And the golden ratio. Golden ratio - mathematics - sacred geometry - science - catalog of articles - rose of the world
Every person who encounters the geometry of objects in space is well acquainted with the golden section method. It is used in art, interior design and architecture. Even in the last century, the golden ratio turned out to be so popular that now many supporters of the mystical vision of the world have given it another name - the universal harmonic rule. The features of this method are worth considering in more detail. This will help to find out why he is interested in several areas of activity at once - art, architecture, design.
The essence of the universal proportion
The principle of the golden section is just a dependence of numbers. However, many are biased towards it, attributing some mystical powers to this phenomenon. The reason lies in the unusual properties of the rule:
- Many living objects have proportions of the torso and limbs that are close to the indications of the golden ratio.
- Dependencies 1.62 or 0.63 determine the size ratios only for living beings. Objects related to inanimate nature very rarely correspond to the meaning of the harmonic rule.
- The golden proportions of the body structure of living beings are an essential condition for the survival of many biological species.
The golden ratio can be found in the structure of the bodies of various animals, tree trunks and shrub roots. Supporters of the universality of this principle are trying to prove that its meaning is vital for representatives of the living world.
You can explain the golden section method using the image of a chicken egg. The ratio of the segments from the points of the shell, equally distant from the center of gravity, is equal to the golden ratio. The most important indicator for the survival of birds is the shape of the egg, and not the strength of the shell.
Important! The golden ratio is calculated based on the measurements of many living objects.
Origin of the golden ratio
The mathematicians of ancient Greece knew about the universal rule. It was used by Pythagoras and Euclid. In the famous architectural masterpiece - the pyramid of Cheops, the ratio of the dimensions of the main part and the length of the sides, as well as the bas-reliefs and decorative details, correspond to the harmonic rule.
The golden section method was adopted not only by architects, but also by artists. The mystery of harmonic proportion was considered one of the greatest mysteries.
The first to document the universal geometric proportion was the Franciscan monk Luca Pacioli. His ability in mathematics was excellent. The golden section gained wide recognition after the publication of Zeising's results on the golden section. He studied the proportions of the human body, ancient sculptures, plants.
How was the golden ratio calculated?
To understand what the golden ratio is, an explanation based on the lengths of the segments will help. For example, inside a large one there are several small ones. Then the lengths of the small segments are related to the total length of the large segment as 0.62. Such a definition helps to figure out how many parts a certain line can be divided into so that it complies with the harmonic rule. Another advantage of using this method is that you can find out what the ratio of the largest segment to the length of the entire object should be. This ratio is 1.62.
Such data can be represented as the proportions of measured objects. At first they were sought out, selecting empirically. However, now the exact ratios are known, so it will not be difficult to build an object in accordance with them. The golden ratio is found in the following ways:
- Construct a right triangle. Split one of its sides, and then draw perpendiculars with secant arcs. When carrying out calculations, it is necessary to build a perpendicular from one end of the segment, equal to ½ of its length. Then a right triangle is completed. If you mark a point on the hypotenuse, which will show the length of the perpendicular segment, then a radius equal to the rest of the line will cut the base into two halves. The resulting lines will be related to each other according to the golden ratio.
- Universal geometric values are also obtained in another way - by building the Durer pentagram. She is a star that is placed in a circle. It contains 4 segments, the lengths of which correspond to the rule of the golden section.
- In architecture, the harmonic proportion is used in a modified form. To do this, a right-angled triangle should be divided along the hypotenuse.
Important! Compared to the classical concept of the golden ratio method, the architect's version has a ratio of 44:56.
If in the traditional interpretation of the harmonic rule for graphics, it was calculated as 37:63, then 44:56 was more often used for architectural structures. This is due to the need to build high-rise buildings.
The secret of the golden ratio
If in the case of living objects the golden ratio, which manifests itself in the proportions of the body of people and animals, can be explained by the need to adapt to the environment, then the use of the rule of optimal proportions in the 12th century to build houses was new.
The Parthenon, preserved from the time of Ancient Greece, was erected using the golden section method. Many castles of the nobles of the Middle Ages were created with parameters corresponding to the harmonic rule.
The golden ratio in architecture
The many buildings of antiquity that have survived to this day serve as confirmation that architects from the Middle Ages were familiar with the harmonic rule. The desire to observe a harmonious proportion in the construction of churches, significant public buildings, residences of royal persons is very clearly visible.
For example, Notre Dame Cathedral was built in such a way that many of its sections correspond to the golden section rule. You can find many works of architecture of the 18th century that were built in accordance with this rule. The rule was also applied by many Russian architects. Among them was M. Kazakov, who created projects for estates and residential buildings. He designed the Senate building and the Golitsyn hospital.
Naturally, houses with such a ratio of parts were erected even before the discovery of the golden section rule. For example, such buildings include the Church of the Intercession on the Nerl. The beauty of the building becomes even more mysterious, given that the building of the Intercession Church was erected in the 18th century. However, the building acquired its modern look after restoration.
In writings about the golden ratio, it is mentioned that in architecture the perception of objects depends on who is observing. The proportions formed using the golden section give the most relaxed ratio of the parts of the structure relative to each other.
A striking representative of a number of buildings that comply with the universal rule is the Parthenon, an architectural monument erected in the fifth century BC. e. The Parthenon is arranged with eight columns on the smaller facades and seventeen on the larger ones. The temple was built of noble marble. Due to this, the use of coloring is limited. The height of the building refers to its length 0.618. If you divide the Parthenon according to the proportions of the golden section, you will get certain ledges of the facade.
All these structures have one thing in common - the harmony of the combination of forms and the excellent quality of construction. This is due to the use of the harmonic rule.
The importance of the golden ratio for a person
The architecture of ancient buildings and medieval houses is quite interesting for modern designers. This is due to such reasons:
- Thanks to the original design of houses, you can prevent annoying clichés. Each such building is an architectural masterpiece.
- Mass application of the rule to decorate sculptures and statues.
- Thanks to the observance of harmonic proportions, the eye is drawn to more important details.
Important! When creating a building project and creating an external appearance, the architects of the Middle Ages used universal proportions, based on the laws of human perception.
Today, psychologists have come to the conclusion that the principle of the golden section is nothing more than a human reaction to a certain ratio of sizes and shapes. In one experiment, a group of subjects were asked to fold a sheet of paper in such a way that the sides turned out with optimal proportions. In 85 results out of 100, people folded the sheet almost exactly according to the harmonic rule.
According to modern scientists, the indicators of the golden section are more in the field of psychology than characterize the laws of the physical world. This explains why there is such interest in him from the hoaxers. However, when constructing objects according to this rule, a person perceives them more comfortably.
Using the golden ratio in design
The principles of using a universal proportion are increasingly used in the construction of private houses. Particular attention is paid to the observance of the optimal proportions of the structure. Much attention is paid to the correct distribution of attention inside the house.
The modern interpretation of the golden section no longer refers only to the rules of geometry and form. Today, the principle of harmonic proportions obeys not only the dimensions of the facade details, the area of rooms or the length of the gables, but also the color palette used to create the interior.
It is much easier to build a harmonious structure on a modular basis. Many departments and rooms in this case are performed as separate blocks. They are designed in strict accordance with the harmonic rule. To erect a building as a set of separate modules is much easier than to create a single box.
Many firms involved in the construction of country houses, when creating a project, follow the harmonic rule. This allows customers to give the impression that the structure of the building has been worked out in detail. Such houses are usually described as the most harmonious and comfortable to use. With the optimal choice of room areas, residents psychologically feel calm.
If the house was built without taking into account harmonic proportions, you can create a layout that will be close to 1: 1.61 in terms of the ratio of wall sizes. To do this, additional partitions are installed in the rooms, or pieces of furniture are rearranged.
Similarly, the dimensions of doors and windows are changed so that the opening has a width that is 1.61 times less than the height value.
Harder to choose colors. In this case, you can observe the simplified value of the golden section - 2/3. The main color background should occupy 60% of the space of the room. Shading shade occupies 30% of the room. The remaining surface area is painted over with tones close to each other, enhancing the perception of the selected color.
The inner walls of the rooms are divided by a horizontal strip. It is located 70 cm from the floor. The height of the furniture should be in harmony with the height of the walls. This rule also applies to the distribution of lengths. For example, a sofa should have dimensions that would be at least 2/3 of the length of the wall. The area of \u200b\u200bthe room, which is occupied by pieces of furniture, should also have a certain value. It refers to the total area of the entire room as 1:1.61.
The golden ratio is difficult to apply in practice due to the presence of only one number. That's why. I design harmonious buildings, use a series of Fibonacci numbers. This provides a variety of options for shapes and proportions of building details. A series of Fibonacci numbers is also called the golden one. All values strictly correspond to a certain mathematical dependence.
In addition to the Fibonacci series, another design method is also used in modern architecture - the principle laid down by the French architect Le Corbusier. When choosing this method, the starting unit of measurement is the height of the owner of the house. Based on this indicator, the dimensions of the building and the interior are calculated. Thanks to this approach, the house is not only harmonious, but also acquires individuality.
Any interior will take on a more complete look if you use cornices in it. When using universal proportions, you can calculate its size. The optimal indicators are 22.5, 14 and 8.5 cm. The eaves should be installed according to the rules of the golden section. The small side of the decorative element should be related to the larger side as it is to the combined values of the two sides. If the large side is equal to 14 cm, then the small one should be made 8.5 cm.
You can give the room comfort by dividing the wall surfaces with the help of gypsum mirrors. If the wall is divided by a curb, the height of the cornice strip should be subtracted from the remaining larger part of the wall. To create a mirror of optimal length, the same distance should be retreated from the curb and cornice.
Conclusion
Houses built according to the principle of the golden section really turn out to be very comfortable. However, the price of building such buildings is quite high, since the cost of building materials increases by 70% due to atypical sizes. This approach is not at all new, since most of the houses of the last century were created based on the parameters of the owners.
Thanks to the use of the golden section method in construction and design, buildings are not only comfortable, but also durable. They look harmonious and attractive. The interior is also decorated according to a universal proportion. This allows you to wisely use the space.
In such rooms, a person feels as comfortable as possible. You can build a house using the principle of the golden section yourself. The main thing is to calculate the loads on the elements of the structure, and choose the right materials.
The golden section method is used in interior design, placing decorative elements of certain sizes in the room. This allows you to give the room comfort. Color solutions are also chosen in accordance with universal harmonic proportions.
GOLDEN RATIO
1. Introduction 2 . Golden Ratio - Harmonic Proportion
3
. The second golden ratio
four . Zo lotus triangle (pentagram)
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History of the golden section
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. Golden ratio and symmetry 7. Fibonacci series 8 . Generalized golden ratio
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. Principles of formation in nature
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. The human body and the golden ratio
1
1
. The Golden Ratio in Sculpture
1
2
. The golden ratio in architecture
1
3
. The golden ratio in music
1
4
. The Golden Ratio in Poetry
1
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. The golden ratio in fonts and household items
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. Optimal physical parameters of the environment
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. The golden ratio in painting
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8
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The Golden Ratio and Image Perception
19.
The Golden Ratio in Photos
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. Golden Ratio and Space 2 1 . Conclusion 2 2 . Bibliography
INTRODUCTION
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.- golden ratio.
Our task is to find out what the golden ratio is and to establish where humanity has found the use of gold. th section. You probably paid attention to the fact that we treat objects and phenomena of the surrounding reality differently. Disorder, shapelessness, disproportion are perceived by us as ugly and produce a repulsive impression. And objects and phenomena that 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 use the golden ratio as their basis.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 so 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 produced a "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". And 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 Pacho l and before Einstein - they will search, but will never find its exact meaning. An endless series after the decimal point - 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 section"?.. What is this ideal, 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. The "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 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 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, proportion is the equality of two ratios: a: b = c: d.
Line segment AB can be divided into two parts in the following ways:
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into two equal parts - AB: AC = AB: BC;
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into two unequal parts in any ratio (such parts do not form proportions);
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thus, when AB: AC = AC: BC.
The last one is the golden division.
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: c or 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 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 as an infinite fraction AE \u003d 0.618 ..., if AB is taken as a unit, BE \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB is taken as 100 parts, then the largest part of the segment is 62, and the smaller one is 38 parts.
The properties of the golden section are described by the equation: x2 - x - 1 = 0. Solution to this equation:
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SECOND GOLDEN SECTION The Bulgarian magazine "Fatherland" 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. This proportion is found in architecture. 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 the ratio 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 rows, you can use the 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° 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 d1 are connected by straight lines with point A. We put the segment dd1 on line Ad1, getting point C. She divided the line Ad1 in proportion to the golden ratio. The lines Ad1 and dd1 are used to build a "golden" rectangle. HISTORY OF THE GOLDEN SECTION
It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician. There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the pyramid of Cheops, temples, 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. 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 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 mean, and conversely, 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 smallest 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 ratio"). Using triangles, Plato builds four regular polyhedra, associating them with the four earthly elements (earth, water, air and fire). And only the last of the five existing regular polyhedra - the dodecahedron, all twelve faces of which are regular pentagons, claims to be a symbolic image of the heavenly world.
Icosahedron and dodecahedron The honor of discovering the dodecahedron (or, as it was supposed, the Universe itself, this quintessence of the four elements, symbolized, respectively, by the tetrahedron, octahedron, icosahedron and cube) belongs to Hippasus, who later died in a shipwreck. This figure really captures many relationships of the golden section, so the latter was assigned the main role in the heavenly world, which was subsequently insisted on by the minor brother Luca Pacioli. In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division. 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 (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. 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. 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 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, Luca Pacioli's book "On Divine Proportion" (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 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.
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. 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." 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 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). If on a straight line of arbitrary length, set aside segment m, next we set aside segment M. Based on these two segments, we build a scale of segments of the golden proportion of the ascending and descending rows 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 approach the golden ratio somewhat closer 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. 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 of an increasing or decreasing series. FIBON ROW AF H And
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 \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 F. 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. 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 fits the F proportion model just like it does in the human body.
GENERALIZED GOLDEN RATIO Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich using Fibonacci numbers solves 10- Yu Hilbert's problem. 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 has been publishing a special journal since 1963. 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 their construction 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 \u003d 2 + 2 ..., in the second - this is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... Is it possible to find a general mathematical formula from which " 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... separated from the previous one by S steps. If we denote the nth member of this series by? S (n), then we get the general formula? S(n) = ? S (n - 1) + ? S (n - S - 1). Obviously, with S = 0, from this formula we will get a "binary" series, with S = 1 - a Fibonacci series, with S = 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers. In general, the golden S-proportion is the positive root of the golden S-section equation x S+1 - x S - 1 = 0. It is easy to show that at S = 0, the division of the segment in half is obtained, and at S = 1, the familiar classical golden section. 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 of 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 "upside down" the historically established hierarchy of relations between rational and irrational numbers. The fact is that at first 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 - 10, 5, 2 - were chosen as a kind of fundamental principle, from which all other natural numbers, as well as rational and irrational numbers, were constructed according to certain rules. A kind of alternative to the existing methods of numbering is a new, irrational system, as the fundamental principle, the beginning of which is chosen as an irrational number (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! - sums of degrees of any of the golden S-proportions. This is one of the reasons why "irrational" arithmetic, having amazing mathematical simplicity and elegance, seems to have absorbed the best qualities of classical binary and "Fibonacci" arithmetic. PRINCIPLES OF SHAPING IN NATURE Everything that took on some form formed, grew, strove to take a place in space and preserve itself. This aspiration finds realization mainly in two variants - upward growth or spreading over the surface of the earth and twisting in a spiral. The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The concept of the golden ratio will be incomplete, if not to say about the spiral. The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and deduced the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering. Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago.
The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. 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." Zo 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. Figure 6 shows the famous Mandelbrot series, a page from a dictionary of infinity of individual patterns called Julian series. Some scientists associate the Mandelbrot series with the genetic code of cell nuclei. A consistent increase in sections reveals amazing fractals in their artistic complexity. And here, too, there are logarithmic spirals! This is all the more important because both the Mandelbrot series and the Julian series are not inventions of the human mind. They arise from the realm of Plato's prototypes. As the doctor R. Penrose said, "they are like Mount Everest." The spiral is closely connected with cycles. The modern science of chaos studies simple cyclic feedback operations and the fractal ones generated by them.
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.
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| Rice. . Chicory |
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.
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| Rice. . viviparous lizard |
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.
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| Rice. . bird egg |
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 coat of the Adeno virus is made up of 252 units of protein cells arranged in a certain sequence. In each corner of the icosahedron are 12 units of protein cells in the form of a pentagonal prism, and spike-like structures extend from these corners.
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| Adeno virus |
Klug's comment once again reminds of the extremely obvious truth: in the structure of even a microscopic organism, which scientists classify as "the most primitive form of life", in this case, a virus, there is a clear plan and a reasonable project has been implemented 16. This project is incomparable in its perfection and accuracy execution with the most advanced architectural designs 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. Moreover, pseudo-elongation-limb and other unusual forms-growths grow from each of its corners. 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. 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. 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. 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.
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
Actually, the exact presence of the golden ratio in the face of a person is the ideal of beauty for the human eye.
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 top of the head is 1:1.618
Face Height / Face Width
The center point of the junction 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
Nose width / distance between nostrils
Pupil distance / Eyebrow distance It is enough just to bring your palm closer to you now and carefully look at your index finger, and you will immediately find the golden section formula in it.
Each finger of our hand consists of three phalanges. The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the golden section number (with the exception of the thumb).
In addition, the ratio between the middle finger and the little finger is alsogolden ratioA 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.
The human inner ear contains an organ Cochlea ("Snail"), which performs the function of transmitting sound vibration. This bone-like structure is filled with fluid and also created in the form of a snail, containing a stable logarithmic spiral shape = 73? 43". Blood pressure changes as the heart beats. It reaches its greatest value in the left ventricle of the heart at the time of its contraction (systole). In the arteries during the systole of the ventricles of the heart, blood pressure reaches a maximum value equal to 115-125 mm Hg in a young, healthy person. At the moment of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of the maximum (systolic) to the minimum (diastolic) pressure is on average 1.6, that is, close to the golden ratio.If we take the average blood pressure in the aorta as a unit, then the systolic blood pressure in the aorta is 0.382, and the diastolic blood pressure is 0.618, that is, their ratio corresponds to the golden ratio. This means that the work of the heart in relation to time cycles and changes in blood pressure are optimized according to the same principle - 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). 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
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GOLDEN SECTION IN SCULPTURE
Sculptural structures, monuments are erected to perpetuate significant events, to preserve 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" give the impression of harmony of beauty, so the sculptors used them in their works. The sculptors claim that the waist divides the perfect human body in relation to the "golden section". For example, the famous statue of Apollo Belvedere consists of parts divided by golden ratios. The great ancient Greek sculptor Phidias often used the "golden section" 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 Athena Parthenos.  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 that if some proportions in a building on one side seem to form the "golden section", then from other points vision they will look different. The "golden section" gives the most relaxed ratio of the sizes of certain lengths.
One of the most beautiful works of ancient Greek architecture is the Parthenon (V century BC).
  The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various degrees of the number 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 Pentile marble. The nobility of the material from which the temple was built made it possible to limit the use of coloring, which was common in Greek architecture, it only emphasizes the details and forms a colored background (blue and red) for the sculpture. The ratio of the height of the building to its length is 0.618. If we divide the Parthenon according to the "golden section", we will get certain protrusions of the facade. On the floor plan of the Parthenon, you can also see the "golden rectangles":  We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris) and in the pyramid of Cheops:   Not only the Egyptian pyramids were built in accordance with the perfect proportions of the golden ratio; the same phenomenon is found in the Mexican pyramids. For a long time it was believed that the architects of Ancient Russia built everything "by eye", without any special mathematical calculations. However, the latest research has shown that Russian architects knew mathematical proportions well, as evidenced by the analysis of the geometry of ancient temples. The famous Russian architect M. Kazakov widely used the "golden section" in his work. His talent was multifaceted, but to a greater extent he revealed himself in numerous completed projects of residential buildings and estates. For example, the "golden section" can be found in the architecture of the Senate building in the Kremlin. According to the project of M. Kazakov, the Golitsyn Hospital was built in Moscow, which is currently called the First Clinical Hospital named after N.I. Pirogov (Leninsky prospect, d.  Petrovsky Palace in Moscow. Built according to the project of M.F. Kazakov.  Another architectural masterpiece of Moscow - the Pashkov House - is one of the most perfect works of architecture by V. Bazhenov.  The wonderful creation of V. Bazhenov has firmly entered the ensemble of the center of modern Moscow, enriched it. The external appearance of the house has survived 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 today. 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." |
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, the art critic L.L. Sabaneev, having analyzed 1770 musical works by 42 authors, showed that the vast majority of outstanding works can be easily divided into parts either by theme, or intonation, or modal system, which are in relation to each other. golden ratio. 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 sonata "Appassionata" and found a number of interesting numerical ratios. In particular, in the development - the central structural unit of the sonata, where themes are intensively developed and keys replace each other - there are two main sections. The first has 43.25 bars, the second has 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 the Golden Section is present.
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 the sister of musical works. The golden ratio in poetry primarily manifests itself as the presence of a certain moment of the poem (climax, semantic turning point, main idea of the work) in the line attributable to the dividing point of the total number of lines of the poem in the golden ratio. So, if the poem contains 100 lines, then the first point of the Golden Section falls on the 62nd line (62%), the second - on the 38th (38%), etc. The works of Alexander Sergeevich Pushkin, including "Eugene Onegin" - the finest correspondence to the golden ratio! The works of Shota Rustaveli and M.Yu. Lermontov are also built on the principle of the Golden Section.
Stradivarius wrote that with the help of
the golden ratio, he determined the places for f -shaped cutouts on the bodies of their famous violins. GOLDEN SECTION IN POETRY Pushkin's poetry Studies of poetic works from these positions are just beginning. And you need to start with the poetry of A.S. Pushkin. After all, his works are an example of the most outstanding creations of Russian culture, an example of the highest level of harmony. With the poetry of A.S. Pushkin, we will begin the search for the golden proportion - 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 the sister of musical works. Each verse has its own musical form - its own rhythm and melody. It can be expected that in the structure of poems some features of musical works, patterns of musical harmony, and, consequently, the golden ratio, will appear. 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 from this point of view 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 like pieces of music; 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": A shoemaker once looked for a picture
And he pointed out the error in the shoes;
Taking the brush at once, the artist corrected himself,
Here, akimbo, the shoemaker continued:
"I think the face is a little crooked...
Isn't that chest too naked?
Here Apelles interrupted impatiently:
"Judge, my friend, not above the boot!"
I have a friend in mind:
I don't know what subject it is.
He was a connoisseur, though strict non-verbally,
But the devil bears him to judge the light:
Try it to judge the boots!
Lermontov's famous poem "Borodino" is divided into two parts: an introduction addressed to the narrator and occupying only one stanza ("Tell me, uncle, it's not without reason ..."), and the main part, representing an independent whole, which is divided into two equivalent parts. In the first of them, the expectation of the battle is described with increasing tension, in the second - 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 by 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 the excited expectation", which completes the first part of the poem (expectation of the battle) and opens its second part (the description of the battle). Thus, the golden ratio plays a very meaningful role in poetry, highlighting the climax of the poem. Poetry of Shota Rustaveli 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 by the 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 half lines are divided into two segments of two types: A - a half line with equal segments and an even number of syllables (4 + 4); B - 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 was born - cinema, which absorbed the dramaturgy of action, painting, music. It is legitimate to look for manifestations of the golden section in outstanding works of cinematography. The first to do this was the creator of the masterpiece of world cinema "Battleship Potemkin", film director Sergei Eisenstein. In the construction of this picture, he managed to embody the basic principle of harmony - the golden ratio. As Eisenstein himself notes, the red flag on the mast of the rebellious 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 A special type of fine art of ancient Greece should be highlighted the manufacture and painting of all kinds of vessels. In an elegant form, the proportions of the golden section are easily guessed.
In painting and sculpture of temples, on household items, the ancient Egyptians most often depicted gods and pharaohs. 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 Sound volume.
It is known that the maximum volume of sound that causes pain is 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.
Air humidity. At a temperature of 18-20®, the humidity range of 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).
Air pressure. At an air pressure of 0.5 MPa, a person experiences unpleasant sensations, his physical and psychological activity worsens. At a pressure of 0.3 - 0.35 MPa, only short-term 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.
Outside air temperature. The boundary parameters of the outdoor air temperature, within which normal existence (and, most importantly, the origin) of a person is possible, is the temperature range from 0 to + (57-58) ® С. Obviously, there is no need to give explanations on the first boundary.
We divide the indicated range of positive temperatures by the golden ratio. This gives us two bounds:
Both boundaries are temperatures characteristic of the human body: the first corresponds to the temperature The second limit corresponds to the maximum possible outdoor temperature for the human body.GOLDEN SECTION IN PAINTING
Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points, and they are located at a distance of 3/8 and 5/8 from the corresponding edges of the plane.
This discovery among the artists of that time was called the "golden section" of the picture. Turning to examples of the "golden section" in painting, one cannot but stop one's attention on the work of Leonardo da Vinci. His identity is one of the mysteries of history. Leonardo da Vinci himself said: "Let no one who is not a mathematician dare to read my works."
He gained fame as an unsurpassed artist, a great scientist, a genius who anticipated many inventions that were not implemented until the 20th century.
There is no doubt that Leonardo da Vinci was a great artist, this was already recognized by his contemporaries, but his personality and activities will remain shrouded in mystery, since he left to posterity not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say "both everyone in the world."
He wrote from right to left in illegible handwriting and with his left hand. This is the most famous example of mirror writing in existence.
The portrait of Monna Lisa (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 de le Giocondo to paint a portrait of a young woman, the banker's wife, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint a portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became alive and interesting.
STORY
Once upon a time there was one poor man, he had four sons: three smart, and one of them this way and that. And then death came for the father. Before he parted 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 happiness. Let each of you learn something, to be able to feed himself." 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 hewn it, made a woman out of it, walked a little and waits. The second brother returned, saw a wooden woman and, since he was a tailor, in one minute dressed her: like a skilled craftsman, he sewed beautiful silk clothes for her. The third son adorned the woman with gold and precious stones - after all, he was a jeweler. Finally, the fourth brother arrived. He did not know how to carpentry and sew, he only knew how to listen to what the earth, trees, herbs, animals and birds were saying, he knew the course of heavenly bodies and also knew how to sing wonderful songs. He sang a song that made the brothers hiding behind the bushes cry. With this song, he revived the woman, she smiled and sighed. The brothers rushed to her and each shouted the same thing: "You must be my wife." But the woman answered: “You created me - be my father. You dressed me, and you decorated me - be my brothers.
And you, who breathed my soul into me and taught me to enjoy life, I need you alone for life".
Having finished the story, Leonardo looked at Monna Lisa, her face lit up with light, her eyes shone. Then, as if awakening from a dream, she sighed, passed her hand over her face, and without a word went to her place, folded her hands and assumed her usual posture. But the deed was done - the artist awakened the indifferent statue; the smile of bliss, slowly disappearing from her face, remained in the corners of her mouth and trembled, giving her face an amazing, mysterious and slightly sly expression, like that of a person who has learned a secret and, keeping it carefully, cannot restrain his triumph. Leonardo worked in silence, afraid to miss this moment, this ray of sunshine that illuminated his boring model...
It is difficult to note what was noticed in this masterpiece of art, but everyone spoke about Leonardo's deep knowledge of the structure of the human body, thanks to which he managed to catch this, as it were, mysterious smile. They talked about the expressiveness of individual parts of the picture and about the landscape, an unprecedented companion of the portrait. They talked about the naturalness of expression, the simplicity of the pose, the beauty of the hands. The artist has done something unprecedented: the picture depicts air, it envelops the figure with a transparent haze. Despite the success, Leonardo was gloomy, the situation in Florence seemed painful to the artist, he got ready to go. Reminders of flooding orders did not help him.
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The golden section in the painting by I. I. Shishkin "Pine Grove" In this famous painting by I. I. Shishkin, the motifs of the golden section are clearly visible. 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 section and further. |
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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.
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 V. I. 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. 1) This is the rise of Morozova's hand with the sign of the cross with two fingers as the highest point. 2) This is a helplessly outstretched hand to the same noblewoman, but this time it is the hand of an old woman - a poor wanderer, a hand from under which, along with the last hope of salvation, the end of the sledge slips out. And what about the "highest point"? At first glance, we have an apparent contradiction: after all, the section A1B1, which is 0.618 ... from the right edge of the picture, does not pass through the hand, not even through the head or eye of the noblewoman, but is somewhere in front of the noblewoman's mouth! |
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| The portrait of Mona Lisa attracts by the fact that the composition of the picture is built on "golden triangles" (more precisely, on triangles that are pieces of a regular star-shaped pentagon). | |
There is no painting more poetic than the painting 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.  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 analyzes a complex drawing. The ingenious combination of two triangles is also built in accordance with the golden ratio: it can be inscribed in a rectangle with an aspect ratio of 5/8. This drawing is surprisingly easy to insert into the upper section of the architecture. The upper corner of the triangle rests against the keystone of the arch in the area closest to the viewer, the lower one - at the vanishing point of perspectives, and the side section indicates the proportions of the spatial gap between the two parts of the arches. Golden spiral in Raphael's "Massacre of the Innocents" |
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Unlike the golden section, the feeling of dynamics, excitement, is perhaps most pronounced in another simple geometric figure - a spiral. The multi-figure composition, made in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, is just distinguished by the dynamism and drama of the plot. Rafael never brought his idea to completion, however, his sketch was engraved by an unknown Italian graphic artist Marcantinio Raimondi, who, based on this sketch, created the Massacre of the Innocents engraving.
If, on Raphael's preparatory sketch, one mentally draws lines running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of a child, a woman clutching him to herself, a warrior with a raised sword, and then along the figures of the same group on the right parts of the 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.
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GOLDEN RATIO AND IMAGE PERCEPTION
The ability of the human visual analyzer to distinguish objects built according to the golden section algorithm as beautiful, attractive and harmonious has long been known. The golden ratio gives the feeling of the most perfect unified whole. The format of many books follows the golden ratio. It is chosen for windows, paintings and envelopes, stamps, business cards. A person may not know anything about the number Ф, but in the structure of objects, as well as in the sequence of events, he subconsciously finds elements of the golden ratio.
Studies have been conducted in which subjects were asked to select and copy rectangles of various proportions. There were three rectangles to choose from: a square (40:40 mm), a "golden section" rectangle with an aspect ratio of 1:1.62 (31:50 mm) and a rectangle with elongated proportions of 1:2.31 (26:60 mm).
 When choosing rectangles in the normal state, in 1/2 cases preference is given to a square. The right hemisphere prefers the golden ratio and rejects the elongated rectangle. On the contrary, the left hemisphere gravitates toward elongated proportions and rejects the golden ratio. When copying these rectangles, the following was observed. When the right hemisphere was active, the proportions in the copies were maintained most accurately. When the left hemisphere was active, the proportions of all the rectangles were distorted, the rectangles were stretched (a square was drawn as a rectangle with an aspect ratio of 1:1.2; the proportions of the stretched rectangle increased sharply and reached 1:2.8). The most strongly distorted proportions of the "golden" rectangle; 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? |
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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.
Here is a photo of a cat, which is located in an arbitrary place in the frame.
Now let's conditionally divide the frame into segments, in the proportion of 1.62 of the total length from each side of the frame. At the intersection of the segments, there will be the main "visual centers" in which it is worth placing the necessary key elements of the image. Let's transfer 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 contrary to the law: there was no planet between Mars and Jupiter. Focused observation of this part 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® from one of the spirals comes another unfolding spiral. ... 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: living 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 to us in a completely different way - mobile, changeable and surprisingly diverse. Life shows us a fantastic carnival of diversity and originality of creative combinations! The world of inanimate nature is, first of all, a world of symmetry, which gives stability and beauty to his creations. The world of nature is, first of all, a world of harmony, in which the "law of the golden section" operates. In the modern world, science is of particular importance due to 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 its perfection of the whole and its parts in art, science, technology and nature. It can be expected that the laws of development of various systems of nature, the laws of growth, are not very diverse and can be traced in the most diverse formations. This is 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. th. 51
There are still many unsolved mysteries in the universe, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for unraveling the world around us, building its shape and optimal visual perception by a person, with the help of which he can feel beauty and harmony.
golden ratio
The principle of determining the size of the golden section underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technology. The doctrine of the golden ratio was founded as a result of research by ancient scientists on the nature of numbers.
It is based on the theory of the proportions and ratios of segment divisions, which was made by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (of the entire segment):
The result is an equation: x 2 - x - 1=0, which is solved as x=(1±√5)/2.
If we consider the ratio 1/x, then it is equal to 1,618…
Evidence of the use of the golden ratio by ancient thinkers is given in the book of Euclid's "Beginnings", written back in the 3rd century. BC, who used this rule to construct regular 5-gons. Among the Pythagoreans, this figure is considered sacred, since it is both symmetrical and asymmetrical. The pentagram symbolized life and health.
Fibonacci numbers
The famous book Liber abaci by the Italian mathematician Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time gives a pattern of numbers, in a series of which each number is the sum of the 2 previous digits. The sequence of Fibonacci numbers is as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.
The scientist also cited a number of patterns:
- Any number from the series, divided by the next, will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as you move from the beginning of the sequence, this ratio will be more and more accurate.
- If you divide the number from the series by the previous one, then the result will tend to 1.618.
- One number divided by the next one will show a value tending to 0.382.
The application of the connection and patterns of the golden section, the Fibonacci number (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction, and in many other sciences.
Spiral of Archimedes and golden rectangle
Spirals, very common in nature, were explored by Archimedes, who even derived her equation. The shape of the spiral is based on the laws of the golden ratio. When it is untwisted, a length is obtained to which proportions and Fibonacci numbers can be applied, the step increase occurs evenly.
The parallel between the Fibonacci numbers and the golden ratio can also be seen by constructing a "golden rectangle" whose sides are proportional as 1.618:1. It is built by moving from a larger rectangle to smaller ones so that the lengths of the sides will be equal to the numbers from the row. Its construction can be done in the reverse order, starting with the square "1". When connecting the corners of this rectangle with lines in the center of their intersection, a Fibonacci or logarithmic spiral is obtained.
The history of the use of golden proportions
Many ancient architectural monuments of Egypt were built using golden proportions: the famous pyramids of Cheops and others. The architects of Ancient Greece widely used them in the construction of architectural objects, such as temples, amphitheatres, stadiums. For example, such proportions were used in the construction of the ancient Parthenon temple (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony based on mathematical patterns.
In later centuries, interest in the golden ratio subsided, and the patterns were forgotten, but again resumed in the Renaissance, along with the book of the Franciscan monk L. Pacioli di Borgo "Divine Proportion" (1509). It included illustrations by Leonardo da Vinci, who fixed the new name "golden section". Also, 12 properties of the golden ratio were scientifically proven, and the author talked about how it manifests itself in nature, in art and called it "the principle of building the world and nature."
Vitruvian Man Leonardo
The drawing by which Leonardo da Vinci illustrated the book of Vitruvius in 1492 depicts a figure of a man in 2 positions with arms extended to the sides. The figure is inscribed in a circle and a square. This drawing is considered to be the canonical proportions of the human body (male), described by Leonardo based on their study in the treatises of the Roman architect Vitruvius.
The center of the body as an equidistant point from the end of the arms and legs is the navel, the length of the arms is equal to the height of a person, the maximum width of the shoulders = 1/8 of the height, the distance from the top of the chest to the hair = 1/7, from the top of the chest to the top of the head = 1/6 etc.
Since then, the drawing has been used as a symbol showing the internal symmetry of the human body.
The term "Golden Ratio" was used by Leonardo to denote proportional relationships in the human figure. For example, the distance from the waist to the feet is related to the same distance from the navel to the top of the head in the same way as the height to the first length (from the waist down). This calculation is done similarly to the ratio of the segments when calculating the golden ratio and tends to 1.618.
All these harmonious proportions are often used by artists to create beautiful and impressive works.
Studies of the golden ratio in the 16th-19th centuries
Using the golden ratio and Fibonacci numbers, research work on the issue of proportions has been going on for more than one century. In parallel with Leonardo da Vinci, the German artist Albrecht Dürer was also developing the theory of the correct proportions of the human body. For this, he even created a special compass.
In the 16th century the question of the connection between the Fibonacci number and the golden section was devoted to the work of the astronomer I. Kepler, who first applied these rules to botany.
A new "discovery" awaited the golden ratio in the 19th century. with the publication of "Aesthetic Research" by the German scientist Professor Zeisig. He raised these proportions to the absolute and announced that they are universal for all natural phenomena. He conducted studies of a huge number of people, or rather their bodily proportions (about 2 thousand), as a result of which conclusions were drawn about statistically confirmed patterns in the ratios of various parts of the body: the length of the shoulders, forearms, hands, fingers, etc.
Art objects (vases, architectural structures), musical tones, sizes when writing poems were also studied - Zeisig displayed all this through the lengths of segments and numbers, he also introduced the term "mathematical aesthetics". After receiving the results, it turned out that the Fibonacci series is obtained.
Fibonacci number and golden ratio in nature
In the plant and animal world, there is a tendency to form in the form of symmetry, which is observed in the direction of growth and movement. The division into symmetrical parts in which golden proportions are observed is a pattern inherent in many plants and animals.
The nature around us can be described using Fibonacci numbers, for example:
- the arrangement of leaves or branches of any plants, as well as the distances, are related to the series of given numbers 1, 1, 2, 3, 5, 8, 13 and so on;
- sunflower seeds (scales on cones, pineapple cells), arranged in two rows in twisted spirals in different directions;
- the ratio of the length of the tail and the entire body of the lizard;
- the shape of the egg, if you draw a line conditionally through its wide part;
- the ratio of the size of the fingers on the human hand.
And, of course, the most interesting forms are the spiraling snail shells, the patterns on the web, the movement of the wind inside a hurricane, the double helix in DNA, and the structure of galaxies - all of which include the Fibonacci number sequence.
The use of the golden ratio in art
Researchers looking for examples of the use of the golden section in art examine in detail various architectural objects and paintings. Famous sculptural works are known, the creators of which adhered to golden proportions - the statues of Olympian Zeus, Apollo Belvedere and
One of the creations of Leonardo da Vinci - "Portrait of Mona Lisa" - has been the subject of research by scientists for many years. They found that the composition of the work entirely consists of "golden triangles", united together into a regular pentagon-star. All the works of da Vinci are evidence of how deep his knowledge of the structure and proportions of the human body was, thanks to which he was able to catch the incredibly mysterious smile of the Mona Lisa.
The golden ratio in architecture
As an example, scientists studied architectural masterpieces created according to the rules of the "golden section": the Egyptian pyramids, the Pantheon, the Parthenon, Notre Dame de Paris Cathedral, St. Basil's Cathedral, etc.
The Parthenon, one of the most beautiful buildings in Ancient Greece (5th century BC), has 8 columns and 17 on different sides, the ratio of its height to the length of the sides is 0.618. The protrusions on its facades are made according to the "golden section" (photo below).
One of the scientists who invented and successfully applied the improvement of the modular system of proportions for architectural objects (the so-called "modulor") was the French architect Le Corbusier. The modulor is based on a measuring system associated with a conditional division into parts of the human body.
The Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the buildings of the Senate in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after N.I. Pirogov), was one of the architects who used laws in the design and construction about the golden ratio.
Applying proportions in design
In fashion design, all fashion designers make new images and models, taking into account the proportions of the human body and the rules of the golden ratio, although by nature not all people have ideal proportions.
When planning landscape design and creating voluminous park compositions with the help of plants (trees and shrubs), fountains and small architectural objects, the patterns of "divine proportions" can also be applied. After all, the composition of the park should be focused on creating an impression on the visitor, who will be able to freely navigate in it and find the compositional center.
All elements of the park are in such proportions that, with the help of geometric structure, mutual arrangement, illumination and light, they give an impression of harmony and perfection to a person.
Application of the golden section in cybernetics and technology
The patterns of the golden section and Fibonacci numbers are also manifested in energy transitions, in processes occurring with elementary particles that make up chemical compounds, in space systems, in the DNA gene structure.
Similar processes occur in the human body, manifesting itself in the biorhythms of his life, in the action of organs, for example, the brain or vision.
Algorithms and patterns of golden proportions are widely used in modern cybernetics and informatics. One of the simple tasks that beginner programmers are given to solve is to write a formula and determine the sum of Fibonacci numbers up to a certain number using programming languages.
Modern research on the theory of the golden ratio
Since the middle of the 20th century, interest in the problems and influence of the laws of the golden proportions on human life has increased dramatically, and from many scientists of various professions: mathematicians, ethnos researchers, biologists, philosophers, medical workers, economists, musicians, etc.
Since the 1970s, The Fibonacci Quarterly has been published in the United States, where works on this topic are published. Works appear in the press in which the generalized rules of the golden section and the Fibonacci series are used in various branches of knowledge. For example, for coding information, chemical research, biological, etc.
All this confirms the conclusions of ancient and modern scientists that the golden ratio is multilaterally connected with the fundamental issues of science and is manifested in the symmetry of many creations and phenomena of the world around us.
Even in ancient Egypt it was known golden ratio, Leonardo da Vinci and Euclid studied its properties.The visual perception of a person is arranged in such a way that he distinguishes in form all the objects that surround him. His interest in an object or its form is sometimes dictated by necessity, or this interest could be caused by the beauty of the object. If in the very basis of the construction of the form, a combination is used golden section and the laws of symmetry, then this is the best combination for visual perception by a person who feels harmony and beauty. The whole whole consists of parts, large and small, and these different sizes of parts have a certain relationship, both to each other and to the whole. And the highest manifestation of functional and structural perfection in nature, science, art, architecture and technology is the Principle golden section. The concept of golden ratio introduced into scientific use the ancient Greek mathematician and philosopher (VI century BC) Pythagoras. But the very knowledge of golden ratio he borrowed from the ancient Egyptians. The proportions of all temple buildings, the pyramids of Cheops, bas-reliefs, household items and decorations from tombs show that the ratio golden section was actively used by ancient masters long before Pythagoras. As an example: the bas-relief from the temple of Seti I at Abydos and the bas-relief of Ramses use the principle golden section in the proportions of the figures. The architect Le Corbusier found this out. On a wooden board recovered from the tomb of the Architect Khesir, a relief drawing is depicted, on which the architect himself is visible, holding measuring instruments in his hands, which are depicted in a position fixing the principles golden section. Knew the principles golden section and Plato (427...347 BC). The Timaeus dialogue is proof of this, since it is devoted to questions golden division, aesthetic and mathematical views of the school of Pythagoras. Principles golden section used by ancient Greek architects in the facade of the Parthenon temple. The compasses that ancient architects and sculptors of the ancient world used in their work were discovered during excavations of the Parthenon temple.
Parthenon, Acropolis, Athens In Pompeii (museum in Naples) proportions golden division are also available.In ancient literature that has come down to us, the principle golden section first mentioned in Euclid's Elements. In the book "Beginnings" in the second part, a geometric principle is given golden section. Euclid's followers were Pappus (3rd century AD), Hypsicles (2nd century BC), and others. To medieval Europe with the principle golden section We met through translations from Arabic of Euclid's "Beginnings". Principles golden section were known only to a narrow circle of initiates, they were jealously guarded, kept in strict secrecy. A renaissance has come and an interest in the principles golden section increases among scientists and artists, since this principle is applicable in science, architecture, and art. And Leonardo Da Vinci began to use these principles in his works, even more than that, he began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, who got ahead of him and published the book "Divine Proportion" after which Leonardo left his the work is not finished. According to historians of science and contemporaries, Luca Pacioli was a real luminary, a brilliant Italian mathematician who lived between Galileo and Fibonacci. As a student of the painter Piero della Francesca, Luca Pacioli wrote two books, On Perspective in Painting, the title of one of them. He is considered by many to be the creator of descriptive geometry. Luca Pacioli, at the invitation of the Duke of Moreau, arrived in Milan in 1496 and lectured there on mathematics. Leonardo da Vinci at this time worked at the Moro court. Luca Pacioli's Divine Proportion, published in Venice in 1509, became an enthusiastic hymn golden ratio, with beautifully executed illustrations, there is every reason to believe that the illustrations were made by Leonardo da Vinci himself. Monk Luca Pacioli, as one of the virtues golden ratio emphasized its "divine essence". Understanding the scientific and artistic value of the golden ratio, Leonardo da Vinci devoted a lot of time to studying it. Performing a section of a stereometric body consisting of pentagons, he obtained rectangles with aspect ratios in accordance with golden ratio. And he gave it a name golden ratio". Which is still holding on. Albrecht Dürer, also studying golden section in Europe, meets with the monk Luca Pacioli. Johannes Kepler, the greatest astronomer of the time, was the first to draw attention to the importance golden section for botany calling it the treasure of geometry. He called the golden ratio self-continuing. “It is so arranged,” he said, “the sum of the two junior terms of an infinite proportion gives the third term, and any two last terms, if added together, give the next term, and the same proportion remains indefinitely.”
Golden Triangle:: Golden Ratio and Golden Ratio:: Golden Rectangle:: Golden Spiral
Golden Triangle
To find segments of the golden ratio of the descending and ascending rows, we will use the pentagram.
Rice. 5. Construction of a regular pentagon and pentagram
In order to build a pentagram, you need to draw a regular pentagon according to the construction method developed by the German painter and graphic artist Albrecht Dürer. If O is the center of the circle, A is a point on the circle, and E is the midpoint of segment OA. The perpendicular to the radius OA, raised at point O, intersects the circle at point D. Using a compass, mark a segment on the diameter CE = ED. Then the length of a side of a regular pentagon inscribed in a circle is equal to DC. We set aside segments DC on the circle and get five points for drawing a regular pentagon. Then, through one corner, we connect the corners of the pentagon with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
Each end of the pentagonal star 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. 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 d1 are connected by straight lines with point A. We put the segment dd1 on line Ad1, getting point C. She divided the line Ad1 in proportion to the golden ratio. The lines Ad1 and dd1 are used to build a "golden" rectangle.
Rice. 6. Building a golden
triangle
Golden Ratio and Golden Ratio
In mathematics and art, two quantities are in the golden ratio if the ratio between the sum of these quantities and the greater is the same as the ratio between the greater and the smaller. 
Expressed algebraically: 
The golden ratio is often denoted by the Greek letter phi (? or?). the figure of the golden ratio illustrates the geometric relationships that define this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.
golden rectangle
The golden rectangle is a rectangle whose side lengths are in the golden ratio, 1:?
(one-to-fi), i.e. 1: or approximately 1:1.618. The golden rectangle can only be built with a ruler 
and a circle: 1. Construct a simple square 2. Draw a line from the middle of one side of the square to the opposite corner 3. Use this line as a radius to draw an arc that defines the height of the rectangle 4. Complete the golden rectangle
golden spiral
In geometry, the golden spiral is a logarithmic spiral whose growth factor b is related to?
, golden ratio. In particular, the golden spiral becomes wider (further away from where it started) by a factor ?
for every quarter turn it makes. 
The successive points of dividing the golden rectangle into squares lie on logarithmic spiral, sometimes known as the golden spiral.
Golden section in architecture and art.
Many architects and artists performed their work in accordance with the proportions of the golden section, especially in the form of a golden rectangle, in which the ratio of the larger side to the smaller one has the proportions of the golden section, believing that this ratio would be aesthetic. [Source: Wikipedia.org ]
Here are some examples:
Parthenon, Acropolis, Athens . This ancient temple fits almost exactly into the golden rectangle.
Vitruvian Man by Leonardo da Vinci you can draw many lines of rectangles in this figure. Then, there are three different sets of golden rectangles: Each set is for the head, torso, and legs area. Leonardo da Vinci's drawing Vitruvian Man is sometimes confused with the principles of the "golden rectangle", however, this is not the case. The construction of the Vitruvian Man is based on drawing a circle with a diameter equal to the diagonal of the square, moving it up so that it touches the base of the square and drawing the final circle between the base of the square and the midpoint between the area of the center of the square and the center of the circle: Detailed explanation about geometric construction >>
Golden ratio in nature.
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio in the arrangement of branches along the stem of the plant and the veins in the leaves. He expanded his studies from plants to animals, studying the skeletons of animals and the branches of their veins and nerves, as well as the proportions of chemical compounds and the geometry of crystals, up to the use of the golden ratio in fine art. In these phenomena, he saw that the golden ratio was being used everywhere as a universal law, Zeising wrote in 1854: The golden ratio is a universal law, which contains the basic principle that forms the desire for beauty and completeness in such areas as nature and art, which permeates, as a paramount spiritual ideal, all structures, shapes and proportions, whether it be a cosmic or physical person, organic or inorganic, acoustic or optical, but the principle of the golden section finds its most complete realization in human form.
Examples:
A cut of the Nautilus shell reveals the golden principle of spiral construction.
Mozart divided his sonatas into two parts, the lengths of which reflect golden ratio, although there is much debate as to whether he did it knowingly. In more modern times, the Hungarian composer Béla Bartók and the French architect Le Corbusier purposefully incorporated the golden ratio into their work. Even today golden ratio surrounds us everywhere in artificial objects. Look at almost any Christian cross, the ratio of vertical to horizontal is the golden ratio. To find the golden rectangle, look in your wallet and you will find credit cards there. Despite this much evidence given in works of art created over the centuries, there is currently a debate among psychologists about whether people really perceive golden proportions, in particular the golden rectangle, as more beautiful than other shapes. In a 1995 journal article, Professor Christopher Green, of York University in Toronto, discusses a number of experiments over the years that did not show any preference for the shape of the golden rectangle, but notes that several others have provided evidence that such a preference does not exist. . But regardless of the science, the golden ratio retains its mystique, in part because it works so well in many unexpected places in nature. Spiral 
shells of the nautilus clam are surprisingly close to golden ratio, and the ratio of the length of the chest and abdomen in most bees is almost golden ratio. Even cross-sections of the most common forms of human DNA fit perfectly into the golden decagon. golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to arouse the interest of mathematical communities. Dr. Steven Marquardt, a former plastic surgeon, used this mysterious proportion golden ratio, in his work, who has long been responsible for beauty and harmony, to make a mask, which he considered the most beautiful form of the human face that can be. 
Mask perfect human face
Egyptian Queen Nefertiti (1400 BC)
The face of Jesus is a copy from the Shroud of Turin and corrected according to the mask of Dr. Stephen Marquardt.
An "averaged" (synthesized) celebrity face. With proportions of the golden section.
Site materials were used: http://blog.world-mysteries.com/
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 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.
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, in particular, to the questions of the golden division.
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 (2nd century BC), Pappus (3rd century AD) and others were engaged in the study of the golden division. Navarre (3rd century). 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 increased among scientists and artists due to its use in both geometry and art, especially architecture. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge. He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the 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 1509, Luca Pacioli's book Divine Proportion was published in Venice, with brilliantly executed illustrations, which is why they are believed to have been made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden ratio, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity of God the Son, God the Father and God the Holy Spirit (it was understood that the small segment is the personification of God the Son, the larger segment is the personification of God the Father, and the entire segment - the god of the holy spirit).
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. And so it continues to this day.
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.” Albrecht Dürer develops in detail the theory of the proportions of the human body. He assigned an important place in his system of ratios to the golden section. 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 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 along with the water.” The golden ratio 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. Zeising considers the golden ratio 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 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 called "Golden division as the basic morphological law in nature and art." In 1876, a small book was published in Russia, outlining this work of Zeising.
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.
Science did not absorb art, but in those historical periods when mathematics and art converged, this gave impetus to the development of both.
The concept of the golden ratio
Let's find out what is common between the ancient Egyptian pyramids, the painting by Leonardo da Vinci "Mona Lisa", a sunflower, a snail, a snowflake, a galaxy and human fingers?
In mathematics, proportion (Latin proportio) is the equality of two ratios: a: b = c: d.
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.
Line segment AB can be divided into two parts by point C 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);
- in the extreme and average ratio in such a way that AB: AC \u003d AC: BC.
The last one is the golden division.
Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden ratio using a compass and ruler. BC = 1/2 AB; 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 as an infinite irrational fraction, if AB is taken as a unit, then AE \u003d 0.618 ..., BE \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB is taken as 100 parts, then the largest part of the segment is 62, and the smaller one is 38 parts.
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 the ratio 56:44.
The line of the second golden section of the rectangle is in the middle between the line of the golden section and the middle line of the rectangle.
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 (1471...1528). 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 in the golden ratio. Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36° at the apex, and the base laid on the lateral side divides it in the golden ratio.
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 section. 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: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.
This series is known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the two previous ones, and the ratio of adjacent numbers of the series approaches the ratio of the golden division. Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series. It was this constant number of division in the Middle Ages that was called the Divine Proportion, and now today it is referred to as the golden section, the golden mean or the golden proportion. In algebra, this number is denoted by the Greek letter φ (phi).
So the golden ratio is 1:1.618
So, 21:34 = 0.617, and 34:55 = 0.618. This ratio is denoted by the symbol φ. This ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden ratio.
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. 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.
golden rectangle and golden spiral
In geometry, a rectangle with a golden ratio of sides began to be called golden. Its long sides are related to the short ones - in a ratio of 1.168: 1.
The golden rectangle also has many amazing properties. By cutting off a square from the golden rectangle whose side is equal to the smaller side of the rectangle, we again get a smaller golden rectangle. This process can be continued ad infinitum. As we keep cutting off the squares, we'll get smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects. The pole of the spiral lies at the intersection of the diagonals of the initial rectangle and the first cut off vertical. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also a golden triangle.
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