Formula golden ratio in pictures. Golden ratio in nature, man, art

Golden ratio in painting

Landscape artists know from experience that half the surface of the canvas cannot be allocated to the sky or to the ground and water. It’s better to take either more sky or more land, then the landscape looks better. .

F.V.Kovalev. Golden ratio in painting

#1
land_driver (Wednesday, 03 February 2016 13:37)

Who seeks will always find!
#2
I knew you'd like it
#3
land_driver (Wednesday, 03 February 2016 18:54)

I especially liked the last section - “what do all the considered examples of the use of the golden ratio in painting prove? Absolutely nothing.”
- What is this film about?
- Nothing about it...
#4
Exposure of favorite myths often causes painful reactions.
#5
Elena (Friday, 12 February 2016 17:36)

I read it with mixed feelings... On the one hand, you can’t argue. On the other hand, there is an obvious option to “check harmony with algebra,” and for some reason this offends. I’ll think about it, thanks for the reason to practice thinking.
#6
land_driver (Friday, 12 February 2016 18:03)

It's always interesting to watch those who expose and those who try to refute those who expose
#7
Elena: Still, the words of Pushkin’s Salieri refer to music. And in music, as in Architecture, “algebra” is present from the very beginning. Another question is how significant this role is. This is written in detail in the article “The Golden Ratio and Pythagoras” on this site. Painting is a completely different matter. The laws of perspective, as we know, are not at all necessary in painting. Just like the laws of reflection and refraction of light. (We will not argue that only realistic painting is possible). All that remains, perhaps, is color theory.
land_driver: It’s much more interesting to participate than just watch.
#8
Maxim Boyko (Monday, 15 February 2016 16:36)

I didn’t understand much, since I’m far from a photographer. But it was interesting to read.
#9
land_driver (Tuesday, 16 February 2016 12:11)

Connecting mathematics with music is like nothing at all
#10
Valera (Tuesday, 16 February 2016 16:51)

Knowledge is bricks that need to be assembled in the right order. A masterpiece is possible everywhere...
#11
Hope (Wednesday, 17 February 2016 04:25)

As they say, you can’t argue with mathematics. It is present everywhere - in life, in music, and in painting. Logically, all creative people should feel mathematics in their gut.
#12
Maxim: Interesting - not bad at all. Thank you.
Land_driver: After Pythagoras, it’s certainly easy.
Valera: Valera is poetic even in prose
Nadezhda: David Hilbert once said about his student who gave up mathematics and became a poet: “He had too little imagination for mathematics.”
#13
Vitaly (Wednesday, 17 February 2016 20:46)

Good practical advice about dividing the canvas into two unequal parts!
I took this rule as a basis when I first became interested in photography, completely intuitively.
And I realized that this was indeed the case, looking at my first surviving photos (early 60s of the last century :)).
#14
Marina (Thursday, 18 February 2016 10:38)

Amazing article - very warm. I have heard about the golden ratio many times and wondered what the essence of this concept is. Your explanation is interesting.
#15
land_driver (Friday, 19 February 2016 12:09)

As for “little imagination” - this is a well-known dispute between physicists and lyricists. It will never stop
#16
land_driver (Saturday, 20 February 2016 19:23)

Today on Tverskaya, right on the street on the façade of a building, we saw a painting that completely contradicts all the rules, including the golden ratio - the horizon line divides the painting exactly in half, and a significant figure is located exactly in the center of the canvas. It's on the opposite side of the street somewhere opposite the Actor Gallery
#17
valera (Saturday, 20 February 2016 19:29)

Since there is only enough imagination for poetry, this leads...
#18
Alexander (Sunday, 21 February 2016 17:04)

I could not even imagine that in those days many artists studied painting so much that methods of the golden section were developed. And in general, if you think about it, painting is a kind of science; in order to paint a beautiful picture, you need to know so much and at the same time understand it well.
P.S. - to be honest, like many other readers of your blog, I’m not well versed in many of the topics that you write on your blog, since speaking is not my element, so excuse me if I write a blizzard in some of the comments, misunderstanding you;) Yours is complicated topic for blogging and you are doing a good job, I rarely meet webmasters like you.
#19
The point is not a dispute between physicists and lyricists, but the fact that all human abilities are connected with each other, physics with lyricism, science with art, knowledge with intuition. Leonardo da Vinci is a brilliant example. And if someone deliberately limits the development of one of these parts, he becomes “crippled.” The greatest breakthroughs of the human spirit have always occurred at the borders of regions, as well as the greatest mistakes and delusions. In particular, those associated with the golden ratio. Mathematicians and artists simply did not understand each other.
#20
land_driver (Thursday, 25 February 2016 13:03)

How can you consciously limit yourself in development? Like, I will deliberately not study mathematics, even though I want it and need it? It seems to me that if a person is lazy, then nothing can be done about it
#24
If everything that is on the ground is more interesting - flowers, streams, a river, a path, etc., and the sky is boring, gray, uniform, then it is more interesting when there is more land in the frame. If the sky is “magical”, if there are some extraordinary clouds in the sky, or a rainbow, or crazy colors, or against the sky there are tall trees, beautiful buildings, but nothing on the ground, then it is more interesting when there is more sky in the frame.
#25
For rest - cross-section, for dynamics - peddling....
#26
Lyudmila (Tuesday, 10 October 2017 21:30)

I saw a medical center with the name Golden Ratio, now I think what the meaning of the name is, in the divine proportion of what to what? I only have associations with a scalpel...
#27
land_driver (Saturday, 14 October 2017 21:31)

This is for sure, when I see a photo divided in half by the horizon line, I immediately feel somehow sad. I just want to cut something off - top or bottom
#28
Eh, it’s been a while since there have been new exciting articles on this wonderful site.
#29
Thank you from the bottom of my heart for the article! Since childhood, I could not understand what the golden ratio is, because all the literature that I came across on this subject gave examples of paintings that very vaguely fit into the rules. I wondered why, if proportion is one very clear constant, there are other proportions where the rectangle is divided not into a square and a rectangle, but into a rectangle and a RECTANGLE. What kind of liberties are these? How does this rule work then? Where is the smooth, beautiful square? And here the face is cut off along the line, the details have moved beyond the edges of the division! Why? – I asked. I also noticed that the situation was aggravated not only by researchers who were wishful thinking, but also by ordinary people who put “snail” on everything, even where it clearly doesn’t fit. It’s as if they themselves don’t understand what the meaning of the golden ratio is, and instead of explaining their examples they say: “Well, you can see it!” In geometry nothing is visible, everything must be calculated and proven :) You are the only author of all the ones I’ve read who not only clearly explained how geometry can work in painting, but also dispelled my bitter thoughts: it’s not me who doesn’t see a clear golden ratio in paintings and with my little mind I can’t understand the meaning of the rule, there is no golden ratio!! In mathematics there is, but in paintings - very rarely :) Thank you very much!

The golden ratio is a mathematical formula, the result of complex calculations made by ancient Greek scientists. The uniqueness and divine nature of the golden ratio is explained by the fact that its use brings an invisible but subconsciously perceptible order to science, music, architecture and even nature.

Golden ratio- this is such a proportional harmonic division of a segment into unequal parts, in which the entire segment is related to the larger part, as the larger part itself is related to the smaller one. It is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and even in nature.

Proportions golden ratio look like this

It is believed that the concept golden ratio"discovered by the ancient Greek philosopher and mathematician Pythagoras. Although, there is an opinion that he finalized the research of more ancient scientists - the Babylonians or Egyptians. This is evidenced by the ideal proportions of the Cheops pyramid and many surviving Egyptian temples correspond golden ratio.

Special attention to the rule golden ratio artists of the Renaissance turned to the heritage of the ancient Greeks. The very concept of this harmonic proportion is “ golden ratio"- belongs to Leonardo da Vinci. In his works its use is quite obvious.

For example, the well-known work “The Last Supper” is an example of use golden ratio.

"The Last Supper" by da Vinci

According to the 19th century French architect Viollet-le-Duc, a form that cannot be explained will never be beautiful.

Vertical golden ratio can also be seen in the painting “Trinity” by Andrei Rublev.

Golden ratio. Rublev "Trinity"

Repeating equal quantities, alternating equal and unequal quantities in proportions golden ratio, artists create a particular rhythm in their paintings, evoke a particular mood in the viewer and involve him in viewing the image. At such moments, a person, even one who is not experienced in art, subconsciously understands that he somehow likes the picture, that it is pleasant to look at.

Line intersections golden ratio form four points on the plane, the so-called visual centers, which are located at a distance of 3/8 and 5/8 from the edges of the picture. It is at these points that it is most advantageous to place the key figures of the picture. This has to do with how the human eye works, how the brain works and our perception.

For example, in the painting “The Appearance of Christ to the People” by Alexander Ivanov, the lines golden ratio intersect clearly on the figure of Christ in the distance. And although the figures in the foreground are much larger in size and drawn out more clearly, it is the blurred figure of Christ that attracts the eye, because it is placed in the visual center.

Golden ratio. Alexander Ivanov. "The Appearance of Christ to the People"

The artist Nikolai Krymov wrote: “They say: art is not science, not mathematics, that it is creativity, mood, and that nothing in art can be explained - look and admire. In my opinion this is not the case. Art is explicable and very logical, you can and should know about it, it is mathematical... You can prove exactly why a painting is good and why it is bad.”

In the visual arts, a simplified rule is more often used golden ratio- the so-called “rule of thirds”, when the picture is conventionally divided into three equal parts vertically and horizontally, forming four key points.

Russian artist Vasily Surikov in his monumental work “Boyaryna Morozova” used one of these four points, placing the head and right hand of the main character of the canvas in the upper left part of the picture. Thus, all points, as well as all lines and views in the picture are directed towards that point.

Now try to identify the points yourself golden ratio in the following pictures.

Konstantin Vasiliev’s work “At the Window” is quite simple for this task. Lines golden ratio they converge exactly on the heroine’s face, in her eyes, which forces the viewer to plunge into thoughts about her experiences.

Golden ratio. Konstantin Vasiliev. "Near the window"

Or another example of focusing our attention is the painting “Luisa San Felice in Captivity” by Giovacchino Tom. Again, it is easy to see that here the lines golden ratio intersect on the heroine's face.

Golden ratio. Giovacchino Tom."Louise San Felice in Captivity"

Now you will probably try to recognize divine harmony golden ratio in every picture you see.

Tibaikina Yulia Vitalievna

(I am a researcher. History of discoveries)

Tibaikina Yulia Vitalievna

Stavropol Territory, Blagodarny

MKOU "Secondary School No. 9", 9th grade

Golden ratio in painting

Abstract of the project.

Project passport.

1. Title: “The Golden Ratio in Painting.”

2. Project manager: Tibaikina N.A.

3. The project is carried out within the framework of the subject elective course “Solving problems of increased complexity in algebra and geometry.”

4. The project addresses issues of the history of mathematics, psychology, philosophy, sociology.

5. Designed for 14–15 years old, 9–11 grades.

6. Project type: research and information. Inside is cool, short term.

7. Project goal: To study the importance of mathematics in human life, its influence on human qualities, to increase interest in mathematics and its study. Develop general study skills.

8. Project objectives:

1. Explore the goals of mathematics education.

2. Get acquainted with the basics of mathematics education.

3. Answer the questions: why do we need mathematics? What can mathematics give to each individual?

4. Study the statements of scientists, politicians, philosophers about the meaning of mathematics.

5. Develop skills of independent work with text, with a questionnaire, communication skills, the ability to analyze and systematize the data received.

6. Develop techniques of critical thinking, the ability to conduct assessments and self-assessment and draw conclusions.

9. Estimated products of the project: student project “Golden Section”, creation of a presentation.

10. Stages of work:

1. Determination of work goals and ways to achieve them, forms and methods of work.

2. Gathering information on the topic.

3. Work in creative groups, processing of results, intermediate results.

4. Preparation and holding of a round table.

5. Discussion of results, preparation of presentation.

This project illustrates the application of mathematics in practice, introduces historical information, shows connections with other areas of knowledge, and emphasizes the aesthetic aspects of the issues being studied.

The project develops competencies in the field of independent activity, based on the assimilation of methods of acquiring knowledge from various sources of information. In the field of civil and social activities, in the field of social and labor activities, in the domestic sphere, in the field of cultural and leisure activities.

The project expands the scope of students' mathematical knowledge: introduces students to the golden ratio and related relationships, develops an aesthetic perception of mathematical facts. Shows the use of mathematics not only in the natural sciences, but also in such areas of the humanities as art. Help you realize the degree of your interest in the subject and evaluate the possibilities of mastering it from the point of view of a future perspective (show the possibilities of applying the acquired knowledge in your future profession as an artist, architect, biologist, civil engineer).

Fundamental question: “Is it possible to measure harmony with algebra?” Problematic questions: what is one of the fundamental principles of nature? Is there a pattern of the “golden ratio”? What ratio is the “golden ratio”? What is the approximate value of the “golden ratio”? Do things that are pleasing to the eye satisfy the “golden ratio”? Where is the “golden ratio” found?

The “Golden Proportion” is aimed at the integration of knowledge, the formation of general cultural competence, the creation of ideas about mathematics as a science that arose from the needs of human practice and develops from them. In the basic course of mathematics, little time is devoted to the golden section; only the mathematical component is presented, and the general cultural aspect is mentioned in passing. Therefore, mathematics is presented in it as an element of the general culture of mankind, which is the theoretical basis of art, as well as an element of the general culture of an individual. At the same time, the course is designed for a basic level of proficiency in very limited mathematical content. The leading approach that was used in developing the course: to show, using extensive material from ancient times to the present day, the ways of interaction and mutual enrichment of two great spheres of human culture - science and art; expand your understanding of the areas of application of mathematics; show that the fundamental laws of mathematics are formative in architecture, music, painting, etc. This project is designed to help students imagine mathematics in the context of culture and history. This project can become an additional factor in the formation of positive motivation in the study of mathematics, as well as students’ understanding of the philosophical postulate about the unity of the world and awareness of the universality of mathematical knowledge. It is assumed that the results of students mastering this course may be the following skills: 1) use mathematical knowledge, algebraic and geometric material to describe and solve problems of future professional activity; 2) apply acquired geometric concepts, algebraic transformations to describe and analyze patterns that exist in the surrounding world; 3) make generalizations and discover patterns based on the analysis of particular examples, experiments, put forward hypotheses and make the necessary tests.

It is expected that the results of students mastering this course may include the following skills:

1) use mathematical knowledge, algebraic and geometric material to describe and solve problems of future professional activity;

2) apply acquired geometric concepts and algebraic transformations to describe and analyze patterns that exist in the surrounding world;

3) make generalizations and discover patterns based on the analysis of particular examples, experiments, put forward hypotheses and make the necessary tests.

Download:

Preview:

Geometry has two treasures, one of them is

the Pythagorean theorem, and the other is the division of a segment in the mean and

extreme respect. The first can be represented by the measure

gold; the second one is painfully reminiscent of a precious stone.

Johannes Kepler

1. Introduction.

The relevance of research.

When studying school subjects, it is possible to consider the relationships between concepts accepted in various fields of knowledge and processes occurring in the natural environment; find out the connection between mathematical laws and the properties and patterns of development of nature. Since ancient times, observing the surrounding nature and creating works of art, people have been looking for patterns that would allow them to define beauty. But man not only created beautiful objects, not only admired them, he increasingly asked himself the question: why is this object beautiful, he likes it, but another, very similar one, is not liked, it cannot be called beautiful? Then from a creator of beauty he turned into its researcher. Already in Ancient Greece, the study of the essence of beauty and beauty was formed into a separate branch of science - aesthetics. The study of beauty has become part of the study of the harmony of nature, its basic laws of organization.

The Great Soviet Encyclopedia gives the following definition of the concept of “harmony”:

“Harmony is the proportionality of parts and the whole, the merging of various components of an object into a single organic whole. In harmony, internal orderliness and measure of being are externally revealed.”

Of the many proportions that people have long used to create harmonic works, there is one, the only and unrepeatable one, which has unique properties. This proportion was called differently - “golden”, “divine”, “golden section”, “golden number”. Classic manifestations of the golden ratio are household items, sculpture and architecture, mathematics, music and aesthetics. In the previous century, with the expansion of the field of human knowledge, the number of areas where the phenomenon of the golden ratio was observed sharply increased. These are biology and zoology, economics, psychology, cybernetics, the theory of complex systems, and even geology and astronomy.

The principle of the “golden proportion” aroused great interest among me and my peers. Interest in this ancient proportion either subsides or flares up with renewed vigor. But in fact, we encounter the golden ratio every day, but we don’t always notice it. In the school geometry course we became acquainted with the concept of proportion. I wanted to learn more about the application of this concept not only in mathematics, but also in our everyday life.

Subject of study:

Display of the “Golden Section” in aspects of human activity:

1.Geometry; 2. Painting; 3. Architecture; 4. Wildlife (organisms); 5. Music and poetry.

Hypothesis:

In his activities, a person constantly encounters objects that are based on the golden ratio.

Tasks:

1. Consider the concept of the “golden ratio” (a little about history), the algebraic determination of the “golden ratio”, the geometric construction of the “golden ratio”.

2. Consider the “golden ratio” as a harmonic proportion.

3. See the application of these concepts in the world around me.

Goals :

1. show on material from ancient times to the present day pathsinteraction and mutual enrichment of two great spheres of human culture - science and art;

2.expand the understanding of the areas of application of mathematics;

3. show that the fundamental laws of mathematics are formative in architecture, music, painting, etc.

Working methods:

Collection and analysis of information.

Independent study (individually and in a group).

Processing of received information and its visual presentation in the form of tables and diagrams.

2.Golden ratio. Application of the golden ratio in mathematics.

2.1 Golden ratio. General information.

In mathematics proportion (lat. proportion)call the equality of two relations: a:b = c:d.

Let's consider a segment. It can be divided into two parts by a point in an infinite number of ways, but only in one case does it result in the golden ratio.

Golden ratio - this is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part as the larger part itself relates to the smaller; or in other words, the smaller segment is to the larger as the larger is to the whole:

a:b = b:c or c:b = b:a. (Fig.1)

Let's find out what number the golden ratio is expressed by. To do this, choose an arbitrary segment and take its length as one. (Fig.2)

Let's divide this segment into two unequal parts. We denote the largest of them by “x”. Then the smaller part is equal to 1's.

In a proportion, as is known, the product of the extreme terms is equal to the product of the middle terms, and we rewrite this proportion in the form: x 2 = (1-x)∙1

The solution to the problem is reduced to the equation x 2 +x-1=0 , the length of the segment is expressed as a positive number, therefore, from the two roots x 1 = and x 2 = a positive root should be chosen.
= 0.6180339.. – an irrational number.

Therefore, the ratio of the length of the smaller segment to the length of the larger one

segment and the ratio of the larger segment to the length of the entire segment is 0.62. This rela-

the sewing will be golden.

The resulting number is denoted by the letter j . This is the first letter in the name of the great ancient Greek sculptor Phidias (born early 5th century BC), who often used the golden ratio in his works. If ≈ 0.62, then 1's ≈ 0.38, so the parts of the “golden ratio” make up approximately 62% and 38% of the entire segment.

2.2. History of the Golden Ratio

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras , ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. At the beginning of the 20th century, in Saqqara (Egypt), archaeologists opened a crypt in which the remains of an ancient Egyptian architect named Hesi-Ra were buried. In literature this name often appears as Hesira. It is assumed that Hesi-Ra was a contemporary of Imhotep, who lived during the reign of Pharaoh Djoser (27th century BC), since the pharaoh's seals were discovered in the crypt. Wooden panels covered with magnificent carvings were recovered from the crypt, along with various material values.(Fig.5)

In the ancient literature that has come down to us, the golden division was first mentioned in the Elements. Euclid . In the 2nd book of the Elements, a geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (2nd century BC), Pappus (3rd century AD), and others. In medieval Europe, they became acquainted with the golden division through Arabic translations of Euclid’s Elements. Translator J.Campano from Navarre (III century) made comments on the translation. The secrets of the golden division were jealously guarded and kept in strict secrecy. They were known only to initiates. During the Renaissance, interest in the golden division increased among scientists and artists due to its use in both geometry and art, especially in architecture.Leonardo da Vinci, an artist and scientist, saw that Italian artists have a lot of empirical experience, but little knowledge. He conceived and began to write a book on geometry, but at that time a monk’s book appeared Luca Pacioli , and Leonardo abandoned his idea. Luca Pacioli was a student of the artistPiero del la Francesca, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry. In 1509 Luca Pacioli's book "The Divine Proportion" 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.

2.4. The golden ratio and related relationships.

Let's calculate the inverse of the number φ:

1:()== ∙=

The reciprocal is usually written asФ = =1.6180339..≈ 1.618.

Number j is the only positive number that turns into its inverse when adding one.

Let us pay attention to the amazing invariance of the golden ratio:

Ф 2 =() 2 ==== and Ф+1=

Such significant transformations as raising to a power could not destroy the essence of this unique proportion, its “soul”.

2.4.1. "Golden" rectangle.

A rectangle whose sides are in the golden ratio, i.e.

the ratio of width to length gives the number φ, calledgolden rectangular

no one

The objects around us provide examples of the golden rectangle:

spoons of many books, magazines, notebooks, postcards, paintings, table covers,

TV screens, etc. close in size to the golden rectangle.

Properties of the “Golden” rectangle.

If from a golden rectangle with sides a and b (where, a>b ) cut a square with side V , then you get a rectangle with sides in and a-c , which is also gold. Continuing this process, each time we will get a smaller rectangle, but again golden.
The process described above results in a sequence of so-called rotating squares. If we connect the opposite vertices of these squares with a smooth line, we get a curve called the “golden spiral”. The point from which it begins to unwind is called a pole. (Fig.7 and Fig.8)

2.4.2. "Golden Triangle".

These are isosceles triangles in which the ratio of the length of the side to the length of the base is equal to F. One of the remarkable properties of such a triangle is that the lengths of the bisectors of the angles at its base are equal to the length of the base itself. (Fig.9)

2.4.3. Pentagram.

A wonderful example of the “golden ratio” is a regular pentagon - convex and star-shaped: (Fig. 10 and Fig. 11)

We connect the corners of the pentagon through one another with diagonals 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 represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio. The star-shaped pentagon is called a pentagram (from the word “pente” - five).

Regular polygons attracted the attention of ancient Greek scientists long before Archimedes. The Pythagoreans chose a five-pointed star as a talisman; it was considered a symbol of health and served as an identification mark.

4.2. The golden ratio and image perception.

The ability of the human visual analyzer to identify objects constructed using the golden ratio algorithm as beautiful, attractive and harmonious has been known for a long time. The golden ratio gives the feeling of the most perfect 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 F, but in the structure of objects, as well as in the sequence of events, he subconsciously finds elements of the golden proportion.

1. The participants in the study were my classmates, who were asked to select and copy rectangles of various proportions. (Fig.12)

From a set of rectangles, they were asked to choose those that the subjects considered the most beautiful in shape. The majority of respondents (23%) pointed to a figure whose sides are in a ratio of 21:34. The neighboring figures (1:2 and 2:3) were also rated highly, respectively 15 percent for the top figure and 17 percent for the bottom, figure 13:23 - 15%. All other rectangles received no more than 10 percent of the votes each. This test is not only a purely statistical experiment, it reflects a pattern that actually exists in nature. (Fig.13 and Fig.14)

2. When drawing your own pictures, proportions close to the golden ratio (3:5), as well as in the ratio 1:2 and 3:4, prevail.

5.Golden ratio in painting.

Back 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; they divide the image size horizontally and vertically in the golden ratio, i.e. they are located at a distance of approximately 3/8 and 5/8 from the corresponding edges of the plane. (Fig.15)

This discovery was called the “golden ratio” of the painting by artists of that time. Therefore, in order to draw attention to the main element of the photograph, the painting must combine this element with one of the visual centers.

Below are various options for grids created according to the Golden Ratio rule for various compositional options.

Basic meshes look like in Fig. 16.

The masters of Ancient Greece, who knew how to consciously use the golden proportion, which, in essence, 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. The entire ancient culture passed under the sign of the golden proportion. They knew this proportion in Ancient Egypt. I will show this using the example of such painters as: Raphael, Leonardo da Vinci, Shishkin.

LEONARDO da VINCI (1452 – 1519)

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. His personality 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 wrote from right to left in illegible handwriting and with his left hand. This is the most famous example of mirror writing in existence.Portrait of Monna Lisa (La Gioconda) Fig. 17For many years, it has attracted the attention of researchers who discovered that the composition of the design is based on golden triangles, which are parts of a regular star-shaped pentagon.

“The Last Supper” (Fig. 18)

- Leonardo's most mature and complete work. In this painting, the master avoids everything that could obscure the main course of the action he depicts; he achieves a rare convincingness of the compositional solution. In the center he places the figure of Christ, highlighting it with the opening of the door. He deliberately moves the apostles away from Christ in order to further emphasize his place in the composition. Finally, for the same purpose, he forces all perspective lines to converge at a point directly above the head of Christ. Leonardo divides his students into four symmetrical groups, full of life and movement. He makes the table small, and the refectory - strict and simple. This gives him the opportunity to focus the viewer’s attention on figures with enormous plastic power. All these techniques reflect the deep purposefulness of the creative plan, in which everything is weighed and taken into account..."

RAPHAEL (1483 – 1520)

In contrast to the golden ratio, the feeling of dynamics and excitement is manifested, perhaps, most strongly in another simple geometric figure - a spiral. The multi-figure composition, executed in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, is precisely distinguished by the dynamism and drama of the plot. Raphael never brought his plan to completion, however, his sketch was engraved by the unknown Italian graphic artist Marcantinio Raimondi, who, based on this sketch, created the engraving “Massacre of the Innocents”.

In Raphael's preparatory sketch, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman holding him close, the warrior with his sword raised, and then along the figures of the same group on the right side sketch. If you naturally connect these pieces with a curved dotted line, then with very great accuracy you get... a golden spiral!

"Massacre of the Innocents" Raphael. (Fig.19)

Conclusion .

The importance of the golden ratio in modern science is very great. This proportion is used in almost all areas of knowledge. Many famous scientists and geniuses tried to study it: Aristotle, Herodotus, Leonardo Da Vinci, but no one completely succeeded. This paper discusses ways to find the “Golden Ratio” and presents examples taken from the fields of science and art that reflect this proportion: architecture, music, painting, sculpture, nature. In my work I wanted to demonstrate the beauty and breadth of the Golden Ratio in real life. I realized that the world of mathematics had revealed one of the amazing secrets to me, which I tried to reveal in my work; in addition, these questions go beyond the scope of the school course, they contribute to the improvement and development of the most important mathematical skills.I am going to continue my research further and look for even more interesting and surprising facts. But when studying the law of the golden ratio, it is important to remember that it is not mandatory in everything that we encounter in nature, but symbolizes the ideal of construction. Small inconsistencies with the ideal are what make our world so diverse.

Bibliography:

Encyclopedia for children. - “Avanta+”. - Mathematics. - 685 pages. - Moscow. - 1998.
Yu.V. Keldysh. – Musical encyclopedia. – Publishing house “Soviet Encyclopedia”. - Moscow. – 1974 – page 958.
Kovalev F.V. Golden ratio in painting. K.: Vyshcha School, 1989.
http://www.sotvoreniye.ru/articles/golden_ratio2.php
http://sapr.mgsu.ru/biblio/arxitekt/zolsech/zolsech2.htm
http://imagemaster.ru/articles/gold_sec.html
Vasyutinsky N. Golden proportion, Moscow “Young Guard”, 1990.
Newspaper "Mathematics", supplement to the teaching aid "First of September". - M.: Publishing House "First of September", 2007.

Depman I.Ya. Behind the pages of a mathematics textbook, - M. Prosveshchenie, 1989 Rice. 2

Fig.4

Rice. 6. Antique golden ratio compass

Figure 5. Hesi-Ra panels.

Fig.7 Fig.8

Fig.9 Fig.10

Fig.11

Fig.12

Fig.13

Fig.14

Fig.15

(Fig. 16)

Fig.17

Fig.18

Now let’s take a look at the visibly geometric “Birch Grove” by Arkhip Kuindzhi, painted in 1879 after the artist’s acquaintance with the Impressionists in Paris. This work is the forerunner of constructivism of the 20th century (let us remember Deineka).

Accent points p lie not only on two of the four golden intersections (the butts of the two central birches), but also on √2 (the yellow grid is the lower horizontal border of the shadow and butt of four more trees, and vertically the trunk of one of the birches) and two horizontals √5 ( highlighted in red - horizontally the far edge of the clearing and the height of distant trees, vertically the border of the crowns of the left group of trees).

It is unlikely that the artist specifically calculated these relationships (he simply does not need it, because the algorithm of his work is from inspiration to harmony, and not from analysis to imitation). But they are harmonious, and the formula of this harmony is not in the golden section, but in the synthesis of the golden section, √5 and √2 and other harmonic constants. In any case, Kuindzhi’s synthesis of transitions of color and geometry is built precisely at the intersection of these irrational quantities.

But perhaps this pattern applies only to the creations of European culture? However, let us turn to Japanese painting.

Now let’s compare it with an ancient Russian miniature:

But here is “The Appearance of Christ to the People” by Alexander Ivanov. The clear effect of the Messiah approaching people arises due to the fact that he has already passed the point of the golden section (the cross of orange lines) and is now entering the point that we will call the point of the silver section (this is a segment divided by the number π, or a segment minus segment divided by the number π).

The figure of A. S. Pushkin in the painting by N. N. Ge “Alexander Sergeevich Pushkin in the village of Mikhailovskoye” was placed by the artist on the line of the golden ratio on the left side of the canvas (Fig. 8). But all other width values are not at all random: the width of the stove is equal to 24 parts of the width of the picture, the shelf is 14 parts, the distance from the shelf to the stove is also 14 parts, etc.

Proportions of the golden division in the linear construction of N. N. Ge’s painting “Alexander Sergeevich Pushkin in the village of Mikhailovskoye”

The golden ratio in I. I. Shishkin’s painting “Pine Grove”
In this famous painting by I. I. Shishkin, the motifs of the golden ratio are clearly visible. A brightly sunlit 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 sunlit hillock. It divides the right side of the picture horizontally according to the golden ratio. To the left of the main pine tree there are many pines - if you wish, you can successfully continue dividing the picture according to the golden ratio further.

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

The Golden Ratio in Leonardo da Vinci's painting "La Gioconda"

The portrait of Mona Lisa is attractive because the composition of the drawing is built on “golden triangles” (more precisely, on triangles that are pieces of a regular star-shaped pentagon).

Golden spiral in Raphael's painting "Massacre of the Innocents"

In contrast to the golden ratio, the feeling of dynamics and excitement is manifested, perhaps, most strongly in another simple geometric figure - a spiral. The multi-figure composition, executed in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, is precisely distinguished by the dynamism and drama of the plot. Raphael never brought his plan to completion, however, his sketch was engraved by the unknown Italian graphic artist Marcantinio Raimondi, who, based on this sketch, created the engraving “Massacre of the Innocents”.
In Raphael's preparatory sketch, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman holding him close, the warrior with his sword raised, and then along the figures of the same group on the right side sketch. If you naturally connect these pieces with a curved dotted line, then with very great accuracy you get... a golden spiral! This can be checked by measuring the ratio of the lengths of the segments cut by a spiral on straight lines passing through the beginning of the curve.
We do not know whether Raphael actually drew the golden spiral when creating the composition “Massacre of the Innocents” or only “felt” it. However, we can say with confidence that the engraver Raimondi saw this spiral. This is evidenced by the new elements of the composition he added, emphasizing the reversal of the spiral in those places where it is only indicated by a dotted line. These elements can be seen in Raimondi's final engraving: the arch of the bridge extending from the woman's head is on the left side of the composition and the reclining body of the child is in its center. Raphael completed the initial composition at the dawn of his creative powers, when he created his most perfect creations. The head of the school of romanticism, the French artist Eugene Delacroix (1798 - 1863), wrote about him: “In the combination of all the wonders of grace and simplicity, knowledge and instinct in composition, Raphael achieved such perfection in which no one has ever compared with him. In the simplest, like in the most majestic compositions everywhere, his mind brings, together with life and movement, perfect order into enchanting harmony.” In the composition “Massacre of the Innocents” these features of the great master are very clearly manifested. It perfectly combines dynamism and harmony. This combination is facilitated by the choice of the golden spiral as the compositional basis of Raphael’s drawing: dynamism is given to it by the vortex character of the spiral, and harmony is given by the choice of the golden ratio as the proportion that determines the deployment of the spiral.

MINISTRY OF EDUCATION AND SCIENCE OF THE RF

Federal State Budgetary Educational Institution

Higher professional education

"Far Eastern State Humanitarian University"

FACULTY OF FINE ARTS AND DESIGN

COURSE WORK

"The Golden Ratio in Art"

2nd year students

P. A. Sorokina

Scientific director

FROM. Titova

Art. teacher

Khabarovsk 2012

Introduction

History of the development of the golden ratio

Antiquity

Middle Ages

Renaissance

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 down on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive. Sitting on the bench, you produced the “golden ratio”.

The goals of the work are, first of all, to study the history of the golden ratio, to study the use of “divine proportion” in art and to get acquainted with the modern use of the golden ratio.

The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. The infinite series after the decimal point is 1.6180339887.

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

So what is the “golden ratio”? What is this perfect, divine combination? Maybe this is the law of beauty? Or is he still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The “golden ratio” is both, and the other, and the third. Only not separately, but simultaneously... And this is his true mystery, his great secret.

Sometimes professional artists, having learned to draw and paint from life, due to their own weak fundamental training, believe that knowledge of the laws of beauty (in particular the law of the golden ratio) interferes with free intuitive creativity. This is a big and deep misconception of many artists who never became true creators. The masters of Ancient Greece, who knew how to consciously use the golden proportion, which, in essence, 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. The entire ancient culture passed under the sign of the golden proportion. They knew this proportion in Ancient Egypt.

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

Now we can say with confidence that the golden proportion is the basis of shape-formation, the use of which provides a variety of compositional forms in all types of art and provides the basis for the creation of a scientific theory of composition and a unified theory of plastic arts.

The work examines the first mentions of the golden ratio, the history of its development, its use in art and the modern vision of the golden ratio.

History of the development of the golden ratio

Antiquity

The history of the “Golden Section” is the history of human knowledge of the world. The concept of the “Golden Section” has passed through all stages of cognition in its development. The first stage of knowledge is the discovery of the “golden ratio” 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 Cheops pyramid, (1) temples, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. At the beginning of the 20th century, in Saqqara (Egypt), archaeologists opened a crypt in which the remains of an ancient Egyptian architect named Hesi-Ra were buried. In literature this name often appears as Hesira. It is assumed that Hesi-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 panels covered with magnificent carvings, executed by the hand of an impeccable craftsman, were recovered. In total, 11 boards were placed in the crypt; Of these, only five have survived, and the remaining panels are completely destroyed. For a long time, the purpose of the panels from the burial of Hesi-Ra was unclear. (2) At first, Egyptologists mistook these panels for false doors. However, starting from the 60s of the 20th century, the situation with the 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 are related to each other as, that is, as a small side and a diagonal with an aspect ratio of 1:2 ("two-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 using the method of proportions, we have sufficient grounds to assert that the Hesi-Ra panels are a system of rules of harmony, encoded in the language of geometry...

So, we have in our hands concrete material evidence that “in clear text” tells about the highest level of abstract thinking of intellectuals from Ancient Egypt. The author, who cut the boards, demonstrated the GS (golden ratio) rule in its widest range of variations with amazing precision, jewelry grace and virtuoso ingenuity. As a result, the GOLDEN SYMPHONY was born, presented by an ensemble of highly artistic works, not only testifying to the genius of their creator, but also convincingly confirming that the author was initiated into the magical mysteries of harmony. This 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.

The entire ancient Greek culture developed under the sign of the golden proportion. The idea of harmony based on the golden ratio could not help but touch Greek art. Nature, taken in a broad sense, included the creative world of man, art, music, where the same laws of rhythm and harmony apply. To take the material and eliminate everything superfluous - this 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 ratio” first became some kind of aesthetic canon.

The basis of art is the theory of proportions. And, of course, issues of proportionality could not pass by Pythagoras. Of the Greek philosophers, Pythagoras, perhaps for the first time, tries to mathematically analyze the essence of harmonic proportions. Pythagoras knew that octave intervals can be expressed by numbers that correspond to the corresponding vibrations of the string, and these numerical relationships were taken 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 ratio. Pythagoras attached special, outstanding significance to the latter, making the pentagram or star-shaped pentagon the distinctive sign 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 Aphrodite of Cnidus by Praxiteles, the most perfect Greek theater in Epidaurus and the oldest surviving theater of Dionysus in Athens - all these are vivid examples of sculpting and creativity, full of deep harmony based on the golden ratio.

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

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

When building temples, man was taken as the basis as “the 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 laid down on the ruler, and a scale was applied to it, rigidly connected with the sequence of six terms of the Fibonacci series: 1, 2, 3, 5, 8, 13 (their sum is 32 = 25) . By adding or subtracting these standard segments, the required proportions of the structure were achieved. A sixfold increase in all sizes set aside on the ruler preserved the harmonic proportion. Temples, theaters or stadiums were built in accordance with this scale.

Plato also knew about the golden division. His dialogue “Timaeus” is devoted to the mathematical and aesthetic views of the Pythagorean school and, in particular, to the issues of the golden division. The façade of the ancient Greek temple of the Parthenon features golden proportions. During its excavations, compasses were discovered that were used by architects and sculptors of the ancient world. The Pompeii compass (museum in Naples) also contains the proportions of the golden division.

Thus, antiquity was completely subordinated to the proportion of the golden section. In architecture, sculpture, painting and music, a proportional division was traced. 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 an admirer 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 them is "Liber abaci". This work was published during Fibonacci's lifetime in two editions in 1202 and 1228. The book consists of 15 sections. Note that Fibonacci conceived his work as a manual for merchants, but in its significance it went far beyond the limits of trading practice and essentially 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 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 number sequence 1, 1, 2, 3, 5, 8, 13, ..., later called the “Fibonacci series”.

Fibonacci was almost two centuries ahead of the Western European mathematicians of his time. Like Pythagoras, who received his "scientific education" from Egyptian and Babylonian priests and then contributed to the transfer of this knowledge to Greek science, Fibonacci received his mathematical education in Arab educational institutions and much of the knowledge acquired there, in particular the Arab-Hindu decimal system , he tried to “introduce” it into Western European science. And like Pythagoras, Fibonacci’s historical role 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 foundations for the further development of Western European mathematics.

So, the Middle Ages learned about the golden ratio in a mathematical form (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 during the Renaissance.

Renaissance

The Renaissance in the cultural history 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 a humanistic worldview and an appeal to the ancient cultural heritage, a kind of “revival” of ancient culture. The Renaissance was marked by major scientific changes in the field of natural science. A specific feature of science of this era was its close connection with art, and this unification was sometimes expressed in the creativity of one person. The most striking example of such a multifaceted personality is Leonardo da Vinci - artist, scientist, engineer.

Along with other achievements of ancient culture, scientists and artists of the Renaissance with great enthusiasm accepted 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 introduced the name “golden ratio” into widespread use, which immediately became the aesthetic canon of the Renaissance.

The idea of harmony was among those conceptual constructs of ancient culture to which the church was very interested. According to Christian doctrine, Velenna was a creation of God and unquestioningly obeyed his will. And the Christian God, when creating 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 the mathematical plan by which God created the Universe.

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

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 \u200b\u200bthe numerical harmony of the Universe that became one of the most important reasons for the huge surge of culture during 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.”

This 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, strict proportionality and harmony of the plot, subject to precise proportions, are manifested. The law of harmony, the law of number, with which the beauty of a work was associated, was most clearly revealed in the works of art and scientific and methodological research of Leonardo, Dürer, and Alberti.

During the Italian Renaissance, research continued in the field of the theory of proportionality in 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 the 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 introduced the 12-note tempered system into music. In a number of his works - "Treatise on Universal 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 ratio” 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 by 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 connections between all known types of matter.

The thesis about the unity of man and nature, consistently carried out in antiquity, was revived again at the end of the 19th century and mainly in the first half of the 20th century in a number of conceptual structures, 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, etc.). The most important direction of research is the search for invariants of being - special stability, found in entire classes of externally 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 knowledge of the objective laws of harmony, the need for an accurate calculation of harmonic relations. Against this background, interest in harmonic proportion, the golden ratio, and 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 works devoted to the problem of proportionality.

Based on the position that proportionality is the relationship 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 a whole into unequal parts is proportional when the relation of the parts of the whole to each other is the same as their relation to the whole, that is, the relation that gives the golden ratio.”

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

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

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

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 corresponds to 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. The Greeks explained the harmony of other consonances in the same way.

Based on the principles that only those combinations of tones are beautiful, the intervals of which are in proportion 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 pleasant consonances for the ear have such intervals that the ratio of frequencies included in the chord is closest to the golden proportion. For example, the connection of the minor third with the octave of the fundamental sound corresponds to a frequency ratio of 3:5, the connection of the major third with the octave of the fundamental sound - 5:8 (3, 5, 8 are Fibonacci numbers!).

Zeising further concludes that since these two combinations of sounds between two-digit ones are the most pleasant to the ear, this apparently explains the fact that only they end musical periods. This also explains why improvised folk chants and simple music of two horns (or English horns) move in sixths and their complements - thirds.

Zeising draws attention to another curious fact. As you know, the major (male) and minor (female) modes are built on the basis of the major and minor triad. A major triad built on 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 “majority”.

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

These conclusions of Zeising with his interpretation of the reasons for the consonance of intervals are confirmed by research in acoustics.

Moving on to the meaning of the law of proportionality in architecture, Zeising points out that architecture in the field of art occupies the same position as the organic world in nature, spiritualizing inert matter on the basis of world laws. At the same time, orderliness, symmetry and proportionality are its indispensable attributes, from which it follows 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 answers to those “eternal” questions that were posed by the ancient Greeks. The conviction has matured that the world is dominated by the “universal law” of number and rhythm, expressing its structural and functional aspects. In this regard, 19th-century science revived interest in the golden ratio.

The meaning of the golden ratio in art

So, before defining the golden ratio, you need to become familiar with the concept of proportion. In mathematics, proportion (lat. proportio) is the equality between two ratios of four quantities: a: b = c: d. Next, as an example, let's look at a straight line segment. Segment AB can be divided into two equal parts (/). This will be a ratio of equal quantities - 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 straight line AB can be divided according to the golden ratio, when AB: AC, like AC: BC. This is the golden division or division in extreme and average ratio. From the above it follows that the golden ratio is such a proportional harmonic division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole, i.e. a: b = b: c or c\b = b: a. The definition - division in extreme and mean ratio - becomes more understandable if we express it geometrically, namely a: b as b: c.

We derive the golden ratio. (6) 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 laid, ending with point D. The segment AD is transferred to the straight line AB. The resulting point f divides the segment AB in the golden proportion. Arithmetically, segments of the golden proportion are expressed as an infinite irrational fraction. AE = 0.618..., if AB is taken as one, ff = 0.382.... In practice, rounding is used: 0.62 and 0.38. If segment AB is taken to be 100 parts, then the larger part of the segment is 62 parts, and the smaller part is 38 parts.

Spirals are very common in nature. The concept of the golden ratio will be incomplete without talking about the spiral.(7)

The shape of the spirally curled shell attracted the attention of the ancient Greek scientist Archimedes. He studied it and came up with an equation for the spiral. The spiral drawn using this equation is called the Archimedes spiral. The increase in her step is always uniform.

So where can we find the golden ratio in art?

Painting

Very often in the same work of painting there is a combination of symmetrical division into equal parts vertically and division into unequal parts along the golden ratio horizontally. Let's look at examples.

In the famous portrait of Monna Lisa (“La Gioconda”) (8), which was completed by Leonardo da Vinci in 1503, an important element of the composition is the cosmically vast landscape, disappearing in a cold haze. The painting by the brilliant artist attracted the attention of researchers, who discovered that the compositional structure of the painting is based on two “golden” triangles, which are parts of a “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 over the square. As a result of this construction, the entire 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 obtained a semicircle at the top of the picture. Below, the arc crossed the axis of symmetry and indicated the size of another golden ratio rectangle at the bottom of the picture. Then a new arc is described with a radius equal to the side of the square, which gave points on the vertical sides of the picture. These points helped to construct an equilateral triangle, which was the framework for constructing the entire group of figures. All proportions in the painting were derived from the height of the painting. They form a series of relationships of the golden ratio 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 artistic works of Raphael, Michelangelo and other Italian artists.

The multi-figure composition “Massacre of the Innocents” (10), executed in 1509-1510 by Raphael, is distinguished by its dynamism and dramatic plot. In Raphael's preparatory sketch, a smooth line is drawn that covers the entire picture. The line begins 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 the child, the woman holding him close, the warrior with the 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 with a curved dotted line, then with very high accuracy you get a “golden” spiral!

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

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

In the icon of the late 15th century. “The Miracle of Flora and Laurel” realized the triple ratio of the golden ratio. First, the master divided the height of the icon into two equal parts. The top one was dedicated to 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, the ratio of the three values of the golden ratio was obtained: a: b, as b: c. In numbers it will look like this: 100, 62, 38, and halved - 50, 31, 19.

Much has been written about the symmetry of Andrei Rublev’s “Trinity” (12). But no one paid attention to the fact that the principle of golden proportions was implemented here along the horizontal lines. The height of the middle angel relates to the height of the side angels, just as their height relates to the height of the entire icon. The line of the golden ratio intersects the axis of symmetry in the middle of the table and the bowl with the sacrificial body. This is a compositional castle of an icon. The figure also shows smaller values of the golden ratio series. Along with the smoothness of lines and color, the proportions of an icon play a significant role in creating the overall impression that the viewer experiences when viewing it.

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

Music

Music is an art form that reflects reality and affects a person through meaningful and specially organized sound sequences consisting of tones. While retaining some semblance of the sounds of real life, musical sounds are fundamentally different from the latter in their strict pitch and temporal (rhythmic) organization (“musical harmony”). Since 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 vibration frequencies of two sounds is described by small numbers, then they give a harmonic sound;

2) to obtain a harmonic triad, you need to add a third sound to a chord of two consonant sounds, the vibration frequency of which is in a harmonic proportional relationship with the first two. The importance of Pythagoras's work on the scientific explanation of the foundations of musical harmony cannot be overestimated. This was the first scientifically based theory of musical harmony.

Any musical work has a temporal extension and is divided by certain milestones (“aesthetic milestones”) into separate parts that attract attention and facilitate the perception of the whole. These milestones can be the dynamic and intonation climaxes of a musical work. Are there any patterns 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 long article, “Chopin's Etudes in the Light of the Golden Ratio” (1925), he shows that the individual time intervals of a musical work, connected by a “climax 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 in the length of the whole that they divide time extensions into separate parts that are in the relationships of the “golden section”. As observations show, the timing of such aesthetic “milestones” to points of division of the general or partial extent in " The golden "respect is often carried out with great precision, which is all the more surprising given that in the absence of poets and music authors of any knowledge about such things, this is all solely a consequence of the internal sense of harmony."

An analysis of a huge number of musical works allowed Sabaneev to conclude that the organization of a musical work is structured in such a way that its cardinal parts, separated by milestones, form rows of the golden ratio. This organization of the work corresponds to the most economical perception of the mass of relations and therefore gives the impression of the highest “harmony” of form. According to Sabaneev, the quantity and frequency of use of the golden ratio in a musical composition depends on the “rank of the composer.” The highest percentage of coincidences is noted among genius composers, that is, “the intuition of form and harmony, as one would expect, is strongest among geniuses of the first class.”

According to Sabaneev’s observations, in the musical works of various composers there is usually not just one golden ratio associated with an “aesthetic event” occurring near it, but a whole series of similar sections. Each such section reflects its own musical event, a qualitative leap in the development of the 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 the golden section is present are by Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin (92%), Schubert (91%).

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

“We must recognize the obvious features of “natural creativity” in those cases when, in the highly spiritualized creations of brilliant authors, generated by the powerful striving of the spirit for truth and beauty, we completely unexpectedly discover some kind of mysterious pattern of numerical relations that is not amenable to direct consciousness.”

E. Rosenov believed that the golden ratio should play an outstanding role in music as a means of bringing homogeneous phenomena into conformity 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 anticipation, combined with culminating points (force, mass, movement of sounds) and with various kinds of outstanding, from the author’s point of view, effects;

3) direct the listener’s attention to those thoughts of the musical work to which the author attaches the most important importance, which he wishes 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 product 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 appears very often. The author pays special attention to Chopin's fantasy, which was created impromptu and was not subject to any editing, which means there was no conscious application of the law of the golden ratio, which is present in this musical work down to small musical formations.

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

Architecture

The principle of the golden ratio can also be observed in architecture. For example, the Church of the Intercession on the Nerl (1165) (13) is considered the most perfect creation of Vladimir architects.

Getting to know the Nerl Temple creates an image of harmony and architectural beauty. And the question involuntarily arises: what “secrets” did the Russian architects who worked eight centuries ago possess?

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 exhibits a proportion, which is the ratio of the larger side to the diagonal of a “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 a “two-adjacent” square and its derivative - the golden proportion. The presence of these proportions determined the beauty of the temple. “The amazing beauty and harmony of the architecture of the Church of the Intercession of the Virgin Mary on the Nerl,” writes architectural theorist K.N. Afanasyev, “is formed by a chain of interconnected relationships 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, freeing Russia forever from the Tatar invasion. To glorify the “capture of Kazan,” which went down in Russian history along with the Battle of Kulikovo, Tsar Ivan the Terrible decided to found the Cathedral of the Intercession on Red Square in Moscow; Later this temple was popularly nicknamed "Basily 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 cathedral buildings is characterized by a harmonious combination of symmetrical and asymmetrical proportions. The temple, symmetrical in its core, contains many geometric "irregularities". Thus, the central volume of the tent is shifted 3 m to the west from the geometric center of the entire composition. However, inaccuracy makes the composition more picturesque, “live” and it wins overall. The architectural decoration of the cathedral is characterized by an upward growth of decorative forms; the forms grow from one another, stretch upward, sometimes rising in large elements, sometimes forming groups consisting of smaller decorative parts.

The proportions of the cathedral were also built in accordance with this compositional idea. Researchers discovered a proportion in it based on the golden ratio series:

where j = 0.618. This division contains the main architectural idea of creating the cathedral, which is common to all domes, uniting them into one proportionate 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 determining 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 mid-17th century, it was forbidden to build single-domed churches as not corresponding to the five-domed rite 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) had 13 domes, and the Transfiguration Church in Kizhi, carved out of wood 2.5 centuries ago, is crowned with 21 domes. Is such an increase in the number of domes “according to Fibonacci” (1, 2, 3, 5, 8, 13, 21) accidental, reflecting the natural law of growth - from simple to complex?

The expression “architecture is frozen music” has become popular. It is not the result of a strict scientific analysis; it is most likely the result of an imaginative, intuitive feeling of some connection between a harmonic architectural form and musical harmony. A musical melody is based on the alternation of sounds of different heights and durations; it is based on the temporal ordering of sounds. The basis 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 evaluate the dimensions of the spatial structure of a geometric figure, we must follow this figure with our gaze from beginning to end, and the greater, for example, its length, the longer the perception will be. Obviously, here lies the organic connection between spatial and temporal perception of objects by humans.

Literature

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

N Vasyutinsky states:

“The culmination of the chapter is Eugene’s declaration of love for Tatyana - the line “To turn pale and fade away... this is bliss!” This line divides the entire eighth chapter into two parts - in the first there are 477 lines, and in the second - 295 lines. Their ratio is 1.617 "! The finest correspondence to the value of the golden proportion! This is a great miracle of harmony, perfected by the genius of Pushkin!"

Much in the structure of poetic works makes this art form similar to music. A clear rhythm, a natural alternation of stressed and unstressed syllables, an ordered meter of poems, and their emotional richness make poetry the 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 proportion 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, which represents an independent whole, which falls into two equal parts. The first of them describes the anticipation of the battle with increasing tension, the second describes the battle itself with a gradual decrease in tension towards the end of the poem. The boundary between these parts is the culmination point of the work and falls exactly at the point of division by the golden section.

The main part of the poem consists of 13 seven-line lines, that is, 91 lines. Having divided it by the golden ratio (91:1.618 = 56.238), we are convinced 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 “culmination point of excited anticipation”, completing the first part of the poem (anticipation of the battle) and opening its second part (description of the battle).

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

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 car design.

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 allow you to easily calculate the golden ratio. You just need to specify 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 like this. The aspect ratio of the iPod Shuffle is 1.59, the iPod Classic is 1.67, and the iPhone4 is 1.7 - sales volume in the first 4 days of trading exceeded 1 million 700 thousand units. These sales results do not surprise fans of Apple products; naturally, the device is evaluated based on other characteristics. But, it seems to me, it was no coincidence that Jonathan Ive settled on 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 wrote notes and made sketches in Moleskine books. This is the real history of humanity in books with proportions 1.57

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

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

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

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

2) Using the golden ratio 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 desired illustration or text there - and it will come to the attention of potential clients.

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

The It's Numbered site applies the principle of the golden ratio not to the entire interface, but only to the content + image combination. (16)
And the MmDesign website uses the golden ratio to display the main visual on the main page.

Using the golden ratio does not guarantee that the website design will be good; there are a number of other, equally important factors that contribute to the development of the correct design. However, the golden ratio can help give balance and completeness to the work, as well as ease of perception for users, which is often not very easy to achieve.

Using the golden ratio rule helps to find balance and 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, and 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 size and shape. Every natural object is something unified, integral. It is not difficult to notice that nature always creates something whole: a person, a tree, a fish, a horse, a dog, etc. Nothing can be taken away or subtracted from this whole without violating the integrity. Nothing can be added. It will be unnecessary and will also violate integrity and harmony. For example, six fingers on a person’s hand, three horns on a bull.

In the 20th century, a huge number of art historical works were carried out, showing the widespread manifestation and use of the “golden section” in all spheres of art: in music (Sabaneev “Etudes of Chopin in the light of the Golden Section), in poetry (academician Tsereteli “The Golden Section in the poem by Shota Rustaveli” "The Knight in the Skin of a Tiger"), cinema (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. Greenbaum on identifying “Fibonacci” patterns in the poetry of A.S. Pushkin and the Russian philosopher A.V. Voloshinov on the study of the mathematical principles of formation in music, architecture, painting and literature.

The whole always consists of parts. Parts of different sizes are in a certain relationship to each other and to the whole. These are the proportions. From a mathematical point of view, we note the repetition of measurable equal and unequal quantities, related to each other as quantities of the golden proportion. These are two types of proportional relationships. All other quantities, if they arose as a result of a violation of shape formation for any reason, do not form proportions. Proportional relationships lead to symmetry, rhythm, harmony and beauty. Disproportional relationships lead to disruption of order, disruption of symmetry and rhythm, which is perceived by a person as unsightly and even ugly.

So, the natural law of divine proportion, manifested in the highest forms of works of art, is revealed in a new, rhythmodynamic form of the aesthetic law. The law of the "golden ratio", known since the times of Ancient Egypt, is one of the most amazing mathematical laws; it was formulated by the great Leonardo and increasingly appears in the rapidly growing stream of natural science and humanities research.

This law is not a forced law, the only or exclusive one, determining the artistic impression; nevertheless, it remains a law directly related to the aesthetic, artistic impact, and has a direct impact on the impression of integrity and beauty. Sensitive to beauty, Pushkin, by artistic instinct alone, 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 sizes of the parts in relation to the whole and, thirdly, he emphasized the culminating points of increasing tension in anticipation, compositionally placing the main thoughts of the narrative in places so noticeable for direct sensory perception.

References

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

Formula golden ratio in pictures. Golden ratio in nature, man, art

Golden ratio in painting

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