admittedMinistry of Culture of the Ukrainian SSR

as a teaching aidfor art institutes

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MAJOR PUBLISHING HOUSE

PUBLISHING ASSOCIATION

"HIGH SCHOOL"

Chapter first

The Golden Ratio and Questions of the Theory of Composition

About the golden ratio

Golden Ratio - Harmonic Proportion

The debate about whether science should or should not invade 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. Thinking artists, theorists and educators who have pondered the problems of educating young people have always come to the conclusion that art cannot develop and flourish without science. The artist and teacher N. P. Krymov wrote: “They say: art is not science, not mathematics, that it is creativity, mood, and that nothing can be explained in art - look and admire. I don't think so. Art is explainable and very logical, it is necessary and possible to know about it, it is mathematical... You can prove exactly why a picture is good and why it is bad” V. I. Surikov argued that there is some immutable law in the composition, when nothing can be removed or added to the picture, even an extra point cannot be put, this is real mathematics.

Famous French architect and theorist of architecture of the XIX 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 of us. Mathematics is hidden behind the seeming simplicity and randomness of the living perception of the surrounding reality. When we listen to music, our brain is doing algebra. When we look at something, our brain is engaged in geometry. A person cannot have an attitude to an object, a feeling, an emotion, until the brain

1 Krymov N. P.-artist and teacher.-M., I960.-S. 32.

Geometry has two treasures: one of them is the Pythagorean theorem, and the other is the division of a segment in the middle and extreme ratio.

I. Kepler (1571-1630)

did not make a "measurement", a comparison of this subject with something similar already in memory. Mathematics comes first, and only then does the feeling arise. The brain performs this work instantly, therefore we do not notice it and do not realize it, and it seems to us that the feeling arises immediately.

Before defining the golden ratio, it is necessary to familiarize yourself with the concept of proportion. In mathematics proportion(lat. proportio) - this isequality between two ratios of fourdisguises: a: b = c:d. Further, for example, let us turn to a straight line segment (Fig. 1). Line segment AB can be divided into two equal parts (/). This will be the ratio of equal values ​​- AB: AC = AB: Sun. This same line (2, 3) can be divided into two unequal parts in any respect. 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 direct AB(4) can be divided according to the golden ratio when AND YOU, how AS: Sun. This is the golden division or division in the extreme and average ratio.

From the above it follows that gold-that section- this is a proportional harmo-nical division of a segment into unequal parts, within 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 related to the larger one as the larger one is to everything, i.e. a: b = b : With or c \b = b: a(Fig. 2). The definition - division in the extreme and average ratio - becomes more understandable if we express it geometrically (Fig. 3), namely a:b how b: s.

From fig. 3 it is clear why the astronomer Johannes Kepler called the golden ratio continuing herself. “It is arranged in such a way,” I. Kepler wrote, “that the two junior terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the following

RICE.I.

Division of a straight line segment into equal parts and according to the golden section:

Geometric and algebraic expression of the golden proportion: a:in = in: with or c: b \u003d b: a

blowing term, and the same proportion is preserved indefinitely” 1 .

As you can see, the construction of a series of segments of the golden ratio can be done both in the direction of increasing (ascending row), and in the direction of decreasing (descending row). In the latter case, it is necessary to subtract the smaller one from the larger segment - we get an even smaller one: b - a = d, etc.

Practical acquaintance with the golden section usually begins with the division of a straight line segment in the golden ratio in a geometric way (Fig. 4).

I-AB; AC=AB: SU(a proportion is formed); 2, 3 - proportion is not formed; 4-AB:AC=AC:BC or BC: AC=AC: AB(golden proportion)

1 Kepler I. About hexagonal snowflakes.-M., 1982.- P. 17.

Average proportional or division of the segment in the extreme and average ratio:

d - b - a; c=a+b

Rice. 4. Fig. 5.

Geometric division of a straight line segment along the golden section Determination of the golden section line in the picture geometrically (developed by A. Dürer): in the following way:

BC = 0.5 AB; CD = sun BC =0.5 AB

Rice . 6.

The use of the golden section in the construction of I. E. Repin’s painting “A. S. Pushkin at the act in the Lyceum on January 8, 1815 "

From a point AT a perpendicular is restored equal to half AB. Received point FROM connected by a line to a dot BUT. A segment is plotted on the resulting line sun, ending with a dot D. Line segment AD transferred to a straight line AB. The resulting point £ divides the segment AB in relation to the golden ratio. Arithmetically cut-

ki golden ratio are expressed as an infinite irrational fraction. AE = 0.618... if AB take as unity, ££ = 0.382.... In practice, rounding is used: 0.62 and 0.38. If the segment AB taken as 100 parts, then the largest part of the segment is 62, and the smaller is 38 parts.

When transferring geometric division method

Rice. 7. Lines of the golden section and diagonals in the picture

for a picture or sketch, they do this: half the length of the picture or sketch is set aside for the height or the continuation of the height, if the sketch is of a narrow format. received point FROM connected to the lower left corner of the picture, etc. (Fig. 5). The line of the golden section on the left side of the picture will be at the same distance from the left edge as it is on the right from the right (shown by the dotted line). The above two proportions of the golden division - equal and unequal, while proportional, are widely used in art.

The figure of A. S. Pushkin in the painting by I. E. Repin “A. S. Pushkin at the act in the Lyceum on January 8, 1815 G." placed by the artist on the line of the golden section on the right side of the picture (Fig. 6). The left side of the picture, in turn, is also divided in proportion to the golden section: from the head of A.S. Pushkin to the head of G.R. Derzhavin and from it to the left edge of the picture. The distance from Derzhavin's head to the right edge of the picture is divided into two equal parts by the golden section line. At the bottom of the picture, the eye catches the division into three equal parts. They form a table on the left side of the picture, Pushkin's leg to the right of the line golden section and the right edge of the picture.

If it is necessary to find the line of the golden section in a picture or a sketch horizontally, then there is no need to make a new division by a geometric method of the height of the picture. It is enough to draw the diagonals of the picture. They are crossed

lines with lines of the golden section along the vertical will indicate the points through which the horizontal lines of the golden section should be drawn (Fig. 7). These lines may be needed when building a landscape. Landscape painters know from experience that one cannot take half of the plane of the canvas under the sky or under the earth and water. It is better to take either more sky, or more land, then the landscape "looks better".

From the proportion of the golden section it follows that if the height or width of the picture is divided into 100 parts, then the larger segment of the golden ratio is 62, and the smaller one is 38 parts. These three values ​​allow us to build a descending series of segments of the golden ratio: 100 - 62 = 38; 62 - - 38 = 24; 38 - 24=14; 24-14=10.

100, 62, 38, 24, 14, 10 - this is a series of gold valuesproportions expressed arithmetically. The segments of the golden ratio are also found in the picture, if the golden section line has already been drawn vertically (Fig. 7). We transfer the line of the golden section to the left edge of the picture. The distance between the lines of the golden section in the middle of the picture is 24 parts. The segment equal to 24 parts is set aside for the segment equal to 38 parts, and we get the remainder equal to 14 parts. The last segment is superimposed on a segment equal to 24 parts, and we get a segment equal to 10 parts. We have received all segments of the descending series of the golden ratio for this picture. We carry out the same operation with the height of the picture. The resulting segments are transferred to a strip of thick paper or cardboard - for the width on the front side and for the height on the back. We call this simple tool proportional line. Such a proportional ruler is suitable only for a given sketch or a sketch of the same size. Its production takes several minutes, but in the future it will facilitate the work on the sketch in search of intervals between figures or groups of figures, between objects, it will help to find their dimensions and, ultimately, to harmonize the linear construction of the picture.

The figure of A. S. Pushkin in N. N. Ge’s painting “Alexander Sergeevich Pushkin in the village of Mikhailovsky” was placed by the artist on the golden section line on the left side of the canvas (Fig. 8). No and all other values ​​​​in width are not at all random: the width of the oven is 24 parts from the width of the picture, the whatnot is 14 parts, the distance from the whatnot to

The proportions of the golden division in the linear construction of the painting by N. N. Ge “Alexander Sergeevich Pushkin in the village of Mikhailovsky

the oven is also equal to 14 parts, etc. The same values ​​\u200b\u200bare in the picture by I. E. Repin (see Fig. 6): from the left edge of the picture to Derzhavin's head - 24 parts; from the table to the toe of the boot of Pushkin's right foot - 24 parts. The same distance from Pushkin's head to the head of a military man, listening with delight to the poet's reading (his head is on the second line of the golden section in the same

turn, like Pushkin's head). From Pushkin's head to the head of a young woman to the right side of the picture, listening with emotion to the recitation, there are also 24 parts, and from her head to the right edge of the picture - 10 parts, etc.

Repetition of equal values, alternationpain them and unequal values ​​in proportions, the golden section creates a certainrhythmically

A series of segments of the golden ratio

skii system, causing the viewer this or thatmood and drawing him into the examinationImages. The order and sequence of this examination are predetermined by the artist.

The advantage of the proportion of the golden section lies in the fact that, once dividing a straight line segment or a side of a picture in a geometric way, segments of any reduction are obtained. In the practical work of the artist, there are enough values ​​corresponding to the numerical values ​​62, 38, 24, 14 and 10 (Fig. 9).

Segments of the golden ratio of the descending series with a known value of the segment AB or the width of a sketch, painting, reproduction - if we want to analyze them, they are obtained by calculation. For example, the width of the sketch is 14 cm. One hundredth of 14 will be 0.14 cm. Multiply 0.14 by 62 and get a larger segment of the golden ratio, equal to 8.68 cm. Therefore, 100 parts \u003d 14.00; 62 parts = 8.68; 38 parts = 5.32; 24 parts = 3.36; 14 parts = 1.96; 10 parts = 1.4 cm.

We plot these values ​​on a proportional ruler, as shown in fig. 7, and further work on the sketch is carried out using this ruler. The intuitive is combined with mathematics and calculation.

It happens that the size of the sketch is 10 cm (100 mm) in width and height (square). Then the golden ratio on the sketch or proportional ruler is plotted along the ruler: 62, 38 and 24 mm. With a picture size of 100x100 cm, proceed in a similar way. If one of the sides of the picture is 100 cm, then, having set aside segments of the golden ratio on it with the help of a ruler, we draw the lines of the golden section. Crossing them

diagonals and obtain data for finding segments of the golden section for the other side of the picture, not equal to 100 cm, as shown in Fig. 7. When the sketch is not very large, apply the method of finding the golden proportions on one of its sides by drawing an auxiliary line measuring 10 cm (100 mm) at an arbitrary angle to the divided line (Fig. 10). On the auxiliary line, which is drawn in the plane of the sketch or outside it,

Auxiliary line 100 mm (10 cm) long for finding segments of the golden ratio on a small sketch

Methods for finding segments of the golden ratio using the “from the square” method: a - square; b - a rectangle of the golden section; in - obtaining points for drawing lines of the golden section horizontally; G - construction

sketch of any format

The values ​​are in millimeters - 62, 38, 24, 14 and 10. The extreme point of the construction line is connected to the edge of the sketch. The remaining lines are drawn parallel to the first. The rest of the construction is carried out as shown in fig. 7. This method was proposed by the artist V. Skubak. The same method is used on a small picture, when an auxiliary line of 100 cm is located on its surface.

If the sketch size is not set, its construction starts with a square (Fig. 11, a). Dividing the lower side of the square into two equal parts and drawing a line from the obtained point to the upper right corner of the square, we take this line as the radius and describe the arc until it intersects with the continuation of the lower side of the square. From the obtained point, we restore the perpendicular until it intersects with the continuation of the upper side of the square. As a result of such a construction, we obtain a rectangular

golden section nickname, or golden rectangle

(Fig. 11, b). If the width of such a rectangle is taken as 100 parts, then its height is 62 parts. The line of the golden section along the vertical is determined by itself. Next, we draw diagonals, we get points for drawing golden section lines along horizontal lines (Fig. 11, in). On the basis of a golden rectangle, a sketch of any format, elongated horizontally or vertically, is constructed (Fig. 11, G).

The Russian Academy of Arts knew about the law of the golden section. There is written evidence of this. In the book "Far - Close" I. E. Repin describes the meeting of the famous critic V. V. Stasov with students of the Academy of Arts. In addition to Repin and Stasov, the meeting was attended by M. M. Antokolsky, G. I. Semiradsky, K. A. Savitsky and others. The conversation was about new realistic art and outdated academism.

Ilya Efimovich notes that Semiradsky flaunted before Stasov his knowledge of Greek art, aesthetic treatises and the golden ratio, and notes that V. V. Stasov knew all this very well.

The golden ratio was used by artists in the compositional construction of paintings. A simplified method was developed, when the plane of the picture was divided into 10 parts vertically and horizontally. The line of the golden section was outlined in relation to 6 and 4 parts (Fig. 12, a). This did not give a ratio of 62:38, but gave a close ratio of 60:40. In practice, this was enough to navigate and place the main figure or group of figures in the most favorable place for this picture.

Academician A.N. Laptev in his article “Some Issues of Composition” writes about the golden section in the following way: “...I would like to mention the law of proportions of the golden section, which has long been known, especially in classical art. Due to some property of our visual perception, these proportions (approximately 6 and 4) are the most harmonious and most consistent with the general concept of beauty, and therefore the most frequently used” 1 .

The same result was obtained by the artists of the Munich Academy by dividing the picture into 5 parts. The golden ratio was taken in a ratio of 3: 2, which is the same, since the reduction is 10; 6 and 4 twice gives 5; 3 and 2. The main figure of the picture or group was placed on the golden section line (Fig. 12b).

In the painting by Giovanni Tiepolo “The Feast of Cleopatra”, the artist placed the head of Cleopatra at the upper right point at the intersection of the lines of golden division vertically and horizontally. This ensures the easiest perception by the eye of the whole picture and its visual-semantic center - the center of attention. The center of attention can be on the right side of the picture or on the left, at the bottom or top. These four points are best places for the location of the main subject of the picture. This is due "to the device of the eye, the work of the brain and the laws of visual perception, which will be discussed below.

On one of the sketches by V. I. Surikov for the painting

Rice. 12.

Division of the picture:

a- for 10 parts in the Russian Academy of Arts: b- in five parts at the Munich Academy of Arts

“Boyarynya Morozova” clearly shows the division of the right vertical edge of the sketch into 10 parts. Then 6 divisions from the bottom or 4 from the top were counted and a golden section line was drawn, which is the proposed horizon. A reproduction of this sketch was published in S. Kaplanova's book "From conception and nature to a finished work" 2 . In the early painting by V. I. Surikov “The Merciful Samaritan” (1874), the head of the wounded man is placed by the artist in the lower right point of the painting, the palm right hand Samaritan - in the upper left, where the servant pours water into her from a jug. Both of these points are on the diagonal. Sustainability

1 Laptev A. M. Some questions of composition//Questions 2 Kaplanova S. From idea and nature to finished

fine arts.- M, 1954.-S. 66-67. work.-M., 1981.-S. 17.

Diagonals, lines of the golden section and the semantic center of the painting by V. I. Surikov "The Merciful Samaritan"

The composition is also given by the fact that the head of the Samaritan is located on the middle line of the picture vertically (Fig. 13).

The disadvantage of dividing the picture into 10 or 5 parts lies in the fact that it gives rather approximate segments of the golden section - 60, 40, 20 (Table 1, row 1). More accurate values ​​of the proportional values ​​of the golden section (62 and 38) make it possible to form 5 values ​​of the golden series (Table 1, row 2), even more accurate initial values ​​-61.8; 38.2 or 61.803 and 38.196 make it possible to continue finding the magnitude

ranks of the descending series of the golden ratio up to 9 values ​​or even to infinity (Table 1, rows 3 and 4). AT practical work the artist over a sketch or a picture, the values ​​\u200b\u200bof the 2nd and 3rd rows are sufficient.

The format of a painting or monumental painting is sometimes set. But more often than not, the artist himself determines the format in accordance with his own idea. For example, an artist begins to develop a sketch of a landscape with a format of 8x12 cm. The sketch has a format of 8X12 cm. To find the line of the golden section along the vertical and segments of the golden section

Rice. fourteen.

Building a landscape according to the golden ratio and finding segments of the golden ratio using an auxiliary line

along the descending row, you can use the drawing of an auxiliary line 10 cm long outside the sketch field (Fig. 14). On the basis of observations, sketches, sketches, the author had an idea: to show the edge of the forest in the picture. The viewer's attention is primarily attracted by spruce. All other trees complement the landscape and form a harmonious whole, easily perceived by the eye. Such a harmonious whole is created due to the location of the spruce on the line of the golden section, and the rest of the trees or groups of trees - in due order. This order (rhythm) is suggested by segments of the descending series of the golden section for the given picture, found with the help of an auxiliary line and plotted on a proportional ruler (for width and height). Further work on the landscape will go "by eye", by feeling. Let the artistic taste of the author, experience and talent lead him to the successful completion of the picture, to the best expression of the idea. Both in architecture and in painting, geometry is used for the needs of proportioning, to create a preliminary scheme, a compositional frame, but no more.

Table 1.values ​​of the descending series of the golden ratio

			1,315 0,813 0,502 0,311 etc.

To find segments of the golden ratio of the ascending and descending series, you can use pentagram(Fig. 15). To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer (1471 - 1528) (Fig. 15, a). Let O be the center of the circle, BUT- a point on the circle and E- middle of the segment OA. Perpendicular to Radius oa, restored at the point O, intersects the circle at the point D. Using a compass, set aside a segment on the diameter CE = ED. The length of a side of a regular pentagon inscribed in a circle is DC. Putting segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram (Fig. 15, b). All diagonals of the pentagon divide each other into segments connected by the golden ratio. We draw a straight line of arbitrary length, plot a segment m , put a cut below M. Based on these two segments, we build a scale of segments of the golden proportion of the ascending and descending series (Fig. 15, in).

If no thumbnail size is specified, take any two scale values ​​as the width or height of the thumbnail and find all other values ​​as shown earlier.

From all that has been said, it follows that an artist who wants to realize harmonic proportion

Construction: a -golden triangle: a:c=F, in =dd 1 ; b- golden rectangle: a: in= F

Rice. fifteen.

Construction of a regular pentagon (a), pentagrams (b) and scale segments (in) golden ratio

the basic structure of his picture on the basis of the golden section, necessarily finds the first two segments of the golden ratio. The solution of this problem contributes to Golden Triangle. Each end of the pentagonal star is a golden

triangle. Its sides form an angle of 36° at the apex, and the base laid on the lateral side divides it in proportion to the golden section. To build a golden triangle "does not even require a protractor (Fig. 16, a). We draw a direct AB. from point BUT lay a segment on it three times O arbitrary value, through the obtained point R draw a perpendicular to the line AB,

on a perpendicular to the right AND to the left From the point R set aside segments O. Received points d and d\ connect with a straight line BUT. Line segment dd\ put aside on the line Ad\, getting a point FROM. She split the line Ad 1 in proportion to the golden ratio. lines Ad\ and dd\ are used to construct a golden rectangle (Fig. 16, b).

Golden ratio and symmetry

The golden ratio cannot be considered in itself, separately, without connection with symmetry. The great Russian crystallographer GV Vul'f (1863-1925) considered the golden section to be one of the manifestations of symmetry.

Golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern ideas, the golden division is an asymmetric symmetry. Now the science of symmetry has included such concepts as static and dynamic symmetry. Static symmetry characterizes rest, balance, and dynamic - movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance, and even stiffness. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Symmetries are characterized by equal segments, equal magnitudes. Dynamic symmetry is characterized by an increase in segments (or their reduction), and it is expressed in the values ​​of the golden section of an increasing or decreasing series.

art form, at the heart of buildingwhich are the proportions of the golden section, and especiallyIt is the combination of symmetry and the golden section, which 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 along the vertical and division into unequal parts along the golden section along the horizontals.

The painting by Leonardo da Vinci “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 the square

rat. But the artist was not content with this format. He completes the construction of a golden section rectangle above the square (Fig. 17, b). As a result of this construction, the whole picture received the format of a golden rectangle placed vertically. With a radius equal to half the side of the square, he described a circle and obtained a semicircle of the upper part of the picture. At the bottom, the arc crossed the axis of symmetry and indicated the size of another golden section rectangle in the lower part of the picture (Fig. 17, in). Then, with a radius equal to the side of the square, a new arc is described, which gave points on the vertical sides of the picture. These points helped to build an equilateral triangle, which was the framework for constructing the whole group of figures. All proportions in the picture were derived from the height of the picture. They form a number of relations of the golden section and serve as the basis for the harmony of forms and rhythm, which carry a hidden charge of emotional impact. Raphael's painting “The Betrothal of Mary” is built in a similar way (Fig. 18).

If we turn to ancient Russian painting, icons of the 15th-16th centuries, we will see the same methods of constructing an image. Vertical format icons are vertically symmetrical, and horizontal divisions are made according to the golden ratio. The icon "Descent into Hell" by Dionysius and the workshop (Fig. 19) is calculated with mathematical accuracy in the proportions of the golden section.

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

Much has been written about the symmetry of the Trinity by Andrei Rublev. But no one paid attention to the fact that the principle of golden proportions was implemented along the horizontal lines (Fig. 20). The height of the middle angel is related to the height of the side angels, just as their height is related to the height of the entire icon. The line of the golden section crosses the axis of symmetry in the middle of the table and the bowl with the sacrificial calf. This is the compositional castle of the icon. The figure also shows smaller values ​​of the golden section series. Along with smooth lines, color

The use of symmetry and the golden section in the painting by Leonardo da Vinci "Madonna in the Grotto": a- proportions of the golden section: b- character placement

paintings in a square; in- scheme of linear construction of the picture

Rice. eighteen.

The use of symmetry and the golden section in Raphael's painting "Betrothal of Mary

Golden proportions in the linear construction of the image on the icon "Descent into Hell" by Dionysius and the workshop (XVI century)

Symmetry and golden proportions in the linear construction of "Trinity" by Andrey Rublev

Goldensection

Symmetry and golden

proportions in linear

image of "Assumption"

Theophan the Greek

Golden proportions in the linear construction of the image on the plate of Pharaoh Narmer (3rd millennium BC)

The proportions of the icon play a significant role in creating the overall impression that the viewer experiences when viewing it.

The icon of Theophan the Greek "Assumption" (Fig. 21) appears to our eyes as a mighty chorale. Symmetry and the golden ratio in construction give this icon such power and harmony that we see and feel when we see Greek temples and listen to Bach's fugues. It is easy to see that the composition of Theophan the Greek's "Assumption" and Andrei Rublev's "Trinity" is one and the same. Researchers of the work of ancient Russian artists note that the merit of Theophan the Greek is not so much that he painted frescoes and icons for Russian cathedrals and churches, but that he taught Andrey Rublev the ancient wisdom.

Let us complete the praise of the community of symmetry and

the golden section by considering the proportions of the victory plate of the Egyptian pharaoh Narmer (3rd millennium BC). The golden section rectangle is the original form of the Narmer plate (Fig. 22). The plate is divided into belts, the height of which is maintained in the proportions of the golden section. The height of the figure of the pharaoh - from the upper belt to the lower one - is equal to 62 parts of the height. The lower part of the plate 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 give an infinite variety of compositional constructions both in nature itself and in works of art of all kinds and types.

History of the golden section

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

In 1911 the French artist Henri Matisse (1869-1954) visited Russia. In Moscow, he saw old Russian icons. “Russians do not even suspect what artistic riches they possess... Your young students here, at home, have incomparably better examples of art... than abroad. French artists should go to study in Russia: 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 indulge in him the stronger, 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 Russia, 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 form a complete picture of antiquity from the wreckage and ruins, the art of painting and architecture in Russia reached great heights.

Arriving in the Soviet Union, the American artist Anton Refregier enthusiastically perceives the surviving murals made by ancient Russian artists. “I look at the majestic paintings of ancient Russian churches, and I am again and again struck by the depth of the humanism of art, which has risen above church dogma to the level of expressing 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 newspaper "Soviet Culture" May 21, 1974. In the same article, Anton

1 Matisse A. Collection of articles about creativity.- M, 1958.- S. 99.

2 Ibid.-S. 104.

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 to the specifics of wall painting, knowledge of geometry, dynamic symmetry, the rules of the “golden mean” ( it is a form) ... An artist, not being knowledgeable in geometry, in the law of dynamic symmetry, the most he can do is arrange everything in a certain order, otherwise, create a collage.” Such a high appreciation of the golden section and its manifestation in Russian art, of course, 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 pyramid of Cheops, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamoia indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values ​​​​of the golden division. The architect Khesira, depicted on the relief of a wooden board from the tomb of his name, holds measuring instruments in his hands, in which the proportions of the golden division are fixed. The slab of Pharaoh Narmer was already mentioned earlier (Fig. 22), built in the proportions of gold division.

The Greeks were skilled geometers. Even arithmetic was taught to their children with the help of geometric figures. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles (Fig. 23, a).

Plato (427-347 BC) also knew about golden division. His dialogue "Timaeus" is devoted to the mathematical and aesthetic views of the school of Pythagoras and, in particular, to the questions of the golden division.

in the facade ancient Greek temple The Parthenon contains golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division (Fig. 23, b).

In the ancient literature that has come down to us, gold

Dynamic rectangles (a) and antique compasses of the golden section (b)

This division is first mentioned in Euclid's Elements. In the 2nd book of the "Beginnings" the geometric construction of the golden division is given. After Euclid, Hypsicles (2nd century BC), Pappus (3rd century AD) and others studied the golden division. In medieval Europe, they got acquainted with the golden division from the 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, kept in strict secrecy. They were known only to the initiates.

The name of the Italian mathematician monk Leonardo from 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, in which all the problems known at that time were collected. One of the tasks 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 lies in the fact that each of its members, starting from the third, is equal to the sum of the two previous ones: 2 + 3=5; 3+5 = 8; 5+8=13; 8+13 = 21; 13 + 21 = 34, etc., and the ratio of the numbers of the series is more and more 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 ratio, its increase or decrease to infinity, when the smaller segment is related to the larger one, as the larger one is to everything. The Fibonacci series could remain only a mathematical incident (case), if it were not for the fact that all researchers of the golden division in flora, as well as in the animal, not to mention art, invariably came to this series as an arithmetic expression of the law of 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 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 in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the painter Piero della Francesca, who wrote two books, one of which was called On Perspective in Painting. He is called the creator of descriptive geometry.

Luca Pacioli was well aware of the importance of science for art. In 1496, at the invitation of Duke Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked at the Moro court in Milan at that time. They are

became friends. In 1509, Luca Pacioli's Divine Proportion was published in Venice, with brilliantly executed illustrations, which is why they are believed to have been 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 understood that the small segment is the personification of God the Son, the larger segment is God father, and the whole segment - the god of the holy spirit). A mystical veil was thrown over the golden ratio.

Leonardo da Vinci also paid much attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in golden division. So he gave this division the name golden section. So it is kept in science until now 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 an introduction to the first draft of a treatise on proportions. Dürer writes: “... It is necessary that the one who knows something should teach it to others who need it. This is what I set out to do."

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

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 ratio for botany (plant growth and structure).

1 Durer A. Diaries, letters, treatises. - L.; M., 1957.- T. 2.- S. 37,

In subsequent centuries, the rule of the golden ratio turned into an academic canon, and when, over time, a struggle began in art with the academic routine, in the heat of the struggle, “they threw the baby out with the water.” The golden ratio was "discovered" again in mid-nineteenth in. In 1855, the German researcher of the golden section, Professor Zeising, published his work Aesthetic Research. What happened to Zeising was exactly what was bound to happen to the researcher who considers a phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions to be "mathematical aesthetics".

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

Zeising tested the validity of his theory on Greek statues. He developed the proportions of the Apollo Belvedere in the most detail. Greek vases, architectural structures of various epochs, plants, animals, bird eggs, musical tones, poetic meters were subjected to research. Zeising defined the golden section, showed how it is expressed in line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction and the other. His next book was entitled "Golden division as the basic morphological law in nature and art." In 1876, a small book, almost a pamphlet, was published in Russia, outlining Zeising's work. The author stole

Xia under the initials Yu.F.V. It is characteristic that not a single work of painting is mentioned in this edition.

AT late XIX- early XX centuries. Many purely formalistic theories have appeared about the use of the golden section in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of machines, furniture, etc.

The anarchy of capitalist production led in the 20th century. to the fact that the products manufactured by one enterprise were very different from similar products of other enterprises. When transporting such products, it often turned out that they did not correspond to the dimensions of the vehicles. The same situation was observed in the construction business.

The French architect Le Corbusier (1887-1965) develops a unified system of values. Was taken as a basis average height a person, equal to 175 cm. A scale of the golden section was built, which gave the required dimensions. Le Corbusier called this scale modulor. Using his "modulor", Le Corbusier built individual buildings and entire complexes of structures.

At the ninth Triennial exhibition in Milan in 1951, three days were devoted to the golden section. These days, the first international conference was held on the topic of proportions in art, and the 1954 Triennale exhibition was entirely devoted to the “divine proportion” and was a praising of the golden section - “the most ancient path of mankind indicated by Pythagoras” (Le Corbusier). Unfortunately, it was mainly about architecture.

The merits of G. B. Borisovsky should be mentioned. In the book "Science. Technique. Art” (Moscow, 1969), the author pays tribute to the golden ratio, but points to its weak side: the golden ratio characterizes only quantitative relationships. He cites the words of Zholtovsky about sausage (said in jest), that if you cut a rotten sausage in the golden ratio, then it will not become tastier. The ratios inherent in the golden ratio, expressed arithmetically or geometrically, really determine only quantitative ratios. But these same relationships, embodied in the living forms of leaves, flowers, animals, give us aesthetic satisfaction,

Dost, we enjoy the beauty of the form. The more they are pleasing to us in the works of human hands: buildings, statues, paintings, carpets, vases, etc., which we do not taste, but look at them with our eyes.

In our country, in the pre-war years, books about the golden section in architecture were published: N. Vrunov. Proportions of antique and medieval architecture. - M., 1935; G. D. Grimm. Proportionality in architecture.- L.; M., 1935. Translated editions were carried out: G. E. Timerding. Golden section. - M., 1924; M. Ghika. Aesthetics of proportions in nature and art. - M., 1936; D. Hambidge. Dynamic Symmetry in Architecture. - M., 1936. And in these books, the manifestation of the law of the golden section in painting was not affected.

In the editorial note to the book by M. Ghick “Aesthetics of Proportions in Nature and Art”, it is indicated that many scientists who have studied the golden ratio do not go beyond a simple statement of fact: “Meanwhile, the task is to explain his reasons. Such an attempt is being made by the Soviet researcher F. I. Zubarev, whose works on the golden ratio are now being prepared for publication. "" It is not known whether the works of F. Zubarev were published or not.

In the post-war years, the attention of scientists of various specialties to the problem of the golden section has been noticeably expanded and deepened. In 1974, I. I. Shafranovsky published the work “Dynamic Symmetry in Crystallography, Mineralogy, Petrography and the Organic World” (Notes of the Leningrad Gorn, Institute named after G. V. Plekhanov. - T. XII, issue 2). In 1977, A.P. Stakhov’s book “Introduction to the algorithmic theory of measurement” was published, and in 1979, his book “Algorithmic theory of measurement” (M., Znanie), which outlined the use of numbers of the Fibonacci series and the golden ratio to improve the performance of analog-to-digital converters. In 1979, I. Shmelev in the journal “Architecture of the USSR” publishes the article “Canon. Rhythm, Proportion, Harmony” (No. 2), which outlines the further development of the idea of ​​“modulor” by Le Corbusier, which allowed him to reveal the mechanism of harmony of rhythmic relationships in the proportions of the male and female bodies, their dynamic complementarity in relation to each other, which removes distrust of the golden section on that

1 Gika M. Aesthetics of proportions in nature and art.- M., 1936.-S. 301.

Second Golden Ratio:

a- geometric construction; 6 - line of the second golden section on

based on the fact that the proportions of the woman's body do not correspond to gold.

Of particular interest is the article by M. A. Marutaev “On Harmony as a Regularity” in the collection “Principle of Symmetry” (Moscow, 1978). He notes that there are three problems in modern science: 1) the nature of the golden section, 2) the enigma of the number 137, and 3) the nature of approximate symmetry, which applies to wildlife, art, and, more recently, to physics. He goes on to show that all three problems are one problem: broken symmetry (approximate symmetry), the number 137, and the golden ratio are interrelated. This confirms, in the author's opinion, the fundamental nature of the principle of the golden section and makes it possible to explain many facts that were previously considered as contradicting the principle of the golden section.

The Bulgarian magazine "Fatherland" (1983.-No. 10) published an article by Tsvetan Tsekov-Karandash about the "second golden section", which follows from the main section and gives a new ratio of 44: 56.

This proportion is found in architecture, and also takes place in the construction of compositions of paintings in an elongated horizontal format.

Line segment AB is divided in proportion to the golden section (Fig. 24, a). From a point FROM the perpendicular is restored SD. Radius AB there is a point D, which is connected by a line to a dot BUT. Straight< ASD is divided in half. From a point FROM a line is drawn until it intersects with a line AD. Dot E divides the segment AD in relation to 56:44.

On fig. 24, b the finding of the line of the second golden section in the picture is shown. It is located in the middle between the line of the golden section and the middle line of the picture.

Natural science foundationscomposition theory

Principles of shaping in nature

Once there were no trees, rivers, fields, mountains. The earth was a fire-breathing ball, where everything boiled, seethed, gradually cooled, something connected with something, disintegrated, synthesized in a new form. And so millions of trial and error. The Earth cooled down, a hard crust was formed. Nature "arranged" the air, stones, water, clay, plants, insects, fish, animals. Man was the highest manifestation of the forces of creative matter. Nature has realized here a combination of symmetry along the vertical and golden section along the horizontals. Nature created, strictly observing its own laws: development (evolution) and conservation of matter. Everything that took on some form formed, grew, strove to take a place in space and preserve itself. This aspiration finds fulfillment mainly in two variants - upward growth or spreading over the surface of the earth and twisting in a spiral.

A living organism, elongated in length, is fraught with many dangers for its owner. The snake dies most often because of its long body. A lizard throws off its tail if it is grabbed by a hawk. The shell is twisted in a spiral. If it is unfolded, then the length is slightly inferior to the length of the snake. A small 10 cm shell has a spiral 35 cm long.

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

30

The shape of the spirally curled shell attracted the attention of the ancient Greek scientist Archimedes. He studied it and deduced the equation of the spiral. The spiral drawn by this equation is called spiral of Archimedes. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering. She played a certain role in the development of television.

Even Goethe emphasized the tendency of nature to spirality. The helical and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. However, only the joint work of botanists and mathematicians shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The researcher of the golden section in the plant world, Yu. Urmantsev, in his article "The Golden Section" came to the following conclusion: "... the golden section reigns in some processes occurring in living nature" ".

S. M. Eisenstein (1898-1948) studied the golden section in detail. He came to the conclusion that if we are talking about organicity, then there is a golden ratio in proportions. It is S. M. Eisenstein who points out the role of the golden section in painting, gives examples of the manifestation of the golden ratio in poetry, and describes in detail the structure of the golden section of his film “Battleship Potemkin”. He also stops at the structure of the golden section spiral, the so-called logarithmic spiral(Fig. 25). The essence of the structure of this spiral is that, starting from point O, its steps each time increase in the proportions of the golden section (increasing series): OA \u003d 10, 0B \u003d 14 RH = 24, OG = 38, OD = 62, etc.

The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The plan of the city of Auroville (India) is evidence of a spiral building. The DNA molecule is twisted in a double helix. Goethe called the spiral "the curve of life."

Among the roadside grasses grows an unremarkable plant - chicory. Let's take a closer look at it. From the main stem formed xia growth k. The first sheet is immediately located.

1 Nature - 1968.-No. 6.-S. 38.

Building a logarithmic (golden) spiral:

a- along segments of the ascending series of values ​​of the golden ratio; b - in the golden rectangle

The process makes a strong ejection into space, stops, releases a leaf, but shorter than the first one, again ejects into space, but with less force, releases an even smaller leaf, and ejection again. If the first outburst is taken as 100 units, then the second is equal to 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden section* (Fig. 26). In a lizard, at first glance, proportions that are pleasing to our eyes are caught: the length of its tail is related to the length of the rest of the body as 62 to 38.

Both in the plant and animal world, the form-building tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio in proportions of parts is perpendicular to the direction of growth. Sleep and wakefulness of a person within a day, heartbeats and his rest, blood pressure is normal - everything tends to manifest itself in the golden ratio.

On fig. 27 shows the golden proportions of a person in the whole figure and parts of the body. In the head

The golden ratio in nature

Human Golden Proportions:

a - in a figure; b - in the head; in- in the hands

kind carried out the division into symmetrical parts and golden proportions. The repetition of the structure of the whole is manifested in the parts.

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

The great French scientist Pierre Curie (1859-1906) at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment.

The Soviet scientist I. I. Shifranovsky, expounding the ideas of the doctrine of symmetry, explains that symmetry manifests itself in everything that surrounds us.

It permeates the Earth and the Universe, creating an amazing harmony of the material world.

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

The artist is most interested in the external forms of natural bodies, visible to the eye and estimated without geometric measurement. From an early age, at school, art school and institute, he is taught to determine by eye the proportions of a person, a person and a building, a building and a tree, etc. He must be able to depict all this on a plane,

to determine the ratio of light and dark, yellow and blue by eye. This is definitely needed. But it is very bad when an artist ends there. The great artists of the past were great also because they were scientists, thinkers, and poets. They saw much more in things than just the proportions and relationships of light and dark.

Summing up the known data on shaping in nature, we can draw the following conclusions:

the “golden number” 1.618 conveys mathematically the original rhythm of functional structures;

phylotaxis (leaf arrangement) demonstrates original forms of symmetry;

Fibonacci numbers mathematically express certain principles of natural development associated with common law conservation; these principles take place both at the organismal and at the molecular level of the development of living systems;

the principle of "golden symmetry" also operates at the level of inanimate nature as a certain tool for its ordering and progressive evolution;

at a time when the Fibonacci series mathematically characterize the progressive trend of natural selection, i.e. the "striving" of nature for the optimal functioning of its systems, the principle of the "golden section" is an extreme (highest) manifestation of the structural and functional perfection of these systems;

The “golden” spiral with module Ф is the mathematical meaning of the secret of life, which optimally reveals itself both in the plant and animal world, because it is a manifestation of the law of harmonic increase of pulsations.

So, we conclude that among the countless variety of forms in nature that the artist encounters, regularity and consistency reign, the connecting thread of which is the proportion of the golden section.

Everything that exists in nature and is perceived by the human eye has a size and shape. Every natural object is something unified, integral. It is easy to see that nature always creates something whole: a person, a tree, a fish, a horse, a dog, etc. Nothing can be taken away from this whole, reduced without violating integrity. Nothing can be added. It will be redundant and

break the integrity and harmony. For example, six fingers on a human hand, three horns on a bull.

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

Thus, five principles of shaping in nature: 1) integrity, 2) proportions, 3) symmetry, 4) rhythm and 5) important in general. These five principles act as laws of shaping. Whatever we turn to in nature, these five principles of shaping are found everywhere.

Patterns of visual perception

Nature created man. She also created his amazing organ - eye, which transmits to the human brain about 90% of all information about the outside world. Naturally, the question arises: does the regularity of proportioning the human body according to the principles of symmetry and the golden section in its visible parts continue to be preserved in the invisible ones, for example, in the structure of the eye, an organ so important for the artist?

Even Leonardo da Vinci noted that the human eye encompasses the beauty of the whole world, that it directs and corrects all human arts, it is the beginning of mathematics, it gave rise to architecture, perspective and painting.

Architecture is the widest area of ​​manifestation of symmetry and the golden ratio, which go hand in hand. In the buildings of antiquity, gothic, renaissance and more recent times, we constantly see symmetry along the vertical and articulation in the ratio of the golden section along the horizontals. And no matter how precise architecture is in proportions, no matter how geometric its forms, it has always been believed that the final judge of integrity, harmony and beauty

honeycomb structure is the human eye. The masters of antiquity already knew that in a strict geometric drawing, geometric accuracy of proportions, it is necessary to introduce barely noticeable adjustments required by the eye. These adjustments are observed in the architectonics of columns, entablature, cornices, steps. During the Renaissance, it was also considered from what point of view a building or sculpture would be viewed. The Italian painter, architect and art historian Giorgio Vasari (1511-1574) in his introduction to the Biographies says that one should not use another better measure than the judgment of the eye, because if any thing is well measured, but the eye it will seem erroneous, then there will be nothing left but to blaspheme it. The eye must, by its judgment, subtract or add so much as to give proportion, grace, and perfection to the work. Renaissance artists were well aware that “... painting as a kind of fine art rests on the laws of visual perception(our italics - E.K.). This explains the completely exceptional interest in the eye, which was so characteristic of the Renaissance. 1 .

And in the following centuries, many conjectures were expressed about the eye as the only infallible judge of proportionality and beauty. G. Grimm writes that Viollet-le-Duc categorically denied the opinion that took root in his time that proportions in architecture are solely the result of flair, intuition. He believed that proportions in architecture are based on laws and geometrical principles that are consistent with the eye. Le Corbusier, who developed the “modulor” and carried out with its help many beautiful buildings, did not consider it a universal and infallible means of determining proportions: sometimes he was shown unsuccessful, poorly arranged projects, justifying himself by the fact that “this was done with the help of "modulora". “If the modulor,” he replied, “leads you to this disgrace, throw it out. Your eyes should be your only judge. Evaluate everything with your own eyes” 2 .

The human eye is not only a receiver of light radiation. The eye prepares information for the brain, arranges it. Therefore, K. Marx called

1 Kotova E. The Eye and the Laws of Beauty // Art.-1966.- No. 12.-S. eight.

2 Le Corbusier. Modulor.-M., 1976.- S. 85.

its productive organ: “The punishment that Rodolphe inflicted on Mastak is the same that Origen himself inflicted. Rodolphe castrates Mastak, deprives him of one productive organ - the eye. The eye is the beacon of the body.

K. Marx noted that the human eye has become an eye that senses the beauty of form. Which form - natural, created by nature, or a form created by human hands? Obviously, first of all, we should talk about natural forms that pleased the human eye, and then about the forms of objects created by man himself. Nature, as we have already seen, does not create forms by chance, but naturally. In the forms of nature, symmetry, proportionality of parts are manifested, and, as the highest manifestation of the organization of growth, the golden section.

It must be assumed that the human eye was also built by nature not by chance, but in the same order that is characteristic of all creative matter. And he is adapted to the perception of what is around him, which is created by the same nature and according to the same laws.

The eye is arranged in such a way that a person can focus on something that is of particular interest to him at that moment. Interest may be dictated by vital necessity, or it may be caused by the beauty of the form.

There is a small indentation in the center of the fundus, the fovea centralis. This is the place of the best vision. The main line of sight is always directed along the axis: the fovea centralis - the center of the lens - the object under consideration. A yellow spot is located around the central fossa. This is the place of daytime vision and the best color perception. The further away from the macula, the fewer cones the retina contains and the more rods. Rods are adapted for twilight vision and for perception of form. At some distance from the yellow spot is the so-called blind spot. There are no cones or rods here, the eye does not see this place. This is the optic nerve.

Why a blind spot? Couldn't all the fibers of the optic nerve leading to cones and rods have been collected somewhere in the depths of the eye, and not on the surface of the retina? And why did nature place a blind spot here, and not somewhere further, because there is still a lot of space in the eyeball ?!

3 Marx K., Engels F. Holy Family // K. Marx, F. Engels - Works - 2nd ed.-T. 2--C. 196,

b

The geometric scheme of the fundus (a) in drawing by Mariotte for finding the blind spot of the eye (b)

But, as you know, nature does nothing without necessity.

In accordance with its structure, the eye not only transmits to the brain the light signals received from outside, does not mirror everything that is in front of it, but prepares information for the brain in a certain order and subordination. The fovea and macula give the sharpest image and the best color perception. The peripheral part of the field of clear vision gives less clear perception and thus ensures the dominant role of the center. The blind spot does not participate in visual perception at all. Behind the blind spot comes an even more distant periphery, which provides only general perception, being, as it were, a background for the field of clear vision, but it is very sensitive to light signals from moving objects, which biologically makes sense and is very important in the struggle for existence (preservation of individual's life).

And what does the farthest periphery of the eyeball do, where light rays do not fall?

A zero-color is created there and serves as a basis for comparing all the color sensations that the retina gives.

As you can see, the eye is arranged reasonably. But the most interesting and surprising will begin when we draw a geometric diagram of the fundus (Fig. 28, a). The macula is slightly elongated horizontally and corresponds to angles of 6° and 8°. At a distance of 12° from the fovea, a blind spot begins, which corresponds to an angle of 6°. To the outer edge of the blind spot from the middle of the fovea 18°. If we describe a circle with such a radius, we obtain the base of the visual cone corresponding to 36°. it field of clear vision.

Of course, the fundus of the eye is not lined with compasses. This is a living tissue, and the boundaries of these elements of the fundus are blurred, indistinct, but they exist. Thus, we can conclude that the outer border of the blind spot is the border of the field of clear vision. A blind spot is, as it were, a signal: I see poorly here, look away, I want to see. The blind spot of the eye was discovered by the French physicist Edme Mariotte in 1668. He used his discovery for the original amusement of the courtiers of King Louis XIV. Mariotte placed two spectators at a distance of two meters opposite each other and asked them to examine with one eye a certain point from the side, then it seemed to each that his counterpart had no head. The head fell into the sector of the blind spot of the looking eye. There is a well-known drawing by Mariotte for finding a blind spot (Fig. 28, b). If you look at the cross with the right eye (the left one should be closed), bringing the drawing closer or further away from the eye, there comes a moment when the black circle is not visible.

Proportional values ​​within the field of clear vision between the elements of the fundus are the values ​​of the golden section(Fig. 29, a). But also the field of clear vision in the general field of vision ABCD occupies a random position. Rice. 29, b shows that here, too, all the quantities are related as the values ​​of the golden section. In addition, in the size and location of the elements of the fundus within the field of clear vision and within the general field of vision, the commonwealth of symmetry and the golden section is preserved. The blind spot of the right eye is mirrored in the blind spot of the left eye. Nature has remained true to itself here. This leads to the conclusion that the eye prepares information for the brain not only in a certain subordination

Golden proportions of the total field of view (a) and fields of clear

vision (b):

BUTBCD- field vision; abWithd - field of clear vision

in terms of clarity and clarity of perception, but also in the proportions and rhythms of the golden section. It is these proportions and rhythms that the brain needs, since it itself actively works precisely in the rhythms of the golden section. This is evidenced by studies conducted at the MPEI bionics laboratory.

Scientific director of the laboratory prof. Sokolov A. writes: “Man has risen above the animal world thanks to meaningful work, mental work. In this state of the brain dominates beta Wave, which therefore must be considered the main

Noah integral part a unified system of all electrical brain waves. The geometric mean frequency for it is 22.13 Hz, and the two bands are 8.13 Hz and 12.87 Hz. The total range, that is, the difference in extreme frequencies, is 21 Hz

And the ratios of these quantities to each other lead us to an amazing result - the golden section...” 1 . 8, 13, 21 are the familiar numbers of the Fibonacci series! And further: “... The beta wave occupies a special place in the system of brain waves. It corresponds to the most "short" activity algorithm. And therefore, it is most often associated with successful activity, with a pleasant feeling, even with joy. And this is the secret of the golden section” 2 .

If the brain in a state of activity works on the beta wave, i.e., in the rhythm of the golden section, and the eye is a part of the brain that is placed on the periphery, then there is nothing surprising in the fact that the retina of the eye (the fundus of the eye, its elements) is permeated with a proportion mi golden ratio. Information about the outside world, going to the brain through the eye, is the best prepared for it.

So, the human eye is the most perfect creation of nature according to the principle of the golden ratio. It contains the harmony of the whole world. The eye is a silent intermediary between creative matter and thinking matter. The brain, eye and heart are united by one common systemic pattern - the proportion of the golden section. Their synchronous work during the perception and experience of beauty gives a person a sense of harmony, aesthetic experience.

Objectification of light impressions

Art history and art pedagogy make a big mistake by underestimating the role of natural science knowledge. There was a conviction that in the teaching about the work of the eye, visual perception, everything is well known, therefore, neither an art historian nor an artist-teacher has anything to do here. It is not surprising, therefore, that I. M. Sechenov's explanation of the objectification of light impressions remained outside the field of view of artists and art historians. In Physiological Essays, he wrote: “In the feeling of pain, hunger, thirst, fatigue,

1 Sokolov A. Secrets of the golden section / / Technique - youth - 1978. - No. 5. - P. 41-42.

2 There.- S. 42.

Objective light impressions:

a- looking at an object through a magnifying glass; b - coincidence of the external object and the rendered image

in the sensation of taste, smell and hearing we do not feel the external cause that caused the sensation - it is felt by us exclusively as a change in the state of our body. In tactile and visual impressions from external objects, on the contrary, we feel not ourselves, not a change in the state of our body, but the object that caused the impression. What I see stands outside me and is called an external object. Meanwhile, it is easy to see that what I am actually seeing is not an external object, but you-carried outside the image of him drawn on the retina. ... In cases of clear vision of objects, we see their actual image on the retina and bring them outside to the very place where the external

ny subject. This is called objectification light impressions.

We bring out the image of an object drawn by light on the retina. This is easy to verify by looking at any object through a magnifying glass (Fig. 30, a). The item stays in place. The magnifying glass increases the angle at which rays enter the eye, and the image on the retina is drawn larger. Then this image is taken out, and we see the object larger than it is. If you look through the theater binoculars, it will be the same. If you put binoculars in front of your eyes on the contrary, the object will be seen smaller and farther away. This technique is used by some artists to look at the picture if there are no conditions in the studio for moving away to the required distance.

Further, I.M. Sechenov explains that the bringing out of the image drawn on the retina occurs along the very lines along which the image of an object in the eye is built. As a result, the rendered image always coincides with the external object, since the eye is adapted to it, the visual axes of both eyes are directed to it (Fig. 30, b). All other objects lying farther or closer to the one being examined, to the right or to the left, are seen less clearly, because their images on the retina are carried by each eye to a different place. I. M. Sechenov gives as an example a drawing that illustrates this situation. Looking at pictures in a stereoscope, we do not see the pictures themselves, but we see their images drawn on the retinas and brought out. We see the object depicted in the pictures in front of us, in front of the stereoscope. On fig. Figure 31 shows why we don't see the ear clearly when we look at the nose.

Most of the paintings by prominent artists are built in such a way that the main subject (object) is shifted from the middle of the canvas to the right or left and is located on the line of the golden section of the rectangle of the picture. Such a shift of the visual (semantic) center from the geometric middle of the canvas is due to the peculiarities of visual perception. This feature, in turn, is associated with the visual pathways of the brain (Fig. 32). The image of the object to which the gaze is directed at the moment is drawn with light on the yellow spot (if the object is small

1 Sechenov I. M. Physiological essays. - M.; Petrograd, 1923.-Ch. 2.-S. 242-243.

Clear and fuzzy vision of objects

shoy). Information about the received irritation is transmitted along the optic nerves to the visual center of the brain and to the right and left hemispheres. Information from the inner part of the retina of the right eye enters only the left hemisphere, and from the outer part of the retina - only to the right. In the same way, the visual paths from the left eye go. As a result, the macula field is represented in both the right and left hemispheres. The right side of the visual field is represented only in the left hemisphere, and the left side - only in the right. The diagram shows that the area of ​​the blind spot, which is not seen by the right eye, sees the left as the periphery of its field of clear vision, and the area of ​​the blind spot, which is not seen by the left eye, sees the right as the periphery of its field of clear vision. As a result, in everyday life, when looking with two eyes, we do not notice voids in the field of vision.

Since the field of the blind spot is presented only in one hemisphere and is rather indistinct - on the periphery of the field of clear vision - it is a signal for the brain about the fuzziness of perception and the need to shift the gaze to this sector in order to obtain more complete visual information. With the movement of the main line of sight, naturally, the imaginary picture plane also moved.

When we consider something in nature, the eye, fixing the main thing, always prefers the right or left part of the field of clear vision, depending on the significance of the visible. If the value of the visible in both halves of the field of clear vision is the same, the result of the choice depends on the dominant eye. But when the artist organizes the image on the plane of the canvas, he himself needs to decide: which part of the field of clear vision to give preference to - right or left. For this preference to be true, some part of the field of clear vision on the right or left must be cut off. This part of the field is the blind spot area, as it is poorly visible. It is enough to cut off this area by the size of the blind spot, i.e., by 6°, or, which is the same, by 1/6 of the field of clear vision, and then the eye easily perceives the center and the rest of the field (Fig. 33 , b). In the drawing “Vase: two profiles” (Fig. 33, in) the human brain puts forward two hypotheses: we see either a vase or two profiles. In the figure "Trough" (Fig. 33, G) we see the trough upside down, downside down. In the figure "Christmas tree" (Fig. 33, a) brain can't decide which part

visual pathways of the brain

The geometric center of the picture and the line of the golden section:

a - the main subject in the geometric center of the picture; b - chief

an object on the line of the golden section; c - drawing "Vase: two profiles";

G- drawing "Trough"

margins, left or right, give preference. By cutting off the field by 1/6 part, we help the brain solve this problem and bring the main object to golden section line(Fig. 33, b). As a result of such an operation, the center of the field of clear vision, where the object under consideration is always located, left the geometric center of the canvas and entered the golden section. Therefore, it is more correct to speak not about shifting the center of attention from the geometric center of the picture, but about cutting off 1/6 of the imaginary picture.

Further, we will not talk about general theoretical issues of composing a picture, but about how the artist should dispose of the surface of the canvas, how to organize the image for the best perception by the viewer, so that the picture is easy to read. In other words, let's call it practical composition. We mean what Goethe was talking about, that it is necessary to study the laws by which we see, to learn how to turn an object into a picture, that is, to transform the visible into filling the plane in the picture.

It has already been said above that the thinkers-artists guessed about the existence of the laws of composition, which directly follow from the laws of vision.

Scientific composition theory

The great masters of painting possessed the secrets of compositional and pictorial mastery. Their works speak for themselves. People's Artist of the USSR B.V. Ioganson advised young artists to copy the works of great masters and not be upset if at first there are mistakes, failures: “... there will definitely be mistakes. These mistakes will lead to reflection, conjecture and, finally, to secret laws of construction(author's italics - F.K.) paintings, similar to the logic of architectural forms "".

There is an opinion among some part of modern artists that there are no laws of composition, there is nothing to teach here and there is no need. But almost every artist secretly knows that the laws of composition still exist. Everyone knows that questions of this kind, in the figurative expression of A. Herzen, "one cannot impose stones on the neck." This invincibility of the subject is very often reminded to the artist of his own practice. Starting to paint a picture on canvas, he sees that something is wrong, the image does not look good, the picture does not enter the eye, it is falling apart. And start painful alterations, rewriting, sometimes exhausting and painful. Instead of successfully completing the picture, the artist exhausts himself with doubts, his strength goes to what should have been decided at the very beginning. It is known that Rubens made a sketch and completed the picture in a week.

Definition of composition

It is believed that the first to use the word “composition” in relation to a work of art was the Italian scientist, art theorist of the Early Renaissance, painter and architect Leon Baggista Alberti (1404-1472), who believed that the composition is such a rational basis for the living scripture, by which the parts of visible things

1 See: School visual arts: Issue. 1.-M., I960.- S. 5.

add up to a picture. In subsequent centuries, and up to the present day, attempts to give an exhaustive definition of the composition do not stop. We present some of them.

Composition is such a comparison of individual forms, in which they are linked into a new whole of a higher order.

F. Schmit

Composition ... in literature and fine arts - the construction of a work, the ratio of individual parts (components) of a work that form a single whole.

Dictionary of foreign words

All kinds of art ... are characterized by the presence of such an important side art form like a composition.

V. Vanslov

Composition is the main form of a work of art.

N. Volkov

In 1961, the People's Artist of the USSR E. A. Kibrik complained that when composition programs were being developed at the Academy of Arts, it was not possible to find a formulation acceptable to everyone that there is a composition.

The development of a scientific definition of the term “composition” is largely hampered by the various use of this word that has become established in the practice of artists. Composition is a subject at an art institute or college. A composition is a thematic painting, as opposed to a portrait, landscape and still life. The definition of composition was further complicated by the desire to combine in one word the concepts of the unity of the content and form of a work of art. In the program on composition for art and graphic faculties of pedagogical institutes, one can read that at present the concept of “composition” is considered as a dialectical phenomenon in its essence, since it has absorbed both the structural organization of the artistic image and the system ideological-thematic and formal-plastic connections and dependencies, and the most important regularities in the construction of a work of art, the process of its creation and perception. These main levels of the concept of “composition” are also the main directions for the formation of skills in compositional activity.

The difficulties of finding the final definition of the term “composition” are also evidenced by the attempts of E. V. Shorokhov to give this definition in the textbook “Composition”, intended for students of art and graphic faculties.

pedagogical institutes. He writes: “... A more complete definition of the concept of “composition” will sound like this: “The composition of a work of fine art is the main artistic form of a work of fine art, uniting all other forms, characterized as a whole with fixed, naturally connected between themselves and with whole parts (elements), in which nothing can be moved or changed, from which nothing can be taken away and to which nothing can be added without damaging the artistic image, this is a whole that is inseparably united with meaning ( idea, content) of the work" ".

A multitude of definitions of a certain concept come up every time, not only in the visual arts, but also in other areas of culture, when there is no clear understanding the content of the concept. In order not to get lost in the wilds of word creation, let us dwell on the academic definition of the concept of “composition”.

Composition- it is the structure of the form of a work of art, aimed at revealing the intentla author. Composition (from Latin compositio - compilation, binding) - this is the construction of a work of art, due to its content, character and purpose, and largely determining its perception.

It is highly advisable to familiarize yourself with the statements about the composition of well-known Soviet artists and art theorists, who feel the pulse of time, insight and striving for truth. KF Yuon notes that a thoughtful and complete composition is considered as a figurative formula in which all parts of the picture are brought to semantic and formal unity. V. Vanslov says that patterns of composition differ significantly in architecture and painting, music and poetry. But composition as a plan and scheme for constructing a work, as the correlation and organization of its sections, parts, characters, and so on, as the subordination of the whole and its elements, is inherent in the artistic form of a work of any kind of art. Composition is a complete integrity, determined by the meaning of the work, says N. Volkov.

1 Shorokhov E.V. Composition.-M., 1986.-S. 10-11.

We give a number of examples on the definition of the concept of composition, expressed by artists and art theorists.

Artists know very well ... that until a composition is found ... it is useless to paint faces. No matter how strongly they are written, they mysteriously lose their power until they find their place in the pictorial space, in the compositional series.

V. Lenyashin

I remember how he (B.N. Zuev) taught me that there must be an odd number of figures in a composition. I remember this - there is something in this rule, at least that it is more natural to compose an “indivisible” composition from an odd number of figures, since an even number is more easily divided into equal parts.

E. Kibrik

Composition is the creative organization of a picture. In no case should it be confused with the placement of objects on the canvas. At first glance, it may seem that a landscape depicting an empty sea below, and a cloudless sky above, is a picture that does not have a composition ... In fact, the ratio of the size of the sky in length and width to the sea is already a composition. The tone of the sky and the sea, the size of the whole picture - all these are elements of the composition.

N. Krymov

Composition is always a construction (color, linear, etc.) that reveals semantic connections. Composition is always an interpretation of the plot.

N. Volkov

Over time, you become convinced that composition is the basis of all creativity, that everything begins and ends with it, that all the components of the craft are focused in it, as it were. It can be elementary, or it can be verified by mathematical logic, subject to the intuition of the artist. ...Composition satisfies me when both the figurative-semantic structure and decorative-plastic tasks are in organic interaction.

V. Sidorov

Our time is the age of composition. If earlier it essentially meant sketching, that is, working on the plot of the picture (it provided for the placement of figures and the interior in the format of the picture), now the meaning and understanding of the composition has changed. It can be in a still life and a drawing, in a sketch. This is not only the ratio of lines, silhouettes, colors - these are mutual influences, and internal connections of elements, persons<ей, линий, форм на основе стилевого единства, чувства цело- стности.

S. Grigoriev

It is important for me, when I start a new canvas, first of all, to feel the whole course of further work. In this process of creating a thing, I consider composition to be the main thing for me. Without it, the picture cannot exist, even when color relationships are correctly found and the state is expressed. Searching for a composition means, for me, first of all, to organize the canvas internally, to determine its structural backbone, its basis.

E. Bragovsky

The question of composition is the most difficult, key issue.

B. Ioganson

...“Questions of composition” still remain only “questions”. ... The right path to a theoretical solution of this complex issue has already been outlined; but there is still no single view of composition, and there is no established methodology for teaching composition.

M. Etkind

The flourishing of the thematic painting is by no means conducive to the slowly improving teaching of composition, especially multi-figure composition, in our universities.

E. Kalinin

The number of examples of definitions of composition given by artists and art theorists can easily be continued. It still follows from them that composition- is the structure, the interconnection of parts,ensuring the integrity of the image, directedlent to the disclosure of the content, ideas of the work.

Content artwork arises in the artist's head as an idea; sometimes it is dictated by the customer, the competition program, and so on, and the composition is created by the artist, the form is built. When there is a unity of content and form, then there is a complete integrity - a picture, a sculpture, an etching. The content of the picture is composed, the composition is built. In the end, it is one whole. To work on the form means to work on the content, and vice versa. The artist's logic is a composition; his intuition is the geometry of the canvas, rhythms, proportions, relationships, because his eye and brain are unconscious geometry, measurement. All knowledge is a sensory dimension.

The search for the laws of composition

Working on a sketch of a painting is always a “ride into the unknown”. It almost always leads the artist to the idea of ​​whether there are any universal patterns of compositional construction and, if so, what determines them. Why did it work in one case and not in the other?

I. E. Repin, already a famous artist, recalled that in his youth the laws of composition haunted him. At the Academy of Arts, they were taught to compose bas-reliefs, arrange figures according to plans, and so on. And only on the Volga, in the bushes on Lysa Gora, did he for the first time comprehend the laws of composition: relief and perspective, space

foreground and background. Nature taught composition.

I. N. Kramskoy knew that they, “these damned laws”, exist apart from the personal taste and temperament of the artist. You need to understand them and obey them. His letter to the artist F. A. Vasiliev is known, in which he points out the need to build a picture in accordance with the laws of vision, because when looking into the distance you cannot see the ground under your feet, and when looking down - the sky above your head.

People's Artist of the USSR B.V. Ioganson was convinced that there are laws of composition, but it is extremely difficult to reveal them, to find general rules. In 1950, in the article “On the Question of Composition,” he wrote: “The question of composition is the most difficult, key issue of fine art ... Is it possible to teach composition, are there laws for composition the same as for drawing ? ... I think you can teach. Be sure to exist and there are regularities in the composition. ... The question of the composition should be well theoretically substantiated. Therefore, in my opinion, it is too early to organize a department of composition in art institutes... For the time being, it is necessary to teach a student composition in the analysis of masterpieces of Russian and world art. There is no other way out, in my opinion.

In 1980, N. Rostovtsev’s book “Methods of teaching fine arts at school” was published, in which the author notes that the old Academy of Arts “taught its students the rules and laws of composition, and today artist-teachers who respect the traditions of realism art, they are trying to restore them” 2 . Further, N. Rostovtsev cites interesting data that speak eloquently about theoretical confusion in matters of composition: Academician M. Manizer indicates ten rules of composition,

A. Laptev - five rules, A. Deineka - nine (the seventh rule is the golden ratio), academician

V. N. Yakovlev - twelve general laws of composition and forty (!) particular rules for the compositional solution of a thematic picture. N. Rostovtsev himself considers eight laws, which include asymmetry, equilibrium, statics, dynamics, etc.

1 Ioganson B.V. For skill in painting.- M., 1952.-

2 Rostovtsev N. N. Methods of teaching fine arts at school.-M., 1980.-S. 172.

N. Rostovtsev does not say that the “old academy” clearly delimited the concepts of composing a picture and its composition. The semantic side of the picture was composed and presented in the program task, taken from myth or the Bible, from domestic history, prompted by real life, and the composition was built. The laws of construction of the form of a picture for a given content, easily and freely perceived by the eye, are the laws of composition.

In 1986, the Prosveshcheniye Publishing House published a much-needed textbook “Composition” by E. V. Shorokhov for students of art and graphic departments of pedagogical institutes.

E. V. Shorokhov divides the laws of composition into basic (general, objective) and particular. To main related laws: integrity, con-trusts, novelties, and subordination all means of composition to the ideological concept. To private we are assigned laws of vitality and influence"frames" on the composition of the image on the plane. Note that in the first edition of E. V. Shorokhov’s textbook “Fundamentals of Composition” (M., 1979), the basic laws included the law of typification (vitality), but there was no law of novelty. The law of wholeness was called the law of wholeness, which is not the same thing. Such laws of shaping as symmetry and rhythm are referred to the rules of composition in the first and second editions. This shuffling of the laws of composition, like cards in a deck, became possible because the concepts of "laws of art", "laws of the picture" and "laws of form (composition)" were allowed to be confused. One was replaced by another. The concept of law and rule is freely interpreted.

It is possible to find the laws of composition and build a scientific theory only when we do not construct the laws of composition on the basis of inferences, but proceed from the really existing objective laws of shaping in nature and art.

What is the scientific theory of composition

Scientific theory of composition, like any other theory, based on scientific knowledge. Even Leonardo da Vinci noted that only that knowledge is true, which is verified by mathematics; there is no certainty in the sciences where none of the mathematical sciences can be applied.

The words of K. Marx are well known that science reaches perfection only when it succeeds in using mathematics. The science of composition is no exception. The theory of composition will become truly scientific and the laws of composition will be irrefutably convincing only when we find the possibility of applying to them at least one of the mathematical sciences, for example, geometry or arithmetic, or both.

The scientific approach to the mysteries of art was laid down in Ancient Greece. Greek artist-educators encouraged their students to master art with the help of science.

Sensually perceiving, a person measures. But feelings can lead to errors (visual illusions). The human mind not only enriches feelings, but can also verify their truth by measuring. Number brings order to the world. The Italian scientist, Renaissance art theorist, painter, architect, poet and musician L. Alberti expressed the wish that the painter should be as knowledgeable as possible in all the liberal arts, but above all, that he should learn geometry. Mathematics is a science that makes up for the imperfection of our senses. A reasonable combination of the sensual, intuitive with the rational, scientific in the artist's work should become the basis for further improvement of the art of painting.

In the middle of the XIX century. In Russia, a direction of aesthetic thought arose and then took shape, which laid the criterion of scientific character as the basis for the approach to the phenomena of art, data from the natural sciences began to be widely involved. Interest in the golden section has especially increased. In relation to architecture, it was interpreted as follows: 1) the golden section dominates architecture; 2) the golden section dominates in nature; 3) the golden section dominates architecture because it dominates nature, and the architect’s creativity is a continuation of nature’s creativity: “What goes against the creativity of nature cannot be beautiful. And vice versa. The laws of organic nature are also the laws of architectural structures, they serve as the basis for the beauty of architecture.

1 Kirichenko E. I. Architectural theories of the 19th century in Russia.-M., 1986 -S. 204.

The deep interest of P. P. Chistyakov and other Russian artists in the scientific data of their time is known. At the same time, they put the thought, the idea, in the first place. P. P. Chistyakov said, what he subordinates everything to the idea. The plot corresponds to the technique, the idea subjugates the technique. As in life, so in art, the idea determines everything. The idea of ​​serving the people inspired the Wanderers to create significant works, close and understandable to the people, expressing their innermost feelings and aspirations. Deep ideology is an inalienable quality of the best works of Soviet fine art.

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INTRODUCTION

In the modern world, and in particular, in the creative fields of contemporary art, such a concept as the "golden section" is widely known. The fact is that this concept has become almost synonymous with the word "harmony". And, of course, the essence of this term is inextricably linked with mathematics, or rather, with its section called "Relations and Proportions", which is studied in the 6th grade mathematics course.

The information presented in the textbook by Vilenkin N.Ya. et al. Math 6 is very concise and intended to be more of an introduction than a learning experience.

The history of the doctrine of proportions is the history of the search for the theory of harmony and beauty. All the efforts of ancient aesthetics and the aesthetics of the Renaissance were aimed at finding the laws of beauty in the commensurability of individual parts, as well as parts and the whole. Even the most perfect creation of nature - man - was created in proportions of continuous division. The most famous historical monuments of art and architecture are said to have been created according to the principle of the "golden section". These are the Parthenon in Greece, Notre Dame de Paris in France, the pyramid of Cheops in Egypt, the Cathedral of the Resurrection of Christ in St. Petersburg, St. Basil's Cathedral in Moscow and many others. What is the essence of this concept and how to apply it?

It is the smallness of the information available in the available source and the desire to learn about the "golden section" that much more prompted the authors of this work to conduct this study.

Target works - to investigate the issue of the influence of the presence of the "golden section" in the paintings of artists on their aesthetic perception.

Respectively, tasks of this work are as follows:

Learn all about the discovery of the concept of the "golden section" and its author;

Understand in detail the essence of the term "golden section";

Highlight the areas of creativity in which the "golden section" is applicable, and how this concept is applied in the visual arts;

Get acquainted with the work of famous artists, including Vladimir ones;

To analyze the works of artists for compliance with the principle of the "golden section";

To explore the question of the importance of using this principle in the manufacture of a picture for its perception by the viewer.

Before carrying out the work, together with the supervisor, a hypothesis was built: in most of the works of artists (both famous and not) the principle of the "golden section" was used. To prove this hypothesis, a sample of paintings was made to study for the presence of lines of the "golden section".

The novelty of this research work, the author considers its practical part, which clearly illustrates the possibility of applying this principle by artists when creating their paintings, and the study of the influence of the presence of the "golden section" on the aesthetic perception of the picture by polling a sample of disinterested persons for sympathy for the presented image.

Methods of theoretical research (in particular, abstraction, axiomatic, analysis and synthesis, induction and deduction, ascent from the abstract to the concrete);

Methods of empirical research (in particular, measurement and comparison).

There is a lot of literature devoted to the "golden section". For the study, Vasyutinskiy N.'s book "The Golden Ratio" was taken as a basis, since the style of presentation of the material is easy to understand, and there is a lot of information about the history of the discovery of the "golden section", its application in various fields. The book consists of four parts.

The first part, "Illumination of Pythagoras", tells about the history of the discovery of the concept, and the amazing facts of the presence of the "golden section" principle in geometry. The second part, “Fibonacci Chemistry”, tells about the connection between the famous Fibonacci numbers and the “golden section”. The third part, "Formula of Beauty", talks about the connection between the structure of the human body and the "golden section", and not only. The last, fourth part, called "Algebra of Music", is devoted to the analysis of harmony in music.

After reading this literary work, it becomes clear that the search for ideal proportions for creating works of art and culture has worried mankind for many centuries and even centuries. After finding this amazing proportion, the leading scientists of their time began to devote their scientific works to the study of the presence of traces of the "golden section" not only in art, but also in wildlife.

The textbook by V.F. Kovalev aroused no less interest among the author of this study. "The Golden Section in Painting", which reveals all aspects of the application of the principle of the "golden section" in the field of fine arts.

"GOLDEN SECTION" OR DIVINE PROPORTION

1. HISTORY OF THE CONCEPT

Like any term, the concept of the “golden section” was once introduced by someone, but sources differ on the issue of the privilege of discovering this concept. Some argue that the ancient Greek mathematician and philosopher Pythagoras 1 was the discoverer of the golden ratio. There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the pyramid of Cheops, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamen testify that the Egyptian masters used the ratios of the golden division when creating them 2 .

In the era of the Italian Renaissance, a new wave of passion for the golden ratio arises. The golden ratio is elevated to the rank of the main aesthetic principle. Leonardo da Vinci calls it "Sectio autea", from where the term "golden section" or "golden number" comes from. Luca Pacioli in 1509 writes the first work on the golden ratio, entitled "De divina Proportione", which means "On the divine proportion." Pacioli found in five Platonic solids - regular polygons (tetrahedron, cube, octahedron, icosahedron and dodecahedron) thirteen manifestations of the "divine" proportion.

The Dutch composer Jacob Obrecht (1430 - 1505) makes extensive use of the golden ratio in his musical compositions, which are likened to "a cathedral created by a brilliant architect."

After the Renaissance, for almost two centuries, the golden ratio was forgotten. In the middle of the 19th century, the German scientist Zeising makes an attempt to formulate a universal law of proportionality and, at the same time, rediscovers the golden section. He shows that this law is manifested in the proportions of the human body and in the body of those animals, the forms of which are distinguished by grace. In the body of ancient statues (in particular, in the statue of Apollo Belvedere) and well-built people, the navel is the point of dividing the height of the body in the golden section. Zeising finds proportional relationships close to the golden ratio in some Hellenic temples (in particular, in the Parthenon), in the configurations of minerals, plants, and music chords.

The golden section arises as a result of solving the following geometric problem. On the segment AB need to find such a point FROM, to AND YOU = FROM YOU.

At the end of the 19th century, the German psychologist Fechner conducted a series of psychological experiments to determine the aesthetic impression of rectangles with different aspect ratios. The experiments turned out to be extremely favorable for the golden section. The essence of the experiment consisted in choosing from ten rectangles, among which there was a “golden” one (with sides whose ratio of lengths gave the golden section), the subject had to choose one. And now, about 22% total number The subjects were chosen precisely by the “golden rectangle”.

In the 20th century, interest in the golden ratio revived with renewed vigor. In the first half of the century, the composer L. Sabaneev formulated the general law of rhythmic balance and, at the same time, substantiated the golden section as a certain norm of creativity, the norm of the aesthetic construction of a musical work.

In the second half of the 20th century, representatives of almost all sciences and arts (mathematics, physics, chemistry, botany, biology, psychology, poetry, architecture, music) turn to Fibonacci numbers and the golden ratio.

The mathematical theory of biological populations goes back to the “rabbit problem”, which is associated with the emergence of Fibonacci numbers. The patterns described by the Fibonacci numbers and the golden ratio are found in many phenomena of the physical and biological world (“magic” nuclei in physics, brain rhythms, etc.)

Soviet mathematician Yu.V. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Academician G.V. Tsereteli discovers the golden ratio in Shota Rustaveli's poem "The Knight with the Panther's Skin". Composer and music theorist M.A. Marutaev, developing the ideas of Zeising, Sabaneev, and using the latest achievements in physics, takes a new step in the development of the concept of harmony as a pattern.

In recent decades, Fibonacci numbers and the golden ratio have unexpectedly shown themselves to be the basis of digital technology. Independently of each other in various areas of digital technology, a number of non-traditional directions in the theory of information coding are emerging.

1. "GOLDEN SECTION" IN PAINTING

Before defining the golden ratio, you need to familiarize yourself with the concept of proportion. Proportion (Latin proportio) is an equality between two ratios of four quantities:

a:b = c:d, and a, b, c, d ≠ 0.

golden ratio- this is such a proportional harmonic division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or, in other words, the smaller segment is related to the larger one in the same way as the larger one is to everything, i.e. c:b = b:a or a:b = b:c(Fig. 1)

Rice. one. Geometric representation of the division of a segment in the golden section

It is believed that the value of the golden ratio when finding the ratio of the largest to the smallest is approximately equal to 1.618.

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

The construction of a series of segments of the golden ratio can be done both in the direction of increase (increasing series) and in the direction of decrease (descending series). In the latter case, it is necessary to subtract the smaller one from the larger segment - we get an even smaller one: b - a \u003d d, etc. (Fig. 2).

Rice. 2. A series of segments of the golden ratio

When considering the search for the golden section line in the picture, each of the sides of the picture (its length and width) is divided into segments in the golden ratio. Then, vertical and horizontal lines are drawn through the found points, and the result is analyzed. The points of intersection of the golden section lines are called golden dot. There are four options for constructing such a point in the picture (Fig. 3).

Fig.3. Golden section lines and diagonals in the picture

The fact is that the length of the picture can be divided in the golden ratio in two ways - by setting aside most of it from the left edge or from the right. Similarly, with width - setting aside from above or below. This gives rise to four options.

It is believed that if you divide a segment equal to 100 in proportion to the golden section, then the larger part will be equal to 62, and the smaller 38 (see Fig. 3).

The golden ratio was used by artists in the compositional construction of paintings. A simplified method was developed when the plane of the picture was divided into 10 parts vertically and horizontally. The line of the golden section was outlined in relation to 6 and 4 parts (Fig. 4, a). This did not give a ratio of 62:38, but gave a close ratio of 60:40. In practice, this was enough to navigate and place the main figure or group of figures in the most favorable place for this picture.

The same result was obtained by the artists of the Munich Academy by dividing the picture into 5 parts. The golden ratio was taken in the ratio 3:2, which is the same, because halving 10, 6 and 4 gives 5, 3 and 2. The main figure of the painting or a group of figures was placed on the line of the golden section (Fig. 4, b).

Rice. four. Division of the picture:

a- 10 parts in the Russian Academy of Arts; b- in 5 parts at the Munich Academy of Arts

Consequently, the principle of the golden ratio has been used and is currently being used by artists around the world when working on a picture for the most successful arrangement of the depicted objects on it.

2.3. THE "GOLDEN SECTION" IN THE WORKS OF FAMOUS VLADIMIR ARTISTS

Britov Kim Nikolaevich (8.01.1925 - 5.01.2010).

Honored Artist of the RSFSR. People's Artist of Russia. In 1997 he was awarded the Gold Medal of the Academy of Arts of Russia. Winner of the I. Levitan Prize. Since 1954 he has been a member of the Union of Artists of the USSR. For 55 years of creative activity he took part in 220 exhibitions in our country and abroad. The artist's works are in the State Tretyakov Gallery, the State Russian Museum, the Vladimir-Suzdal Historical, Architectural and Art Museum-Reserve, in many Russian regional museums, in the Easton Academy of Arts (USA), the Kim Il Sung Museum (DPRK), the New Munich Gallery (Germany). ), as well as in numerous public and private collections in Europe, Asia, North and Latin America. Honorary resident of the city of Vladimir (2003) 3 .

Painting «Village Lyubets. The snow has fallen." Original image dimensions 16.1 cm by 11.9 cm (2002) 5

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.35: 4.55 ~ 1.615

11,9: 7,35 ~ 1,619

Painting "Sunflowers" (2007). Original image dimensions 16.1 cm by 12.7 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.85: 4.85 ~ 1.618

12,7: 7,85 ~ 1,618

Painting "Nerl blue" (2009) Dimensions of the original image 8.5 cm by 6.3 cm

Calculations of the lines of the golden section:

By length 5.25: 3.25 ~ 1.615

8,5: 5,25 ~ 1,619

By width 3.9: 2.4 ~ 1.625

6,3: 3,9 ~ 1,615

Kokurin Valery Grigorievich(born 1930, Vladimir).

(photo taken on the website of the gallery of modern Vladimir painting "Britov. Yukin. Kokurin" http://www.britov.ru/authors/kokurin_valerij/)

Member of the Union of Artists of Russia (1960)

Awarded the first prize of the Central Committee of the Komsomol (1962)

Laureate of the Regional Komsomol Prize named after Gerasim Feigin (1979)

People's Artist of the Russian Federation (1998)

Diploma of the Russian Academy of Arts (1999)

Gold medal of the Russian Academy of Arts (2005)

Laureate of the Prize of the Union of Artists of Russia named after A.P. Gritsaya (2006) 4

Gold medal to them. IN AND. Surikov (2010) VTOO "Union of Artists of Russia"

The artist's paintings are in the collections of the State Tretyakov Gallery, the State Russian Museum, the Murom Historical and Art Museum, the Vladimir Historical and Art Museum-Reserve, as well as in private collections in many countries of the world 5 .

Painting "Village in the Carpathians" (1984) Dimensions of the original image 16.1 cm by 12.7 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.85: 4.85 ~ 1.618

12,7: 7,85 ~ 1,618

Painting «Rostov. Toward Evening (1989) Original image dimensions 16.1 cm by 11.6 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.17: 4.43 ~ 1.618

11,6: 7,17 ~ 1,618

Painting "Autumn in Snovitsy" (1975) Dimensions of the original image 16.1 cm by 11.7 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.23: 4.45 ~ 1.617

11,7: 7,23 ~ 1,618

Yukin Vladimir Yakovlevich(1920, Mstera - 2000, Vladimir).

(photo taken on the website of the Vladimir regional branch of the VTOO "Union of Artists of Russia" http://www.vshr.ru/)

Member of the Union of Artists of Russia (1952)

People's Artist of the Russian Federation (1995)

Silver medal of the Academy of Arts of the USSR (1991)

Laureate of the State Prize of the RSFSR (1992)

Member of the Great Patriotic War.

State awards:

Order of the Patriotic War II degree (1985)

Medal "For the victory over Germany" (1945)

Medal "For the Liberation of Prague"

Medal "XX Years of Victory"

Medal "XXX Years of Victory"

Medal "40 Years of Victory"

Medal "50 Years of Victory"

Painting "Birches" (1952) Dimensions of the original image 16.1 cm by 11.4 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.05: 4.35 ~ 1.620

11,4: 7,05 ~ 1,617

Bridge painting (1950-1990s) Dimensions of the original image 16.1 cm by 13.2 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 8.16: 5.04 ~ 1.619

13,2: 8,16 ~ 1,618

Painting «Vladimir. Knyaginin Monastery "The dimensions of the original image are 16.1 cm by 12.9 cm

Calculations of the lines of the golden section:

Lengthwise 9.95: 6.15 ~ 1.618

16,1: 9,95 ~ 1,618

Width 7.97: 4.93 ~ 1.617

12,9: 7,97 ~ 1,618

The painting "Boats are floating on the river" Dimensions of the original image 17.8 cm by 11.9 cm

Calculations of the lines of the golden section:

Length 11: 6.8 ~ 1.618

17,8: 11 ~ 1,618

Width 7.35: 4.55 ~ 1.615

11,9: 7,35 ~ 1,619

Conclusion: in most of the paintings presented, the application of the principle of the golden ratio can be traced.

2.4. "GOLDEN SECTION" IN THE WORKS OF DOMESTIC AND FOREIGN ARTISTS

I. I. Shishkin

Painting "Rye". Original image dimensions 12.8 cm by 7.3 cm

Calculations of the lines of the golden section:

By length 7.9: 4.9 ~ 1.612

12,8: 7,9 ~ 1,620

Width 4.5: 2.8 ~ 1.607

7,3: 4,5 ~ 1,622

Lubomir Kolarov

Painting "Ship Dreams". Original image dimensions 13.1 cm by 8.5 cm

Calculations of the lines of the golden section:

By length 8.1: 5 ~ 1.620

13, 1: 8,1 ~ 1,617

Width 5.25: 3.25 ~ 1.615

8,5: 5,25 ~ 1,619

Thomas Kinkade

Painting "Magic Landscape". Original image dimensions 13.35 cm by 10 cm

Calculations of the lines of the golden section:

By length 8.25: 5.1 ~ 1.617

13, 35: 8,25 ~ 1,618

Width 6.18: 3.82 ~ 1.617

10: 6,18 ~ 1,618

Painting "Hare" Dimensions of the original image 7.1 cm by 6.4 cm

Calculations of the lines of the golden section:

By length 4.39: 2.71 ~ 1.619

7,1: 4,39 ~ 1,617

Width 6.18: 3.82 ~ 1.617

10: 6,18 ~ 1,618

Leonardo da Vinci

Painting "Last Supper". Original image dimensions 15.5 cm by 7.1 cm

Calculations of the lines of the golden section:

By length 9.58: 5.92 ~ 1.618

15,5: 9,58 ~ 1,617

Width 4.39: 2.71 ~ 1.619

7,1: 4,39 ~ 1,617

I. I. Shishkin

Painting "Ship Grove". Original image dimensions 14.7 cm by 9.2 cm

Calculations of the lines of the golden section:

By length 9.08: 5.62 ~ 1.615

14,7: 9,08 ~ 1,618

By width 5.7: 3.5 ~ 1.628

9,2: 5,7 ~ 1,614

William Turner

Name unknown. Original image dimensions 15.5 cm by 9.9 cm

Calculations of the lines of the golden section:

By length 9.57: 5.93 ~ 1.613

15,5: 9,57 ~ 1,619

Width 6.11: 3.79 ~ 1.612

9,9: 6,11 ~ 1,620

Leonardo da Vinci

Painting "Saint Anna and Mary with the Child". Original image dimensions 10.4 cm by 7 cm

Calculations of the lines of the golden section:

By length 6.42: 3.98 ~ 1.613

10,4: 6,42 ~ 1,619

Width 4.32: 2.68 ~ 1.611

A. K. Savrasov

Painting "The Rooks Have Arrived". Original image dimensions 9.5 cm by 7.3 cm

Calculations of the lines of the golden section:

By length 5.87: 3.63 ~ 1.617

9,5: 5,87 ~ 1,618

Width 4.51: 2.79 ~ 1.616

7,3: 4,51 ~ 1,618

Conclusion: in all the presented paintings, the application of the principle of the "golden proportion" can be traced.

2.5. INFLUENCE OF COMPLIANCE WITH THE PRINCIPLE OF THE "GOLDEN SECTION" ON THE PERCEPTION OF THE PICTURE

After finalizing the previous paragraph, the author of the research work, together with the supervisor, conducted a survey among others in order to clarify their attitude to the paintings (“like - dislike”) and analyzed the result.

Painting "Birch Grove". Original image dimensions 10.9 cm by 6.3 cm

Calculations of the lines of the golden section:

Length 6.75: 4.15 ~ 1.626

10,8: 6,75 ~ 1,614

By width 3.9: 2.4 ~ 1.625

6,3: 3,9 ~ 1,615

Painting "Golden Autumn". Original image dimensions 16.3 cm by 8.1 cm

Calculations of the lines of the golden section:

By length 10.1: 6.2 ~ 1.629

16,3: 10,1 ~ 1,613

Width 5: 3.1 ~ 1.612

In this survey, the percentage of people who liked the first picture, possibly having a "golden ratio" (in our opinion), was 50%. The percentage of people who chose the second picture in the survey, which definitely has a "golden ratio", was 50%. This is proved by the fact that two paintings that have a "golden ratio" are equally pleasing to contemplators.

Painting "Golden Autumn". The dimensions of the original image are 16.1 cm by 10 cm.

Calculations of the lines of the golden section:

By length 9.9: 6.2 ~ 1.600

16,1: 9,9 ~ 1,620

Width 6.2: 3.8 ~ 1.631

Painting "Streets of St. Petersburg". The dimensions of the original image are 15.2 cm by 11.6 cm.

Calculations of the lines of the golden section:

By length 9.4: 5.8 ~ 1.620

15,2: 9,4 ~ 1,617

Width 7.2: 4.4 ~ 1.636

11,6: 7,2 ~ 1,611

In this survey, the percentage of people who liked the first picture, which has a "golden ratio" (in our opinion), was 65%. This proves the fact that the "golden ratio" affects perception.

Painting "Gulf of Naples". The dimensions of the original image are 15.8 cm by 9.8 cm.

Calculations of the lines of the golden section:

By length 9.8: 6 ~ 1.633

15,8: 9,8 ~ 1,612

Width 7.5: 4.6 ~ 1.630

12,1: 7,5 ~ 1,613

Painting "Sonnet". The dimensions of the original image are 15.4 cm by 11.4 cm.

Calculations of the lines of the golden section:

By length 9.5: 5.9 ~ 1.610

15,4: 9,5 ~ 1,621

Width 7.04: 4.36 ~ 1.614

11,4: 7.04 ~ 1,619

In this survey, the percentage of people who liked the first picture, which has a "golden ratio" (in our opinion), was 75%. This proves the fact that the "golden ratio" affects perception.

Painting "Magic Landscape". The dimensions of the original image are 13.35 cm by 10 cm.

Calculations of the lines of the golden section:

By length 8.25: 5.1 ~ 1.617

13, 35: 8,25 ~ 1,618

Width 6.18: 3.82 ~ 1.617

10: 6,18 ~ 1,618

Painting "Autumn mood". The dimensions of the original image are 8.7 cm by 6.4 cm.

Calculations of the lines of the golden section:

By length 5.4: 3.3 ~ 1.636

8,7: 5,4 ~ 1,611

Width 3.95: 2.45 ~ 1.612

In this survey, the percentage of people who liked the second picture, which does not have the lines of the "golden section" (in our opinion), was 60%. In this case, the author believes that such an unobvious choice is due to the difference in the subject matter of these paintings, the types of objects depicted, the color palette, and, in general, the areas of fine art in which these paintings are written.

Based on the presented statistical data, the author came to the conclusion that when the artist uses the principle of the "golden ratio" when creating a picture, its aesthetic perception by the contemplator leaves a more favorable impression compared to the perception of an artwork in which this principle was not respected.

3. CONCLUSION

When posing a problematic issue, the author, together with the supervisor, planned to dedicate the work to calculating the compliance of the architectural monuments of the city of Vladimir with the principle of the golden ratio. However, the work was not carried out due to the lack of initial statistical data - it was not possible to find the real dimensions of the architectural structures.

In the process of working on the study, the author studied various sources of information on the relevant topics. A lot of interesting facts were analyzed together with the head of the work. After getting acquainted with the principle of applying the golden section in painting, the main part of the research work was carried out.

Information about modern famous artists of the Vladimir land was obtained by the author from open sources on the Internet. Images of all paintings are taken there. The selection of paintings was made on the basis of objects of images - these are paintings with landscapes of Vladimir and the Vladimir region, and paintings, presumably based on the principle of the golden ratio. Then the author of the work examined the paintings of both domestic and foreign artists for the presence of lines of the "golden section", the images of which were taken from open sources on the Internet. Assumptions were put forward by the author of the work.

In the process of working on finding the lines of the golden section above the paintings, the author measured the dimensions of the latter on their reduced image in electronic form. In general, if we take the real sizes of the paintings, and their scaled versions, there should be no discrepancies in the location of the golden section lines, because The principle of the golden section is based on division into parts, regardless of size.

In general, the author's assumptions about the presence of image objects on the lines of the golden section in the paintings were confirmed. In some pictures, this is seen more, in some, the presence of the principle of the golden ratio is only guessed. The hypothesis that in all the works of famous and not so famous artists the principle of the golden ratio is used, put forward by the author at the beginning of the research work, was partially confirmed, since it is not possible to check absolutely all the paintings.

After the practical part, the author grouped several paintings in pairs in order to conduct a survey among others to study the aesthetic perception of paintings with and without the presence of lines of the "golden section". After processing the percentage of selections of the most liked paintings, it was quite expected that the paintings with the observance of the principle of the "golden ratio" were chosen by the respondents more often than the paintings without this principle. The selection of paintings and respondents was made by the author independently.

In general, in the process of conducting the study, the author achieved the set goal: to investigate the issue of the influence of the presence of the "golden section" in the paintings of artists on their aesthetic perception. In the process of achieving this goal, the author solved the following tasks:

learned everything about the discovery of the concept of the "golden section" and its author;

understood in detail the essence of the term "golden section";

highlighted the areas of creativity in which the "golden section" is applicable, and how this concept is applied in the visual arts;

got acquainted with the work of famous artists, including Vladimir ones;

conducted an analysis of the works of artists in compliance with the principle of the "golden section";

investigated the question of the importance of using this principle in the manufacture of a picture for its perception by the viewer.

In the process of conducting this study, the author learned a lot about the principle of the "golden section", its use in artistic creativity and the impact on the perception of works of art by contemplators.

4. LIST OF USED LITERATURE

Belyaev M.I. About the secret of the golden ratio / article from open sources on the Internet http://www.milogiya2007.ru/uzakon2_2.htm/

Bendukidze A.D. Golden section. Magazine "Quantum", No. 8, 1973.

Vasyutinskiy N. Golden proportion. - M .: Publishing house "Young Guard", 1990.

Kovalev V.F. Golden section in painting. - K .: Vyscha school. Head publishing house, 1989.

Lavrus V. Golden section /article from open sources Internet http://n-t.ru/tp/iz/zs.htm/

Website of the Vladimir regional branch of the VTOO "Union of Artists of Russia" http://www.vshr.ru/

The website of the Gallery of Modern Vladimir Painting “Britov. Yukin. Kokurin" http://www.britov.ru/

Stakhov A.P. Golden ratio codes. - M.: "Radio and communication", 1984.

Tsvetkov V.D. The Heart, the Golden Ratio, and Symmetry / Open Source Article Online http://314159.ru/tsvetkov/tsvetkov2.htm/

Shevelev I.Sh., Marutaev M.A., Shmelev I.P. Golden section. - M.: Publishing house "Stroyizdat", 1990.

1 Vasyutinskiy N. Golden proportion. - M .: Publishing house "Young Guard", 1990.

2 Lavrus V. Golden section (Internet publication http://n-t.ru/tp/iz/zs.htm).

3 Based on materials from the website of the gallery of modern Vladimir painting “Britov. Yukin. Kokurin" http://www.britov.ru/authors/britov_kim/

4 According to the materials of the website of the Vladimir regional branch of the VTOO "Union of Artists of Russia" http://www.vshr.ru/

5 Based on materials from the website of the Gallery of Modern Vladimir Painting “Britov. Yukin. Kokurin"http://www.britov.ru/authors/kokurin_valerij/)

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

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

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

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

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

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

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

And you, who breathed my soul into me and taught me to enjoy life, I need you alone for life.

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

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

The golden section in the painting by I. I. Shishkin "Pine Grove"

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

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

The golden ratio in the painting by Leonardo da Vinci "La Gioconda"

The portrait of Mona Lisa attracts by the fact that the composition of the picture is built on "golden triangles" (more precisely, on triangles that are pieces of a regular star-shaped pentagon).

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

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

On the preparatory sketch of Raphael, 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 clutching him to her, the warrior with a raised sword and then along the figures of the same group on the right side sketch. If you naturally connect these pieces of the curve with a dotted line, then with very high accuracy you get ... a golden spiral! This can be checked by measuring the ratio of the lengths of the segments cut by the spiral on the straight lines passing through the beginning of the curve.

We do not know if Raphael actually painted 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 turn of the spiral in those places where it is indicated only 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 lying body of the child is in its center. Raphael completed the original 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 the composition, Raphael achieved such perfection in which no one else could compare with him. In the simplest, like in the most majestic, compositions everywhere, his mind brings, together with life and movement, perfect order into an 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 section as a proportion that determines the deployment of the spiral.

"It is necessary for a beautiful building to be built like a well-built person" (Pavel Florensky)

Is it possible to “verify harmony with algebra”? “Yes,” Leonardo thought, and pointed out how to do it. The “golden section” is not a middle, but a proportion - a simple mathematical ratio that contains the “star law and the flower formula”, a pattern on the chitinous cover of animals, the length of tree branches, the proportions of the human body. You see a harmonious composition, a proportionate physique or a building pleasing to the eye - measure it and you will come to the same formula. During the Renaissance, ancient statues were measured to check the “law of harmony”, and a century and a half ago, the proportions of the “golden section” were checked by correlating the length of the legs and torso of guards soldiers - everything is absolutely accurate.

Artist Alexander Pankin explores the laws of beauty... on the famous squares of Kazimir Malevich.

- In the early 80s, at a lecture about Malevich, they were asked to show a slide of Black Square. After the image appears on the screen, the lecturer says sternly: “Turn it over, please.” We laughed: it's hard for a simple person to understand why draw something like that. It is beautiful?

– Examining Malevich’s paintings with a compass and a ruler, I came to the conclusion that they are surprisingly harmonious. There is not a single random element here. Taking a single segment, say, the size of a canvas or the side of a square, one can build the whole picture according to one formula. There are squares, all the elements of which are correlated in the proportion of the “golden section”, and the famous “Black Square” is drawn in the proportion of the square root of two.

- Do you draw these proportions in the margins for complete resemblance to the school task in geometry?

– What I do can be called “objective art”. At first glance, what kind of creativity is this if the task is not to express one's individuality? There is even such an expression - "the artist is recognizable." But I discovered a surprising pattern: the less the desire to express yourself, the more creativity. Where the frames are too wide, where everything is possible, we gradually come to the point where people begin to spoil the canvases (say, Brener approached a painting by Malevich with a can of paint), some icons are cut and say: “But I see it that way.” Canon is important. It is no coincidence that in icon painting it is so strictly observed. For creativity, it is better not to open doors wide open, but to crawl through a gap. I am interested in the form, how it is formed and develops by itself.

- This is a computer algorithm, what does painting have to do with it?

- In 1918, Malevich said that painting was over, - only geometry remained. That year he painted a white square on a white background. But then Malevich’s “return to Earth” happened, his painting became objectified. Science did not absorb art, but in those historical periods when geometry and art converged, this gave impetus to the development of both. So it was during the Renaissance, when Leonardo explored the proportions of the "golden section", and at the beginning of the twentieth century, when Paul Cezanne said: "Treat nature through a cylinder, a ball, a cone." If the Impressionists painted something personal, changeable, then the Cubists, on the contrary, were interested in the shaping element - the frame. Now there are conferences “Mathematics and Art” and seminars where scientists and artists meet, real discoveries happen. Since the time of Leonardo, the so-called Fibonacci number series has been known: 0,1,1,2,3,5,8,13,21,34... This is a “golden” sequence of numbers, according to this law, flower leaves and seeds are arranged in a sunflower. I depicted this series on the plane in the form of triangles. It turned out to be an amazing thing. The terms of the Fibonacci series grow very quickly: the triangle turned into an arrow, two sides go to infinity, and one of the legs always remains equal to five! Before that, I did not understand what “finite infinity” is! Looking at this picture, Professor Alexander Zenkin mathematically proved that such a system of triangles is the core of the Fibonacci series. A new mathematical object has been discovered!

- Pankin's triangles?

- At one seminar there were proposals to name them that way, because for some reason no one had noticed this mathematical regularity before.

– Maybe you study Malevich’s harmony not because you see a special meaning in his work, but because other paintings are more difficult to fit into the formula?

– Why! Recently, I also want to explore the "Stranger" Kramskoy. I looked: there, too, the “golden section” is at the heart of it. The same rules and patterns that I found in Malevich's paintings can be applied to other paintings, very interesting things will turn out. Malevich's paintings are the cornerstone of shaping, you can't go past him. The “Black Square” is a reference point, a cosmic funnel where art enters and exits changed. New spaces are emerging. For the Wanderers or for naturalists like Shilov, a picture is a window behind which three-dimensional objects are located in the usual direct perspective. In Cezanne, spaces lie on the canvas. There are two points of view at the same time in the icons: you look from your place and at the same time you seem to be inside what is happening. The space is objectified, and it is not for nothing that icons do not need frames. It seems to me that in the future the space of the picture will lie not behind the canvas, but in front of it ...

- Recently in the store I saw a poster with the "Black Square". I was delighted and bought it, I wanted to hang it at home, and then I changed my mind. It is uncomfortable to sleep when the “Black Square” hangs over the bed. Would you like to hang a Malevich square over your bed?

– To be honest, my paintings hang above my bed, they hang everywhere with me. And I would like ... probably Ivanova - “The Appearance of Christ to the People”. An amazing composition - the figure of Christ in the center and from it, as if the rays diverge. For some reason I didn't notice this before...

Since ancient times, people have been worried about the question of whether such elusive things as beauty and harmony are subject to any mathematical calculations. Of course, all the laws of beauty cannot be contained in a few formulas, but by studying mathematics, we can discover some terms of beauty - the golden ratio. Our task is to find out what the golden section is and to establish where mankind has found the use of the golden section.

You have probably noticed that we treat objects and phenomena of the surrounding reality differently. Be h decency, be h uniformity, disproportion are perceived by us as ugly and produce a repulsive impression. And objects and phenomena, which are characterized by measure, expediency and harmony, are perceived as beautiful and cause us a feeling of admiration, joy, cheer up.

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

The golden ratio was known in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the "golden section" was studied. Euclid applied it, creating his geometry, and Phidias - his immortal sculptures. Plato said that the universe is arranged according to the "golden section". Aristotle found the correspondence of the "golden section" to the ethical law. The highest harmony of the "golden section" will be preached by Leonardo da Vinci and Michelangelo, because beauty and the "golden section" are one and the same. And Christian mystics will draw pentagrams of the "golden section" on the walls of their monasteries, escaping from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. Be h the final row 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 section" is. But you will certainly see this proportion in the curves of sea shells, and in the form of flowers, and in the form of beetles, and in a beautiful human body. Everything living and everything beautiful - everything obeys the divine law, whose name is the "golden section". So what is the "golden ratio"? What is this perfect, divine combination? Maybe it's the law of beauty? Or is it still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. "Golden section" is both that, and another, and the third. Only not separately, but at the same time ... And this is his true mystery, his great secret.

It is probably difficult to find a reliable measure for an objective assessment of beauty itself, and logic alone will not do here. However, the experience of those for whom the search for beauty was the very meaning of life, who made it their profession, will help here. First of all, these are people of art, as we call them: artists, architects, sculptors, musicians, writers. But these are people of the exact sciences, first of all, mathematicians.

Trusting the eye more than other sense organs, Man first of all learned to distinguish the objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The shape, which is based on a combination of symmetry and the golden section, contributes to the best visual perception and the appearance of a sense of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

GOLDEN SECTION - HARMONIC PROPORTION

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

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

• into two equal parts - AB:AC \u003d AB: BC;
• into two unequal parts in any ratio (such parts do not form proportions);
• thus, when AB:AC=AC:BC.

The latter is the golden division (section).

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

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

Geometric representation of the golden ratio

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

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

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

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

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

Solution to this equation:

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

SECOND GOLDEN SECTION

This proportion is found in architecture.

Construction of the second golden section

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

Dividing a rectangle with a line of the second golden ratio

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

GOLDEN TRIANGLE (pentagram)

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

Construction of a regular pentagon and pentagram

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

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

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

Construction of the golden triangle

HISTORY OF THE GOLDEN SECTION

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

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

Dynamic Rectangles

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

icosahedron and dodecahedron

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

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

Antique golden ratio compasses

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

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

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

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

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

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

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

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

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

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

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

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

Building a scale of segments of the golden ratio

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

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

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

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

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

GOLDEN RATIO AND SYMMETRY

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

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

FIBONACCCI SERIES

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

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

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

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

GENERALIZED GOLDEN RATIO

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

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

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

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

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

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

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

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

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

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

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

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

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

PRINCIPLES OF SHAPING IN NATURE

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

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

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

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

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

Mandelbrot series

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

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

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

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

Chicory

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

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

viviparous lizard

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

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

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

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

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

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

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

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

Adeno virus

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

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

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

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

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

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

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

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

THE HUMAN BODY AND THE GOLDEN SECTION

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

Golden proportions in parts of the human body

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

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

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

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

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

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

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

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

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

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

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

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

The structure of the helix section of the DNA molecule

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

GOLDEN SECTION IN SCULPTURE

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

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

GOLDEN SECTION IN ARCHITECTURE

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

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

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

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

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

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

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

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

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

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

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

Pashkov House

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

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

GOLDEN RATIO IN MUSIC

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

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

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

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

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

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

GOLDEN SECTION IN POETRY

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

Much in the structure of poetic works makes this art form related to music. A clear rhythm, a regular alternation of stressed and unstressed syllables, an ordered dimensionality of poems, their emotional richness make poetry a sister of musical works. Each verse has its own musical form, its rhythm and melody. It can be expected that in the structure of poems some features of musical works will appear, patterns musical harmony and hence the golden ratio.

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

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

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

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

I do not value high-profile rights, From which not one is dizzy. I do not grumble about the fact that the gods refused I'm in the sweet lot of challenging taxes Or prevent the kings from fighting with each other; And little grief to me, is the press free Fooling boobies, or sensitive censorship In magazine plans, the joker is embarrassing. All this, you see, words, words, words. Other, better, rights are dear to me: Another, better, I need freedom: Depend on the king, depend on the people - Don't we all care? God is with them. Do not give a report, only to yourself Serve and please; for power, for livery Do not bend either conscience, or thoughts, or neck; At your whim to wander here and there, Marveling at the divine beauty of nature, And before the creatures of art and inspiration Trembling joyfully in delights of tenderness, Here is happiness! That's right...

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

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

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

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

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

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

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

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

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

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

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

GOLDEN RATIO IN FONTS AND HOUSEHOLD ITEMS

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

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

OPTIMUM PHYSICAL PARAMETERS OF THE EXTERNAL ENVIRONMENT

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

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

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

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

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

GOLDEN SECTION IN PAINTING

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

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

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

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

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

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

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

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

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

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

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

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

pine grove

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

IN AND. Surikov. "Boyar Morozova"

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

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

The golden ratio really cuts here on the most important thing. In it, and it is in it - greatest power Morozova.

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

Venus

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

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

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

Massacre of the innocents

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

GOLDEN RATIO AND IMAGE PERCEPTION

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

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

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

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

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

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

THE GOLDEN RATIO IN PHOTOGRAPHY

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

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

GOLDEN RATIO AND SPACE

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

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

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

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

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

CONCLUSION

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

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

In the modern world, science acquires special meaning due to the increasing human impact on nature. Important tasks for present stage are the search for new ways of coexistence of man and nature, the study of philosophical, social, economic, educational and other problems facing society.

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

about ways to "rive" the viewer's eye to the work on the example of the classics of Russian painting, also a visually simplified rule of thirds, which formed the basis of composition in modern photography.

Starting a new work, each artist always begins by mentally trying to determine on the canvas the main point where all the storylines of the picture will have to be pulled together, as if to an invisible magnet. The same point - the main and semantic - should be present in the photograph, as if unfolding the action around the main object in the frame.

At artistic canvas and photography has one thing in common - they are both static and non-volumetric art forms, limited by two coordinate axes: X and Y.

Unlike, for example, sculpture or architecture, which "live" in space, or - music - which "moves" in time. The artists learned to give the picture "volume" through the use of different plans - near and far. Photographers went even further - they can designate these plans with sharpness or blur, forcing the viewer to psychologically focus on the focused object against the background of a blurry background and / or foreground, thus conditionally and visually creating "depth" in the frame, the third coordinate " Z".

As for the transfer of "movement" - technically, artists and photographers solve this problem in different ways: the artist conveys movement due to the internal tension of the hero in a frozen pose, and the photographer actually transfers to the photograph the movement that occurs during a long exposure (for example, a trace of headlights when shooting in the evening: the car manages to drive some segment of the path - i.e. there is a "movement in time" - and its trace remains worked out from the beginning of its movement to the end.)

However, both artists and photographers understand that the real value of their work will be given by the fact that if the viewer, passing by, suddenly stops and begins to examine the picture (photo), think it over, empathize with the events with the depicted characters. Thus, the viewer becomes a participant in the creative process, and the author achieves the highest form, when his static work, as it were, "develops in time" due to the viewer's internal understanding and the time he spends on it.

This is where the mechanism turns on when correctly placed accents in the work affect the viewer and his perception. Since ancient times, there is a formula of the so-called "golden section". Psychologists have proven that the artist's observance of this rule leads to the establishment of a harmonious dialogue with the viewer - i.e. on a subconscious level, a trained (!) viewer understands what it is about.

The golden section rule is a mathematical formula that has rather complex calculations and is derived in ancient times(still from Euclid, 3000 BC). However, as Wikipedia aptly points out: "The "rule of the golden ratio" in art usually refers to asymmetrical compositions that do not necessarily contain the golden ratio mathematically."

Those. in relation to art, we are talking about a simplified rule of the golden ratio - the rule of thirds, which has become widespread precisely in relation to photography.

The rule of thirds is calculated simply: you need to conditionally divide the image into three equal parts vertically and horizontally - the intersection points of these lines - and there are the most important semantic points in the image. Especially culminating of them is the upper right point, because. the eye "moves through the picture" (according to psychologists) from the lower left corner to the upper right.

A classic example of this is the outstanding giant, 7.5-meter canvas A. Ivanova "The Appearance of Christ to the People", which he painted in Italy for 20 years (from 1837 to 1857)

N.V. Gogol wrote: "A great creation, such as The Appearance of Christ, raises, educates, creates the artist himself: over the years of work, his talent, nature become deeper, more significant - you need to morally, ideologically rise to your plan."

Please note that the figure of Christ is not only on the line of intersection of thirds, but also all geometric lines, turns of bodies, movement of views - everything is directed towards Him. Not only that - the artist had to think through the whole perspective and the ratio of proportions in the picture with his inner vision!

Now an important question that concerns photography as well - Where should the horizon line be??

It is traditionally believed that the horizon line runs along the upper line of thirds, if the artist (photographer) depicts what is happening "on the ground" to a greater extent, or along the lower semantic line - if the sky is most important to him. All this has a long history and is associated with deep symbolism, which is inevitably present in the soul of every artist.

This picture is also no exception - here the horizon line runs strictly along the upper semantic line, behind the figure of Christ, as if once again pedantically emphasizing the author's position that all events related to Christ take place here on earth.

And the most interesting. Despite the huge and bright, almost human-sized (in the original) figures in the foreground, our gaze is involuntarily constantly riveted to the lonely figure of Christ, located in the distance and drawn in less detail. This is precisely the answer to many questions related to the psychology of image perception.

Or, another example - an almost six-meter painting IN AND. Surikov "Boyarynya Morozova"(1887)

It is authentically known that the artist began to write it from the finger. Despite the fact that the point of the "golden section" falls strictly on the head of the main character, her hand raised with two fingers is also included in the so-called. "region of the golden section". I want to recall the above - in relation to art, we operate with the concept of a "simplified", not a mathematical rule of the golden section. Therefore, many artists, and - everywhere - photographers, in order not to appear pedants and scholastics in art, often "blur" the point itself to a certain conditional area around it.

A few more words about the direction of movement in the picture. Here it is the opposite of what was described above, and from the so-called. psychology of the gaze - the movement in the picture (and in the frame) from right to left symbolizes the “leaving”, “leaving” the canvas by the heroes. Brief history: along with Archpriest Avvakum, the boyar Fedosya Morozova went against the tsar and Patriarch Nikon, defending the old faith - one of the symbols of which is the sign of the cross with two fingers - she herself became a symbol of the schism of the Russian Orthodox Church and a favorite of the common people. In November 1671, she was taken to prison past the Chudov Monastery, where the complex images of commoners symbolize a close relationship with their heroine. Despite the bright image of the noblewoman, her "burning gaze" is, alas, not "Freedom leading to the barricades" - this picture is of leaving the battlefield, breaking the external and moving the semantic "tension of the spirit" into the internal.

Also pay special attention to all the geometric lines in the picture - the lines of snow, the lines of roofs and ledges, the lines of the sleigh, the lines of gazes and poses - everything is directed towards the face and to the raised hand of the heroine.

Now a few words about something else. As we already know, the points and zones of the golden section are conflicting places in the image, which are the sources of dramatic development, the state of "restlessness", some kind of constant oppositions and unresolved problems highlighted by the artist (photographer) in his work.

How much does he have the right to life? the presence of symmetry in the frame?

As the great Russian crystallographer G. V. Vul'f (1863-1925) believed, the golden section is one of the manifestations of symmetry and the golden section cannot be considered on its own, separately, without connection with symmetry.

According to Kovalev F.V. in his book "Golden section in painting":

According to modern ideas, the golden division is an asymmetric symmetry. Now the science of symmetry has included such concepts as static and dynamic symmetry. Static symmetry characterizes rest, balance, and dynamic symmetry characterizes movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance, and even stiffness. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Symmetries are characterized by equal segments, equal magnitudes. Dynamic symmetry is characterized by an increase in segments (or their decrease), and it is expressed in the values ​​of the golden section of an increasing or decreasing series.

art form, 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 along the vertical and division into unequal parts along the golden section along the horizontals.

As a first example, I will cite the most important, the greatest Andrey Rublev's creation "Trinity"(1420s).

It turns out that despite the fact that the angels from the Old Testament Trinity are given an equal vertical third of the image, thus symbolizing the equality of Persons in Holy Trinity, the emphasis of the great icon painter - made on the other - on the bowl. Thus he introduced into the Old Testament history already new symbols- symbols of Christianity. Please note that the bowl is on an even, light, contrasting background in relation to the entire icon. It is located in the center of the image vertically - being an unshakable support and center - and at the same time - in a conflict point (a third of the composition) horizontally. Moreover, the conflict point is not the upper one - which would put the cup, as, for example, the symbol of the Grail, "at the forefront." Thus, all attention would be directed to the bowl, which would be on a dais. No. The chalice is below, "in this world" - it is here that the Sacrament takes place - as the path of "deification" of a person. (If we digress for a moment into symbolism - angels do not receive communion - their luminous nature does not need the sacrifice of Christ, which was performed solely for the sake of people. That is why the cup is at the lower semantic point. Although, if you look closely at the inner contour of the angels and the table - we will see another, more symbolic Chalice in the size of the entire icon).

Much has been written about the symmetry of the Trinity by Andrei Rublev. - Reported by Kovalev V.F. - But no one paid attention to the fact that the principle of golden proportions is also implemented here horizontally. The height of the middle angel is related to the height of the side angels, just as their height is related to the height of the entire icon. The line of the golden section crosses the axis of symmetry in the middle of the table and the bowl with the sacrificial calf. This is the compositional castle of the icon.

Thus, the author, by combining symmetry and asymmetry, was able to achieve the embodiment in the icon of his complex worldview and the canons of the Church. However, the main question that concerns our topic is that, in the 15th century, Andrei Rublev managed (by simple limited means) to convey to his viewer the entire versatility of dogmatic teaching exclusively in the language of symbolism and the relationship of symbols in space.

A simpler example of combining the rule of thirds and symmetry we see in the example Vladimir icon.

The gaze of the Mother of God falls simultaneously on the center of the composition in vertical orientation and strictly on a third - in horizontal orientation. This is just what it is prime example states of "rest" and "poise", centering and non-conflict of the image with respect to the whole. However, the horizontal dot, as if raised to the top of the image to the place of conflict (third), speaks of "fundamentality", "elevation", "separation from the ground".

Now the most difficult thing - on the example of a textbook picture Vasily Pukirev "Unequal Marriage"(1862)

Vasily Vladimirovich Pukirev(1832-1890), came from a peasant family, studied at the Moscow School of Painting, then taught there, lived hard and died in poverty. For a domestic genre, his picture was unusually huge: the figures were almost life-size. Obviously, he wanted to draw attention to an issue that is sore for society.

Wedding ceremony. The bride is just a girl. Eyes humbly lowered, crying, just look - drop the candle. The groom holds himself in an emphatically youthful manner and looks sternly at the young chosen one, who is fit for his granddaughter.

The groom is the buyer. The bride is a commodity. O scandalous picture argued and called one of the most tragic paintings of the Russian school.

Even Ilya Repin wrote that Pukirev spoiled a lot of blood for more than one old general, and N. Kostomarov, having seen the picture, took back his intention to marry a young lady.

Let's now look at lines, dots and accents.

The most active culminating point of the golden ratio falls on the girl's head - and not just on her head - but on her crown. (as if an allusion to her martyrdom). The girl's face is illuminated to the maximum, in addition, all eyes are directed to her, which undoubtedly makes her a "magnet" in the picture.

Where is the groom? Strictly in the center. The order on his chest generally falls into the very center of the picture, and the posture and the candle in his hand emphasize the centricity of his position - his weight in society, his confidence in himself and his actions - nothing can violate his fundamentalism. His head - the second in terms of illumination, however, is in the conflict place of the third, cutting off the line on which there are other witnesses of the event - all of whose portraits are different. The centricity of his raised candle conflicts with the bride's lowered candle, which is also in the golden ratio zone.

But there is another hero, very important, he is in the shade, lit only by backlight - this is the priest. Please note that the picture shows that part of the ceremony when the betrothal takes place and the priest puts a ring on the bride's finger. The bride does not even look at the ring. But the level of her eyes is in exact but conflicting (dynamic) symmetry with respect to her hand and the hand of the priest with the ring (highlighted by rectangles). Not only that - this invisible line passes directly through the center of the composition and through the order of the groom. The order symbolizes not only his status and power, but also the right, the unconditional right, to receive a "reward" for his merits.

Pay attention to the place of the priest. The church is out of conflict - it occupies the central symmetrical third of the left edge. In general, it has nothing to do with it, therefore it is fundamentally not illuminated by frontal light - thus, it is a "pure" symbol, without a face, but with a clearly defined outline. It is by his "blessing" that the greatest injustice will occur.

The zone of the golden section, where his hand and the hand of the bride is located, “crosses” with a half-lowered candle (a symbol of extinction before the term of life) and a crown on the bride’s head - all this happens against the background of two symmetrical rods - the groom’s figure vertically and the priest’s figure horizontally .

Well, of course, if we are talking about symbolism, one cannot fail to mention the only hero - he does not participate in the conflict geometry of the picture - but his only look directly at us (this is the friend of the best man, according to legend - the beloved bride) - is, as it were, mute reproach to all of us, witnesses of what is happening.

Summing up the above, I would like to smoothly move directly to the art of photography. I hope that a carefully conducted analysis on the example of Russian painters will help you easily and accurately, using the tips on the right, determine the meanings and accents in the photographs below.

As an example, I have prepared several photographs of an outstanding Moscow master, master of Russian photography,

I especially want to emphasize that, despite the different tools of the artist and photographer - in terms of symbolism and polyphony (diversity), photography is in no way inferior to a painting.

For example, "Christmas night in Bethlehem" G.Rozov.

The plot is simple: two female pilgrims are waiting in the temple. But pay attention to the whole system of contrasts!

One of them sits in a bright strip of light, which conditionally occupies a third of the frame, the other - despite two-thirds - is in the shade. The one on the right is sitting humbly with her head bowed in dark monastic robes (a sign of repentance). The one on the left - with a proudly raised head in light clothes and a waddle pose. On the right - attention is concentrated, because. she is in focus, the left one is her background contrast out of focus.

And now the main thing. The humbly folded, well-lit hands of the right woman are strictly in the horizontal center of the image, as if "reconciling" the two worlds - and all this despite the fact that relative to the vertical - they are strictly in the third allotted to them and right at the mathematical intersection of the lines - a conflict "light" and "darkness", opposition and tension of "spaces".

Therefore (including) there is a feeling that despite the light and shadow sides - the woman on the right occupies the central and b about most of the composition, while the woman on the left (even despite the difference in levels in height) is actually isolated by an uninformative 1/6 of the frame.

Or, for example, a work from the series "Kazan leaving".

Already in the title of the series is the word "outgoing". The movement of the gaze, all geometric lines - from right to left (the same technique as in "Boyaryna Morozova" by Surikov, the same direction). The girl is turned away from the viewer strictly at the point of the golden section - she is a "part" of this plot - it is not the center - then the author would have cropped the photo from above and the girl "raised" higher in the frame - but a part, a fragment. This is also evidenced by her uncertain posture and casually dressed dress - plus, in fact, above her is a massive contrasting space of the door, and of the entire building as a whole. The whole image breathes with "abandonment", even a little girl - does not "charge" everyone around with her energy, but dutifully and slightly ridiculously complements the overall picture.

The next photo is an example of peace, peace and solitude. Nothing disturbs the balance and quiet water surface. Undoubtedly, the horizon line passing through the middle of the image is an eloquent proof of this!

A few words about the next, seemingly simple, work. As you can see, it contains several plans, meanings and symbols. I want to focus on just one. Above, we mentioned the symbolism inherent in all artists, traditionally assigning the upper part of the frame to the sky, and the lower part to the earth. At the intersection of these worlds, most of the plot "dramas" take place. Knowing this common truth, the author, as if "playfully" - "turned" the accents - shifted the line of conflict to the vertical. Now the "sky" occupies strictly the left third of the frame, and the "earth" - "advancing" the right two thirds.

Why "sky" and not traffic lights and road signs? Because, having chosen the lower shooting point, the author, as it were, "passed" through these obstacles with his eyes. Yes, and the lines of gleaming glass fragments, by their very forms, also "rush" into the sky.

I am sure that the following photographs and a small schematic analysis will easily make it possible to understand the design and accents.

And in conclusion, a few words about the use of various symbolism in plots similar in form and even content. As illustrations, I will give two photographs - Georgy Rozov and my own. There is no question in comparing these two pictures, the photo of G. Rozov was taken earlier - and mine is partly a replica of his plot, but with a changed meaning.

1. Both photos are divided by the horizon line in half - the symmetrical composition here is a symbol of the fact that the newlyweds are not self-sufficient in the frame, but are part of a whole, "peaceful" ("festive") world.

Therefore, the sky and the rest of the landscape play an equally expressive role in both subjects.

2. In both photographs there is an alley ("path"), looking into the distance - and all the geometric lines in the pictures tend to this "distance".

3. In the upper plot, the "distance" falls on the slightly displaced climax center of the entire frame, which is undoubtedly the main "ideological" basis. This also proves that young people have their backs to us and go to this "center", despite the fact that they fall into the zone of the third - i.e. the beginning of their movement from asymmetry to symmetry. If you look closely, they are not alone in the alley - there are also people walking ahead. This means that the WAY itself is important to the author - as a way of life, the road along which they are already walking together. Here the PATH is the main meaning of the plot.

In the lower work, the accents are somewhat shifted. The culminating point of the "dali" (arch) is not in the center, but in the conflict zone of the third. As well as a counterbalance to it - at the opposite point of conflict - the faces of the young, not even the faces themselves, but "the air between them." They do not follow the path, although they stand on it. Here is a clear opposition - equivalent in accents - "distance" and "two". Those. a path that they still HAVE to choose and go through. Here the "path" is just a possible perspective of their future movement - figurative "symbolism".

"Photography is like a trace of life" (documentary interview).

"The camera is a finely tuned instrument" (author's article).

Other master classes by Zoriya Fine.

Photo Fine Study - learning photography.

Photo school graduates gallery.

Video reviews of graduates of the photography school.

User: Denis Date: 03/30/2011 E-mail: [email protected] Good article, everything is clearly shown! I do not think that this article is for beginners, it is for those who want to constantly know something new and develop. Is everyone really versed in painting and knows all the nuances of photography ??? Of course, after reading the article, someone will say ** I know it **! and compare himself to one of the great artists...??? In fact, we know little, when a person says that he knows this, he thereby stops his path ... why should he go ahead if he knows everything??? thanks for the good article, for the material provided for comparison. I wish you inspiration in the implementation of your future projects!!!

User: Serbian Date: 21.04.2011 E-mail: [email protected] If I express my diametrically opposite opinion about your article on the application of the golden section rule in photography, this will not find its application, will it? I present my own research on this aspect. I think that the theory is simply far-fetched, it’s always like this with us - a person invents something new, ingenious, there are interpreters right there pushing another dissertation ... (( I do not think that Ivanov and Surikov knew the golden section formula. And why does it only apply to large canvases? There are no rules and laws here - they just SEE the way all people see. Our field of view is about 140 degrees horizontally, this is individual. The central part, about 45 degrees. we see entirely and at once (this is one third), we see one third on the left and on the right with peripheral vision, not clearly. The same goes for the vertical, but the angles are smaller there. When developing a standard for cinema, they proceeded from the same principles - from the visual places in the hall, approximately from the center of the hall, the viewer should see the screen at an angle of 45 degrees as well. From here the standard for 35 mm film cameras with a frame format of 24x36mm was born, the lens should be 45-50mm. Therefore, the photograph should be viewed at an appropriate distance. Approaching the picture at the recommended distance, the viewer immediately sees 1/9 of the image, which, as a rule, is dominant, for the rest it is necessary to shift the gaze. What, in my opinion, is the strength and "focus of the picture"? In the dominant part, the upper part of the picture at first does not carry information - a stupid dark contrasting bush at first attracts the attention of an unreasoned look and leaves without attention the figure on the upper right, painted in the background color - n *** and mountains. The gaze descends to the brightest figure in the picture, then to people who are clearly not united by one idea, a crowd staring at different sides. There is some bewilderment as to what the author wanted to show, the empty horizon still does not interest us, we consider the main figure in the frame - he is the only meaningful person in the picture, he points to something ... and then a miracle happens, where a minute ago no one it was not, suddenly, as in a live movie, the figure of Christ appears from nowhere! This is the strength of karatina - knowledge of the laws of perception of vision and psychology. In Morozova's emphasis on the figure, based on contrast, this is not the brightest part of the picture, on the contrary, the darkest, it's just that there is no one in the foreground, and there is simply nowhere for the eye to look, but at her, and the diagonals point like an arrow at her. But the double-fingeredness is not immediately detected, it goes beyond the horizon and disguises itself, like Christ. Attention is drawn to a half-naked beggar in the snow, he is bright and naked, unlike the others, and seeing his two-fingeredness, you begin to understand what is the matter here and find the same sign in Morozova. This is the strength of the picture. After all, the Old Believers, going against the reforms of the church, burned themselves or went to Siberia. And where does the golden ratio with its formula? The icon with the Trinity simply must be a multiple of three ... The Mother of God does not seem to be aloof, but rather sad about the future of her child, because she was warned in advance that he is the Messiah ... and the plan of the "half-length portrait" is justified by the fact that it is in this perspective that we see the interlocutor when communicating with him. As for "Unequal Marriage", I mostly agree, and the diagonal is an arrow that indicates the direction ... Further. Photo. The modern literate population (I don’t know how it was before) and this is known for a long time a close image, say a photograph on a table, looks differently, this is determined by the movement of the pupil and the nerve impulse. First, there is a momentary evaluative look along the trajectory: the upper left corner, the upper right corner, then obliquely down diagonally to the lower left, then the lower right. It is the dark spot in the upper left (and upper right) corner that acts depressingly, defining the first relation to the image. Then, in several stages from left to right, as if in a wide line, going down, we look through the entire image (we consider a vertical sheet longer). Then the eye stops at what attracted it - a bright or contrasting object. This rule is common for photography and cinema, and it is a good idea for a photographer to keep this in mind when creating their work. The photograph leaves an impression regardless of the polarity of the emotions evoked, the main thing is that they be. In general, a dark top and a light bottom irritates the perception (photographers who are accustomed to negatives do not care (they are already adapted))) As for the rest of the photographs of the article, every photographer, even without a camera, “frames” with his eyes, and then through the lens such an arrangement of objects in the frame that seems to him the most harmonious, balanced in terms of illumination, and through the viewfinder it is more in terms of lines and brightness, and through the SLR - takes into account the colors in the subject very much. Agree, would it be stupid to place the visible end of the river in the upper right corner or cut off the textured sky? And the white bushes with clouds? Move the frame to the left - the upper right corner is clearly out of place, but the impudent tall bush on the right, although it interferes, brightens the corner, and the black on the right would fit into the frame ... Don't you do that? The frame with the bench is obviously unballanced, but this was done to please the reflection in the floor... Next - a building with a lantern - try to take a step to the right, to the left - it will only get worse, the horizon is broken, but then we will push the lantern and the spire to the edge, and a lot of black will climb from below - the author chose the optimal shooting point - this is exactly what A. Gordievich taught me with examples ... (thanks to him for the science!)) Well, and so on, without any science - the OPTIMUM shooting point was chosen under these conditions, from other points without any hints it would be only worse!

User: Zoriy - Serbia Date: 21.04.2011 I am very glad that I have such a friend in the virtual world - and the old school, who at the same time finds more time and patience to express his thoughts clearly and clearly! Your TK about the golden ratio in personal correspondence "in Contact", I did not read right away, because. I was banned for a few days, and then I did not unsubscribe. I apologize!) But when I read it, I realized that it must be published because of the brightness and originality of the vision! As for the text itself, it has now become clear to me why filmmakers (a moving picture) and photographers (a static picture) fundamentally differ in their views. Some of the nuances you described very clearly and they are very specific. Incl. I understood for myself why the art of cinematography is not close to me and I basically do not take the camera in my hands. Despite this - I would not say that all this contradicts the article - rather, it supplements with new information. If you look objectively, the article itself is not the fruit of my scientific research - it is just a form of popularizing some general knowledge, first of all, to my students. For their elementary orientation and formation. Rozov, after all, wrote in the commentary: a hackneyed topic and it seems that you can’t say anything new? but well written :) As for the very idea of ​​the "tightness" of this theory as such, I partly disagree. And the question is not even in the movement of the gaze over the image. The fact is that, in my basic education (composition department of the Gnessin Academy), I constantly encountered a special form of distribution of culminations in time - moreover, a form that is a multiple of three. Maybe partly because of this - I internally, as it were, synthesized these generally non-intersecting types of art. I spent a lot of time as a student in my beloved Tretyakov Gallery (it is from there that the paintings are given in the article), in the Hermitage, in Pushkin. I studied the plasticity of Rodin in sculpture. One of my old friends, with whom I disappeared for years in the workshop on Sukharevka - People's Artist of Russia, member of the Presidium of the Academy of Arts - Andrei Andreevich Tutunov - a classic of the Soviet school. I am writing all this not for the sake of bragging, but only because to explain that the idea of ​​this "simple"-looking article is in a colossal personal experience and personal experience of form in art.

User: Serbia Date: 04/22/2011 Thanks for the "thick" answer! But I want to highlight. The end of the 19th century, disputes about whether photography will now replace painting, as later there were disputes about the theater with the advent of cinema, and in our time newspapers - the Internet ... They say how Repin, having acquired a camera, decided to take a group portrait of friends, but this turned out to be so technically difficult that, having carried long time, he painted a portrait by hand in half an hour)) Camera obscura had previously been used by artists to correctly convey perspective when sketching architecture - parchment was superimposed on the back wall, contours were outlined with a pencil and then the same contours were transferred to the canvas, but this limited the angle of view. Later, a number of artists completely switched to the format of the picture with an angle of view of 45 degrees, which has already become familiar to our eyes. Mistakes in the transfer of perspective are the scourge of such paintings, let's take, for example, the famous engraving of the Vinnitsa Moores, which is very stretched horizontally. At first, photographs were treated as a mechanical substitute for a picture. But the essential difference is that, for example, the picture "The Appearance of Christ ..." is written for perception with an angle of view of about 120 degrees, as we see everything around, and the golden section is the angle at which the camera sees (and the zone of increased clarity of the retina) - this is where the focus appears with the appearance of Christ in the picture. A photograph, on the other hand, is just a little thing and it must be compared with miniatures, and the perception of photographs different size- different. And it is located entirely in the zone of the "golden section", in 1/9 of the area accessible to vision. Look at the front wheels of a truck slowly moving towards you: its tread merges into stripes, take a closer look, and suddenly the eye clearly sees the entire tread pattern for a fraction of a second - these are those micro-movements of concentration of a person’s gaze, without them we do not perceive the world. A frog, for example, does not see motionless objects at all, animals at a higher level turn their heads and only mammals can see without rotating their pupils, our optic nerve must be constantly irritated in order for it to perceive anything. I say this to the fact that it is necessary to take a closer look at the movement of the eye in order to obtain an effect that corresponds to the intention of the photo artist. So the cinema in the hall from the first rows is perceived terribly unnatural, then the brain makes a correction, how automatically, without our participation, it makes adjustments to the white balance when moving from evening natural lighting to a room with incandescent lamps ...))

User: Zoriy - Serbia Date: 22.04.2011 This is what else! In academic circles, it is generally not accepted to "see" the figure of Christ in the picture. They consider it generally insignificant. And the image of Christ itself is allegedly presented by the author in large-scale foreground faces. According to the principle "retinue plays the king." This, of course, explains why the author spent 20 years honing these portraits and figures - to which three halls are dedicated in the Tretyakov Gallery. But my article is about something completely different. It's not about angles and proportions - it's about climaxes and distributions of accents. And besides, not all painting is so huge - for the most part, it is also chamber, like a high-quality photograph. Here they have equal chances in front of the viewer ... :)

	User: Irina R. Date: 05/03/2011 You nibi read my thoughts. The theme of the composition chimed - the article “The Golden Ratio Rule” appeared, I wanted to discuss the topic of beauty, ethics and permissibility in the picture - “is the article “The Original Sin of Photography”. Like mysticism))) Anonymous authors, even more works, and tієї, hochab one, yak wanted to hang on the wall mute. About testing for an hour on that same wall, I started talking. But the best of all are those who, looking over from this amount of material taken up, the picture does not please you, but on the other hand - confuses you. Stilki brudu, vіdsutnostі relish... Shorazu in such a situation I guess you - the artist is guilty of showing beauty. I’ll turn again to your robots, maybe through those who are closer to you, maybe through those who work effectively, by an order of magnitude, more for others ... I’ll guess the words again: “No one understands, no one appreciates.”