms in nature that the artist encounters, regularity and consistency reign, the connecting thread of which is the proportion of the golden section. Everything that exists in nature and is perceived by the human eye has a size and shape. Every natural object is something unified, integral. It is easy to see that nature always creates something whole: a person, a tree, a fish, a horse, a dog, etc. Nothing can be taken away from this whole, reduced without violating integrity. Nothing can be added. It will be superfluous and also violate the integrity and harmony. For example, six fingers on a human hand, three horns on a bull.

In the 20th century, a huge number of works on art history were made, showing the wide manifestation and use of the "golden section" in all areas of art: in music (Sabaneev "Chopin's Etudes in the illumination of the Golden Section), in poetry (academician Tsereteli "The Golden Section in the poem of Shota Rustaveli "The Knight in the Panther's Skin"), cinematography (film director Einstein), architecture (Grimm G.D. "Proportionality in architecture), painting (Kovalev F.V.), architecture (Shevelev I.Sh.), music (Marutaev M. A.) Of great interest are the studies of the Russian philologist O. N. Grinbaum to identify "Fibonacci" patterns in the poetry of A. S. Pushkin and the Russian philosopher A. V. Voloshinov to study the mathematical principles of shaping in music, architecture, painting and literature.

The whole is always made up of parts. Parts of different sizes are in a certain relationship to each other and to the whole. This is the proportions. From a mathematical point of view, we note the repetition of measurable equal quantities and unequal ones, correlating with each other as quantities of the golden ratio. These are two kinds of proportional relations. All other quantities, if they arose as a result of a violation of shaping for any reason, do not constitute proportions. Proportional relationships lead to symmetry, rhythm, harmony and beauty. Disproportionate relationships lead to a violation of order, a violation of symmetry and rhythm, which is perceived by a person as ugly and even ugly.

Thus, the natural law of divine proportion, which manifests itself in the highest forms of works of art, is found in a new, rhythmodynamic form of aesthetic law. The law of the "golden section", known since ancient Egypt, is one of the most amazing mathematical laws; it was formulated by the great Leonardo and is increasingly featured in the rapidly growing stream of natural science and humanities research.

This law is not a coercive, sole or exclusive law that determines the artistic impression; nevertheless, it remains a law, directly related to aesthetic, artistic impact, has a direct impact on the impression of wholeness and beauty. Sensitive to beauty, Pushkin, with only one artistic instinct, firstly, guessed the moments of the "golden section" in the development of his narrative with an intuition that was amazing in its mathematical accuracy; secondly, he established the proportional dimensions of the parts in relation to the whole and, thirdly, he emphasized the climax points of the expectation growing in tension, compositionally placing the main thoughts of the narrative in places so noticeable to direct sensory perception.

References

1. Bendukidze, A. B. Golden section: textbook / A. B. Bendukidze; M, 1973. - 53-55s.

GOLDEN SECTION - HARMONIC PROPORTION

In mathematics, a proportion is the equality of two ratios:

Line segment AB can be divided into two parts in the following ways:

into two equal parts - AB: AC = AB: BC;
into two unequal parts in any ratio (such parts do not form proportions);
thus, when AB:AC=AC:BC.

The latter is the golden division (section).

The golden section is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part in the same way as the larger part itself is related to the smaller one, in other words, the smaller segment is related to the larger one as the larger one is to everything

a:b=b:c or c:b=b:a.

Geometric representation of the golden ratio

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden ratio using a compass and ruler.

Division of a line segment according to the golden ratio. BC=1/2AB; CD=BC

From point B, a perpendicular equal to half AB is restored. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is plotted, ending with point D. The segment AD is transferred to the straight line AB. The resulting point E divides the segment AB in the ratio of the golden ratio.

Segments of the golden ratio are expressed without h final fraction AE=0.618..., if AB is taken as a unit, BE=0.382... For practical purposes, approximate values of 0.62 and 0.38 are often used. If the segment AB is taken as 100 parts, then the largest part of the segment is 62, and the smaller 38 parts.

The properties of the golden section are described by the equation:

Solution to this equation:

The properties of the golden ratio created around this number a romantic aura of mystery and almost a mystical generation. For example, in a regular five-pointed star, each segment is divided by a segment crossing it in proportion to the golden ratio (i.e. the ratio of the blue segment to green, red to blue, green to purple, is 1.618).

SECOND GOLDEN SECTION

This proportion is found in architecture.

Construction of the second golden section

The division is carried out as follows. The segment AB is divided in proportion to the golden section. From point C, the perpendicular CD is restored. Radius AB is point D, which is connected by a line to point A. Right angle ACD is bisected. A line is drawn from point C to the intersection with line AD. Point E divides segment AD in relation to 56:44.

Division of a rectangle by a line of the second golden ratio

The figure shows the position of the line of the second golden section. It is located in the middle between the golden section line and the middle line of the rectangle.

GOLDEN TRIANGLE (pentagram)

To find segments of the golden ratio of the ascending and descending rows, you can use the pentagram.

Construction of a regular pentagon and pentagram

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer. Let O be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, raised at point O, intersects with the circle at point D. Using a compass, mark the segment CE=ED on the diameter. The length of a side of a regular pentagon inscribed in a circle is DC. We set aside segments DC on the circle and get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36 0 at the top, and the base laid on the side divides it in proportion to the golden section.

Draw straight line AB. From point A we lay off on it a segment O of arbitrary size three times, through the resulting point P we draw a perpendicular to the line AB, on the perpendicular to the right and left of point P we put off segments O. The resulting points d and d 1 are connected by straight lines with point A. Segment dd 1 we put it on the line Ad 1, getting point C. She divided the line Ad 1 in proportion to the golden ratio. The lines Ad 1 and dd 1 are used to build a "golden" rectangle.

Construction of the golden triangle

HISTORY OF THE GOLDEN SECTION

Indeed, the proportions of the pyramid of Cheops, temples, household items and decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from the tomb of his name, holds measuring instruments in his hands, in which the proportions of the golden division are fixed.

The Greeks were skilled geometers. Even arithmetic was taught to their children with the help of geometric figures. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles.

Dynamic Rectangles

Plato also knew about the golden division. The Pythagorean Timaeus, in Plato's dialogue of the same name, says: “It is impossible for two things to be perfectly united without a third, since a thing must appear between them that would hold them together. Proportion can best accomplish this, for if three numbers have the property that the mean is related to the lesser as the greater is to the mean, and vice versa, the lesser is to the mean as the mean is to the greater, then the last and the first will be the middle, and middle - first and last. Thus, everything necessary will be the same, and since it will be the same, it will make a whole. Plato builds the earthly world using triangles of two types: isosceles and non-isosceles. He considers the most beautiful right-angled triangle to be one in which the hypotenuse is twice the smaller of the legs (such a rectangle is half an equilateral, the main figure of the Babylonians, it has a ratio of 1: 3 1/2, which differs from the golden ratio by about 1/25, and is called Timerding "rival of the golden ratio"). Using triangles, Plato builds four regular polyhedra, associating them with the four earthly elements (earth, water, air and fire). And only the last of the five existing regular polyhedra - the dodecahedron, all twelve faces of which are regular pentagons, claims to be a symbolic image of the heavenly world.

icosahedron and dodecahedron

The honor of discovering the dodecahedron (or, as it was supposed, the Universe itself, this quintessence of the four elements, symbolized, respectively, by the tetrahedron, octahedron, icosahedron and cube) belongs to Hippasus, who later died in a shipwreck. This figure really captures many relationships of the golden section, so the latter was assigned the main role in the heavenly world, which was subsequently insisted on by the minor brother Luca Pacioli.

In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division.

Antique golden ratio compasses

In the ancient literature that has come down to us, the golden division was first mentioned in Euclid's Elements. In the 2nd book of the "Beginnings" the geometric construction of the golden division is given. After Euclid, Hypsicles (2nd century BC), Pappus (3rd century AD) and others studied the golden division. In medieval Europe, they got acquainted with the golden division from Arabic translations of Euclid's "Beginnings". The translator J. Campano from Navarre (3rd century) commented on the translation. The secrets of the golden division were jealously guarded, kept in strict secrecy. They were known only to the initiates.

In the Middle Ages, the pentagram was demonized (as, indeed, much that was considered divine in ancient paganism) and found shelter in the occult sciences. However, the Renaissance again brings to light both the pentagram and the golden ratio. Thus, a scheme describing the structure of the human body gained wide circulation in that period of the assertion of humanism.

Leonardo da Vinci also repeatedly resorted to such a picture, in fact, reproducing a pentagram. Its interpretation: the human body has divine perfection, because the proportions inherent in it are the same as in the main celestial figure. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge. He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Francesca, who wrote two books, one of which was called On Perspective in Painting. He is considered the creator of descriptive geometry.

Luca Pacioli was well aware of the importance of science for art.

In 1496, at the invitation of Duke Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked at the Moro court in Milan at that time. In 1509, Luca Pacioli's De divina proportione, 1497, published in Venice in 1509, was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. There is only one such proportion, and uniqueness is the highest attribute of God. It embodies the holy trinity. This proportion cannot be expressed by an accessible number, remains hidden and secret, and is called irrational by mathematicians themselves (so God can neither be defined nor explained by words). God never changes and represents everything in everything and everything in each of his parts, so the golden ratio for any continuous and definite quantity (regardless of whether it is large or small) is the same, cannot be changed or changed. otherwise perceived by the mind. God called into being heavenly virtue, otherwise called the fifth substance, with its help four other simple bodies (four elements - earth, water, air, fire), and on their basis called into being every other thing in nature; so our sacred proportion, according to Plato in the Timaeus, gives formal being to the sky itself, for it is attributed to the form of a body called the dodecahedron, which cannot be built without the golden section. These are Pacioli's arguments.

Leonardo da Vinci also paid much attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in golden division. Therefore, he gave this division the name of the golden section. So it is still the most popular.

At the same time, in northern Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches an introduction to the first draft of a treatise on proportions. Dürer writes: “It is necessary that the one who knows something should teach it to others who need it. This is what I set out to do."

Judging by one of Dürer's letters, he met with Luca Pacioli during his stay in Italy. Albrecht Dürer develops in detail the theory of the proportions of the human body. Dürer assigned an important place in his system of ratios to the golden section. The height of a person is divided in golden proportions by the belt line, as well as a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face - by the mouth, etc. Known proportional compass Dürer.

Great astronomer of the 16th century Johannes Kepler called the golden ratio one of the treasures of geometry. He is the first to draw attention to the significance of the golden ratio for botany (plant growth and structure).

Kepler called the golden ratio self-continuing. “It is arranged in such a way,” he wrote, “that the two junior terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains until infinity."

The construction of a series of segments of the golden ratio can be done both in the direction of increase (increasing series) and in the direction of decrease (descending series).

If on a straight line of arbitrary length, postpone the segment m , put aside a segment M . Based on these two segments, we build a scale of segments of the golden proportion of the ascending and descending rows.

Building a scale of segments of the golden ratio

In subsequent centuries, the rule of the golden ratio turned into an academic canon, and when, over time, a struggle began in art with an academic routine, in the heat of the struggle, “they threw the child out with the water.” The golden section was “discovered” again in the middle of the 19th century.

In 1855, the German researcher of the golden section, Professor Zeising, published his work Aesthetic Research. With Zeising, exactly what happened was bound to happen to the researcher who considers the phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions to be "mathematical aesthetics".

Zeising did a great job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden section. The proportions of the male body fluctuate within the average ratio 13:8=1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8:5=1.6. In a newborn, the proportion is 1: 1, by the age of 13 it is 1.6, and by the age of 21 it is equal to the male. The proportions of the golden section are also manifested in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, poetic meters were subjected to research. Zeising defined the golden ratio, showed how it is expressed in line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction and the other. His next book was entitled "Golden division as the basic morphological law in nature and art." In 1876, a small book, almost a pamphlet, was published in Russia, outlining Zeising's work. The author took refuge under the initials Yu.F.V. Not a single painting is mentioned in this edition.

At the end of the 19th - beginning of the 20th centuries. a lot of purely formalistic theories appeared about the use of the golden section in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

GOLDEN RATIO AND SYMMETRY

The golden ratio cannot be considered in itself, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulff (1863-1925) considered the golden ratio to be one of the manifestations of symmetry.

Golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the golden division is an asymmetric symmetry. The science of symmetry includes such concepts as static and dynamic symmetry. Static symmetry characterizes rest, balance, and dynamic symmetry characterizes movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments, equal magnitudes. Dynamic symmetry is characterized by an increase in segments or their decrease, and it is expressed in the values of the golden section of an increasing or decreasing series.

FIBONACCCI SERIES

The name of the Italian mathematician monk Leonardo from Pisa, better known as Fibonacci, is indirectly connected with the history of the golden section. He traveled a lot in the East, introduced Europe to Arabic numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, in which all the problems known at that time were collected.

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2+3=5; 3+5=8; 5+8=13, 8+13=21; 13+21=34, etc., and the ratio of adjacent numbers of the series approaches the ratio of the golden division. So, 21:34=0.617, and 34:55=0.618. This ratio is denoted by the symbol Ф. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden ratio, its increase or decrease to infinity, when the smaller segment is related to the larger one as the larger one is to everything.

As shown in the figure below, the length of each knuckle of the finger is related to the length of the next knuckle in a F-proportion. The same relationship is seen in all fingers and toes. This connection is somehow unusual, because one finger is longer than the other without any visible pattern, but this is not accidental, just as everything in the human body is not accidental. The distances on the fingers, marked from A to B to C to D to E, are all related to each other in the proportion F, as are the phalanges of the fingers from F to G to H.

Take a look at this frog skeleton and see how each bone conforms to the F-ratio pattern just like it does in the human body.

GENERALIZED GOLDEN RATIO

Scientists continued to actively develop the theory of Fibonacci numbers and the golden section. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. There are methods for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden section. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

One of the achievements in this area is the discovery of generalized Fibonacci numbers and generalized golden ratios.

The Fibonacci series (1, 1, 2, 3, 5, 8) and the “binary” series of weights 1, 2, 4, 8 discovered by him are completely different at first glance. But the algorithms for constructing them are very similar to each other: in the first case, each number is the sum of the previous number with itself 2=1+1; 4=2+2..., in the second - this is the sum of the two previous numbers 2=1+1, 3=2+1, 5=3+2... Is it possible to find a general mathematical formula from which "binary » series, and the Fibonacci series? Or maybe this formula will give us new numerical sets with some new unique properties?

Indeed, let's set a numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... and separated from the previous one by S steps. If we denote the nth member of this series by? S (n), then we get the general formula? S(n)=? S(n-1)+? S(n-S-1).

Obviously, with S=0 from this formula we will get a "binary" series, with S=1 - a Fibonacci series, with S=2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

In general, the golden S-proportion is the positive root of the golden S-section equation x S+1 -x S -1=0.

It is easy to show that when S=0, the division of the segment in half is obtained, and when S=1, the familiar classical golden section is obtained.

The ratios of neighboring Fibonacci S-numbers with absolute mathematical accuracy coincide in the limit with the golden S-proportions! Mathematicians in such cases say that golden S-sections are numerical invariants of Fibonacci S-numbers.

The facts confirming the existence of golden S-sections in nature are given by the Belarusian scientist E.M. Soroko in the book "Structural Harmony of Systems" (Minsk, "Science and Technology", 1984). It turns out, for example, that well-studied binary alloys have special, pronounced functional properties (thermally stable, hard, wear-resistant, oxidation-resistant, etc.) only if the specific weights of the initial components are related to each other by one from golden S-proportions. This allowed the author to put forward a hypothesis that golden S-sections are numerical invariants of self-organizing systems. Being confirmed experimentally, this hypothesis can be of fundamental importance for the development of synergetics, a new field of science that studies processes in self-organizing systems.

Using golden S-proportion codes, any real number can be expressed as a sum of degrees of golden S-proportions with integer coefficients.

The fundamental difference between this method of encoding numbers is that the bases of new codes, which are golden S-proportions, turn out to be irrational numbers for S>0. Thus, the new number systems with irrational bases, as it were, put the historically established hierarchy of relations between rational and irrational numbers “upside down”. The fact is that at first the natural numbers were "discovered"; then their ratios are rational numbers. And only later, after the Pythagoreans discovered incommensurable segments, irrational numbers appeared. For example, in decimal, quinary, binary and other classical positional number systems, natural numbers were chosen as a kind of fundamental principle: 10, 5, 2, from which all other natural numbers, as well as rational and irrational numbers, were constructed according to certain rules.

A kind of alternative to the existing methods of numbering is a new, irrational, system, as the fundamental principle of the beginning of the calculation of which an irrational number is chosen (which, we recall, is the root of the golden section equation); other real numbers are already expressed through it.

In such a number system, any natural number is always representable as a finite number - and not infinite, as previously thought! are the sums of powers of any of the golden S-proportions. This is one of the reasons why "irrational" arithmetic, having amazing mathematical simplicity and elegance, seems to have absorbed the best qualities of classical binary and "Fibonacci" arithmetic.

PRINCIPLES OF SHAPING IN NATURE

Everything that took on some form, formed, grew, strove to take a place in space and preserve itself. This aspiration finds realization mainly in two variants: upward growth or spreading over the surface of the earth and twisting in a spiral.

The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The concept of the golden ratio will be incomplete, if not to say about the spiral.

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and deduced the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.

Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago.

The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."

Mandelbrot series

The golden spiral is closely related to cycles. The modern science of chaos studies simple cyclic feedback operations and the fractal forms generated by them, which were previously unknown. The figure shows the well-known Mandelbrot series - a page from the dictionary h limbs of individual patterns, called Julian series. Some scientists associate the Mandelbrot series with the genetic code of cell nuclei. A consistent increase in sections reveals amazing fractals in their artistic complexity. And here, too, there are logarithmic spirals! This is all the more important because both the Mandelbrot series and the Julian series are not inventions of the human mind. They arise from the realm of Plato's prototypes. As the doctor R. Penrose said, "they are like Mount Everest"

Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf.

The appendage makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of lesser force, releases a leaf of an even smaller size and ejection again.

If the first outlier is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden ratio.

Chicory

In many butterflies, the ratio of the size of the thoracic and abdominal parts of the body corresponds to the golden ratio. Having folded its wings, the night butterfly forms a regular equilateral triangle. But it is worth spreading the wings, and you will see the same principle of dividing the body into 2, 3, 5, 8. The dragonfly is also created according to the laws of the golden ratio: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail.

In the lizard, at first glance, proportions that are pleasant to our eyes are captured - the length of its tail relates to the length of the rest of the body as 62 to 38.

viviparous lizard

Both in the plant and in the animal world, the form-building tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.

Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.

Of great interest is the study of the forms of bird eggs. Their various forms fluctuate between two extreme types: one of them can be inscribed in a rectangle of the golden section, the other in a rectangle with a module of 1.272 (the root of the golden ratio)

Such forms of bird eggs are not accidental, since it has now been established that the shape of eggs described by the ratio of the golden section corresponds to higher strength characteristics of the egg shell.

The tusks of elephants and extinct mammoths, the claws of lions, and the beaks of parrots are logarithmic forms and resemble the shape of an axis that tends to turn into a spiral.

In wildlife, forms based on "pentagonal" symmetry (starfish, sea urchins, flowers) are widespread.

The golden ratio is present in the structure of all crystals, but most crystals are microscopically small, so that we cannot see them with the naked eye. However, snowflakes, which are also water crystals, are quite accessible to our eyes. All the figures of exquisite beauty that form snowflakes, all axes, circles and geometric figures in snowflakes are also always, without exception, built according to the perfect clear formula of the golden section.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous. For example, many viruses have a three-dimensional geometric shape of an icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a certain sequence. At each corner of the icosahedron are 12 protein cell units in the shape of a pentagonal prism, and spike-like structures extend from these corners.

Adeno virus

The golden ratio in the structure of viruses was first discovered in the 1950s. scientists from London's Birkbeck College A. Klug and D. Kaspar. The first logarithmic form was revealed in itself by the Polyo virus. The form of this virus turned out to be similar to that of the Rhino virus.

The question arises: how do viruses form such complex three-dimensional forms, the device of which contains the golden ratio, which is quite difficult to construct even with our human mind? The discoverer of these forms of viruses, the virologist A. Klug, makes the following comment: “Dr. Kaspar and I have shown that for the spherical shell of the virus, the most optimal shape is symmetry like the shape of the icosahedron. Such an order minimizes the number of connecting elements... Most of Buckminster Fuller's geodesic hemispherical cubes are constructed according to a similar geometric principle. The installation of such cubes requires an extremely precise and detailed explanation scheme, while unconscious viruses themselves construct such a complex shell of elastic, flexible protein cell units.

Klug's comment once again reminds of an extremely obvious truth: in the structure of even a microscopic organism, which scientists classify as "the most primitive form of life", in this case, a virus, there is a clear plan and a reasonable project has been implemented. This project is incomparable in its perfection and precision of execution with the most advanced architectural projects created by people. For example, projects created by the brilliant architect Buckminster Fuller.

Three-dimensional models of the dodecahedron and icosahedron are also present in the structure of the skeletons of unicellular marine microorganisms radiolarians (beamers), the skeleton of which is made of silica.

Radiolarians form their body of a very exquisite, unusual beauty. Their shape is a regular dodecahedron, and from each of its corners a pseudo-elongation-limb and other unusual forms-growths grow.

The great Goethe, a poet, naturalist and artist (he painted and painted in watercolor), dreamed of creating a unified doctrine of the form, formation and transformation of organic bodies. It was he who introduced the term morphology into scientific use.

Pierre Curie at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment.

The patterns of "golden" symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the gene structures of living organisms. These patterns, as indicated above, are in the structure of individual human organs and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

THE HUMAN BODY AND THE GOLDEN SECTION

All human bones are in proportion to the golden section. The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered to be ideally built.

Golden proportions in parts of the human body

If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.

the distance from the level of the shoulder to the crown of the head and the size of the head is 1:1.618;
the distance from the point of the navel to the crown of the head and from the level of the shoulder to the crown of the head is 1:1.618;
the distance of the navel point to the knees and from the knees to the feet is 1:1.618;
the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1.618;
in fact, the exact presence of the golden proportion in the face of a person is the ideal of beauty for the human gaze;
the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618;
face height/face width;
the central point of connection of the lips to the base of the nose / length of the nose;
face height/distance from the tip of the chin to the center point of the junction of the lips;
mouth width/nose width;
width of the nose/distance between the nostrils;
distance between pupils / distance between eyebrows.

It is enough just to bring your palm closer to you now and carefully look at your index finger, and you will immediately find the golden section formula in it.

Each finger of our hand consists of three phalanges. The sum of the lengths of the first two phalanges of the finger in relation to the entire length of the finger gives the golden ratio (with the exception of the thumb).

In addition, the ratio between the middle finger and the little finger is also equal to the golden ratio.

A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence.

It should also be noted that in most people the distance between the ends of the spread arms is equal to height.

The truths of the golden ratio are within us and in our space. The peculiarity of the bronchi that make up the lungs of a person lies in their asymmetry. The bronchi are made up of two main airways, one (left) is longer and the other (right) is shorter. It was found that this asymmetry continues in the branches of the bronchi, in all smaller airways. Moreover, the ratio of the length of short and long bronchi is also the golden ratio and is equal to 1:1.618.

In the human inner ear there is an organ Cochlea ("Snail"), which performs the function of transmitting sound vibration. This osseous structure is filled with fluid and also created in the form of a snail, containing a stable logarithmic spiral shape =73 0 43".

Blood pressure changes as the heart beats. It reaches its greatest value in the left ventricle of the heart at the time of its contraction (systole). In the arteries during the systole of the ventricles of the heart, blood pressure reaches a maximum value equal to 115-125 mm Hg in a young, healthy person. At the moment of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of the maximum (systolic) to the minimum (diastolic) pressure is on average 1.6, that is, close to the golden ratio.

If we take the average blood pressure in the aorta as a unit, then the systolic blood pressure in the aorta is 0.382, and the diastolic 0.618, that is, their ratio corresponds to the golden ratio. This means that the work of the heart in relation to time cycles and changes in blood pressure are optimized according to the same principle of the law of the golden ratio.

The DNA molecule consists of two vertically intertwined helices. Each of these spirals is 34 angstroms long and 21 angstroms wide. (1 angstrom is one hundred millionth of a centimeter).

The structure of the helix section of the DNA molecule

So 21 and 34 are numbers following one after another in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic helix of the DNA molecule carries the formula of the golden section 1: 1.618.

GOLDEN SECTION IN SCULPTURE

Sculptural structures, monuments are erected to perpetuate significant events, to preserve in the memory of descendants the names of famous people, their exploits and deeds. It is known that even in ancient times the basis of sculpture was the theory of proportions. Relationships of parts of the human body were associated with the formula of the golden section. The proportions of the "golden section" create the impression of harmony, beauty, so the sculptors used them in their works. Sculptors claim that the waist divides the perfect human body in relation to the "golden section". So, for example, the famous statue of Apollo Belvedere consists of parts that are divided according to golden ratios. The great ancient Greek sculptor Phidias often used the "golden ratio" in his works. The most famous of them were the statue of Olympian Zeus (which was considered one of the wonders of the world) and the Athena Parthenon.

The golden proportion of the statue of Apollo Belvedere is known: the height of the depicted person is divided by the umbilical line in the golden section.

GOLDEN SECTION IN ARCHITECTURE

In books on the "golden section" one can find the remark that in architecture, as in painting, everything depends on the position of the observer, and if some proportions in a building on the one hand seem to form the "golden section", then from other points of view they will look different. The "golden section" gives the most relaxed ratio of the sizes of certain lengths.

One of the most beautiful works of ancient Greek architecture is the Parthenon (V century BC).

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various degrees of the number Ф = 0.618 ...

The Parthenon has 8 columns on the short sides and 17 on the long ones. The ledges are made entirely of squares of Pentilean marble. The nobility of the material from which the temple was built made it possible to limit the use of coloring, common in Greek architecture, it only emphasizes the details and forms a colored background (blue and red) for the sculpture. The ratio of the height of the building to its length is 0.618. If we divide the Parthenon according to the "golden section", we will get certain protrusions of the facade.

On the floor plan of the Parthenon, you can also see the "golden rectangles".

We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris) and in the pyramid of Cheops.

Not only the Egyptian pyramids were built in accordance with the perfect proportions of the golden ratio; the same phenomenon is found in the Mexican pyramids.

For a long time it was believed that the architects of Ancient Rus' built everything “by eye”, without any special mathematical calculations. However, the latest research has shown that Russian architects knew mathematical proportions well, as evidenced by the analysis of the geometry of ancient temples.

The famous Russian architect M. Kazakov widely used the "golden section" in his work. His talent was multifaceted, but to a greater extent he revealed himself in numerous completed projects of residential buildings and estates. For example, the "golden section" can be found in the architecture of the Senate building in the Kremlin. According to the project of M. Kazakov, the Golitsyn Hospital was built in Moscow, which is currently called the First Clinical Hospital named after N.I. Pirogov.

Petrovsky Palace in Moscow. Built according to the project of M.F. Kazakova

Another architectural masterpiece of Moscow - the Pashkov House - is one of the most perfect works of architecture by V. Bazhenov.

Pashkov House

The wonderful creation of V. Bazhenov has firmly entered the ensemble of the center of modern Moscow, enriched it. The external view of the house has remained almost unchanged to this day, despite the fact that it was badly burned in 1812. During the restoration, the building acquired more massive forms. The internal layout of the building has not been preserved either, which only the drawing of the lower floor gives an idea of.

Many statements of the architect deserve attention in our days. About his favorite art, V. Bazhenov said: “Architecture has three main subjects: beauty, calmness and strength of the building ... To achieve this, the knowledge of proportion, perspective, mechanics or physics in general serves as a guide, and all of them have a common leader is reason.”

GOLDEN RATIO IN MUSIC

Any piece of music has a time span and is divided into some "aesthetic milestones" into separate parts that attract attention and facilitate perception as a whole. These milestones can be dynamic and intonational culmination points of a musical work. Separate time intervals of a piece of music, connected by a "climactic event", as a rule, are in the ratio of the Golden Ratio.

Back in 1925, art critic L.L. Sabaneev, having analyzed 1770 pieces of music by 42 authors, showed that the vast majority of outstanding works can be easily divided into parts either by theme, or by intonation, or by modal system, which are in relation to the golden section. Moreover, the more talented the composer, the more golden sections were found in his works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition. This result was verified by Sabaneev on all 27 Chopin etudes. He found 178 golden sections in them. At the same time, it turned out that not only large parts of the etudes are divided by duration in relation to the golden section, but parts of the etudes inside are often divided in the same ratio.

Composer and scientist M.A. Marutaev counted the number of measures in the famous Appassionata sonata and found a number of interesting numerical relationships. In particular, in development, the central structural unit of the sonata, where themes are intensively developed and keys replace each other, there are two main sections. In the first - 43.25 cycles, in the second - 26.75. The ratio 43.25:26.75=0.618:0.382=1.618 gives the golden ratio.

Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Chopin (92%), Schubert (91%) have the largest number of works in which there is a Golden Section.

If music is the harmonic ordering of sounds, then poetry is the harmonic ordering of speech. A clear rhythm, a regular alternation of stressed and unstressed syllables, an ordered dimensionality of poems, their emotional richness make poetry a sister of musical works. The golden ratio in poetry primarily manifests itself as the presence of a certain moment of the poem (climax, semantic turning point, main idea of the work) in the line attributable to the dividing point of the total number of lines of the poem in the golden ratio. So, if the poem contains 100 lines, then the first point of the Golden Ratio falls on the 62nd line (62%), the second - on the 38th (38%), etc. The works of Alexander Sergeevich Pushkin, including "Eugene Onegin", are the finest correspondence to the golden ratio! The works of Shota Rustaveli and M.Yu. Lermontov are also built on the principle of the Golden Section.

Stradivari wrote that he used the golden ratio to determine the locations for f-shaped notches on the bodies of his famous violins.

GOLDEN SECTION IN POETRY

Studies of poetic works from these positions are just beginning. And you need to start with the poetry of A.S. Pushkin. After all, his works are an example of the most outstanding creations of Russian culture, an example of the highest level of harmony. From the poetry of A.S. Pushkin, we will begin the search for the golden ratio - the measure of harmony and beauty.

Let's start with the size of the poem, that is, the number of lines in it. It would seem that this parameter of the poem can change arbitrarily. However, it turned out that this was not the case. For example, the analysis of poems by A.S. Pushkin showed that the sizes of verses are distributed very unevenly; it turned out that Pushkin clearly prefers sizes of 5, 8, 13, 21 and 34 lines (Fibonacci numbers).

Many researchers have noticed that poems are like pieces of music; they also have climactic points that divide the poem in proportion to the golden ratio. Consider, for example, a poem by A.S. Pushkin "Shoemaker":

Let's analyze this parable. The poem consists of 13 lines. It highlights two semantic parts: the first in 8 lines and the second (the moral of the parable) in 5 lines (13, 8, 5 are the Fibonacci numbers).

One of Pushkin's last poems "I do not value high-profile rights ..." consists of 21 lines and two semantic parts are distinguished in it: in 13 and 8 lines:

I do not value high-profile rights,

From which not one is dizzy.

I do not grumble about the fact that the gods refused

I'm in the sweet lot of challenging taxes

Or prevent the kings from fighting with each other;

And little grief to me, is the press free

Fooling boobies, or sensitive censorship

In magazine plans, the joker is embarrassing.

All this, you see, words, words, words.

Other, better, rights are dear to me:

Another, better, I need freedom:

Depend on the king, depend on the people -

Don't we all care? God is with them.

Do not give a report, only to yourself

Serve and please; for power, for livery

Do not bend either conscience, or thoughts, or neck;

At your whim to wander here and there,

Marveling at the divine beauty of nature,

And before the creatures of art and inspiration

Trembling joyfully in delights of tenderness,

Here is happiness! That's right...

It is characteristic that the first part of this verse (13 lines) is divided into 8 and 5 lines in terms of semantic content, that is, the entire poem is built according to the laws of the golden ratio.

N. Vasyutinsky states: “The culmination of the chapter is Evgeny’s declaration of love for Tatyana - the line “Pale and fade ... that’s bliss!” This line divides the entire eighth chapter into two parts: the first has 477 lines, and the second has 295 lines. Their ratio is 1.617! The subtlest correspondence to the value of the golden ratio! This is a great miracle of harmony, accomplished by the genius of Pushkin!

E. Rosenov analyzed many poetic works by M.Yu. Lermontov, Schiller, A.K. Tolstoy and also discovered the "golden section" in them.

Lermontov's famous poem "Borodino" is divided into two parts: an introduction addressed to the narrator, occupying only one stanza ("Tell me, uncle, it's not without reason ..."), and the main part, representing an independent whole, which is divided into two equivalent parts. The first of them describes, with increasing tension, the expectation of a battle, the second describes the battle itself with a gradual decrease in tension towards the end of the poem. The border between these parts is the climax of the work and falls exactly on the point of dividing it by the golden section.

The main part of the poem consists of 13 seven lines, that is, 91 lines. Dividing it with the golden ratio (91:1.618=56.238), we make sure that the division point is at the beginning of the 57th verse, where there is a short phrase: “Well, it was a day!” It is this phrase that represents the “culminating point of excited expectation”, which concludes the first part of the poem (expectation of the battle) and opens its second part (description of the battle).

Thus, the golden ratio plays a very meaningful role in poetry, highlighting the climax of the poem.

Many researchers of Shota Rustaveli's poem "The Knight in the Panther's Skin" note the exceptional harmony and melody of his verse. These properties of the poem Georgian scientist, academician G.V. Tsereteli attributes it to the conscious use of the golden ratio by the poet both in the formation of the form of the poem and in the construction of her poems.

Rustaveli's poem consists of 1587 stanzas, each of which consists of four lines. Each line consists of 16 syllables and is divided into two equal parts of 8 syllables in each half line. All hemistiches are divided into two segments of two types: A - a hemistich with equal segments and an even number of syllables (4 + 4); B is a half-line with an asymmetrical division into two unequal parts (5+3 or 3+5). Thus, in the half line B, the ratios are 3:5:8, which is an approximation to the golden ratio.

It has been established that out of 1587 stanzas in Rustaveli's poem, more than half (863) are constructed according to the principle of the golden section.

In our time, a new kind of art has been born - cinema, which has absorbed the dramaturgy of action, painting, music. It is legitimate to look for manifestations of the golden section in outstanding works of cinematography. The first to do this was the creator of the masterpiece of world cinema “Battleship Potemkin”, film director Sergei Eisenstein. In the construction of this picture, he managed to embody the basic principle of harmony - the golden ratio. As Eisenstein himself notes, the red flag on the mast of the rebellious battleship (the apogee point of the film) flies at the point of the golden ratio, counted from the end of the film.

GOLDEN RATIO IN FONTS AND HOUSEHOLD ITEMS

A special type of fine art of ancient Greece should be highlighted the manufacture and painting of all kinds of vessels. In an elegant form, the proportions of the golden section are easily guessed.

In painting and sculpture of temples, on household items, the ancient Egyptians most often depicted gods and pharaohs. The canons of the image of a standing person, walking, sitting, etc. were established. Artists were required to memorize individual forms and schemes of images from tables and samples. Ancient Greek artists made special trips to Egypt to learn how to use the canon.

OPTIMUM PHYSICAL PARAMETERS OF THE EXTERNAL ENVIRONMENT

It is known that the maximum sound volume, which causes pain, is equal to 130 decibels. If we divide this interval by the golden ratio of 1.618, we get 80 decibels, which are typical for the loudness of a human scream. If we now divide 80 decibels by the golden ratio, we get 50 decibels, which corresponds to the loudness of human speech. Finally, if we divide 50 decibels by the square of the golden ratio of 2.618, we get 20 decibels, which corresponds to a human whisper. Thus, all the characteristic parameters of sound volume are interconnected through the golden ratio.

At a temperature of 18-20 0 C interval humidity 40-60% is considered optimal. The boundaries of the optimal humidity range can be obtained if the absolute humidity of 100% is divided twice by the golden ratio: 100 / 2.618 = 38.2% (lower limit); 100/1.618=61.8% (upper limit).

At air pressure 0.5 MPa, a person experiences discomfort, his physical and psychological activity worsens. At a pressure of 0.3-0.35 MPa, only short-term operation is allowed, and at a pressure of 0.2 MPa, it is allowed to work for no more than 8 minutes. All these characteristic parameters are interconnected by the golden ratio: 0.5/1.618=0.31 MPa; 0.5/2.618=0.19 MPa.

Boundary parameters outdoor temperature, within which the normal existence (and, most importantly, the origin) of a person is possible, is the temperature range from 0 to + (57-58) 0 C. Obviously, the first limit of explanations can be omitted.

We divide the indicated range of positive temperatures by the golden ratio. In this case, we obtain two boundaries (both boundaries are temperatures characteristic of the human body): the first corresponds to the temperature, the second boundary corresponds to the maximum possible outside air temperature for the human body.

GOLDEN SECTION IN PAINTING

Even in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has horizontal or vertical. There are only four such points, and they are located at a distance of 3/8 and 5/8 from the corresponding edges of the plane.

This discovery among the artists of that time was called the "golden section" of the picture.

Turning to examples of the "golden section" in painting, one cannot but stop one's attention on the work of Leonardo da Vinci. His identity is one of the mysteries of history. Leonardo da Vinci himself said: "Let no one who is not a mathematician dare to read my works."

He gained fame as an unsurpassed artist, a great scientist, a genius who anticipated many inventions that were not implemented until the 20th century.

There is no doubt that Leonardo da Vinci was a great artist, his contemporaries already recognized this, but his personality and activities will remain shrouded in mystery, since he left to posterity not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “both everything in the world."

He wrote from right to left in illegible handwriting and with his left hand. This is the most famous example of mirror writing in existence.

The portrait of Monna Lisa (Gioconda) has attracted the attention of researchers for many years, who found that the composition of the drawing is based on golden triangles that are parts of a regular star pentagon. There are many versions about the history of this portrait. Here is one of them.

Once Leonardo da Vinci received an order from the banker Francesco del Giocondo to paint a portrait of a young woman, the banker's wife, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint a portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became alive and interesting.

FAIRY TALE. Once upon a time there was one poor man, he had four sons: three smart, and one of them this way and that. And then death came for the father. Before parting with his life, he called his children to him and said: “My sons, soon I will die. As soon as you bury me, lock up the hut and go to the ends of the world to make your own fortune. Let each of you learn something so that you can feed yourself.” The father died, and the sons dispersed around the world, agreeing to return to the glade of their native grove three years later. The first brother came, who learned to carpentry, cut down a tree and hewed it, made a woman out of it, walked away a little and waits. The second brother returned, saw a wooden woman and, since he was a tailor, in one minute dressed her: as a skilled craftsman, he sewed beautiful silk clothes for her. The third son adorned the woman with gold and precious stones - after all, he was a jeweler. Finally, the fourth brother arrived. He did not know how to carpentry and sew, he only knew how to listen to what the earth, trees, herbs, animals and birds were saying, he knew the course of heavenly bodies and also knew how to sing wonderful songs. He sang a song that made the brothers hiding behind the bushes cry. With this song, he revived the woman, she smiled and sighed. The brothers rushed to her and each shouted the same thing: "You must be my wife." But the woman replied: “You created me - be my father. You dressed me, and you decorated me - be my brothers. And you, who breathed my soul into me and taught me to enjoy life, I need you alone for life.

Having finished the tale, Leonardo looked at Monna Lisa, her face lit up with light, her eyes shone. Then, as if awakening from a dream, she sighed, passed her hand over her face, and without a word went to her place, folded her hands and assumed her usual posture. But the deed was done - the artist awakened the indifferent statue; the smile of bliss, slowly disappearing from her face, remained in the corners of her mouth and trembled, giving her face an amazing, mysterious and slightly sly expression, like that of a person who has learned a secret and, keeping it carefully, cannot restrain his triumph. Leonardo worked in silence, afraid to miss this moment, this ray of sunshine that illuminated his boring model...

It is difficult to note what was noticed in this masterpiece of art, but everyone spoke about Leonardo's deep knowledge of the structure of the human body, thanks to which he managed to catch this, as it were, mysterious smile. They talked about the expressiveness of individual parts of the picture and about the landscape, an unprecedented companion of the portrait. They talked about the naturalness of expression, the simplicity of the pose, the beauty of the hands. The artist has done something unprecedented: the picture depicts air, it envelops the figure with a transparent haze. Despite the success, Leonardo was gloomy, the situation in Florence seemed painful to the artist, he got ready to go. Reminders of flooding orders did not help him.

The golden section in the picture of I.I. Shishkin "Pine Grove". In this famous painting by I.I. Shishkin, the motives of the golden section are clearly visible. The brightly lit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a hillock illuminated by the sun. It divides the right side of the picture horizontally according to the golden ratio. To the left of the main pine there are many pines - if you wish, you can successfully continue dividing the picture according to the golden ratio and further.

pine grove

The presence in the picture of bright verticals and horizontals, dividing it in relation to the golden section, gives it the character of balance and tranquility in accordance with the artist's intention. When the artist's intention is different, if, say, he creates a picture with a rapidly developing action, such a geometric scheme of composition (with a predominance of verticals and horizontals) becomes unacceptable.

IN AND. Surikov. "Boyar Morozova"

Her role is assigned to the middle part of the picture. It is bound by the point of the highest rise and the point of the lowest fall of the plot of the picture: the rise of Morozova's hand with the sign of the cross with two fingers, as the highest point; helplessly outstretched hand to the same noblewoman, but this time the hand of an old woman - a beggar wanderer, a hand from under which, along with the last hope of salvation, the end of the sledge slips out.

And what about the "highest point"? At first glance, we have a seeming contradiction: after all, the section A 1 B 1, which is 0.618 ... from the right edge of the picture, does not pass through the arm, not even through the head or eye of the noblewoman, but turns out to be somewhere in front of the noblewoman's mouth.

The golden ratio really cuts here on the most important thing. In him, and precisely in him, is Morozova's greatest strength.

There is no painting more poetic than that of Sandro Botticelli, and the great Sandro has no painting more famous than his Venus. For Botticelli, his Venus is the embodiment of the idea of \u200b\u200bthe universal harmony of the "golden section" that prevails in nature. The proportional analysis of Venus convinces us of this.

Venus

Raphael "School of Athens". Raphael was not a mathematician, but, like many artists of that era, he had considerable knowledge of geometry. In the famous fresco "The School of Athens", where the society of the great philosophers of antiquity is held in the temple of science, our attention is attracted by the group of Euclid, the largest ancient Greek mathematician, who disassembles a complex drawing.

The ingenious combination of two triangles is also built in accordance with the golden ratio: it can be inscribed in a rectangle with an aspect ratio of 5/8. This drawing is surprisingly easy to insert into the upper section of the architecture. The upper corner of the triangle rests against the keystone of the arch in the area closest to the viewer, the lower one - at the vanishing point of perspectives, and the side section indicates the proportions of the spatial gap between the two parts of the arches.

The golden spiral in Raphael's painting "The Massacre of the Innocents". Unlike the golden section, the feeling of dynamics, excitement, is perhaps most pronounced in another simple geometric figure - the spiral. The multi-figure composition, made in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, is just distinguished by the dynamism and drama of the plot. Rafael never brought his idea to completion, however, his sketch was engraved by an unknown Italian graphic artist Marcantinio Raimondi, who, based on this sketch, created the Massacre of the Innocents engraving.

Massacre of the innocents

If, on Raphael's preparatory sketch, one mentally draws lines running from the semantic center of the composition - the points where the warrior's fingers closed around the child's ankle, along the figures of the child, the woman clutching him to her, the warrior with a raised sword, and then along the figures of the same group on the right side sketch (in the figure, these lines are drawn in red), and then connect these pieces of the curve with a dotted line, then a golden spiral is obtained with very high accuracy. This can be checked by measuring the ratio of the lengths of the segments cut by the spiral on the straight lines passing through the beginning of the curve.

GOLDEN RATIO AND IMAGE PERCEPTION

The ability of the human visual analyzer to distinguish objects built according to the golden section algorithm as beautiful, attractive and harmonious has long been known. The golden ratio gives the feeling of the most perfect unified whole. The format of many books follows the golden ratio. It is chosen for windows, paintings and envelopes, stamps, business cards. A person may not know anything about the number Ф, but in the structure of objects, as well as in the sequence of events, he subconsciously finds elements of the golden ratio.

Studies have been conducted in which subjects were asked to select and copy rectangles of various proportions. There were three rectangles to choose from: a square (40:40 mm), a "golden section" rectangle with an aspect ratio of 1:1.62 (31:50 mm) and a rectangle with elongated proportions of 1:2.31 (26:60 mm).

When choosing rectangles in the normal state, in 1/2 cases preference is given to a square. The right hemisphere prefers the golden ratio and rejects the elongated rectangle. On the contrary, the left hemisphere gravitates toward elongated proportions and rejects the golden ratio.

When copying these rectangles, the following was observed: when the right hemisphere was active, the proportions in the copies were maintained most accurately; with the activity of the left hemisphere, the proportions of all the rectangles were distorted, the rectangles were stretched (the square was drawn as a rectangle with an aspect ratio of 1:1.2; the proportions of the stretched rectangle increased sharply and reached 1:2.8). The proportions of the "golden" rectangle were most strongly distorted; its proportions in copies became the proportions of the rectangle 1:2.08.

When drawing your own drawings, proportions close to the golden ratio and elongated prevail. On average, the proportions are 1:2, while the right hemisphere prefers the proportions of the golden section, the left hemisphere moves away from the proportions of the golden section and stretches the pattern.

Now draw some rectangles, measure their sides and find the aspect ratio. Which hemisphere do you have?

THE GOLDEN RATIO IN PHOTOGRAPHY

An example of the use of the golden ratio in photography is the location of the key components of the frame at points that are located 3/8 and 5/8 from the edges of the frame. This can be illustrated by the following example: a photograph of a cat, which is located in an arbitrary place in the frame.

Now let's conditionally divide the frame into segments, in the proportion of 1.62 of the total length from each side of the frame. At the intersection of the segments, there will be the main "visual centers" in which it is worth placing the necessary key elements of the image. Let's move our cat to the points of "visual centers".

GOLDEN RATIO AND SPACE

It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using this series, found regularity and order in the distances between the planets of the solar system.

However, one case that seemed to be against the law: there was no planet between Mars and Jupiter. Focused observation of this area of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century. The Fibonacci series is widely used: with its help, they represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.

The two Golden Spirals of the galaxy are compatible with the Star of David.

Pay attention to the stars emerging from the galaxy in a white spiral. Exactly 180 0 from one of the spirals, another unfolding spiral comes out ... For a long time, astronomers simply believed that everything that is there is what we see; if something is visible, then it exists. They either did not notice the invisible part of the Reality at all, or they did not consider it important. But the invisible side of our Reality is actually much larger than the visible side and, probably, more important... In other words, the visible part of the Reality is much less than one percent of the whole - almost nothing. In fact, our true home is the invisible universe...

In the Universe, all galaxies known to mankind and all bodies in them exist in the form of a spiral, corresponding to the formula of the golden section. In the spiral of our galaxy lies the golden ratio

CONCLUSION

Nature, understood as the whole world in the variety of its forms, consists, as it were, of two parts: animate and inanimate nature. Creations of inanimate nature are characterized by high stability, low variability, judging by the scale of human life. A person is born, lives, grows old, dies, but the granite mountains remain the same and the planets revolve around the Sun in the same way as in the time of Pythagoras.

The world of wildlife appears before us completely different - mobile, changeable and surprisingly diverse. Life shows us a fantastic carnival of diversity and originality of creative combinations! The world of inanimate nature is, first of all, a world of symmetry, which gives stability and beauty to his creations. The world of nature is, first of all, a world of harmony, in which the “law of the golden section” operates.

In the modern world, science is of particular importance, due to the increased impact of man on nature. Important tasks at the present stage are the search for new ways of coexistence of man and nature, the study of philosophical, social, economic, educational and other problems facing society.

In this paper, the influence of the properties of the "golden section" on living and non-living nature, on the historical course of the development of the history of mankind and the planet as a whole was considered. Analyzing all of the above, one can once again marvel at the grandeur of the process of cognition of the world, the discovery of its ever new patterns and conclude: the principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature. It can be expected that the laws of development of various systems of nature, the laws of growth, are not very diverse and can be traced in the most diverse formations. This is the manifestation of the unity of nature. The idea of such unity, based on the manifestation of the same patterns in heterogeneous natural phenomena, has retained its relevance from Pythagoras to the present day.

During the Renaissance, the search for ideal proportions brought artists and scientists together. Mathematicians studied perspective relationships, and artists used projective geometry to depict realistic three-dimensional scenes. In these innovations, Raphael, Dürer and Leonardo da Vinci especially distinguished themselves.

The presence of F in The Birth of Venus by Botticelli and in The Flagellation of Christ by Piero della Franceschi- one of the secrets of these paintings.

In 1435 Leon Battista Alberti's "Treatise on Painting" was published, proclaiming that "the first requirement for an artist is knowledge of geometry" and formulated the first scientific definition of perspective. A little later, da Vinci actively continued to study the perspective.
There is no direct evidence that Leonardo used the golden ratio in his works. But the compositions of his work contain an astonishing array of golden proportions, especially "golden" rectangles.

"Last Supper"

Even in the portrait of Mona Lisa, the researchers found a sequence of "golden" rectangles of different sizes. Some talk about triangles and even pentagons and spirals. Indeed, different artists unconsciously used different "golden" figures in the basis of compositions.

The Holy Family by Michelangelo

Leonardo da Vinci was also a theorist of painting and a supporter of its unity with mathematics. His Treatise on Painting (circa 1498) begins with the phrase “Let no one who is not a mathematician dare to read my works”.
Leonardo applied scientific knowledge about the proportions of the human body to the theories of Pacioli and Vitruvius about beauty. In the famous drawing "Vitruvian Man", a male figure, inscribed in a circle and a square at the same time, is placed in the center of the universe. The image corresponds to the recommendations of Vitruvius, an architect of the 1st century BC. under Julius Caesar. He became popular during the Renaissance due to the translation of his works.