F in Kovalev golden ratio composition. Golden ratio in painting - F

Golden ratio in painting. Kovalev F.V.

Kyiv: 1989 - 1 44 p.

The laws of shape formation in nature and art, visual perception and compositional construction of the image are outlined. The role of the golden ratio is shown. Recommendations are given for the practical application of the golden proportion in creating a holistic harmonious form that most fully expresses the content of a work of art and satisfies the human need for beauty. For art institutes and schools.

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Table of contents
Preface 3
Introduction 4
Chapter One The Golden Ratio and Questions of Composition Theory About the Golden Ratio 8
Golden ratio - harmonic proportion 8
Golden ratio and symmetry 19
History of the golden ratio 26
Natural scientific foundations of composition theory 30
Principles of shape formation in nature 30
Patterns of visual perception 33
Objectification of light impressions 36
Scientific theory of composition 39
Definition of composition 39
Search for the laws of composition 41
What is the scientific theory of composition 42
Human creativity 43
Laws, rules, techniques and means of composition 46
Chapter Two Practical Composition
Composition when working from life 56
Point of view 56
Distance to object. The size of the image on the retina.
Transmitting distance to an object 58
The imaginary picture and the real picture 59
Methods for determining viewing angles when working from nature 63
Techniques for mechanical image acquisition 66
Techniques of compositional constructions 67
Analysis of painting 67
Composition of still life and interior 69
Landscape composition 70
About the portrait. Live performances 73
The artist's place in front of the painting 75
Image integrity 79
Chapter Three: Working on a painting
Golden ratio in the linear construction of a picture 80
Idea, format, rhythm and golden ratio 80
Sketch of the painting. Distance calculations and solving the “inverse problem” 85
The geometric center of the picture and the line of the golden ratio. Harmonization of form 91
The main line of sight in painting 91
Compositional algorithm for linear construction of a picture 94
Golden ratio and composition of tones 94
Light and eye 94
Overall light tone 98
The law of three components and the principle of close relationships 98
Composition of light tones 102
Golden ratio and color composition 106
General color tone of the picture 109
Palette limitation 111
Color systems and models 112
Symmetry of color. Contrast and nuance 113
Color harmony 114
Construction and development of color. Complete compositional algorithm of painting 115
Chapter Four Scientific and intuitive in the artist’s work
Painting size 118
Artistic structure of the painting 118
Beautiful and mysterious 118
Subject - visual image - artistic image 119
Artists and scientists. Scientific and everyday terminology 121
Subject and color 123
Two poles of painting 123
Draw with form, write with color 127
Chapter Five Explicable and Sensual in Painting About Color 128
The language of painting is a special language of art 128
Color is singular, color is multiple. Psychological assessment of color 129
What does color depend on 130
Good tradition 132
Theories of painting - scientific basis 132
A word about the author 134
Painter's Memo 136
In the world of wise thoughts 137
List of used and recommended literature 142

The experience of teaching drawing, painting and composition at art institutes and schools convincingly shows that the theoretical principles of these disciplines, especially composition, are not sufficiently developed. Many concepts and definitions underlying their teaching are interpreted ambiguously. Each artist-teacher relies on his personal experience, on the experience of his teachers and inevitably introduces a share of subjectivity into teaching.
Scientific knowledge in mathematics (geometry and perspective), physics (light and color), human anatomy and physiology (structure and functioning of the eye and brain), psychology (perception of color and its effect on humans), obtained by students in classes in general education subjects, is small are linked or not linked at all with training as an artist.
In the process of preparing an artist, great hopes are placed on the natural talent of students, the degree of mastery and completion of tasks in special subjects, but the role of developing a coherent system of scientific knowledge and a materialistic worldview is underestimated. Therefore, when leaving the walls of an educational institution, a young artist does not receive a solid foundation in the system of concepts of a professional and ideological nature, which negatively affects his creative work.

About the golden ratio.

The debate about whether science should or should not intrude into the reserved areas of art has been going on for a long time. And this dispute is clearly scholastic in nature. In all eras of prosperity, art has entered into an alliance with science. Artist-thinkers, theorists and teachers who reflected on the problems of educating young people always came to the conclusion that without science art cannot develop and prosper. The artist and teacher N.P. 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 need and can know about it, it is mathematical... You can prove exactly why a picture is good and why it is bad” 1 V. I. Surikov argued that in composition there is some kind of immutable law, when in a picture You can’t remove or add anything, you can’t even add an extra point, this is real mathematics. Famous French architect and architectural theorist of the 19th century. Viollet-le-Duc believed that a form that cannot be explained will never be beautiful. On the doors of the Sikyon School of Drawing in Ancient Greece it was written: “People who do not know geometry are not allowed here.” Artists should not be afraid of mathematics; it is outside and inside us. Behind the apparent simplicity and randomness of the living perception of the surrounding reality lies mathematics. When we listen to music, our brain does algebra. When we look at something, our brains do geometry.

Table of contents
Preface
Introduction
Chapter first
Golden ratio and questions of composition theory
About the golden ratio
Golden ratio - harmonic proportion
Golden ratio and symmetry
History of the golden ratio
Natural scientific foundations of composition theory
Principles of formation in nature
Patterns of visual perception
Objectification of light impressions
Scientific theory of composition
Definition of composition
Search for the laws of composition
What is the scientific theory of composition
Human creativity
Laws, rules, techniques and means of composition
Chapter two
Practical composition
Composition when working from life
Point of view
Distance to object. The size of the image on the seed. Transferring the distance to an object
The imaginary picture and the real picture
Methods for determining viewing angles when working from nature
Techniques for mechanical image acquisition
Techniques of compositional constructions
Analysis of the picture
Composition of still life and interior
Landscape composition
About the portrait. Full-scale productions
The artist's place in front of the painting
Integrity of the image
Chapter Three
Working on a painting
The golden ratio in the linear construction of a picture
Idea, format, rhythm and golden ratio
Sketch of the painting. Distance calculations and solving the “inverse problem”
The geometric center of the picture and the line of the golden ratio. Harmonization of form
The main line of sight in the picture
Compositional algorithm for linear construction of a picture
Golden ratio and composition of tones
Light and eye
General light tone
The law of three components and the principle of close relationships
Composition of light tones
Golden ratio and color composition
General color tone of the painting
Palette limitation
Color systems and models
Symmetry of color. Contrast and nuance
Color harmony
Construction and development of color. Complete compositional algorithm of the painting
Chapter Four
Scientific and intuitive in the artist’s work
Painting size
Artistic structure of the painting
Beautiful and mysterious
Subject - visual image - artistic image
Artists and scientists. Scientific and everyday terminology
Subject and color
Two poles of painting
Draw with form, write with color
Chapter Five
Explicable and sensual in painting
About color
The language of painting is a special language of art
Color is singular, color is multiple. Psychological assessment of color
What does color depend on?
Good tradition
Theories of painting - scientific basis
A word about the author
Painter's Memo
In the world of wise thoughts
List of used and recommended literature.

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Golden ratio in painting - F. V. Kovalev

Preface

I want to shorten the path of long and difficult searches for you.

N. P. Krymov

Scientific knowledge in mathematics (geometry and perspective), physics (light and color), human anatomy and physiology (structure and functioning of the eye and brain), psychology (perception of color and its effect on humans), obtained by students in classes in general education subjects, is small are linked or not linked at all with training as an artist.

In the process of preparing an artist, great hopes are placed on the natural talent of students, the degree of mastery and completion of tasks in special subjects, but the role of developing a coherent system of scientific knowledge and a materialistic worldview is underestimated. Therefore, when leaving an educational institution, a young artist does not receive a solid foundation in the system of concepts of a professional and ideological nature, which negatively affects his creative work.

For many years, the efforts of the author of this textbook were aimed at searching for and justifying scientific, objectively determined concepts of the theory of composition and methods of image construction. Great attention was paid to the need for a natural fusion of knowledge and artistic skills, the artist's instincts and reason. A prominent place was occupied by the consideration of the laws of visual artistic perception arising from the structure and work of the human eye and brain, the associated laws of proportion, symmetry and rhythm, the harmonization of the form of a work of art, the unity of form and content as a manifestation of the unity of the materiality of the world, when thoughts, ideas, feelings and the deeds of great scientists and artists, in the figurative expression of the writer V. A. Chivilikhin 1, are considered as a radiant flowering of matter. When considering the role of proportion and symmetry in the harmonization of form, special attention is paid to the golden ratio.

The basis of this book is a series of lectures “The Golden Ratio in Painting”, given at the Kiev State Art Institute and formulated into a course on the theory of composition.

The author is deeply grateful to the leadership of the Kyiv State Art Institute, vice-rector for scientific work, associate professor Mikhail Aleksandrovich Krivolapov, for the invaluable assistance provided in the preparation of the textbook “The Golden Ratio in Painting.”

1 Chivilikhin V. A. Memory // Roman-newspaper. - 1985. - No. 4. - P. 94.

Introduction

The art education system in the USSR has its own history and traditions, established by the Academy of Arts in the 18th and 19th centuries. Russian fine art, represented by a number of such outstanding masters as A. A. Ivanov, V. I. Surikov, I. E. Repin and others, has won worldwide recognition. But by the end of the 19th century. The academic education system is becoming outdated, acquiring the features of routine, and does not meet the new requirements that life itself places on art. Artists sought to expand the range of themes and subjects of their paintings, to answer burning questions of social life. This required an expansion of the means of fine art, a search for new artistic expression. The academic system of art education could no longer respond to these demands. Methods of teaching drawing, painting, and especially composition needed to be improved. Natural sciences at that time provided an explanation for many theoretical questions: about light and color, the work of the brain and the structure of the eye, color perception, etc. Professors at the Academy of Arts of that time could not use them to improve art education due to their own weak fundamental training. Only Pavel Petrovich Chistyakov, this teacher of teachers, attended lectures on general physiology by the famous scientist I.R. Tarkhanov, so he explained many issues of artistic practice based on the anatomy of the eye and the physiology of vision.

Fine art of the late 19th - early 20th centuries, which was experiencing a serious crisis, could not solve the issues of improving art education. This state of affairs contributed to the emergence of all kinds of formalistic theories and trends, even the absurd. All this affected the art school. The fundamental principle set forth by the outstanding artist-teacher P.P. Chistyakov was forgotten, which stated that teaching should not proceed according to the arbitrariness of each artist, but according to the laws that lie in the very nature that surrounds us, and with complete evidence. The teacher has no right to mislead the student with his subjective reasoning, or to subordinate him to his personal artistic preferences. On the contrary, the concept, fueled by idealistic aesthetics, was cultivated that inspiration and feeling were primary for the artist. After visiting an exhibition of paintings by young artists in 1893, I. E. Repin wrote: “This is the kingdom of cheeky mediocrity, anarchists in painting... This legion of daubers especially liked the new rules. There is no need to study; anatomy is nonsense; drawing - outdated calligraphy; achieve, rewrite - self-destruction; compiling, searching - old routine idiocy..." 1.

The formation of Soviet art in the first decade after October took place in conditions of intense struggle between various creative groups, proclaiming their declarations as the only correct ones. This struggle of directions and currents penetrated into the sphere of art pedagogy, having a strong influence on the training of young artists in drawing, painting and composition. There was a theory of “free learning”, which in practice led to a lack of learning. Realist artists waged a persistent and consistent struggle to establish the principles of realism in the school. D. N. Kordovsky, V. N. Meshkov, K. F. Yuon and others made a great contribution to the formation of the Soviet art school. V. A. Favorsky gave lectures on the theory of composition, which were a great success among students and teachers of Vkhutemas. Art schools were organized in the union and autonomous republics. There is an urgent need to develop a unified system of art education.

In 1926, the All-Union Conference on Art Education was held. It took place in Moscow. The report on the state of art education in the country was made by the People's Commissar of Education A.V. Lunacharsky. He noted that music has a coherent teaching system based on scientific theory. In the field of fine arts, things are much worse. Complete physical and mathematical theories have not been developed here. It was emphasized that when developing a theory, it is necessary to take into account the physical side, considering the elements of this art using all experimental and mathematical methods, as well as the physiological side, including the structure of the corresponding organs, and the study of the functions of the nervous system and brain. The guidelines of this report were aimed at developing a scientific theory of fine arts in general and a scientific theory of composition in particular.

The foundations of music theory were laid in ancient times. Even the ancient Greek philosopher and mathematician Pythagoras and his students, who studied harmony with great passion and persistence, noticed that the pitch of the sound for a given string tension depends on its length. By shortening the strings in half, a higher sound was obtained, by shortening them even more, an even higher sound, and so on. This is how the scale was formed. Knowing the length of the string (mathematical basis), the number of vibrations per second (physical basis), it is not difficult to create the physical and mathematical foundations of music theory. The anatomical structure of the organ of hearing was also studied earlier than the structure of the organ of vision. The German physicist, mathematician and physiologist Hermann Helmholtz (1821 - 1894) built a model of the ear, which made it possible to study the nature of the impact of sound waves on the organ of hearing. He developed a physical and physiological theory of the perception of musical sounds, and in 1862 he published the book “The Doctrine of Auditory Sensations as a Physiological Basis for the Theory of Music.”

In music, a fundamental knowledge of the science of harmony necessarily precedes the art of composition. In the fine arts, there is still an opinion that theory is harmful to the artist, that familiarity with the science of proportions is not necessary. Moreover, many believe that knowledge of the law of the golden ratio interferes with free intuitive creativity. In addition, too much hope is placed on nature (draw from life, write from life, look for a composition in nature). This is a big and deep misconception. Having learned to draw and write from life, but not knowing the theory of composition, many artists never become “composers,” 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 doctrine of proportions call it, helps the artist to create consciously and freely. Using the laws of the golden division, 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.

The Yugoslav researcher of continuous division D. Petrovich in the book “Theorists of Proportions” notes that recently there has been a growing completely natural need to create a unified theory of proportions, and the golden ratio, in connection with this, has aroused general interest. D. Petrovich lists 133 sources that deal with proportions and refers the reader to a bibliographic reference book on the problem of proportionation, which contains about 900 titles. He concludes: summary research on the phenomenon of continuous division (golden section) in nature and art predicts a great future for him in the field of composition 2.

Now we can confidently say that the golden proportion is the basis of form-building, the use of which ensures 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 natural sciences have already given us sufficient scientific information about light, the eye and brain, and visual perception. Therefore, the creation of such a theory on strictly scientific grounds has become quite possible.

The history of the doctrine of proportions is the history of the search for a theory of harmony and beauty. All ancient aesthetics, as well as the aesthetics of the Renaissance, sought the laws of beauty in the commensurability of individual parts, as well as parts and the whole. This commensurability in form is provided by the proportions of the golden section and symmetry.

We will not sin against the truth if we say: all roads lead to the golden ratio. Man - the best, most perfect creation of nature - was created in the proportions of continuous division. It realized the proportion of the golden ratio both as a whole and in parts: in the work of the brain and heart, the structure of the eye, the proportions of parts of the face, arm, hand and the whole body.

We are accustomed to repeating that nature is a great teacher, and the artist is a student of nature. How to understand these words? Should an artist continually draw from life, write hundreds of sketches, take everything ready from nature and copy it, look for suitable scenes for composition, or should he study the compositional techniques of nature and the laws of shape-formation? Should he understand how nature can create various forms from a single matter according to a single plan? Should an artist create according to the known laws of nature, continue her great work, speak her language and create works according to the laws of harmony and beauty? With his work, the author tries to answer these questions.

The book is devoted to the golden ratio in painting, its theoretical understanding and role in the practical work of the painter. The development of our fine arts and its theory have been formed in such a way that artists know very little about the golden ratio in painting, mostly by hearsay or not at all.

Quite a lot has been written about the use of the golden ratio in architecture. P.E. Shpara in the book “Technical Aesthetics and the Fundamentals of Artistic Design” points out that in the theory of architecture, issues of composition are developed in the most detail and constitute a special subject of the theory of architectural composition and the golden ratio in this theory takes its rightful place 3.

The rules of the golden number are studied not only by architects, but also by designers, mathematicians and other specialists. In 1979, the publishing house "Prosveshchenie" (M.) published a textbook "Artistic Design" edited by B.V. Naumov and E.D. Shchedrin for pedagogical institutes, where the issue of proportions and the golden section was given some attention. But the authors cannot say why we are pleased to look at works of art in which the proportions of the golden section are maintained, why they produce an aesthetic impression.

It is gratifying to note that the golden ratio is increasingly being paid attention to by authors of works on painting, aesthetics and science, where it finds a positive assessment. “One of the types of proportion, the so-called “golden section”, is widespread in nature and is used by people,” points out V. S. Kornienko in the book “On the Laws of Beauty” 4. O. Ya. Kochik, noting the desire of the artist V. E. Borisov-Musatov to construct a picture in such a way that it would be logically organized, subject to strict laws of form, gives examples of the use of the proportions of the golden section in his paintings “Tapestry”, “Pond” and sketch of the painting “Autumn Evening” 5.

P.I. Gavrilyuk paid a lot of attention to the consideration of the golden section in the book “The Problem of Aesthetics and Control Theory” 6 . In O. Butkevich’s major work “Beauty” there are the words: “We are captivated by harmony and balance, but not just mechanical symmetry... we are fascinated by the golden ratio...” 7 .

In 1982, the publishing house “Vishcha Shkola” (K.) published the textbook “Fundamentals of Artistic Design”. The author of the manual, I.T. Volkotrub, in the section “Regularities and means of composition,” speaks of the great importance of proportions, which determine the proportionality and harmony of the elements of form, and emphasizes that the “golden ratio” stands out among them. A form organized using the proportions of the golden section evokes the impression of beauty, pleasantness, consistency, proportionality, and harmony. He also notes that the secret of the golden ratio was known in ancient times: Egyptian temples, pyramids, the ancient Greek temple of the Parthenon and the statues of Phidias are all the result of the practical application of the golden ratio by architects and artists. Albrecht Durer noticed it in the proportionality of the human body. The great master Antonio Stradivari was well acquainted with it, who used this ratio in the manufacture of his famous violins 8 .

An idea of the time limits of the practical use of the golden number when creating works of art is given by N. A. Pomerantseva’s book “Aesthetic Foundations of the Art of Ancient Egypt.” The book shows that “... the system of proportions was the basis that ensured the synthesis of all types of art and brought all elements within the composition into harmonious balance” 9.

In 1986, I. Shevelev’s book “The Principle of Proportion” (M.) was published, which thoroughly examined the principles of shape formation in nature and showed the role of the golden section.

The motivating reason for studying the role of the golden ratio in painting was the need for my pedagogical and creative practice. Following the pedagogical principles of P. P. Chistyakov encouraged me and my students to give themselves and their students a clear and intelligible answer to all questions related to working on a painting: choice of point of view, perspective and compositional construction of the image, construction of color, organization of the center of attention and, as a result, the subordination of all pictorial - expressive means of painting to the most perfect expression of the concept - the idea of the painting. Connected with this is the possibility of the painting having the most powerful impact on the viewer.

Studying literature about the golden ratio can take a young artist not only many hours and days, but also months and years, which are so necessary for him to improve his artistic skills. At the same time, without knowledge of the doctrine of the golden ratio, an artist can become like a person blindly looking for a door in a dark room. Without knowledge of the laws of the golden ratio, he will never be able to rise to the heights of artistic perfection to which the classical artists rose. An artist who is not armed with knowledge will always be captivated by all sorts of accidents and miscalculations, which will inevitably reduce the artistic merit of his work.

The rules of the golden proportion are manifested not only in the linear construction of an image on a plane. They extend to the entire surface of the canvas when deciding the overall light tone of the picture and when constructing color, and ultimately to all components of the artistic form. This book shows that the manifestation of the laws of golden proportion, symmetry and rhythm in works of art is not due to their autonomy, but is always subordinated to identifying the meaning, idea of the work, the most complete and clear disclosure of its content. The patterns of visual perception and emotional impact on the viewer are considered. The laws and rules of composition, techniques for constructing an image are given, and a scheme for the general theory of fine art is proposed.

1 Repin I. E. Distant - close. - M., 1937. - P. 537.

2 Petrovich D. Theorists of proportions. - M., 1979. - 193 p.

3 Shpara P. E. Technical aesthetics and the fundamentals of artistic design. - K., 1984. - P. 28.

4 Kornienko V. S. About the laws of beauty. - Kharkov, 1970. - P. 154.

5 Kochik O. Ya. Painting system of V. E. Borisov-Musatov. - M., 1980. - P. 30, 128 and 139.

6 Gavrilyuk P.I. The problem of aesthetics and management theory. - K., 1970.-P. 29, 60-65.

7 Butkevich O. Beauty. - L., 1979. - P. 114.

8 Volkotrub I. T. Fundamentals of artistic design. - K., 1982. - P. 38.

9 Pomerantseva N. A. Aesthetic foundations of the art of Ancient Egypt. - M., 1985. - P. 111.

I. Golden ratio and questions of composition theory

About the golden ratio

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 ratio.

I. Kepler (1571-1630)

Golden ratio and symmetry

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

The golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the golden division is an asymmetrical symmetry. Now the science of symmetry has included such concepts as static and dynamic symmetry. Static symmetry characterizes peace and balance, while dynamic symmetry characterizes movement and growth. Thus, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance, and even rigidity. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Symmetry is characterized by equal segments and equal sizes. Dynamic symmetry is characterized by an increase in segments (or their decrease), and it is expressed in the values of the golden section of an increasing or decreasing series.

An artistic form, the construction of which is based on the proportions of the golden section, and especially the combination of symmetry and the golden section, is a highly organized form that contributes to the clearest expression of content, the easiest visual perception and the appearance of a sense of beauty in the viewer.

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.

Leonardo da Vinci's painting “Madonna in the Grotto” is not strictly symmetrical, but its construction is based on symmetry (Fig. 17, a). The entire content of the picture is expressed in the figures that are located in its lower part. They fit into a square. But the artist was not content with this format. He completes the golden ratio rectangle above the square (Fig. 17, b). 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. At the bottom, the arc crossed the axis of symmetry and indicated the size of another rectangle of the golden ratio in the lower part of the picture (Fig. 17, c). 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 (Fig. 18).

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 the workshop (Fig. 19) 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”. But no one paid attention to the fact that the principle of golden proportions was implemented horizontally here too (Fig. 20). 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.

Rice. 17. The use of symmetry and the golden section in Leonardo da Vinci’s painting “Madonna in the Grotto”: a - proportions of the golden section: b - placement of the characters in the picture in a square; c - diagram of the linear construction of the picture.

Rice. 18. The use of symmetry and the golden ratio in Raphael’s painting “The Betrothal of Mary”

Rice. 19. Golden proportions in the linear construction of the image on the icon “The Descent into Hell” by Dionysius and the workshop (XVI century)

Rice. 20. Symmetry and golden proportions in the linear construction of “The Trinity” by Andrei Rublev

Rice. 21. Symmetry and golden proportions in the linear depiction of “The Assumption” by Theophanes the Greek

Rice. 22. Golden proportions in the linear construction of the image on the slab of Pharaoh Narmer (3rd millennium BC)

The icon of Theophanes the Greek “Assumption” appears to our eyes as a mighty chorale (Fig. 21). 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.

Let us complete our praise of the commonwealth of symmetry and the golden ratio by considering the proportions of the victory slab of the Egyptian pharaoh Narmer (3rd millennium BC). The golden ratio rectangle is the original shape of the Narmer plate (Fig. 22). The slab is divided into belts, the height of which is maintained in the proportions of the golden ratio. The height of the pharaoh's figure - from the upper belt to the lower - is equal to 62 parts of the height. The lower part of the slab from the girdle to the edge is equal to 24 parts, and the upper part, from the upper girdle to the upper edge, is 14 parts. The rhythmic structure of the reverse side of the slab is somewhat different, because the content of the image required a different comparison of proportional values. The proportions of the golden section and symmetry provide an endless variety of compositional structures both in nature itself and in works of art of all kinds and types.

History of the golden section

The history of the golden ratio is interesting and fascinating. She once again confirms that the secrets of nature are hidden and jealously guarded by her. The mystery of the golden ratio is no exception.

In 1911, the French artist Henri Matisse (1869 - 1954) visited Russia. In Moscow he saw ancient Russian icons. “Russians have no idea what artistic treasures they own... Your student youth have here, at home, incomparably better examples of art... than abroad. French artists should go to Russia to study: Italy gives less in this area,” the artist later wrote 1 .

Many years later, Matisse recalled how “touched” he was by ancient Russian art and what impact it had on his work: “You surrender to it the more the more clearly you see that his achievements are supported by tradition - an ancient tradition” 2. Matisse undoubtedly had in mind the art traditions of classical Greece. He saw that Rus', through Byzantium, inherited the living tradition of ancient art and, in its historical and national conditions, continued it. While Italy was reviving antiquity, trying to create a coherent idea of antiquity from the wreckage and ruins, the art of painting and architecture in Rus' reached great heights.

Arriving in the Soviet Union, the American artist Anton Refregier enthusiastically perceives the surviving paintings made by ancient Russian artists. “I look at the majestic paintings of ancient Russian churches, and again and again I am shocked by the depth of humanism of art, which rose above church dogma to the level of expression of the emotional spirit of the people. And I look with amazement at the construction of the composition, at the proportions of the friezes on the walls. Here we can also learn the knowledge of the law of dynamic symmetry, the absolute faith of artists in these laws, revealed by the ancient Greeks and confirmed in all the great periods of architecture and painting,” he wrote in the article “In a language understandable to the masses,” published in the newspaper “Soviet Culture” "May 21, 1974. In the same article, Anton Refregier notes the merits of the works of Renaissance artists: “I would name two such qualities - deep humanism (this is the content) and a responsible, respectful attitude towards the specifics of wall painting, knowledge of geometry, dynamic symmetry, rules" “golden mean” (this is form)... An artist, not being aware of geometry, the law of dynamic symmetry, the most he can do is arrange everything in a certain order, otherwise he can create a collage.” Such a high appreciation of the golden ratio and its manifestations in Russian art certainly encourages us to study this phenomenon.

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an 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 Tutankhamoia indicate that Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded. The slab of Pharaoh Narmer (Fig. 22), built in the proportions of the golden division, was already mentioned earlier.

The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for constructing dynamic rectangles (Fig. 23, a).

Plato (427 - 347 BC) 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 (Fig. 23, b).

Rice. 23. Dynamic rectangles (a) and antique golden ratio compass (b)

In the ancient literature that has come down to us, the golden division was first mentioned in Euclid’s Elements. In the 2nd book of the Elements, a geometric construction of the golden division is given. After Euclid, Hypsicles (2nd century BC), Pappus (3rd century AD), and others studied the golden division. In medieval Europe, they became familiar with the golden division from Arabic translations of Euclid’s Elements. The 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.

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly woven into the history of the golden ratio. He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read: “How many pairs of rabbits will be born from one pair in one year?” Reflecting on this topic, Fibonacci built the following series of numbers:

The series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. became known in science as the Fibonacci series. Its peculiarity is that each of its terms, starting from the third, is equal to the sum of the previous two: 2 + 3 = 5; 3+5 = 8; 5 + 8=13; 8+13 = 21; 13+21 = 34, etc., and the ratio of the numbers in the series is increasingly approaching the ratio of the golden division. So, 21:34 = 0.617, and 34:55 = 0.618. This ratio is denoted by the symbol F. Only this ratio - 0.618:0.382 - gives a continuous division of a straight line segment in the golden proportion, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to everything. The Fibonacci series could have remained only a mathematical incident (case), if not for the fact that all researchers of the golden division in the plant world, as well as in the animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden division .

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 had a lot of empirical experience, but little knowledge. He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is called the creator of descriptive geometry.

Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of Duke Moreau, he came to Milan, where he gave lectures on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. They became friends. 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. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity: God the son, God the father and God the holy spirit (it was implied that the small segment is the personification of God the son, the larger segment is the god of the father, and the whole segment - God of the Holy Spirit). A mystical veil was thrown over the golden ratio.

Leonardo da Vinci also paid a lot of attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in the golden division. Therefore, he gave this division the name golden ratio. This is how it remains in science to this day as the most popular.

It is characteristic that at the same time in the north of Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches the introduction to the first version of the treatise on proportions. Dürer writes: “...It is necessary that someone who knows how to do something should teach it to others who need it. This is what I set out to do." 3

Dürer complains that the secrets of the ancients have been lost, that the church fathers should not so violently destroy everything that remains of the ancients. Judging by one of Dürer's letters, he met with Luca Pacioli while in Italy. Albrecht Durer develops in detail the theory of proportions of the human body. Dürer assigned an important place in his system of relationships to the golden section. A person's height is divided in golden proportions by the line of the belt, as well as by a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is known.

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

In subsequent centuries, the rule of the golden proportion turned into an academic canon, and when, over time, the struggle against academic routine began in art, in the heat of the struggle “they threw out the baby with the bathwater.” The golden ratio was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden ratio professionally