A person distinguishes objects around him by form. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden ratio, 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 relation to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

Golden ratio - harmonic proportion

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

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

into two equal parts - AB : AS = AB : Sun;

into two unequal parts in any ratio (such parts do not form proportions);

this way when AB : AS = AS : Sun.

The latter is the golden division or division of the segment in the extreme and average ratio.

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part as much as the larger part itself refers to the smaller one; or in other words, a smaller segment refers to a larger one as a larger one to everything

a : b = b : c or with : b = b : and.

Figure: one. Geometric image of the golden ratio

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

Figure: 2. Division of a straight line segment along the golden ratio. BC = 1/2 AB; CD = BC

From point IN a perpendicular equal to half is restored AB... Received point WITH connected by a line to a dot AND... A segment is laid on the resulting line Sunending with a dot D... Line segment AD transferred to a straight line AB... The resulting point E divides the segment AB in the golden ratio.

The segments of the golden ratio are expressed by an infinite irrational fraction AE \u003d 0.618 ... if AB take as a unit, BE \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB taken as 100 parts, then most of the segment is 62, and the smaller part is 38 parts.

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

x 2 - x - 1 = 0.

Solution to this equation:

The properties of the golden ratio have created a romantic halo of mystery and almost mystical worship around this number.

Second golden ratio

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

This proportion is found in architecture, and also occurs when building compositions of images of an elongated horizontal format.

Figure: 3. Construction of the second golden ratio

The division is carried out as follows (see Fig. 3). Line segment AB divided in the proportion of the golden ratio. From point WITH the perpendicular is restored CD... Radius AB there is a point Dwhich is connected by a line to a point AND... Right angle ACD divided in half. From point WITH draw a line until it intersects the line AD... Dot E divides the segment AD in the ratio 56:44.

Figure: 4. Dividing a rectangle with a line of the second golden ratio

In fig. 4 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

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

Figure: five. Constructing 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 Durer (1471 ... 1528). Let be O - the center of the circle, A is a point on a circle and E - the middle of the segment OA... Perpendicular to radius OArestored at the point ABOUT, intersects the circle at the point D... Using a compass, we postpone the segment on the diameter CE = ED... The side length of a regular pentagon inscribed in a circle is DC... Putting the segments on the circle DC and we get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden proportion.

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

Figure: 6. Building the golden triangle

We draw a straight line AB... From point AND postpone a segment on it three times ABOUT arbitrary value through the obtained point R draw a perpendicular to the line AB, perpendicular to the right and left of the point R postpone the segments ABOUT... Obtained points d and d 1 we connect with straight lines with a point AND... Line segment dd 1 put aside on the line Ad 1, getting a point WITH... She split the line Ad 1 in the proportion of the golden ratio. Lines Ad 1 and dd 1 is used to construct a "golden" rectangle.

The history of the golden ratio

It is believed that the concept of gold division was introduced into scientific use by Pythagoras, the ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and ornaments from the tomb of Tutankhamun indicate that Egyptian craftsmen used the golden division ratios 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 \u200b\u200bof the golden division. The architect Khesira, depicted on the relief of a wooden board from the tomb of his name, is holding measuring instruments in which the proportions of the golden division are fixed.

The Greeks were skilled geometers. They even taught arithmetic to their children using geometric shapes. The Pythagorean square and the diagonal of this square were the basis for constructing dynamic rectangles.

Figure: 7. Dynamic rectangles

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

The facade of the ancient Greek temple of the Parthenon has golden proportions. During its excavations, compasses were discovered, which were used by architects and sculptors of the ancient world. In the Pompeii compass (a museum in Naples), the proportions of the golden division are also laid.

Figure: 8. Antique compass of the golden ratio

In the ancient literature that has come down to us, the gold division was first mentioned in the "Elements" of Euclid. In the second book of the "Beginnings" the geometric construction of the gold division is given. After Euclid, Gipsicles (II century BC), Pappus (III century AD) and others were engaged in the study of gold division. In medieval Europe with the gold division we met through the Arabic translations of Euclid's Elements. Translator J. Campano from Navarra (III century) made comments on the translation. The secrets of the gold division were jealously guarded, kept in strict secrecy. They were known only to the initiates.

During the Renaissance, interest in the golden division among scientists and artists increased in connection with its application both in geometry and in art, especially in architecture Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge ... He conceived and began to write a book on geometry, but at this time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his venture. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the painter Piero della Francesca, who wrote two books, one of which was entitled 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 the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the court of Moro at that time. In 1509, Luca Pacioli's book Divine Proportion was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was a rapturous hymn to the golden ratio. Among the many virtues of the golden ratio, the monk Luca Pacioli did not fail to name its "divine essence" as an expression of the divine trinity of God the Son, God the Father and God the Holy Spirit (it was understood that the small segment is the personification of the God of the Son, the larger segment is the God of the Father, and the entire segment - the god of the holy spirit).

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

At the same time, in northern Europe, in Germany, Albrecht Durer was working on the same problems. He sketches the introduction to the first draft of a treatise on proportions. Durer writes. “It is necessary that someone who knows how to 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 Durer develops in detail the theory of the proportions of the human body. Durer assigned an important place in his system of ratios to the golden ratio. Human height is divided in golden proportions by the belt line, as well as by the line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is known.

The great astronomer of the XVI century. Johannes Kepler called the golden ratio one of the treasures of geometry. He was the first to draw attention to the significance of the golden ratio for botany (plant growth and structure).

Kepler called the golden proportion of the continuation of itself “It is arranged like this,” he wrote, “that the two lowest terms of this endless proportion add up to the third term, and any two last terms, if added, 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 upward (increasing row) and in the direction of decreasing (descending row).

If on a straight line of arbitrary length, postpone the segment m, next to postpone the segment M... Based on these two segments, we build a scale of segments of the golden ratio of the ascending and descending series

Figure: nine. Building a scale of segments of the golden ratio

In the following centuries, the rule of the golden ratio turned into an academic canon, and when, over time, the struggle with the academic routine began in art, in the heat of the struggle “the child was thrown out along with the water”. The golden section was again "discovered" in the middle of the 19th century. In 1855, the German researcher of the golden ratio, Professor Zeising, published his work Aesthetic Research. With Zeising, exactly what happened was what should inevitably happen to a researcher who considers the phenomenon as such, without any connection with other phenomena. He absolutized the proportion of the golden ratio, 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 "mathematical aesthetics."

Figure: ten. Golden proportions in parts of the human body

Zeising did a tremendous 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 ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 \u003d 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 \u003d 1.6. In a newborn, the ratio 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 ratio are also manifested in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

Figure: eleven. Golden proportions in the human figure

Zeising tested the validity of his theory on Greek statues. In most detail, he developed the proportions of Apollo Belvedere. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic dimensions were subjected to research. Zeising gave a definition of the golden ratio, showed how it is expressed in line segments and in numbers. When the numbers expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction or the other. His next book was titled “The Golden Division as the Basic Morphological Law in Nature and Art”. In 1876, a small book, almost a brochure, was published in Russia, with a presentation of Zeising's work. The author took refuge under the initials Yu.F.V. No painting is mentioned in this edition.

In the late XIX - early XX centuries. a lot of purely formalistic theories appeared about the use of the golden ratio in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

Fibonacci series

The name of the Italian mathematician monk Leonardo from Pisa, better known as Fibonacci (son of Bonacci), is indirectly associated with 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:

Row 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 two previous ones 2 + 3 \u003d 5; 3 + 5 \u003d 8; 5 + 8 \u003d 13, 8 + 13 \u003d 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the gold division. So, 21: 34 \u003d 0.617, and 34: 55 \u003d 0.618. This relationship is indicated by the symbol F... Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in golden proportion, its increase or decrease to infinity, when the smaller segment relates to the larger one as the larger one to everything.

Fibonacci also dealt with the practical needs of trading: what is the smallest amount of weights to weigh a commodity? Fibonacci proves that the following system of weights is optimal: 1, 2, 4, 8, 16 ...

Generalized golden ratio

The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of golden division.

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

One of the advances 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" row of weights 1, 2, 4, 8, 16 ... at first glance are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 \u003d 1 + 1; 4 \u003d 2 + 2 ..., in the second it is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... Is it possible to find a general mathematical formula from which and " binary "series, and the Fibonacci series? Or maybe this formula will give us new numerical sets with some new unique properties?

Indeed, let us set the numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5 ... Consider a number series, S + 1 of the first members of which are units, and each of the subsequent ones is equal to the sum of two members of the previous one and spaced from the previous one by S steps. If n-th term of this series we denote by φ S ( n), then we obtain the general formula φ S ( n) \u003d φ S ( n - 1) + φ S ( n - S - 1).

Obviously, for S \u003d 0 from this formula we obtain a "binary" series, for S \u003d 1 - Fibonacci series, for S \u003d 2, 3, 4.new series of numbers, which are called S- Fibonacci numbers.

In general, gold S- the proportion is the positive root of the golden equation S-sections x S + 1 - x S - 1 \u003d 0.

It is easy to show that for S \u003d 0, the segment is divided in half, and when S \u003d 1 - the familiar classic golden ratio.

Relations of neighbors S-Fibonacci numbers coincide with absolute mathematical precision in the limit with gold S-proportions! Mathematicians in such cases say that gold S-sections are numerical invariants S- Fibonacci numbers.

Facts supporting the existence of gold S-sections in nature, the Belarusian scientist E.M. Forty 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 linked to each other by one of the gold S- proportions. This allowed the author to put forward the hypothesis that gold S-sections are numerical invariants of self-organizing systems. Having been confirmed experimentally, this hypothesis may be of fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.

With codes golden S-proportions, you can express any real number as the sum of the degrees of gold S-proportions with integer coefficients.

The fundamental difference between this method of coding numbers is that the bases of new codes, which are golden S- proportions, at S \u003e 0 turn out to be irrational numbers. 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 natural numbers were "discovered"; then their relations are rational numbers. And only later - after the discovery of incommensurable segments by the Pythagoreans - irrational numbers appeared. For example, in decimal, pentary, binary and other classical positional number systems, natural numbers - 10, 5, 2 - were chosen as a kind of fundamental principle, from which all other natural numbers, as well as rational and irrational numbers were constructed according to certain rules.

A kind of alternative to the existing methods of numbering is a new, irrational system, as a fundamental principle, the beginning of which is an irrational number (which, 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 in the form of a finite - and not infinite, as previously thought! - the sums of the degrees of any of the gold S- proportions. This is one of the reasons why "irrational" arithmetic, possessing amazing mathematical simplicity and elegance, seems to have absorbed the best qualities of classical binary and "Fibonacci" arithmetic.

The principles of shaping in nature

Everything that took some form, formed, grew, sought to take a place in space and preserve itself. This striving finds implementation mainly in two versions - growing up or spreading along 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 golden ratio would be incomplete, if not the spiral.

Figure: 12. Archimedes spiral

The shape of the spirally curled shell caught the attention of Archimedes. He studied it and derived the spiral equation. The spiral drawn from this equation is named after him. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

Even Goethe emphasized the tendency of nature to spiral. 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. 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 weaves the web in a spiral manner. A hurricane spins like a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral the "life curve".

Among the roadside grasses grows an unremarkable plant - chicory. Let's take a closer look at him. A process has formed from the main stem. The first sheet is located right there.

Figure: 13. Chicory

The shoot makes a strong ejection into space, stops, releases a leaf, but is shorter than the first, again ejects into space, but with less force, releases a leaf of an even smaller size, and ejects again. If the first emission is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 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. The impulses of its growth were gradually decreasing in proportion to the golden section.

Figure: 14. Viviparous lizard

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

In both the plant and the animal world, the formative tendency of nature is persistently breaking 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 the parts, the repetition of the structure of the whole is manifested.

Figure: fifteen. Bird egg

The great Goethe, a poet, naturalist and artist (he painted and painted in watercolors), dreamed of creating a unified teaching about 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 this century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without considering 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 genetic structures of living organisms. These patterns, as indicated above, are in the structure of individual organs of a person and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

Golden ratio and symmetry

The Golden Ratio cannot be considered by itself, separately, without a connection with symmetry. The great Russian crystallographer G.V. Wolfe (1863 ... 1925) considered the golden ratio to be one of the manifestations of symmetry.

The gold division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the gold division is an asymmetric symmetry. The science of symmetry includes concepts such 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 immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments, equal values. Dynamic symmetry is characterized by an increase or decrease in segments, and it is expressed in the values \u200b\u200bof the golden section of an increasing or decreasing series.

The Bulgarian journal Otechestvo (# 10, 1983) published an article by Tsvetan Tsekov-Karandash "On the second golden section", which follows from the main section and gives a different ratio of 44: 56.

This proportion is found in architecture, and also occurs when building compositions of images of an elongated horizontal format.

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

Golden Triangle

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

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

We draw a straight line AB... From point AND we postpone on it three times a segment O of an arbitrary value, through the resulting point R draw a perpendicular to the line AB, perpendicular to the right and left of the point R postpone the segments ABOUT... Obtained points d and d1 connect with straight lines AND... Line segment dd1 put off on the line Ad1, getting point WITH... She split the line Ad1 in the proportion of the golden ratio. Lines Ad1 and dd1 use to build a "golden" rectangle.

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

Figure: 5. 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 Durer (1471 ... 1528). Let O be the center of the circle, A a point on the circle, and E the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, we postpone the segment CE \u003d ED on the diameter. The side length of a regular pentagon inscribed in a circle is DC. We put aside the segments DC on the circle and get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden proportion.

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

Figure: 6. Construction of the golden triangle

We draw a straight line AB. From point A, we lay on it three times a segment O of arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the obtained points d and d1 with straight lines to point A. line Ad1, getting point C. She divided line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to draw a "golden" rectangle.

The history of the golden ratio

It is believed that the concept of gold division was introduced into scientific use by Pythagoras, the ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamun indicate that Egyptian masters used the golden division ratios 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 \u200b\u200bof the golden division. The architect Khesira, depicted on the relief of a wooden board from the tomb of his name, is holding measuring instruments in which the proportions of the golden division are fixed.

Figure: 7. Dynamic rectangles

Figure: 8. Antique compasses of the golden section

In the ancient literature that has come down to us, the golden division was first mentioned in the "Elements" of Euclid. In the second book of the "Beginnings" the geometric construction of the gold division is given. After Euclid, Gipsicles (II century BC), Pappus (III century AD) and others were engaged in the study of gold division. In medieval Europe with the gold division we met from the Arabic translations of Euclid's Elements. Translator J. Campano from Navarra (III century) made comments on the translation. The secrets of the gold division were jealously guarded, kept in strict secrecy. They were known only to the initiates.

During the Renaissance, interest in the gold division among scientists and artists increased in connection with its application both in geometry and in art, especially in architecture Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge ... He conceived and began to write a book on geometry, but at this time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his venture. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the painter Piero della Francesca, who wrote two books, one of which was entitled On Perspective in Painting. He is considered the creator of descriptive geometry.

Leonardo da Vinci also paid a lot of attention to the study of the gold division. He produced sections of a stereometric solid formed by regular pentagons, and each time he received rectangles with aspect ratios in gold division. Therefore, he gave this division the name Golden Ratio. So it still holds as the most popular.

At the same time, in the north of Europe, in Germany, Albrecht Durer was working on the same problems. He sketches an introduction to the first draft of a treatise on proportions. Durer writes. “It is necessary that someone who knows how to 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 Durer develops in detail the theory of the proportions of the human body. Durer assigned an important place in his system of ratios to the golden ratio. A person's height is divided in golden proportions by the belt line, as well as by the line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is known.

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

If on a straight line of arbitrary length, postpone segment m, next to it we lay off segment M. Based on these two segments, we build a scale of segments of the golden ratio of the ascending and descending series

Figure: 9. Building a scale of segments of the golden ratio

Figure: 10. Golden proportions in parts of the human body

Figure: 11. Golden proportions in the human figure

Zeising did a tremendous 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 ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 \u003d 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 \u003d 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 ratio are also manifested in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

In the late XIX - early XX centuries. a lot of purely formalistic theories appeared on the use of the golden section in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

Since ancient times, people have been worried about whether such elusive things as beauty and harmony are subject to any mathematical calculations. Of course, it is impossible to fit all the laws of beauty into several formulas, but by studying mathematics, we can discover some of the components of beauty - the golden ratio. Our task is to find out what the golden ratio is and establish - where humanity has found the application of the golden ratio.

You have probably noticed that we have different attitudes towards objects and phenomena of the surrounding reality. Be sdecency, be suniformity, 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, and lift our spirits.

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

The golden section was known even 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 ratio". Aristotle found the correspondence of the "golden section" to the ethical law. The highest harmony of the "golden ratio" will be preached by Leonardo da Vinci and Michelangelo, because beauty and the "golden ratio" are one and the same. And Christian mystics will paint pentagrams of the "golden section" on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but never find its exact meaning. Be sthe 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 ratio" is. But you will certainly see this proportion in the curves of sea shells, in the form of flowers, in the appearance of beetles, and in a beautiful human body. Everything alive and everything beautiful - everything obeys the divine law, the name of which is the "golden ratio". So what is the golden ratio? What is this perfect, divine combination? Maybe this is the law of beauty? Or is he a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The "golden ratio" is both the one and the other, and the third. Only not separately, but simultaneously ... And this is its true mystery, its great secret.

Probably, it is difficult to find a reliable measure for an objective assessment of the beauty itself, and one cannot do with logic alone. 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. These are, first of all, people of art, as we call them: artists, architects, sculptors, musicians, writers. But these are people of exact sciences, first of all, mathematicians.

Trusting the eye more than other senses, Man first of all learned to distinguish objects around him in shape. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden ratio, 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 relation to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

GOLDEN RATIO - HARMONIC PROPORTION

In mathematics, proportion is the equality of two ratios:

A straight 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 \u003d AC: BC.

The latter is the golden division (section).

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part as the larger part itself refers to the smaller one, in other words, the smaller segment refers to the larger one as the larger to the whole

a: b \u003d b: c or c: b \u003d b: a.

Geometric image of the golden ratio

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

Division of a straight line segment along the golden ratio. BC \u003d 1 / 2AB; CD \u003d BC

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

The segments of the golden ratio are expressed without sfinal fraction AE \u003d 0.618 ..., if AB is taken as a unit, BE \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB is taken as 100 parts, then most of the segment is 62, and the smaller is 38 parts.

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

Solution to this equation:

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

SECOND GOLDEN SECTION

This proportion is found in architecture.

Construction of the second golden ratio

The division is carried out as follows. The segment AB is divided in the proportion of the golden ratio. The perpendicular CD is restored from point C. The radius AB is point D, which is connected by a line to point A. The right angle ACD is divided in half. From point C a line is drawn up to the intersection with line AD. Point E divides the segment AD in the ratio 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 ratio. It is located in the middle between the golden section line and the middle line of the rectangle.

GOLDEN TRIANGLE (pentagram)

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

Constructing 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 Durer. Let O be the center of the circle, A a point on the circle, and E the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, we postpone the segment CE \u003d ED on the diameter. The side length of a regular pentagon inscribed in a circle is DC. We put aside the segments DC on the circle and get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden proportion.

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

We draw a straight line AB. From point A, we lay on it three times a segment O of an arbitrary value, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the obtained points d and d 1 with straight lines to 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. Lines Ad 1 and dd 1 are used to draw a "golden" rectangle.

Building the golden triangle

HISTORY OF THE GOLDEN SECTION

Indeed, the proportions of the Cheops pyramid, temples, household items and ornaments from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratio 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 \u200b\u200bof the golden division. The architect Khesira, depicted on the relief of a wooden board from the tomb of his name, holds in his hands measuring instruments in which the proportions of the golden division are fixed.

Dynamic rectangles

Plato also knew about the gold division. The Pythagorean Timaeus in the dialogue of the same name by Plato says: “It is impossible for two things to be perfectly combined without the third, since a thing must appear between them that would hold them together. The proportion can do this in the best way, for if three numbers have the property that the average relates to the smaller as much as the larger to the average, and, conversely, the smaller relates to the average as the average to the larger, then the last and the first will be the average, and middle - first and last. Thus, everything that is necessary will be the same, and since it will be the same, it will be the whole. " Plato builds the earthly world using triangles of two kinds: isosceles and not 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 of the 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"). With the help of triangles, Plato builds four regular polyhedrons, associating them with the four earthly elements (earth, water, air and fire). And only the last of the five existing regular polyhedrons - the dodecahedron, all twelve faces of which are regular pentagons, claims to be a symbolic representation of the celestial world.

ICOSAhedron and dodecahedron

The honor of discovering the dodecahedron (or, as it was believed, 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. In this figure, many relationships of the golden ratio are really captured, so the latter was assigned the main role in the heavenly world, on which the brother of the minority Luca Pacioli later insisted.

Antique compasses of the golden ratio

In the ancient literature that has come down to us, the gold division was first mentioned in the "Elements" of Euclid. In the second book of the Beginnings, the geometric construction of the golden division is given. After Euclid, Gipsicles (II century BC), Pappus (III century AD), and others were engaged in the study of gold division. In medieval Europe, they became acquainted with the gold division from the Arabic translations of Euclid's Elements. Translator J. Campano from Navarra (III century) made comments on the translation. The secrets of the gold 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, and 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. So, in that period, the assertion of humanism was widely adopted by a diagram describing the structure of the human body.

Such a picture, in fact, reproducing the pentagram, was repeatedly used by Leonardo da Vinci. Her interpretation: the human body has divine perfection, because the proportions inherent in it are the same as in the main heavenly figure. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience and little knowledge. He conceived and began to write a book on geometry, but at this time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his venture. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the painter Piero della Francesca, who wrote two books, one of which was entitled 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 the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the court of Moro at that time. In 1509, Luca Pacioli's book De divina proportione (1497, published in Venice 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 a rapturous hymn to the golden ratio. There is only one such proportion, and uniqueness is the highest quality of God. The holy trinity is embodied in it. This proportion cannot be expressed by an accessible number, it remains hidden and secret and is called irrational by the mathematicians themselves (so God can neither be defined nor explained in words). God never changes and represents everything in everything and everything in every part, so the golden ratio for every continuous and definite quantity (regardless of whether it is large or small) is one and the same, it can neither be changed nor changed. to another is perceived by reason. God called into being the heavenly virtue, otherwise called the fifth substance, with its help, and four other simple bodies (four elements - earth, water, air, fire), and on their basis he called into being every other thing in nature; so our sacred proportion, according to Plato in 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 a lot of attention to the study of the gold division. He made sections of a stereometric body formed by regular pentagons, and each time he received rectangles with aspect ratios in gold division. Therefore, he gave this division the name of the golden ratio. So it still holds as the most popular.

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

Kepler called the golden proportion of the continuation of itself “It is arranged like this,” he wrote, “that the two younger members of this endless proportion add up to the third term, and any two last terms, if added, 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 upward (increasing row) and in the direction of decreasing (descending row).

If on a straight line of arbitrary length, postpone the segment m , next to postpone the segment M ... Based on these two segments, we build a scale of segments of the golden ratio of the ascending and descending series.

Building a scale of segments of the golden ratio

In 1855, the German researcher of the golden ratio, Professor Zeising, published his work Aesthetic Research. With Zeising, exactly what happened was what should inevitably happen to a researcher who considers the phenomenon as such, without any connection with other phenomena. He absolutized the proportion of the golden ratio, 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 "mathematical aesthetics."

GOLDEN SECTION AND SYMMETRY

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

The golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the gold division is asymmetric symmetry. The science of symmetry includes concepts such 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 immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments, equal values. Dynamic symmetry is characterized by an increase or decrease in segments, and it is expressed in the values \u200b\u200bof the golden section of an increasing or decreasing series.

FIBONACCI RANGE

The name of the Italian mathematician monk Leonardo from Pisa, better known as Fibonacci, is indirectly associated 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, which collected all the problems known at that time.

Row 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 two previous ones 2 + 3 \u003d 5; 3 + 5 \u003d 8; 5 + 8 \u003d 13, 8 + 13 \u003d 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the gold division. So, 21: 34 \u003d 0.617, and 34: 55 \u003d 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 golden proportion, its increase or decrease to infinity, when a smaller segment refers to the larger as the larger to everything.

As shown in the lower figure, the length of each finger joint is related to the length of the next joint in proportion F. The same relationship is observed in all fingers and toes. This connection is somehow unusual, because one finger is longer than the other without any visible regularity, 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 of 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 matches the F-proportion model, just like it does in the human body.

GENERALIZED GOLDEN SECTION

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

One of the advances 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 their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 \u003d 1 + 1; 4 \u003d 2 + 2 ..., in the second it is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 ... Is it possible to find a general mathematical formula from which one obtains “binary "Series, and Fibonacci series? Or maybe this formula will give us new numerical sets with some new unique properties?

Indeed, let us set a numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5 ... Consider a numerical series, S + 1, the first members of which are ones, and each of the subsequent ones is equal to the sum of two members of the previous and spaced S steps from the previous one. If we denote the nth term of this series by? S (n), then we get the general formula? S (n) \u003d? S (n-1) +? S (n-S-1).

Obviously, for S \u003d 0 from this formula we get a "binary" series, for S \u003d 1 - a Fibonacci series, for S \u003d 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-ratio equation x S + 1 -x S -1 \u003d 0.

It is easy to show that when S \u003d 0, the segment is divided in half, and when S \u003d 1, the familiar classical golden ratio.

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

The facts confirming the existence of golden S-sections in nature are cited by the Belarusian scientist E.M. Forty 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 linked to each other by one from golden S-proportions. This allowed the author to put forward a hypothesis that the golden S-sections are numerical invariants of self-organizing systems. Confirmed experimentally, this hypothesis may be of fundamental importance for the development of synergetics, a new field of science that studies processes in self-organizing systems.

Using golden S-ratio codes, you can express any real number as the sum of the degrees of golden S-proportions with integer coefficients.

The fundamental difference between this method of coding numbers is that the bases of new codes, which are golden S-proportions, for S\u003e 0 turn out to be irrational numbers. 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 natural numbers were "discovered"; then their relations are rational numbers. And only later, after the discovery of incommensurable segments by the Pythagoreans, irrational numbers appeared. For example, in decimal, pentary, binary and other classical positional number systems, natural numbers were chosen as a kind of fundamental principle: 10, 5, 2, of which all other natural numbers, as well as rational and irrational numbers were constructed according to certain rules.

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

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

PRINCIPLES OF FORMATION IN NATURE

Everything that took some form, formed, grew, sought to take a place in space and preserve itself. This striving finds implementation mainly in two versions: growing up or spreading along the surface of the earth and twisting in a spiral.

The shape of the spirally curled shell caught the attention of Archimedes. He studied it and deduced the spiral equation. The spiral drawn from this equation is named after him. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

Even Goethe emphasized the tendency of nature to spiral. 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. 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 weaves the web in a spiral manner. A hurricane spins like a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral the "life curve".

Mandelbrot series

The golden spiral is closely related to cycles. The modern science of chaos studies simple cyclic feedback operations and the fractal forms they generate that were previously unknown. The figure shows the famous Mandelbrot series - a page from the dictionary slimbs of individual patterns called Julian rows. Some scientists associate the Mandelbrot series with the genetic code of cell nuclei. The sequential increase in cross-sections reveals fractals of amazing artistic complexity. And here, too, there are logarithmic spirals! This is all the more important since both the Mandelbrot series and the Julian series are not an invention of the human mind. They arise from the area of \u200b\u200bthe prototypes of Plato. 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 him. A process has formed from the main stem. The first sheet is located right there.

If the first emission is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 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. The impulses of its growth were gradually decreasing in proportion to the golden section.

Chicory

In many butterflies, the ratio of the sizes of the chest and abdominal parts of the body corresponds to the golden ratio. Having folded its wings, the 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 a lizard, at first glance, proportions pleasant to our eyes are caught - the length of its tail is as much related to the length of the rest of the body as 62 to 38.

Viviparous lizard

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

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

Such shapes of bird eggs are not accidental, since it has now been established that the shape of eggs described by the ratio of the golden ratio 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 shapes and resemble the shape of an axis that tends to turn into a spiral.

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

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 exquisite beauty of the figures that form the snowflakes, all the axes, circles and geometric shapes in the snowflakes are also always, without exception, built according to the perfect clear formula of the golden ratio.

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

Adeno virus

For the first time, the golden ratio in the structure of viruses was discovered in the 1950s. scientists from London Birkbeck College A. Klug and D. Kaspar. The first to appear in a logarithmic form was the Polyo virus. The form of this virus was found to be similar to that of the Rhino virus.

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

Klug's commentary 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. 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, radiolaria (ray beetles), the skeleton of which is made of silica.

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

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

HUMAN BODY AND GOLDEN SECTION

All human bones are sustained in the proportion of the golden ratio. 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 perfectly folded.

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 a person's foot and the navel point as a unit of measurement, then a person's height is equivalent to 1.618.

distance from shoulder level to crown of head and head size is 1: 1.618;
the distance from the navel point to the crown of the head and from shoulder level 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;
the exact presence of the golden ratio in a person's face is the ideal of beauty for the human eye;
the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618;
face height / face width;
the center point 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 nose / distance between nostrils;
distance between pupils / distance between eyebrows.

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

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 number of the golden ratio (excluding the thumb).

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

A person has 2 hands, fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, only 10, but with the exception of two biphalangeal 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 their arms is equal to height.

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

In the inner ear of a person there is an organ called Cochlea ("snail"), which performs the function of transmitting sound vibration. This bone-like structure is filled with liquid and is also created in the form of a snail, containing a stable logarithmic spiral shape \u003d 73 0 43 ".

Blood pressure changes as the heart works. It reaches its greatest value in the left ventricle of the heart at the time of its compression (systole). In the arteries during the systole of the ventricles of the heart, the 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 maximum (systolic) to minimum (diastolic) pressure is 1.6 on average, 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 pressure is 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 spirals. The length of each of these spirals is 34 angstroms, the width is 21 angstroms. (1 angstrom is one hundred millionth of a centimeter).

The structure of the spiral section of the DNA molecule

So 21 and 34 are numbers following each other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic spiral of the DNA molecule carries the golden ratio formula 1: 1.618.

GOLDEN SECTION IN SCULPTURE

Sculptural structures, monuments are erected to perpetuate significant events, to preserve the names of famous people, their exploits and deeds in the memory of descendants. It is known that even in antiquity, sculpture was based on the theory of proportions. The relationship of parts of the human body was associated with the golden ratio formula. The proportions of the "golden section" create the impression of harmony and beauty, so the sculptors used them in their works. The sculptors claim that the waist divides the perfect human body in terms of the "golden ratio". 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 these were the statue of Olympian Zeus (which was considered one of the wonders of the world) and the Parthenon Athena.

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

GOLDEN SECTION IN ARCHITECTURE

In books about the "golden ratio" 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 ratio", 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 (5th century BC).

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed in terms of various powers of the number Ф \u003d 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 Pentilian marble. The nobility of the material from which the temple was built made it possible to limit the use of the usual coloring 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 make the division of the Parthenon according to the "golden ratio", then we 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 are Egyptian pyramids built according to the perfect proportions of the golden ratio; the same phenomenon is found in the Mexican pyramids.

For a long time, it was believed that the architects of Ancient Russia built everything "by eye", without special mathematical calculations. However, recent studies have shown that Russian architects were well aware of mathematical proportions, as evidenced by an analysis of the geometry of ancient temples.

The famous Russian architect M. Kazakov widely used the “golden ratio” in his work. His talent was multifaceted, but to a greater extent he was revealed in numerous completed projects of residential buildings and estates. For example, the "golden ratio" 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 now 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 - Pashkov's house - is one of the most perfect pieces of architecture by V. Bazhenov.

Pashkov House

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

Many of the architect's statements deserve attention today. V. Bazhenov said about his favorite art: “The most important architecture has three subjects: beauty, tranquility and strength of a building ... Knowledge of proportion, perspective, mechanics or physics in general serves as a guide to achieve this, and reason is their common leader”.

GOLDEN SECTION IN MUSIC

Back in 1925, art critic L.L. Sabaneev, having analyzed 1,770 musical works by 42 authors, showed that the overwhelming majority of outstanding works can be easily divided into parts either by theme, or by intonational structure, or by modal structure, which are in relation to the golden ratio. Moreover, the more talented the composer, the greater the number of his works found golden sections. According to Sabaneev, the golden ratio leads to the impression of a special harmony of the musical composition. Sabaneev checked this result on all 27 etudes by Chopin. He found 178 golden sections in them. At the same time, it turned out that not only large parts of the studies are divided according to their duration in relation to the golden ratio, but also parts of the studies inside are often divided in the same ratio.

Composer and scientist M.A. Marutaev calculated the number of measures in the famous sonata "Appassionata" and found a number of interesting numerical ratios. In particular, there are two main sections in the development - the central structural unit of the sonata, where themes develop intensively and replace each other. In the first - 43.25 bars, in the second - 26.75. The ratio 43.25: 26.75 \u003d 0.618: 0.382 \u003d 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 a harmonious ordering of sounds, then poetry is a harmonious ordering of speech. A clear rhythm, a regular alternation of stressed and unstressed syllables, an ordered dimension of poems, their emotional richness make poetry a sister of musical works. The golden ratio in poetry is primarily manifested as the presence of a certain moment of the poem (culmination, semantic break, the main idea of \u200b\u200bthe work) in the line corresponding to the division point of the total number of lines of the poem in the golden proportion. So, if the poem contains 100 lines, then the first point of the Golden Section 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 of the golden ratio! The works of Shota Rustaveli and M.Yu. Lermontov are also built according to the principle of the Golden Section.

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

THE GOLDEN SECTION IN POETRY

Studies of poetry are just beginning from these positions. 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 our search for the golden ratio - the measure of harmony and beauty.

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

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 be changed arbitrarily. However, it turned out that this is not the case. For example, N. Vasyutinsky's analysis of poems by A.S. Pushkin showed that the sizes of the verses are distributed very unevenly; it turned out that Pushkin clearly prefers sizes of 5, 8, 13, 21 and 34 lines (Fibonacci numbers).

It has been noticed by many researchers that poems are like musical works; they also have climax points that divide the poem in the proportion of the golden ratio. Consider, for example, a poem by A.S. Pushkin's "Shoemaker":

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

One of the last poems of Pushkin "I do not value high-profile rights ..." consists of 21 lines and two semantic parts stand out 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 what the gods refused

It's my sweet lot to challenge taxes

Or prevent the kings from fighting with each other;

And little sorrow for me, is the seal free

Fools fools, or sensitive censorship

In magazine designs, he is embarrassed by the joker.

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

Some, better, rights are dear to me:

Another, better, freedom I need:

To depend on the king, to depend on the people -

Isn't it all the same to us? God is with them.

Do not give a report, only to yourself

Serve and please; for power, for livery

Do not bend neither conscience, nor thoughts, nor neck;

To wander here and there on a whim,

Marveling at the beauty of the divine nature,

And before the creatures of arts and inspiration

Trembling joyfully in delight 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 semantic content, that is, the entire poem is built according to the laws of the golden ratio.

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

N. Vasyutinsky states: "The culmination of the chapter is Evgeny's explanation of his love for Tatiana - the line" Turn pale and fade ... here is 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 finest correspondence to the size of the golden ratio! This is a great miracle of harmony, accomplished by the genius of Pushkin! "

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

Lermontov's famous poem "Borodino" is divided into two parts: the introduction, addressed to the narrator, occupying only one stanza ("Tell me, uncle, it's not for nothing ..."), and the main part, which is an independent whole, which splits into two equal parts. The first of them describes, with increasing tension, the expectation of a fight, in the second - the fight itself with a gradual decrease in tension towards the end of the poem. The border between these parts is the culmination point of the work and falls exactly on the point of dividing by its golden ratio.

The main part of the poem consists of 13 seven lines, that is, 91 lines. Dividing it with the golden ratio (91: 1.618 \u003d 56.238), we make sure that the division point is at the beginning of the 57th verse, where there is a short phrase: "Well, there was a day!" It is this phrase that represents the “culmination point of excited anticipation”, which concludes the first part of the poem (expectation of a fight) and opens its second part (description of the fight).

Thus, the golden ratio plays a very meaningful role in poetry, highlighting the culminating point 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 are the Georgian scientist, academician G.V. Tsereteli attributes it to the poet's conscious use of the golden section both in the formation of the form of the poem and in the construction of its poems.

Rustaveli's poem consists of 1,587 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 hemistich. All hemistichs are divided into two segments of two types: A - a hemistich with equal segments and an even number of syllables (4 + 4); B is a hemistich with asymmetric division into two unequal parts (5 + 3 or 3 + 5). Thus, in hemistich B, the ratios are 3: 5: 8, which is an approximation to the golden ratio.

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

In our time, a new kind of art has been born - cinema, which has absorbed the drama of action, painting, and music. It is legitimate to look for the manifestations of the golden ratio in outstanding works of cinematography. The creator of the masterpiece of world cinema "Battleship Potemkin", film director Sergei Eisenstein, was the first to do this. In the construction of this picture, he managed to embody the main principle of harmony - the golden ratio. As Eisenstein himself notes, the red flag on the mast of the rebel battleship (the film's apogee) hovers at the golden ratio, measured from the end of the film.

GOLDEN SECTION IN FONTS AND HOUSEHOLD ITEMS

A special type of fine art in Ancient Greece should be highlighted in the manufacture and painting of all kinds of vessels. In a graceful form, the proportions of the golden ratio are easily guessed.

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

OPTIMAL PHYSICAL PARAMETERS OF THE EXTERNAL ENVIRONMENT

It is known that the maximum sound volume, which causes pain, is 130 decibels. If we divide this interval by the golden ratio of 1.618, then we get 80 decibels, which are characteristic of the loudness of a human cry. If now 80 decibels are divided by the golden ratio, then 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 person's whisper. Thus, all the characteristic parameters of the sound volume are interconnected through the golden proportion.

At a temperature of 18-20 0 C interval humidity 40-60% is considered optimal. The limits of the optimum humidity range can be obtained if the absolute humidity of 100% is divided twice by the golden ratio: 100 / 2.618 \u003d 38.2% (lower limit); 100 / 1.618 \u003d 61.8% (upper bound).

When air pressure 0.5 MPa, a person experiences unpleasant sensations, his physical and psychological activity worsens. At a pressure of 0.3-0.35 MPa, only short-term work is allowed, and at a pressure of 0.2 MPa, it is allowed to work no more than 8 minutes. All these characteristic parameters are related to each other by the golden ratio: 0.5 / 1.618 \u003d 0.31 MPa; 0.5 / 2.618 \u003d 0.19 MPa.

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

Let us divide the indicated range of positive temperatures by the golden ratio. In this case, we get two boundaries (about the boundaries are the 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 CROSSING IN PAINTING

Back in the Renaissance, artists discovered that any painting has certain points that involuntarily rivet our attention, the so-called visual centers. In this case, it does not matter at all what format the picture is 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 by the artists of that time was called the "golden section" of the picture.

Moving on to examples of the "golden ratio" in painting, one cannot help but focus on the work of Leonardo da Vinci. His personality is one of the mysteries of history. Leonardo da Vinci himself said: "Let no one, not being a mathematician, dare to read my works."

He gained fame as an unsurpassed artist, great scientist, 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 activity 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 “about 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, which are parts of a regular star-shaped 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 wife of a banker, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint the portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became lively and interesting.

STORY... Once upon a time there was one poor man, he had four sons: three smart ones, and one of them this and that. And then death came for my father. Before parting with his life, he called his children to him and said: “My sons, I will soon die. As soon as you bury me, lock the hut and go to the ends of the world to seek your happiness. Let each of you learn something so that he can feed himself. " The father died, and the sons dispersed around the world, agreeing three years later to return to the clearing of their native grove. The first brother came, who learned to do carpentry, cut down a tree and hewn it, made a woman out of it, walked away a little and waited. The second brother returned, saw a wooden woman and, since he was a tailor, dressed her in one minute: 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 came. He did not know how to carpentry and sew, he only knew how to listen to what the earth, trees, grasses, animals and birds say, knew the course of heavenly bodies and also knew how to sing wonderful songs. He began to sing 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 a father to me. You dressed me, and you decorated - be my brothers. And you, who breathed soul into me and taught me to enjoy life, I need you alone for life. "

Finishing the tale, Leonardo looked at Monna Lisa, her face lit up with light, her eyes shining. Then, as if awakening from a dream, she sighed, ran her hand over her face and without a word went to her place, folded her arms and assumed the usual position. 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 a person who has learned a secret and, carefully keeping it, cannot contain the triumph. Leonardo worked in silence, afraid to miss this moment, this ray of sun that illuminated his boring model ...

It is difficult to note what they noticed in this masterpiece of art, but everyone talked about Leonardo's deep knowledge of the structure of the human body, thanks to which he was able to catch this, as it were, a mysterious smile. They talked about the expressiveness of individual parts of the picture and about the landscape, an unprecedented companion of a 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 shows air, it envelops the figure in a transparent haze. Despite his success, Leonardo was gloomy, the situation in Florence seemed painful to the artist, he got ready to go. He was not helped by the reminders of the surging orders.

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

Pine grove

The presence of bright verticals and horizontals in the picture, dividing it in relation to the golden ratio, gives it the character of poise 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 compositional scheme (with a predominance of verticals and horizontals) becomes unacceptable.

IN AND. Surikov. "Boyarynya 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 decline of the plot of the picture: the rise of Morozova's hand with the two-fingered sign of the cross, as the highest point; a hand helplessly outstretched to the same boyar, but this time the hand of an old woman - a beggar wanderer, a hand from under which, together 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, spaced 0.618 ... from the right edge of the picture, does not pass through the hand, not even through the head or eye of the boyaryn, but appears somewhere in front of the boyaryn's mouth.

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

There is no painting more poetic than the painting of Botticelli Sandro, and the great Sandro does not have a painting more famous than his "Venus". For Botticelli, his Venus is the embodiment of the idea of \u200b\u200buniversal harmony of the "golden ratio" 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 a considerable knowledge of geometry. In the famous fresco "The School of Athens", where the society of the great philosophers of antiquity is to be found in the temple of science, our attention is drawn to the group of Euclid, the greatest ancient Greek mathematician, who analyzes a complex drawing.

The clever combination of two triangles is also built according to 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 corner - at the vanishing point of the perspectives, and the side area denotes the proportions of the spatial gap between the two parts of the arches.

The golden spiral in Raphael's painting "The Beating of the Babies". Unlike the golden ratio, the feeling of dynamics, excitement is manifested, perhaps, most strongly in another simple geometric figure - the spiral. The multi-figured composition, executed 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 plan to completion, but his sketch was engraved by an unknown Italian graphic artist Marcantinio Raimondi, who based on this sketch created the engraving "Beating of Babies".

Massacre of the innocents

If, on Raphael's preparatory sketch, mentally draw lines going 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 holding him to her, the warrior with the sword brought up and then along the figures of the same group on the right side sketch (in the figure, these lines are drawn in red), and after that connect these pieces with a curved dotted line, then a golden spiral is obtained with very high accuracy. This can be verified 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.

GOLD SECTION AND IMAGE PERCEPTION

It has long been known about the ability of the human visual analyzer to distinguish objects constructed according to the golden section algorithm as beautiful, attractive and harmonious. The golden ratio gives the feeling of the most perfect single 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 were 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 an elongated aspect ratio of 1: 2.31 (26:60 mm).

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

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

When drawing your own drawings, proportions close to the golden ratio and elongated prevail. On average, the proportions are 1: 2, with the right hemisphere giving preference to the proportions of the golden ratio, the left hemisphere moving away from the proportions of the golden ratio and drawing out the pattern.

Now draw some rectangles, measure their sides and find the aspect ratio. Which hemisphere is dominant in you?

GOLDEN SECTION IN PHOTOS

An example of using 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 at an arbitrary place in the frame.

Now let's conditionally divide the frame into segments, in proportion of 1.62 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 transfer our cat to the points of the "visual centers".

GOLDEN SECTION AND SPACE

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

However, one case that seemed to contradict the law: there was no planet between Mars and Jupiter. Concentrated observation of this region of the sky led to the discovery of the asteroid belt. This happened after the death of Titius in the early 19th century. The Fibonacci series is widely used: it is used to 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.

Note the stars emerging from the galaxy in a white spiral. Exactly 180 0 from one of the spirals another unfolding spiral emerges ... 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 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 Reality is much less than one percent of the whole - almost nothing. In fact, our real home is the invisible universe ...

In the Universe, all galaxies known to mankind and all bodies in them exist in the form of a spiral, corresponding to the golden ratio formula. 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: living and inanimate nature. Creations of inanimate nature are characterized by high stability, weak variability, judging by the scale of human life. A person is born, lives, grows old, dies, but the granite mountains remain the same and the planets revolve around the Sun in the same way as in the time of Pythagoras.

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

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

In this work, 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 cognizing the world, discovering more and more of its regularities and conclude: the principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature. It can be expected that the laws of development of various systems of nature, the laws of growth are not very diverse and can be traced in the most diverse formations. This is where the unity of nature is manifested. The idea of \u200b\u200bsuch a unity, based on the manifestation of the same patterns in heterogeneous natural phenomena, has retained its relevance from Pythagoras to the present day.

Airbrushing is based on the same "pillars" as other forms of art.

Our entire world can be described with numbers. Many numbers play such a significant role in this description that they have their own names: pi, expanent (e), etc. Among these "nominal" numbers, there is something quite remarkable. Mathematicians, artists, architects at different times called it "golden number", "divine number", "divine section". The term "golden ratio" was coined by Claudius Ptolemy, and it became popular thanks to Leonardo Da Vinci, who widely used it in his works. Artists have noticed that the proportions of forms, which are especially pleasing to the eye for perception, are based on the "golden ratio".

So what is this number? The golden ratio is called the Phi number (Phi) equal to 1.61803. The number is named after the great ancient Greek sculptor Phidius, who used it in his sculptures. How can you clearly demonstrate the principle of the "golden ratio"? Let's take a simple example. If you build a rectangle, one side of which is 1.618 times longer than the other, then the resulting aspect ratio is the "golden ratio". The most common "golden rectangles" in the modern world are credit cards. The human body is considered beautiful, and its proportions are ideal if the ratio between the smaller and larger parts of the body is equal to the ratio between the larger and the whole, that is, equal to the Phi number.

***
The most famous mathematical work of ancient science is the "Principles" of Euclid. It was from the "Beginnings" that the geometric problem "about dividing a segment in extreme and average ratio" came to us. Which is the "Golden Section" itself.
The essence of the problem is as follows:
Let us divide the segment AB by the point C in such a ratio that the greater part of the segment CB relates to the smaller part of the segment AC, as the segment AB to its most part CB, i.e.

Let us denote proportion (1.1) by x. Then, taking into account that AB \u003d AC + CB, the proportion (1.1) can be written in the following form:

Whence follows the following algebraic equation for calculating the desired proportion x:

x * \u003d x + 1. (1.2)
x * - squared

From the "physical meaning" of proportion (1.1) it follows that the sought-for solution of equation (1.2) must be a positive number, whence it follows that the solution to the problem of dividing a segment in the extreme and mean ratio is the positive root of equation (1.2), which we denote by, i.e

The approximate value of the golden ratio is:
= 1,61803 39887 49894 84820 45868 34365 63811 77203…

GOLDEN GEOMETRIC FIGURES

Based on the above proportions in geometry, the following concepts of golden geometric shapes are defined:
- golden rectangle (in which the ratio of the larger side to the smaller one is equal to the golden ratio);
- golden right-angled triangle;
- golden ellipse;
- golden isosceles triangle.

A rectangular triangle with sides 3: 4: 5 is called "perfect", "sacred" or "Egyptian".
The creators of the Egyptian pyramids chose a golden right-angled triangle for the Cheops pyramid, and the “sacred” triangle for the Khafre pyramid.

Pentagon ("pentagonon" - Greek), regular pentagon. If we draw all the diagonals in the pentagon, then as a result we get a pentagonal star called a pentagram (“pentagrammon” - Greek: “pente” - five and “grammon” - a line) or a pentacle.

The pentagram, called in popular beliefs "the witch's foot", played a large role in all magical sciences and was considered as a means of protection from evil spirits.
Every eight years, the planet Venus describes an absolutely correct pentacle along a large circle of the celestial sphere.
The building of the Pentagon, the US military, is shaped like the Pentagon.

The Pentagon and Pentacle include a number of remarkable figures that have been used extensively in art. In ancient art, the so-called law of the golden bowl is widely known, which was used by antique sculptors and goldsmiths. The shaded portion of the pentagon gives a schematic representation of the golden bowl.

Once upon a time in the Soviet Union there was a State quality mark, in which the motives of the golden cup are clearly visible.

In living nature, forms based on pentagonal symmetry are widespread - starfish, sea urchins, flowers ..

HARMONY GOLDEN SECTION
(brief overview of art history)

The great creations of Greek sculptors: Phidias, Polyktetus, Myron, Praxiteles, are considered the standard of beauty of the human body, a model of a harmonious physique. In their creations, Greek masters used the principle of the golden ratio. One of the highest achievements of classical Greek art is the statue of Doryphoros carved by Polyktetus in the 5th century BC. e. This statue is considered the best example for analyzing the proportions of the ideal human body, established by ancient Greek sculptors, and is directly related to the Golden Ratio. M \u003d 0.618 ...
Venus de Milo, the statue of the goddess Aphrodite and the standard of female beauty, is one of the finest monuments of Greek sculpture.

Leonardo Da Vinci used the proportions of the Golden Ratio in many of his most famous works, and in particular in The Last Supper and the famous La Gioconda.
The researchers of the painting "La Gioconda" have found that the compositional construction of the painting is based on two golden triangles, turned towards each other by their bases. Harmonic analysis of the picture shows that the pupil of the left eye, through which the vertical axis of the canvas passes, is at the intersection of two bisectors of the upper golden triangle, which, on the one hand, bisect the angles at the base of the golden triangle, and on the other hand, at the points of intersection with the thighs of the golden triangles divide them in the proportion of the Golden Ratio. Thus, Leonardo Da Vinci used in his painting not only the principle of symmetry, but also the Golden Ratio.

The painting "The Holy Family" by Michelangelo is recognized as one of the masterpieces of Western European art of the Renaissance. Harmonic analysis showed that the composition of the painting is based on a pentacle.

The proportions of the statue of David (by Michelangelo) are based on the Golden Ratio.

A striking example of baroque architecture, the Smolny Cathedral in St. Petersburg makes a lasting impression. Its main proportions are also seen as the Golden Ratio.

The famous painting "Ship Grove" by Ivan Shishkin shows the motives of the Golden Section. A pine tree brightly lit by the sun (standing in the foreground) divides the picture horizontally with the Golden Section. To the right of the pine is a sunlit hillock. He divides the picture vertically by the Golden Ratio. To the left of the main pine tree there are many pines - you can continue dividing by the Golden Section horizontally on the left side of the picture. The presence in the picture of bright verticals and horizontals, dividing it in relation to the Golden Section, gives it the character of poise and tranquility.

The construction of the UN headquarters in New York was completed in 1943. At that time, the building attracted general attention not only as a public building, created using the latest architectural means, but also as the first example of the use of a solid solar-modulating screen on one of the facades. This building also features the motives of the Golden Section. In the composition of the building, three golden rectangles, placed on top of each other, are clearly distinguished, which are its main architectural idea.

Any piece of music has a time span and is divided by some "aesthetic milestones" into separate parts that attract attention and facilitate perception as a whole. These milestones can be dynamic and intonational climaxes of a piece of music. Separate time intervals of a piece of music, connected by a "culminating event", as a rule, are in the ratio of the Golden Section. In the musical works of various composers, not one Golden Section is usually stated, but a whole series of similar sections. Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin (92%), Schubert (91 %).

If music is a harmonious ordering of sounds, then poetry is a harmonious ordering of speech. A clear rhythm, a regular alternation of stressed and unstressed syllables, an ordered dimension of poems, their emotional saturation make poetry a sister of musical works. The golden ratio in poetry is primarily manifested as the presence of a certain moment of the poem (culmination, semantic break, the main idea of \u200b\u200bthe work) in a line corresponding to the dividing point of the total number of lines of the poem in the golden proportion. So, if the poem contains 100 lines, then the first point of the Golden Section falls on the 62nd line (62%), the second - on the 38th (38%), etc. The works of Alexander Sergeevich Pushkin, including “Eugene Onegin "- the finest correspondence to the golden ratio! The works of Shota Rustaveli and M.Yu. Lermontov are also built according to the principle of the Golden Section.

One of the modern forms of art is cinematography, which has absorbed the drama of action, painting, and music. It is right to look for manifestations of the Golden Section in the outstanding works of cinema. The creator of the masterpiece of world cinema "Battleship Potemkin", film director Sergei Eisenstein, was the first to do this. In the construction of this picture, he managed to embody the basic principle of harmony - the Golden Section. As Eisenstein himself notes, the red flag on the mast of the rebel battleship (the film's apogee) hovers at the golden ratio, measured from the end of the film.

For many millennia, the Golden Ratio has been the object of admiration and worship of prominent scientists and thinkers: Pythagoras, Plato, Euclid, Luca Pacioli, Johannes Kepler, Pavel Florensky ...
Currently, the Golden Ratio turns out to be a source of new fruitful ideas in mathematics and theoretical physics, biology and botany, economics and computer science ...

The material is based on the book "The Da Vinci Code and Fibonacci Series" by A. Stakhov, A. Sluchenkova, I. Shcherbakov, published in 2007, by the Peter publishing house.

Golden ratio - harmonic proportion

Second golden ratio

Golden Triangle

The history of the golden ratio

Fibonacci series

Generalized golden ratio

The principles of shaping in nature

Golden ratio and symmetry

Golden Triangle

The history of the golden ratio

GOLDEN RATIO - HARMONIC PROPORTION

SECOND GOLDEN SECTION

GOLDEN TRIANGLE (pentagram)

HISTORY OF THE GOLDEN SECTION

ICOSAhedron and dodecahedron

GOLDEN SECTION AND SYMMETRY

FIBONACCI RANGE

GENERALIZED GOLDEN SECTION

PRINCIPLES OF FORMATION IN NATURE

HUMAN BODY AND GOLDEN SECTION

GOLDEN SECTION IN SCULPTURE

GOLDEN SECTION IN ARCHITECTURE

GOLDEN SECTION IN MUSIC

THE GOLDEN SECTION IN POETRY

GOLDEN SECTION IN FONTS AND HOUSEHOLD ITEMS

OPTIMAL PHYSICAL PARAMETERS OF THE EXTERNAL ENVIRONMENT

GOLDEN CROSSING IN PAINTING

GOLD SECTION AND IMAGE PERCEPTION

GOLDEN SECTION IN PHOTOS

GOLDEN SECTION AND SPACE

CONCLUSION

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