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Compasses of the golden section. Fibonacci scriber and its use in the manufacture of furniture

Dynamic Rectangles

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

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

Antique golden ratio compasses

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

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

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

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

Judging by one of Dürer's letters, he met with Luca Pacioli during his stay in Italy. Albrecht Dürer develops in detail the theory of the proportions of the human body. Dürer assigned an important place in his system of ratios to the golden section. The height of a person is divided in golden proportions by the belt line, as well as 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. Known proportional compass Dürer.

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

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

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

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

Building a scale of segments of the golden ratio

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

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

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

The golden ratio was known in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the "golden section" was studied. Euclid applied it, creating his geometry, and Phidias - his immortal sculptures. Plato said that the universe is arranged according to the "golden section". Aristotle found the correspondence of the "golden section" to the ethical law. The highest harmony of the "golden section" will be preached by Leonardo da Vinci and Michelangelo, because beauty and the "golden section" are one and the same. And Christian mystics will draw pentagrams of the "golden section" on the walls of their monasteries, escaping from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. Be h the final row after the decimal point is 1.6180339887... A strange, mysterious, inexplicable thing - this divine proportion mystically accompanies all living things. Inanimate nature does not know what the "golden section" is. But you will certainly see this proportion in the curves of sea shells, and in the form of flowers, and in the form of beetles, and in a beautiful human body. Everything living and everything beautiful - everything obeys the divine law, whose name is the "golden section". So what is the "golden ratio"? What is this perfect, divine combination? Maybe it's the law of beauty? Or is it still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. "Golden section" is both that, and another, and the third. Only not separately, but at the same time ... And this is his true mystery, his great secret.

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

Trusting the eye more than other sense organs, Man first of all learned to distinguish the objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The 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 relationship to each other and to the whole. The principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

GOLDEN SECTION - HARMONIC PROPORTION

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

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

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

The latter is the golden division (section).

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

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

Geometric representation of the golden ratio

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

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

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

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

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

Solution to this equation:

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

SECOND GOLDEN SECTION

This proportion is found in architecture.

Construction of the second golden section

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

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

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

GOLDEN TRIANGLE (pentagram)

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

Construction of a regular pentagon and pentagram

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

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

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

Construction of the golden triangle

HISTORY OF THE GOLDEN SECTION

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

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

Dynamic Rectangles

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

icosahedron and dodecahedron

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

Antique golden ratio compasses

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

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

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

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

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

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

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

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

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

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

Building a scale of segments of the golden ratio

In subsequent centuries, the rule of the golden ratio turned into an academic canon, and when, over time, a struggle began in art with an academic routine, in the heat of the struggle, “they threw the child out with the water.” The golden ratio was "discovered" again in mid-nineteenth V.

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

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

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

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

GOLDEN RATIO AND SYMMETRY

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

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

FIBONACCCI SERIES

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

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

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

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

GENERALIZED GOLDEN RATIO

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

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

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

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

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

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

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

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

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

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

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

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

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

PRINCIPLES OF SHAPING IN NATURE

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

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

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

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

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

Mandelbrot series

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

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

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

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

Chicory

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

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

viviparous lizard

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

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

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

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

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

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

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

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

Adeno virus

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

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

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

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

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

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

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

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

THE HUMAN BODY AND THE GOLDEN SECTION

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

Golden proportions in parts of the human body

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

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

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

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

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

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

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

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

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

Blood pressure changes as the heart beats. It reaches its greatest value in the left ventricle of the heart at the time of its contraction (systole). In the arteries during the systole of the ventricles of the heart, 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 the maximum (systolic) to the minimum (diastolic) pressure is on average 1.6, that is, close to the golden ratio.

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

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

The structure of the helix section of the DNA molecule

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

GOLDEN SECTION IN SCULPTURE

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

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

GOLDEN SECTION IN ARCHITECTURE

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

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

Seen in the drawings whole line patterns associated with the golden ratio. The proportions of a building can be expressed in terms of various degrees numbers Ф=0.618...

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

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

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

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

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

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

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

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

Pashkov House

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

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

GOLDEN RATIO IN MUSIC

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

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

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

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

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

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

GOLDEN SECTION IN POETRY

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

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

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

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

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

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

I do not value high-profile rights,

From which not one is dizzy.

I do not grumble about the fact that the gods refused

I'm in the sweet lot of challenging taxes

Or prevent the kings from fighting with each other;

And little grief to me, is the press free

Fooling boobies, or sensitive censorship

In magazine plans, the joker is embarrassing.

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

Other, better, rights are dear to me:

Another, better, I need freedom:

Depend on the king, depend on the people -

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

Do not give a report, only to yourself

Serve and please; for power, for livery

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

At your whim to wander here and there,

Marveling at the divine beauty of nature,

And before the creatures of art and inspiration

Trembling joyfully in delights of tenderness,

Here is happiness! That's right...

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

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

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

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

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

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

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

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

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

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

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

GOLDEN RATIO IN FONTS AND HOUSEHOLD ITEMS

special kind visual arts Ancient Greece it is necessary to highlight the manufacture and painting of various vessels. In an elegant form, the proportions of the golden section are easily guessed.

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

OPTIMUM PHYSICAL PARAMETERS OF THE EXTERNAL ENVIRONMENT

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

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

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

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

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

GOLDEN SECTION IN PAINTING

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

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

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

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

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

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

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

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

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

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

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

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

pine grove

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

IN AND. Surikov. "Boyar Morozova"

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

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

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

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

Venus

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

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

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

Massacre of the innocents

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

GOLDEN RATIO AND IMAGE PERCEPTION

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

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

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

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

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

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

THE GOLDEN RATIO IN PHOTOGRAPHY

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

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

GOLDEN RATIO AND SPACE

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

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

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

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

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

CONCLUSION

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

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

IN modern world science is of particular importance, in connection with the increasing impact of man on nature. Important tasks at the present stage are the search for new ways of coexistence of man and nature, the study of philosophical, social, economic, educational and other problems facing society.

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

golden ratio - universal principle harmony

"Tastes do not argue" - how many times each of us has heard this formula, and even pronounce it. By agreeing with it, we are thereby ready to defend any disgrace that the human imagination can afford. A person who is deeply selfish, fussy, passionate, unaccustomed to listening to the world in big and small, simply has no reason to develop taste and comprehend harmony, and therefore he is able to generate the most monstrous aesthetics, while calling it beauty. "You can't forbid a beautiful life," the inhabitant spits out through greasy lips, defending his tastes and forbidding others to argue about them. "Of course, of course, we will not argue about tastes! Everyone is right in his own way, so long as he does not harm us," animals in the form of people echo, not understanding themselves deeper than bodily needs. And they are settled in squalid dwellings, they are stuffed with destructive music, they are school bench they feed wretchedness, serving it under the sauce of inevitability. The decline of aesthetics, the inattention to beauty is always the decline of humanity, which no longer wants to dream or strive for beauty. It is suffering and death.

It is difficult for an individual person to resist the whole system of vulgarity, and he is doomed to submit to it and perish if he does not have sufficient knowledge. I would like to believe that the feeling of beauty, the harmony of the world lives in every person - you just need to show it, learn how to use it.

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

Trusting the eye more than other sense organs, a person first of all learned to distinguish the objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden section, contributes to the best visual perception and the appearance of a sense of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature. This idea was shared and shared by many prominent modern scientists, proving in their studies that true beauty is always functional. Among them are aircraft designers. And architects, and anthropologists, and many others.

History of the golden ratio

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from the tomb of his name, holds measuring instruments in his hands, in which the proportions of the golden division are fixed.

The German professor G.E. Timerding, who wrote a book on the golden ratio in the first quarter of the twentieth century, states: "Among the Pythagoreans<...>the thought of mysterious forces and properties was associated with the regular pentagon, but these properties are revealed only when, next to the ordinary regular pentagon, that star is considered, which is obtained by sequentially connecting through one of all the vertices of an ordinary pentagon, composed by the diagonals of the pentagon "- and further notes: the pentagram played a large role in all magical sciences.The five-pointed star, as Timerding shows, is literally stuffed with the proportions of the golden section.

Plato (427...347 BC) also knew about the golden division. The Pythagorean Timaeus in Plato’s dialogue of the same name says: “It is impossible for two things to be perfectly connected without a third, since a thing must appear between them that would hold them together. This can best be done by proportion, because if three numbers have the property that the average so is to the lesser as the greater is to the middle, and vice versa, the lesser is to the mean as the mean is to the greater, then the last and the first will be the middle, and the middle the first and the last. since it will be the same, it will make a whole." Plato builds the earthly world using triangles of two types: isosceles and non-isosceles. He considers the most beautiful right-angled triangle to be one in which the hypotenuse is twice the smaller of the legs (such a rectangle is half an equilateral, the main figure of the Babylonians, it has a ratio of 1: 3 1/2, which differs from the golden ratio by about 1/25, and is called Timerding "opponent of the golden ratio"). Using triangles, Plato builds four regular polyhedra, associating them with the four earthly elements (earth, water, air and fire). And only the last of the five existing regular polyhedra - the dodecahedron, all twelve faces of which are regular pentagons, claims to be a symbolic image of the heavenly world.

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

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

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

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

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

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

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

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

In subsequent centuries, the rule of the golden ratio turned into an academic canon, and when, over time, a struggle began in art with the academic routine, in the heat of the struggle "they threw out the child with the water." The golden section was "discovered" again in the middle of the 19th century. In 1855, the German researcher of the golden section, Professor Zeising, published his work "Aesthetic Research". With Zeising, exactly what happened was bound to happen to the researcher who considers the phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions to be "mathematical aesthetics".

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

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

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

A bit of geometry

In mathematics proportion(lat. 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: AC = AB: BC;

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

so when AB: AC = AC: BC.

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

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

a:b = b:cor c: b = b: a.

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

From a point IN a perpendicular is restored equal to half AB. Received point WITH connected by a line to a dot A. A segment is drawn on the resulting line sun, ending 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.

Segments of the golden ratio are expressed by an infinite irrational fraction AE= 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 the largest part of the segment is 62, and the smaller is 38 parts.

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

x2 - x - 1 = 0.

Solution to this equation:

The second golden ratio

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

Such a proportion is found in architecture, and also takes place in the construction of compositions of images of an elongated horizontal format.

The division is carried out as follows. Line segment AB is divided according to the golden ratio. From a point WITH the perpendicular is restored CD. Radius AB there is a point D, which is connected by a line to a point A. Right angle ACD is divided in half. From a point WITH a line is drawn until it intersects with a line AD. Dot E divides the segment AD in relation to 56:44.

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

Golden Triangle

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

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer (1471...1528). Let O- the center of the circle A- a point on the circle and E- middle of the segment OA. Perpendicular to Radius OA, restored at the point ABOUT, intersects the circle at a point D. Using a compass, set aside a segment on the diameter CE = ED. The length of a side of a regular pentagon inscribed in a circle is DC. Putting segments on the circle DC and get five points to draw a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

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

We draw a straight line AB. from point A set aside on it three times a segment O of arbitrary size, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside segments ABOUT. Received points d And d1 connect with a straight line A. Line segment dd1 put on the line Ad1, getting a point WITH. She split the line Ad1 in proportion to the golden ratio. lines Ad1 And dd1 used to build a "golden" rectangle.

Fibonacci series

The name of the Italian mathematician monk Leonardo from Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected 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 in one year from one pair will be born." Reflecting on this topic, Fibonacci built the following series of numbers:

Months

etc.

Pairs of rabbits

etc.

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers of the series approaches the ratio of the golden division. So, 21:34 = 0.617, and 34:55 = 0.618. This ratio is denoted by the symbol Ф. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden ratio, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to everything.

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

The Fibonacci series could have remained only a mathematical incident if it were 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 golden division law.

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 elegant methods for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden section. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

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

Principles of shaping in nature

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

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

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

Rice. 12. Chicory

The process makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of less force, releases a leaf of an even smaller size and ejection again. If the first outlier is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden ratio.

Rice. 13.viviparous lizard

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

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

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

Rice. 14. bird egg

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

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

Golden ratio and symmetry

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

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

Observe and apply

Understanding and using the principle of the golden section should not be the lot of some elite - this is the most basic knowledge from which the infinitely complex laws of harmony and proportion begin. There are no limits to the meaningful application of these laws in the life of every day. The allocation of the main and secondary in relation to the whole can concern anything. This is the distribution of one's time, and any creative process, including all types of art, literature, music, and the formation of one's own attitude to any processes and phenomena. This is the Golden, middle way, which the ancients spoke about.

Every artist, every director, every advertising specialist knows how to make an image pleasing to the eye, how to build it according to the laws of harmony and psychology. human perception. Sometimes the worst enemies of culture achieve significant victories using knowledge of the laws of Nature. Thus, under the guise of something pleasant and endearing, we often allow the strongest poisons to enter our hearts. So many people talk about freedom, while they themselves poison themselves voluntarily, wondering later where their illnesses and misfortunes come from.

There can be no freedom in ignorance. Roughness and illegibility of taste must be overcome. Let this be the concern of both individuals and communities and states.

Compiled by R. Annenkov

Often you have to deal with a situation when the element you have drawn "does not ring"? Something wrong? Wrong proportions?

It should not be argued that there is no ideal in nature, because it exists and was deduced long ago with the help of mathematics and geometry. The name of the person who first introduced the term "golden section" is unknown, although many are accustomed to believing that this is Leonardo Da Vinci. The earliest appearance of this term is in 1835 thanks to Martin Ohm, in a footnote to the second edition of his Pure Elementary Mathematics.

What does the golden section formula look like?

This is a harmonious ratio of two quantities b and a, a > b, when a/b = (a+b)/a is true. A number equal to the ratio a/b is usually denoted by an uppercase Greek letter

(\displaystyle \phi )

In honor of the ancient Greek sculptor and architect Phidias.

For practical purposes, they are limited to an approximate value of = 1.618 or = 1.62. In a rounded percentage, the golden ratio is the division of a value in relation to 62% and 38%.

Sometimes the number is called the "golden number"

So that you and I do not bother with mathematics, smart people came up with such a circle. With it, you can already check finished projects on the ratio of parts, and build new ones, taking into account the principle of the "golden section"

Let your projects remain in the world cultural heritage!

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