It was known even in ancient Egypt Golden ratio, Leonardo da Vinci and Euclid studied its properties.A person’s visual perception is designed in such a way that he distinguishes by shape 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 at the very basis of form construction, a combination is used golden ratio 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 parts of different sizes 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 ratio. Concept of golden ratio introduced into scientific use by 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 Cheops pyramid, bas-reliefs, household items and decorations from tombs show that the ratio golden ratio was actively used by ancient masters long before Pythagoras. As an example: the bas-relief from the temple of Seti I in Abydos and the bas-relief of Ramses used the principle golden ratio 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, there is a relief drawing on which the architect himself is visible, holding measuring instruments in his hands, which are depicted in a position fixing the principles golden ratio. Knew about the principles golden ratio and Plato (427...347 BC). The dialogue "Timaeus" is proof of this, since it is devoted to questions golden division, aesthetic and mathematical views of the Pythagorean school. Principles Golden ratio 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 also available.In ancient literature that has come down to us, the principle golden ratio mentioned for the first time in Euclid's Elements. In the book "Beginnings" in the second part the geometric principle is given golden ratio. The followers of Euclid were Pappus (III century AD), Hypsicles (II century BC), and others. To medieval Europe with the principle golden ratio We got acquainted through translations of Euclid’s “Principles” from Arabic. Principles golden ratio were known only to a narrow circle of initiates, they were jealously guarded and kept in strict confidence. The era of renaissance and interest in the principles has arrived golden ratio 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 moreover, he began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, who preceded him and published the book “Divine Proportion”, after which Leonardo left his work unfinished. According to historians of science and contemporaries, Luca Pacioli was a real luminary, a brilliant Italian mathematician who lived in the period between Galileo and Fibonacci. As a student of the artist 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 Moro, came to Milan in 1496 and lectured there on mathematics. Leonardo da Vinci worked at the Moro court at this time. Luca Pacioli's book The 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 done by Leonardo da Vinci himself. Monk Luca Pacioli, as one of the virtues golden ratio highlighted its “divine essence.” Understanding the scientific and artistic value of the golden ratio, Leonardo da Vinci devoted a lot of time to studying it. Carrying out a section of a stereometric body consisting of pentagons, he obtained rectangles with aspect ratios in accordance with golden ratio. And he gave it the name “ golden ratio" Which still holds up to this day. Albrecht Dürer, also studying golden ratio in Europe, meets with the monk Luca Pacioli. Johannes Kepler, the greatest astronomer of his time, was the first to draw attention to the meaning golden ratio for botany calling it the treasure of geometry. He called the golden proportion self-continuing. “It is structured this way,” he said, “the sum of the two junior terms of an infinite proportion gives the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum.”

Golden Triangle:: Golden Ratio and Golden Ratio:: Golden Rectangle:: Golden Spiral

Golden Triangle

To find the segments of the golden proportion of the descending and ascending rows, we will use a 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 Durer. If O is the center of the circle, A is a point on the circle and E is the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, mark a segment on the diameter CE = ED. Then the side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw 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 represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio. We draw straight AB. From point A we lay down on it three times a segment O of an arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the resulting points d and d1 with straight lines to point A. We lay off the segment dd1 on line Ad1, obtaining point C. She divided line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to construct a “golden” rectangle.

Rice. 6. Building gold

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 larger is the same as the ratio between the larger 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

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1:? (one-to-fi), that is 1: or approximately 1:1.618. The golden rectangle can only be constructed with a ruler and a compass: 1. Construct a simple square 2. Draw a line from the middle of one side of the area 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 from its origin) by a factor ? for every quarter turn it makes.

Consecutive points of dividing the golden rectangle into squares lie on logarithmic spiral, which is sometimes known as the golden spiral.

Golden ratio in architecture and art.

Many architects and artists executed their works 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 side has the proportions of the golden section, believing that this ratio would be aesthetically pleasing. [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 make 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 Vitruvian Man drawing is sometimes confused with the Golden Rectangle principles, 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 upward so that it touches the base of the square and drawing a 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 proportion in the arrangement of branches along the stem of a plant and the veins in the leaves. He expanded his research and moved 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 the visual arts. In these phenomena, he saw that the golden ratio was used everywhere as a universal law, Zeising wrote in 1854: The Golden Ratio is a universal law, which contains the basic principle shaping the desire for beauty and completeness in such areas as nature and art, which permeates, as a primary spiritual ideal, all structures, forms and proportions, whether cosmic or physical, organic or inorganic, acoustic or optical, but the principle of the golden ratio finds its most complete realization in the human form.

Examples:

Cutting through 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 this deliberately. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier deliberately incorporated the principle of the golden ratio into their works. Even today golden ratio surrounds us everywhere in artificial objects. Look at almost any Christian cross, the ratio of the vertical part to the horizontal part is the golden proportion. To find the golden rectangle, look in your wallet and you will find credit cards there. Despite this abundant evidence from works of art created over the centuries, there is currently debate among psychologists about whether people actually perceive golden proportions, particularly 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 have not shown any preference for the golden rectangle shape, 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 has excellent applications in many unexpected places in nature. Spiral Nautilus shells 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 a cross-section of the most common forms of human DNA fits perfectly into the golden decagon. Golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to attract the interest of mathematical communities. Dr. Steven Marquardt, a former plastic surgeon, used this mysterious proportion golden ratio, in his work, which had long been responsible for beauty and harmony, to make a mask that he considered the most beautiful form of the human face that could be.

Mask perfect human face

Egyptian Queen Nefertiti (1400 BC)

The face of Jesus is a copy of the Shroud of Turin and corrected to match the mask of Dr. Stephen Marquardt.

“Average” (synthesized) celebrity face. With golden ratio proportions.

Website materials used: http://blog.world-mysteries.com/

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

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

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

Golden Triangle

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

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O- center of the circle, A- a point on a circle and E- the middle of the segment OA. Perpendicular to radius OA, restored at the point ABOUT, intersects the circle at the point D. Using a compass, plot a segment on the diameter C.E. = ED. The side length of a regular pentagon inscribed in a circle is DC. Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another 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 represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

We carry out a direct AB. From point A we plot on it three times a segment O of an arbitrary size, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside the segments ABOUT. Received points d And d1 connect with straight lines to a point A. Segment dd1 put on line Ad1, getting a point WITH. She split the line Ad1 in proportion to the golden ratio. Lines Ad1 And dd1 used to construct a “golden” rectangle.

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

Rice. 5. Construction of a regular pentagon and pentagram

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, restored at point O, intersects the circle at point D. Using a compass, plot the segment CE = ED on the diameter. The side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw a regular pentagon. We connect the corners of the pentagon through one another 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 represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

Rice. 6. Construction of the golden triangle

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

History of the golden ratio

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

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

Rice. 7. 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 Pythagorean school and, in particular, to the issues of the golden division.

The façade of the ancient Greek temple of the Parthenon features golden proportions. During its excavations, compasses were discovered that were used by architects and sculptors of the ancient world. The Pompeian compass (museum in Naples) also contains the proportions of the golden division.

Rice. 8. Antique golden ratio compass

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

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

Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. In 1509, Luca Pacioli’s book “The Divine Proportion” was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity - God the Son, God the Father and God the Holy Spirit (it was implied that the small segment is the personification of God the Son, the larger segment is the God of the Father, and the entire segment - God of the Holy Spirit).

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

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

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

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

Kepler called the golden proportion self-continuing. “It is structured in such a way,” he wrote, “that the two lowest terms of this never-ending proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion is maintained until infinity."

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

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

Rice. 9. Construction of a scale of segments of the golden ratio

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

Rice. 10. Golden proportions in parts of the human body

Rice. 11. Golden proportions in the human figure

Zeising did a tremendous job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man. The proportions of the golden ratio also appear 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 the most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic meters were studied. Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the numbers expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction or the other. His next book was titled “The Golden Division as the Basic Morphological Law in Nature and Art.” In 1876, a small book, almost a brochure, was published in Russia outlining this work of Zeising. The author took refuge under the initials Yu.F.V. This edition does not mention a single work of painting.

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

Any person who has at least indirectly encountered the geometry of spatial objects in interior design and architecture is probably well aware of the principle of the golden ratio. Until recently, several decades ago, the popularity of the golden ratio was so high that numerous supporters of mystical theories and the structure of the world call it the universal harmonic rule.

The essence of universal proportion

Surprisingly different. The reason for the biased, almost mystical attitude towards such a simple numerical dependence was several unusual properties:

A large number of objects in the living world, from viruses to humans, have basic body or limb proportions very close to the value of the golden ratio;
The dependence of 0.63 or 1.62 is typical only for biological creatures and some types of crystals; inanimate objects, from minerals to landscape elements, have the geometry of the golden ratio extremely rarely;
Golden proportions in body structure turned out to be the most optimal for the survival of real biological objects.

Today, the golden ratio is found in the structure of the body of animals, the shells and shells of mollusks, the proportions of leaves, branches, trunks and root systems of a fairly large number of shrubs and herbs.

Many followers of the theory of the universality of the golden section have repeatedly made attempts to prove the fact that its proportions are the most optimal for biological organisms in the conditions of their existence.

The structure of the shell of Astreae Heliotropium, one of the marine mollusks, is usually given as an example. The shell is a coiled calcite shell with a geometry that practically coincides with the proportions of the golden ratio.

A more understandable and obvious example is an ordinary chicken egg.

The ratio of the main parameters, namely, the large and small focus, or the distances from equidistant points of the surface to the center of gravity, will also correspond to the golden ratio. At the same time, the shape of a bird's egg shell is the most optimal for the survival of the bird as a biological species. In this case, the strength of the shell does not play a major role.

For your information! The golden ratio, also called the universal proportion of geometry, was obtained as a result of a huge number of practical measurements and comparisons of the sizes of real plants, birds, and animals.

Origin of universal proportion

The ancient Greek mathematicians Euclid and Pythagoras knew about the golden ratio of the section. In one of the monuments of ancient architecture - the Cheops pyramid, the ratio of sides and base, individual elements and wall bas-reliefs are made in accordance with universal proportion.

The golden section technique was widely used in the Middle Ages by artists and architects, while the essence of universal proportion was considered one of the secrets of the universe and was carefully hidden from the common man. The composition of many paintings, sculptures and buildings was built strictly in accordance with the proportions of the golden ratio.

The essence of universal proportion was first documented in 1509 by the Franciscan monk Luca Pacioli, who had brilliant mathematical abilities. But real recognition took place after the German scientist Zeising conducted a comprehensive study of the proportions and geometry of the human body, ancient sculptures, works of art, animals and plants.

In most living objects, some body dimensions are subject to the same proportions. In 1855, scientists concluded that the proportions of the golden section are a kind of standard for the harmony of body and form. We are talking, first of all, about living beings; for dead nature, the golden ratio is much less common.

How to get the golden ratio

The golden ratio is most easily represented as the ratio of two parts of the same object of different lengths, separated by a point.

Simply put, how many lengths of a small segment will fit inside a large one, or the ratio of the largest part to the entire length of a linear object. In the first case, the golden ratio is 0.63, in the second case the aspect ratio is 1.618034.

In practice, the golden ratio is just a proportion, the ratio of segments of a certain length, sides of a rectangle or other geometric shapes, related or conjugate dimensional characteristics of real objects.

Initially, the golden proportions were derived empirically using geometric constructions. There are several ways to construct or derive harmonic proportion:

For your information! Unlike the classic golden ratio, the architectural version implies an aspect ratio of 44:56.

If the standard version of the golden ratio for living beings, paintings, graphics, sculptures and ancient buildings was calculated as 37:63, then the golden ratio in architecture from the end of the 17th century began to be increasingly used as 44:56. Most experts consider the change in favor of more “square” proportions to be the spread of high-rise construction.

The main secret of the golden ratio

If the natural manifestations of the universal section in the proportions of the bodies of animals and humans, the stem base of plants can still be explained by evolution and adaptability to the influence of the external environment, then the discovery of the golden section in the construction of houses of the 12th-19th centuries came as a certain surprise. Moreover, the famous ancient Greek Parthenon was built in compliance with universal proportions; many houses and castles of wealthy nobles and wealthy people in the Middle Ages were deliberately built with parameters very close to the golden ratio.

Golden ratio in architecture

Many of the buildings that have survived to this day indicate that the architects of the Middle Ages knew about the existence of the golden ratio, and, of course, when building a house, they were guided by their primitive calculations and dependencies, with the help of which they tried to achieve maximum strength. The desire to build the most beautiful and harmonious houses was especially evident in the buildings of residences of reigning persons, churches, town halls and buildings of special social significance in society.

For example, the famous Notre Dame Cathedral in Paris has many sections and dimensional chains in its proportions that correspond to the golden ratio.

Even before the publication of his research in 1855 by Professor Zeising, at the end of the 18th century the famous architectural complexes of the Golitsyn Hospital and the Senate building in St. Petersburg, the Pashkov House and the Petrovsky Palace in Moscow were built using the proportions of the golden section.

Of course, houses have been built in strict compliance with the golden ratio rule before. It is worth mentioning the ancient architectural monument of the Church of the Intercession on the Nerl, shown in the diagram.

All of them are united not only by a harmonious combination of forms and high quality of construction, but also, first of all, by the presence of the golden ratio in the proportions of the building. The amazing beauty of the building becomes even more mysterious if we take into account its age. The building of the Church of the Intercession dates back to the 13th century, but the building received its modern architectural appearance at the turn of the 17th century as a result of restoration and reconstruction.

Features of the golden ratio for humans

The ancient architecture of buildings and houses of the Middle Ages remains attractive and interesting for modern people for many reasons:

An individual artistic style in the design of facades allows us to avoid modern cliches and dullness; each building is a work of art;
Massive use for decorating and decorating statues, sculptures, stucco moldings, unusual combinations of building solutions from different eras;
The proportions and composition of the building draw the eye to the most important elements of the building.

Important! When designing a house and developing its appearance, medieval architects applied the rule of the golden ratio, unconsciously using the peculiarities of perception of the human subconscious.

Modern psychologists have experimentally proven that the golden ratio is a manifestation of a person’s unconscious desire or reaction to a harmonious combination or proportion in sizes, shapes and even colors. An experiment was conducted in which a group of people who did not know each other, did not have common interests, different professions and age categories, were offered a series of tests, among which was the task of bending a sheet of paper in the most optimal proportion of sides. Based on the testing results, it was found that in 85 cases out of 100, the sheet was bent by the subjects almost exactly according to the golden ratio.

Therefore, modern science believes that the phenomenon of universal proportion is a psychological phenomenon, and not the action of any metaphysical forces.

Using the universal section factor in modern design and architecture

The principles of using the golden proportion have become extremely popular in the construction of private houses in the last few years. The ecology and safety of building materials have been replaced by harmonious design and proper distribution of energy inside the house.

The modern interpretation of the rule of universal harmony has long spread beyond the usual geometry and shape of an object. Today, the rule is subject to not only the dimensional chains of the length of the portico and pediment, individual elements of the facade and the height of the building, but also the area of rooms, window and door openings, and even the color scheme of the interior of the room.

The easiest way to build a harmonious house is on a modular basis. In this case, most departments and rooms are made in the form of independent blocks or modules, designed in compliance with the rule of the golden ratio. Constructing a building in the form of a set of harmonious modules is much easier than building one box, in which most of the facade and interior must be within the strict framework of the golden ratio proportions.

Many construction companies designing private households use the principles and concepts of the golden ratio to increase the cost estimate and give clients the impression that the design of the house has been thoroughly worked out. As a rule, such a house is declared to be very convenient and harmonious to use. A correctly selected ratio of room areas guarantees spiritual comfort and excellent health of the owners.

If the house was built without taking into account the optimal ratios of the golden section, you can redesign the rooms so that the proportions of the room correspond to the ratio of the walls in the proportion 1:1.61. To do this, furniture can be moved or additional partitions installed inside rooms. In the same way, the dimensions of window and door openings are changed so that the width of the opening is 1.61 times less than the height of the door leaf. In the same way, planning of furniture, household appliances, wall and floor decoration is carried out.

It is more difficult to choose a color scheme. In this case, instead of the usual ratio of 63:37, followers of the golden rule adopted a simplified interpretation - 2/3. That is, the main color background should occupy 60% of the space of the room, no more than 30% should be given to the shading color, and the rest is allocated to various related tones, designed to enhance the perception of the color scheme.

The interior walls of the room are divided by a horizontal belt or border at a height of 70 cm; installed furniture should be commensurate with the height of the ceilings according to the golden ratio. The same rule applies to the distribution of lengths, for example, the size of the sofa should not exceed 2/3 of the length of the partition, and the total area occupied by the furniture relates to the area of the room as 1:1.61.

The golden proportion is difficult to apply in practice on a large scale due to just one cross-sectional value, therefore, when designing harmonious buildings, they often resort to a series of Fibonacci numbers. This allows you to expand the number of possible options for proportions and geometric shapes of the main elements of the house. In this case, a series of Fibonacci numbers interconnected by a clear mathematical relationship is called harmonic or golden.

In the modern method of designing housing based on the principle of the golden ratio, in addition to the Fibonacci series, the principle proposed by the famous French architect Le Corbusier is widely used. In this case, the height of the future owner or the average height of a person is chosen as the starting unit of measurement by which all parameters of the building and interior are calculated. This approach allows you to design a house that is not only harmonious, but also truly individual.

Conclusion

In practice, according to reviews from those who decided to build a house according to the golden ratio rule, a well-built building actually turns out to be quite comfortable for living. But the cost of the building due to individual design and the use of building materials of non-standard sizes increases by 60-70%. And there is nothing new in this approach, since most buildings of the last century were built specifically for the individual characteristics of their future owners.

Secret golden ratio tried to comprehend Plato, Euclid, Pythagoras, Leonardo da Vinci, Kepler. The Golden Ratio, created long ago, still excites the minds of many scientists.

Since ancient times, people have sought to understand how our world is organized and structured by nature.

Pythagoras believed that the world is organized according to strict geometric laws and the basis of the universe is number. There are suggestions that he borrowed his knowledge of the golden division from the Egyptians and Babylonians. This is evidenced by the proportions of the Cheops pyramid, temples, household items and decorations from the tomb of Tutankhamun.

One of the tasks of the ancients was to divide a segment into 2 equal parts so that the length of the larger segment was related to the length of the smaller one in the same way as the length of the entire segment was to the length of the larger one.

Or this proportion can be inverted and find the ratio of smaller to larger. As a result, it was calculated that the ratio of larger to smaller = 1.61803..., and smaller to larger = 0.61803...

In Ancient Greece, such a division was called a harmonic ratio. In 1509, an Italian mathematician and monk Luca Pacioli wrote a whole book " About divine proportion».

2. Golden triangle and pentagram

« Gold" triangle is an isosceles triangle, the ratio of the side to the base is 1.618 ( Appendix 1).

Golden ratio can also be seen in the pentagram - this is what the Greeks called the star polygon.

A pentagon with drawn diagonals forming a five-pointed star was called a pentagram, which has been considered a revered figure since ancient times.

It was an ancient magical sign of goodness and the brotherhood of the five principles underlying the world of fire, earth, water, wood and metal. A pentagram is a regular pentagon, on each side of which isosceles triangles of equal height are built.

The five-pointed star is very beautiful; it is not for nothing that many countries place it on their flags and coats of arms. The perfect shape of this figure pleases the eye.

The pentagon is literally woven from proportions, and above all the golden proportion ( appendix 2).