The Rule of the Golden section applies. Golden section in nature, man, art

The gold cross section is a universal manifestation of structural harmony. It meets in nature, science, art - in everything, with which a person can come with. One day, having acquainted with the golden rule, humanity did not change him anymore.

Definition.
The most capacious definition of the golden section states that a smaller part refers to a greater, as big - to all. Approximate value is 1, 6180339887. In a rounded percentage value of the proportion of parts of the whole will be correlated as 62% by 38%. This ratio in the forms of space and time is valid.

Ancient saw in the golden section reflection of cosmic order, and Johann Kepler called him one of the treasures of geometry. Modern science Considers the golden cross section as "asymmetric symmetry", calling it in a broad sense by a universal rule reflecting the structure and order of our world order.

History.
The ancient Egyptians had an idea of \u200b\u200bthe gold proportions, they knew about them in Russia, but for the first time a scientific golden cross section explained the monk of Luka Pachet in the book "Divine proportion" (1509), the illustrations to which he allegedly made Leonardo da Vinci. Pachet saw divine trinity in the golden section: a small segment personified his son, a big father, and the whole - the Holy Spirit.

Directly with the Rule of the Golden section connected the name of the Italian mathematics Leonardo Fibonacci. As a result of a solution to one of the tasks, the scientist reached the sequence of numbers, known now as a number of Fibonacci: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc., on the attitude of this sequence to the golden proportion drew the attention of Kepler : "She works so that two younger member of this endless proportion in the amount give the third dick, and any two last member, if they folded, give the following member, and the same proportion remains to infinity." Now the Fibonacci series is an arithmetic basis for calculating the proportions of the golden section in all its manifestations

Fibonacci numbers - harmonic division, beauty measure. Golden section in nature, man, art, architecture, sculpture, design, mathematics, music https://psihologiyaotnoshenij.com/stati/zolotoe-sechenie-kak-To-rabotaet

Leonardo da Vinci also devoted a lot of time to study the features of the golden section, most likely, it is the term himself. Its drawings of the stereometric body formed by the correct pentagons prove that each of the rectangles obtained during the cross section gives the ratio of the parties in the gold division.

Over time, the Rule of the Golden section turned into an academic routine, and only the Adolf Ceasing philosopher in 1855 returned to his second life. He brought to the absolute of the proportion of the Golden section, making them versatile for all the phenomena of the surrounding world. However, his "mathematical aesthetics" caused a lot of criticism.

Nature.
Even without going into calculations, the golden cross section can be easily discovered in nature. So, the ratio of tail and body lizard fall under it, the distance between the leaves on the branch, there is a golden cross section and in the shape of an egg, if the conditional line is carried out through its widest part.

The Belarusian scientist Eduard Sorokh, who studied the forms of gold divisions in nature, noted that everything is growing and striving to take their place in space, endowed with the proportions of the golden cross section. In his opinion, one of the most interesting forms is spiral twisting.
More Archimedes, paying attention to the helix, brought the equation based on its form, which is now applied in the technique. Later Guete noted the nature of nature to spiral forms, calling the spiral of the "curve of life." Modern scientists were established that such manifestations of spiral forms in nature as a snail shell, sunflower seed location, web patterns, hurricane movement, structure of DNA and even the structure of galaxies enter into a series of fibonacci.

Human.
Fashion designers and designers of clothing All calculations are made based on the proportions of the golden section. Man is a universal form to verify the laws of the golden section. Of course, from nature, not all people have the proportions are ideal, which creates certain difficulties with the selection of clothing.

Adolf Ceying, exploring the proportionality of a person, has done a colossal work. He measured about two thousand human bodies, as well as many ancient statues and led that the golden cross section expresses the average law. In a person, almost all parts of the body are subordinated to him, but the main indicator of the golden section is the division of the body of the Pup point.
As a result of measurements, the researcher found that the proportions of the male body 13: 8 closer to the golden section than the proportions of the female body - 8: 5.

Art spatial Forms.
The artist Vasily Surikov said, "that there is an immutable law in the composition, when in the picture it is impossible not to remove anything, nor add, even the extra point is impossible to put it, this is a real mathematics." For a long time, artists followed this law intuitively, but after Leonardo Da Vinci, the process of creating a picturesque web can no longer be without solving geometric tasks. For example, Albrecht Durer to determine the points of the golden section used the proportional circuit invented.

Art historian F. in. Kovalev, examining in detail the picture of Nikolai Ge "Alexander Sergeevich Pushkin in the village of Mikhailovsky," notes that every detail of the canvas, be it a fireplace, shelf, chair or poet himself, strictly inscribed in golden proportions.

The investigors of the golden section without tired and measured the masterpieces of architecture, claiming that they became such because they were created on gold canons: in their list, the Great Pyramids Giza, the Cathedral of the Parisian Mother of God, the Temple of Basil Blessed, Parfenon.
And today, in any art, the spatial forms are trying to follow the proportions of the golden section, since they, according to art historians, facilitate the perception of the work and form an aesthetic feeling at the viewer.

Word, sound and film.
Forms temporarily? Arts in their own way demonstrate to us the principle of golden division. Literary critics, for example, noticed that the most popular number of lines in the poems of the late period of Pushkin's creativity corresponds to the Fibonacci row - 5, 8, 13, 21, 34.

There is a Rule of the Golden section and in separately taken works of Russian classics. So the culmination point of the "peak lady" is the dramatic scene of Herman and Countess, ending with the death of the latter. In the lead of 853 lines, and the climax falls on 535 row (853: 535 \u003d 1, 6) - this is the point of the golden section.

Soviet musicologist er K. Rosenov notes the striking accuracy of the ratios of the golden section in the strict and free forms of the works of Johanne Sebastian Baha, which corresponds to a thoughtful, focused, technically verified style of the master. This is true in relation to the outstanding creations of other composers, where the most bright or unexpected musical solution is usually accounted for on the golden section.
Film director Sergey Eisenstein Scenario of his film "Bremen All Potemkin" deliberately coordinated with the Rule of Golden Section, dividing the tape for five parts. In the first three sections, the action unfolds on the ship, and in the last two - in Odessa. Go to the scene in the city and there is a golden middle of the film.

Gold cross section Examples. How got a golden cross section

So, the golden cross section is a gold proportion, which is also a harmonious division. In order to explain this more clearly, consider some features of the form. Namely: the form is something entire, but the whole, in turn, always consists of some parts. These parts most likely have different characteristics at least in different sizes. Well, such dimensions are always in a certain relationship both among themselves and in relation to the whole.

So, in other words, we can argue that the golden cross section is the ratio of two values, which has its own formula. Using this ratio when creating a form helps to make it as beautiful as possible and harmonious for the human eye.

In the tattoo, the spirals laid much more sense than it seems at first glance. Such a simple pattern is built on the so-called golden section principle, which is found in nature everywhere. Moreover, this principle is known from ancient times, which is confirmed by its presence in the foundation of the Egyptian pyramids.

Symbols of tattoos with spirals

In tattoos, Ta-Moko or in the same Celtic patterns, the spirals are found very often, and this is not surprising. The lack of direct angles of this figure symbolizes communication with nature, which does not like straight angles, trying to always smooth them. Spiral tattoo means unity with nature, as a rule, such a tattoo make calm, reasonable people.

But this is only a general meaning, often people are trying to learn about the meaning of the tattoo in the form of a spiral, in fact, it is puzzled with other tattoos. Often the tattoo of the spiral shell introduces people in confusion, she is in lately Very popular. One meaning is absolutely different, it is suitable for closed people, a single, who usually undergoing some shock and not want to share it, and in his honor they make such a knack.

It is extremely similar to the spiral of a wave tattoo, which symbolizes the love of the sea or a tattoo of the Black Sun, the meaning of which we wrote in detail.

Often, the spiral tattoo is done as a guard, as this is a symbol of cyclical life, it transmits the energy of peace and existence. You can apply the image of the spiral on the shoulders, forearm, chest and back. More tattoo is suitable for women, since another tattoo value is a feminine.

It is believed that the concept of the golden section of Pythagores introduced the first. The Euclides have come to this day (with the help of a golden section built the right pentagons, which is why such a pentagon is called "gold"), and the number of golden section is named after the ancient Greek architect Fidia. That is, this is our number "FI" (denoted by the Greek letter φ), and equally it 1.6180339887498948482 ... Naturally, this value is rounded: φ \u003d 1,618 or φ \u003d 1.62, and in the percentage ratio, the gold cross section looks like 62% and 38%.

What is the uniqueness of this proportion (and if she, believe me)? Let's first try to figure out the example of the segment. So, we take a segment and divide it on unequal parts in such a way that his smaller part belongs to more, as big to everything. I understand not very clear what I will try to illustrate clearly on the example of segments:

So, we take a segment and divide it into two others, so that a smaller cut A, relate to a larger segment B, as well as the segment B refers to a whole, that is, the whole line (A + B). Mathematically it looks like this:

This rule works infinitely, you can divide the seglings for how long. And see how simple it is. The main thing is to understand it once.

But now consider a more complex example that comes across very often, since the golden cross section is still represented in the form of a golden rectangle (the aspect ratio of which is φ \u003d 1.62). This is a very interesting rectangle: if you "cut off" from it, then we will get a golden rectangle again. And so infinite many times. See:

But mathematics would not be mathematics if there was no formulas in it. So, friends, now will be a little "hurt". The golden proportion solution hidden under the spoiler, a lot of formulas, but without them I do not want to leave an article.

The principle of the golden section. Successful creation or golden section rule

Imprinting the moment - this is the moment of the artist or photographer's creation. In addition to inspiration, the master must follow strictly defined rules, which appear: contrast, accommodation, equilibrium, the rule of adherence to the third and many others. But the priority is still recognized as the Rule of the Golden section, it is also a rule of thirds.

Just about the difficult

If in a simplified form to present the basis of the rules of the golden section, then actually this is the division of the reproducible moment on nine equal parts (three vertically three horizontally). For the first time, Leonardo da Vinci introduced him specifically, building all his compositions in this peculiar grid. It was he who practically confirmed that the key elements of the image should be concentrated at the intersection points of vertical and horizontal lines.

The golden section rule in the photo is subject to a certain correction. In addition to the nine-segment grid, it is recommended to use so-called triangles. The principle of their construction is based on the rules of the third. To do this, the bottom of the upper point is carried out a diagonal to the bottom, and from the opposite top - a beam, which divides an existing diagonal in one of the internal points of intersection of the grid. The key element of the composition should be displayed on average largest of the resulting triangles. It is worth making a remark: the reduced scheme for the construction of triangles displays only their principle, and, it means, it makes sense to experiment with the manual given.

How to use a grid and triangles?

The gold cross section rule in the photo is valid for certain standards, depending on what is depicted on it.

Horizon factor. According to the rule of the third, it should be placed in horizontal lines. In this case, if the imprint object is above the level of the horizon, the factor passes through the bottom line, and vice versa.

Location of the main object. The classic is considered to be the location in which the central element is in one of the intersection points. If the photographer sends two objects, then they must be diagonally or in parallel points.

The use of triangles. The golden cross section rule in the case under consideration retreats from the canons, but slightly. The object is not obliged to be located at the point of intersection, but is located as close as possible on the middle triangle.

Direction. This principle of shooting is used in a dynamic photo and is that two thirds of the snapshot space should remain before the driven object. This will ensure the effect of moving forward and indicate the target. Otherwise, the photo can remain unacceptable.

Correction of the Rules of the Golden section

Despite the fact that the rule of the third in the existing theory of constructing the composition is considered classic, more and more photographers are inclined to abandon him. Motivation is simple: the analysis of pictures of famous artists shows that the rule of the golden section is not withstanding. With this statement you can argue.

Consider the well-known JoconDu, which opponents of using the Rule of thirds lead as an example (forgetting that yes Vinci himself stood at the origins of his practical use). Their arguments are the fact that the master did not consider it necessary to arrange the key elements of the picture at the intersection points, as the classic image requires it. But they are missing the factor of horizontal lines, according to which the head and the torso depicted are located in such a way that the silhouette in general does not "cut the eye." In addition, in this product, a spiral was most used, which in most cases forget photograph theorists. And so it is possible to refute the assertions with respect to almost every creation, which is driven as an example.

The golden section rule can be used, and you can refuse it if it is required to emphasize the disharmony of the composition. However, it is impossible to argue that it is not key in the formation of an art object.

Golden cross section in architecture. How got a golden cross section

The proportion of the golden section is easiest to imagine as the ratio of two parts of one object of different lengths, separated by a point.

Simply put, how many lengths of the small segment fit inside the large, or the ratio of the most of the parts to the entire length of the linear object. In the first case, the ratio of the golden section is 0.63, in the second embodiment, the aspect ratio is 1.618034.

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

Initially, gold proportions were derived empirically by geometric constructions. There are several ways to build or remove the harmonic proportion:

The classic splitting of one of the sides of the rectangular triangle and the construction of perpendicular and the sectional arcs. To do this, from one end of the segment it is necessary to restore the perpendicular height in ½ of its length and build right triangleas in the scheme.
If the hypotenuse is postponing the height of the perpendicular, then the radius equal to the remaining segment, the base dissects into two segments with lengths proportional to the golden section;
The method of constructing a Dürer's pentagram, a brilliant German graph and a geometer. Today we know the method of the golden section of Durera, as a way to build a star or pentagram, inscribed in a circle, in which at least four segments of the harmonious proportion;
In architecture and construction, the gold cross section is more often used in an improved form. In this case, the splitting of the rectangular triangle is not on the cathelet, but on hypotenuse, as a scheme.

For your information! Unlike the classic gold relation, the architectural version implies the aspect ratio of the segment in the proportion of 44:56.

If the standard version of the golden section for living beings, painting, graphics, sculptures and antique buildings was calculated as 37:63, then the golden cross section in architecture from the end of the XVII century became more commonly used 44:56. Most experts consider the change in favor of more "square" proportions to the spread of high-rise construction.

Many are dreaming perfect appearance, but not everyone has a clear idea of \u200b\u200bwhich proportions can be considered harmonious. The formula of the golden cross section of the face is inextricably linked with the number of 1.618 and other relations. So, the proportions of beauty can be described as follows:

the ratio of height and width of the face should be equal to 1.618;
if you split the mouth length and wing width of the nose, then it turns out 1.618;
when dividing distances between pupils and eyebrows, again, it turns out 1.618;
the length of the eyes must coincide with the distance between them, as well as with the width of the nose;
persons of the face from the hair growth line to the eyebrows, from the nose to the tip of the nose, and the lower part before the chin should be equal;
if you hold the vertical lines to the corners of the lips from the pupils, then three equal in the width of the site.

It should be understood that in nature the coincidence of all parameters is rare enough. But there is nothing wrong with that. This does not mean that persons who do not match the ideal proportions can be called ugly or nonimely. On the contrary, it is the "defects" sometimes give the person an unforgettable charm.

Golden section in compositions of drawings in Paint.Net
Mathematically, the "golden section" can be described like this - the ratio of the whole to the greater part should be equal to the ratio of most to the smaller. We illustrate on the example of the segment.

In our case, the entire segment is divided into two parts - greater A and less B. Then, if V / A is equal to A / B, the division of the segment will be implemented according to the principle called the "golden cross section".
Not quite accurately, but close to the "golden section", for example, a ratio of 2/3 or 5/8. Numbers in similar relations are often called "golden".
Why do we draw to us for drawing in paint.net? "Golden section" is important for the composition. It is believed that the objects containing the "golden cross section" are perceived by people as the most harmonious. It was in such relations that the sizes of hosts for their paintings are famous artists.
Consider a simplified version of the construction of the "golden section" for the composition of the drawing, or the "third" rule. The rule of the third is that we mentally divide the frame, three parts horizontally and vertical and at the intersection points of imaginary lines, we place key and important details of our drawing or photocollage.

The principle of "golden section" can be used when cropping the image. So, for example, a frame formed by the Golden section rule, from a large photo may have the following form.

Golden section in music. Golden section method in musical works

"Golden section" is a concept, rather, mathematical and study - the task of science. This division of some kind of two parts in such a relation when the biggest part is so relate to the smaller, as a whole. This ratio turns out to be equal to the transcendent number F \u003d 1.6180339 ... with amazing properties.

The gold section method is to search for values \u200b\u200bof the function on a given segment. This method is based on the principle of dividing the segment in the so-called gold proportion. It received the greatest distribution to search for extreme values \u200b\u200bwhen solving tasks related to optimization. In addition to mathematics, the gold section method is used in a variety of spheres, ranging from architecture, art and ending with astronomy. So, for example, the well-known Soviet director Sergei Eisenstein used him in his picture "Bremenos Potemkin", and Leonardo da Vinci - when writing them with the famous "Jokonda".

The gold section method is applied in music. It turned out that in the musical works, this golden proportion is very often found. At the beginning of the 20th century, a meeting was made at a meeting of the Moscow music mug containing information about which application finds a golden cross section in music. Member of the musical circle composers S. Rachmaninov, S. Taneyev, R. Gliere and others listened with great interest. Report of the musicologist Rosenova E.K. "The law of the golden section in music and poetry" laid the beginning of research on mathematical patterns associated with the gold proportion in music. He analyzed the musical works of Mozart, Bach, Beethoven, Wagner, Chopin, Glinka and other composers and showed that this "divine proportion" is present in their works.

The climax of many musical works is not located in the center, but a little shifted by the end of the work in the ratio of 62:38 - this is the point of the golden proportion. Doctor of Art History, Professor L. Mazel noted, studying the eight-dimensional melodies of Chopin, Beethoven, Scriabin, that in many creations of these composers, the climax, as a rule, falls on a weak fraction of the fifth, that is, to the point of the golden section - 5/8. L. Mazel believed that almost every composer was a harmonic style adherent to a similar musical structure: five closures of the lift and three clutches. This suggests that the golden section method was actively applied by composers consciously or unconsciously. It is likely that such a structural location of the climax of the climax gives the musical product a harmonic sound and emotional color.

A serious study of musical works for the manifestation of the golden proportion in them was taken by composer and musicologist L. Sabaneev. He studied about two thousand creations of various composers and came to the conclusion that at about 75% of cases, the golden cross section was present in the musical product at least once. The largest number of works in which the Golden Proportion is found, it noted in such composers as an amar (95%), Beethoven (97%), Haidn (97%), Mozart (91%), Scriabin (90%), Chopin ( 92%), Schubert (91%). He was most closely investigated by Etudes Chopin and came to the conclusion that the golden cross section was determined in 24 etudes from 27. Only in three etudes of Chopin Golden proportion was not discovered. Sometimes the structure of the musical product included simultaneously and symmetry, and the golden cross section. For example, Beethoven has many works are divided into symmetric parts, and in each of them a golden cross section is manifested.

So, it can be said that the presence of a golden section in a musical work is one of the criteria for harmony of the musical composition.

Anyone who, at least indirectly faced with the geometry of spatial objects in interior design and architecture, is probably well known for the principle of the golden section. More recently, several dozen years ago, the popularity of the golden section was so high that numerous supporters of mystical theories and the device of the world are called a universal harmonious rule.

Essence of universal proportion

Surprisingly different. The cause of biased, almost a mystical attitude towards such a simple numerical dependence was somewhat unusual properties:

A large number of objects of the living world, from the virus to humans, have basic proportions of the body or limbs, very close to the value of the golden section;
The dependence of 0.63 or 1.62 is characteristic only for biological creatures and some varieties of crystals, non-living objects, from minerals to landscape elements, have a golden section geometry extremely rarely;
Gold proportions in the structure of the body turned out to be the most optimal for the survival of real biological objects.

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

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

Usually, as an example, the Astreae Heliotropium sink device is given, one of the marine clams. The shell is a rolled spiral calcite shell with geometry, almost coinciding with the proportions of the golden section.

An ordinary chicken egg is more understandable and obvious example.

The ratio of the main parameters, namely, large and small focus, or distances from the equidalized points of the surface to the center of gravity, will also correspond to the Golden section. At the same time, the shape of a bird egg shell is the most optimal for the survival of the bird, as a biological species. At the same time, the strength of the shell plays far from a major role.

For your information! The gold cross section, called another universal proportion of geometry, was obtained as a result of a huge number of practical measurements and comparisons of the size of real plants, birds, animals.

Origin of universal proportion

Ancient Greek Mathematics Euclidean and Pythagoras knew about the golden proportion of the section. In one of the monuments of the ancient architecture - the Pyramid of Heops, the aspect ratio of the parties and bases, individual elements and wall bas-reliefs are made in accordance with the universal proportion.

The Methodology of the Golden section was widely used in the Middle Ages with artists and architects, while the essence of the universal proportion was considered one of the secrets of the Universe and was carefully hidden from a simple town man. The composition of many paintings, sculptures and buildings lined up strictly in accordance with the proportions of the golden section.

For the first time, the essence of the universal proportion was documented in 1509 g by the monk-Franciscanian Luke Pachet, who had brilliant mathematical abilities. But this recognition took place after conducting the German scientist to issue a comprehensive study of proportions and geometry of the human body, ancient sculptures, works of art, animals and plants.

Most of the living objects have some body sizes obey the same proportions. In 1855, scientists were concluded that the proportions of the gold section are a kind of standard of body harmony and form. It is, first of all, about living beings, for the dead nature, the golden cross section is much less common.

How got a golden cross section

The proportion of the golden section is easiest to imagine as the ratio of two parts of one object of different lengths, separated by a point.

Initially, gold proportions were derived empirically by geometric constructions. There are several ways to build or remove the harmonic proportion:

For your information! Unlike the classic gold relation, the architectural version implies the aspect ratio of the segment in the proportion of 44:56.

Main Secret of the Golden Section

If natural manifestations of universal cross section in the proportions of animal and human bodies, the storm base of plants can still be explained by evolution and adaptability to external environment, the opening of the golden section in the construction of houses of the XII-XIX century became a certain surprise. Moreover, the famous ancient Greek Parthenon was built in compliance with universal proportions, many houses and locks of wealthy nobles and wealthy people in the Middle Ages were consciously with respect to the parameters, very close to the golden cross section.

Golden section in architecture

Many of the buildings preserved until today's days show that the Middle Ages architects knew about the existence of a golden section, and, of course, during the construction of the house were guided by their primitive calculations and dependencies, with which they tried to achieve maximum strength. Especially the desire to build the most beautiful and harmonious houses in the construction of the residences of the reigning persons, churches, the town hall and buildings, which are of special social importance in society.

For example, the famous Paris Cathedral of Our Lady in their proportions has many areas and dimensional chains corresponding to the Golden section.

Even before the publication of its research in 1855, the famous architectural complexes of the Golitsyn hospital and the Senate Building in St. Petersburg, Pashkov and the Palace House in Moscow, were built at the end of the 18th century.

Of course, at home with accurate observance of the Rules of the Golden section built before. It is worth mentioning the monument to the ancient architecture of the church of the Pokrov on the nerve, depicted in the scheme.

All of them unite not only the harmonious combination of forms and high quality Construction, but also, first of all, the presence of a gold section in the proportions of the building. The amazing beauty of the building becomes even more mysterious, if you take into account the age, the building of the Pokrov Church dates back to the XIII century, but the modern architectural appearance has received the construction at the turn of the XVII century as a result of restoration and restructuring.

Golden Section Feature for Man

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

Individual artistic style in the design of facades avoids a modern stamp and serness, each building is a work of art;
Mass use for decorating and decorating statues, sculptures, stucco, unusual combinations of building solutions of different eras;
The proportions and composition of the building attract gaze to the most important elements of the building.

Important! When designing the house and developing the appearance, medieval architects used the Rule of the Golden section, unconsciously using the features of the perception of a person's subconscious.

Modern psychologists experimentally proved that the golden cross section is a manifestation of unconscious desire or a person's reaction to a harmonious combination or proportion in size, forms and even colors. An experiment was carried out, during which a group of people unfamiliar among themselves, who had no common interests, different professions and age categories, proposed a number of tests, among whom was the task of bending a sheet of paper in the most optimal proportion of the parties. According to the test results, it was found that in 85 cases out of 100 sheets, the tests were backed by almost exactly the golden cross section.

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

Using a factor of universal cross section in modern design and architecture

The principles of applying a gold proportion in the past few years have become unusually popular in the construction of private houses. The harmonicity of the design and the correct distribution of energy inside the house came to change the ecology and safety of building materials.

Modern interpretation of the rules of universal harmony has long been distributed beyond the limits of the usual geometry and the form of the object. Today, not only the size chains of the length of the portico and the front length, individual elements of the facade and the height of the building, but also the area of \u200b\u200bthe rooms, window and doorways, and even the color gamut of the indoor interior of the room are observed.

The easiest way to build a harmonious house on a modular basis. In this case, most departments and rooms are manufactured in the form of independent blocks or modules designed in compliance with the Rules of the Golden Section. Build a building in the form of a set of harmonious modules is much easier than to build one box, in which most of the facade and interior can be in the rigid framework of the proportions of the golden section.

Many construction companies performing the design of private households use the principles and concepts of the golden section to increase the estimates and creating an impression of the deep study of the construction of the house. As a rule, such a house is declared as very convenient and harmonious in use. The correctly selected ratio of the area of \u200b\u200bthe room guarantees the spiritual comfort and the excellent health of the owners.

If the house was built without taking into account the optimal ratios of the golden section, it is possible to redevelop the rooms so that the premises proportions correspond to the ratio of the walls in proportion 1: 1.61. To do this, furniture can be moved or additional partitions inside the rooms can be installed. Similarly, the dimensions of the window and door openings are changed so that the width of the opening is less than the height of the door canvase 1.61 times. In the same way, furniture, home appliances, wall and floor decoration are performed.

It is more difficult to choose a color design. In this case, instead of the usual ratio of 63:37, a simplified interpretation was adopted by the gold order for the gold order - 2/3. That is, the main color background should occupy 60% of the space of the room, the extension color gives no more than 30%, and the rest is given to various related tones, designed to strengthen the perception of the color solution.

The inner walls of the room are divided by a horizontal belt or a border at an altitude of 70 cm, the installed furniture should make measurement with the height of the ceilings at the ratio of the golden cross section. The same rule concerns the distribution of lengths, for example, the size of the sofa should not exceed 2/3 of the length of the simplest, and the total area occupied by furniture belongs to the room area, like 1: 1.61.

The gold proportion is difficult to use in practice due to just one section of the section, so when designing harmonious buildings, Fibonacci numbers are often resorted to a number. This allows you to expand the number. possible options proportions and geometric shapes of the main elements of the house. In this case, a number of Fibonacci numbers interconnected with a clear mathematical dependence are called harmonic or gold.

In the modern method of designing housing based on the principle of the Golden section, except for a number of Fibonacci, the principle proposed by the famous French architect Le Corbusier is widely used. In this case, as a starting unit of measurement, on which all the parameters of the building and the internal interior are calculated, the growth of the future owner or the average height of the person is selected. This approach allows us to design the house not only harmonious, but also in a true individual.

Conclusion

In practice, according to the reviews of those who decided to build a house according to the rule of the Golden section, a highly built building is really quite convenient for living. But the cost of the structure due to the individual design and application of building materials of non-standard sizes increases by 60-70%. And in this approach there is nothing new, since most of the buildings of the last century were based on the individual features of future owners.

Definition

The most capacious definition of the golden section states that a smaller part relates to a greater, as large for all. Its approximate value is 1,6180339887. In a rounded percentage value of the proportion of parts of the whole will be correlated as 62% by 38%. This ratio acts in the forms of space and time. Ancient saw in the golden section reflection of cosmic order, and Johann Kepler called him one of the treasures of geometry. Modern science considers the golden cross section as an "asymmetric symmetry", calling it in a broad sense by a universal rule reflecting the structure and order of our world order.

History

It is believed that the concept of gold division introduced into scientific Pythagoras, ancient Greek philosopher and mathematician (VI century BC). There is a suggestion that Pythagoras his knowledge of the golden division borrowed from Egyptians and Babylonian. And indeed, the proportions of the pyramid of Heops, temples, bas-reliefs, the objects of life and jewelry from the tombs of Tutankhamon show that the Egyptian masters used the ratios of gold division when creating them. The French architect le Corbüzenashel, which in the relief from the pharaoh of Network I in ABIDOS and in the relief, depicting the Pharaoh Ramses, the proportions of the figures correspond to the values \u200b\u200bof gold division. The architect hashire depicted on the relief of a wooden board from the tomb of his name, keeps in the hands of measuring instruments in which the proportions of gold division are fixed.

The Greeks were skilled geometers. Even arithmetic trained their children with geometric figures. The square of Pythagora and the diagonal of this square were the basis for building dynamic rectangles.

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

In the facade of the ancient Greek temple of Parfenon there are golden proportions. With its excavations, the circuses were found, which were used by architects and sculptors of the ancient world. In Pompeary Circle (Museum in Naples), the proportions of gold division are also laid.

Fig. Antique Circle Golden Section

In the antique literature that came to us, the gold division was first mentioned in the "Beginning" Euclida. In the 2nd book "Beginning" gives a geometric construction of gold division. After Euclide, the Golden Delival study was engaged in the gypsy (II century BC), PAPP (III century. AD) and others. In medieval Europe with gold division, they met the Arab translation "began" Euclida. Translator J. Campano from Navarre (III century) Made to transfer comments. The secrets of the golden division were jealously worried, stored in a strict mystery. They were known only dedicated.

The idea of \u200b\u200bthe golden proportions was in Russia, but for the first time a scientific golden cross section explained monk Luka Pacholi. In the book "Divine proportion" (1509), the illustrations to which Leonardo da Vinci allegedly did. Pachet saw divine trinity in the golden section: a small segment personified his son, a big father, and the whole - the Holy Spirit. According to contemporaries and historians of science, Luka Pacheti was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileem. Luka Pacheli was a student of the artist Piero della Franni, who wrote two books, one of which was called "about the future in painting." He is considered the creator of the descriptive geometry.

Luka Pacheti perfectly understood the importance of science for art. In 1496, at the invitation of the Duke of Moro, he comes to Milan, where he lectures in mathematics. In Milan, at the courtyard of Mora, Leonardo da Vinci worked at that time.

Directly with the Rule of the Golden section associated with the name of Italian mathematics Leonardo Fibonacci. As a result of a solution to one of the tasks, the scientist entered the sequence of numbers now known as a number of Fibonacci: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. At the ratio of this sequence to the golden proportion, the Kepler drew attention to: "It works so that two younger members of this endless proportion in the amount give the third dick, and any two of the last member, if they are folded, give the following member, and the same proportion is preserved to infinity " Now Fibonacci series is an arithmetic basis for calculating the proportions of the golden section in all its manifestations.

Leonardo da Vinci Also, he devoted a lot of time to study the peculiarities of the golden section, most likely it belongs to the term itself. Its drawings of the stereometric body formed by the correct pentagons prove that each of the rectangles obtained during the cross section gives the ratio of the parties in the gold division.

Over time, the Rule of the Golden section turned into an academic routine, and only a philosopher Adolf Ceying In 1855 he returned his second life. He brought to the absolute of the proportion of the Golden section, making them versatile for all the phenomena of the surrounding world. However, his "mathematical aesthetics" caused a lot of criticism.

Nature

Astronomer XVI century. Johann Kepleler. called the golden section by one of the treasures of geometry. It first draws attention to the value of the golden proportion for botany (plant growth and their structure).

Kepler called the gold proportion to continuing himself "it works so," he wrote, - that two younger member of this endless proportion in the amount give the third dick, and any two last member, if they are folded, give the following member, and the same proportion is preserved Infinity. "

The construction of a series of segments of the golden proportion can be made both towards the increase (increasing row) and in the direction of the reduction (downward row).

If on a straight arbitrary length, postpone the segment m., Next lay off the segment M.. Based on these two segments, we build the scale of segments of the gold proportion of the ascending and downward rows.

Fig. Building a gold proportion segment scale

Fig. Chicory

Even without going into calculations, the golden cross section can be easily discovered in nature. So, the ratio of tail and body lizard fall under it, the distance between the leaves on the branch, there is a golden cross section and in the shape of an egg, if the conditional line is carried out through its widest part.

Fig. Lizard niphelistic

Fig. Egg bird

Yet Archimedes, paying attention to the helix, brought the equation based on its form, which is now applied in the technique. Later Goethe noted the nature of nature to spiral forms, calling spiral "Curvation of Life". Modern scientists were established that such manifestations of spiral forms in nature as a snail shell, sunflower seed location, web patterns, hurricane movement, structure of DNA and even the structure of galaxies enter into a series of fibonacci.

Human

Fashion designers and designers of clothing All calculations are made based on the proportions of the golden section. Man is a universal form to verify the laws of the golden section. Of course, from nature, not all people have the proportions are ideal, which creates certain difficulties with the selection of clothing.

In the diary, Leonardo da Vinci has a picture inscribed in the circumference of a naked person, located in two positions superimposed on each other. Relying on the studies of the Roman architect Vitruvia, Leonardo was similar way to establish the proportions of the human body. Later, the French architect Le Corbusier, using the "Vitruvian man" Leonardo, created his own scale of "harmonic proportions", which influenced the aesthetics of the architecture of the XX century. Adolf Ceying, exploring the proportionality of a person, has done a colossal work. He measured about two thousand human bodies, as well as many ancient statues and led that the golden cross section expresses the average law. In a person, almost all parts of the body are subordinated to him, but the main indicator of the golden section is the division of the body of the Pup point.

As a result of measurements, the researcher found that the proportions of the male body 13: 8 closer to the golden section than the proportions of the female body - 8: 5.

Art of spatial shapes

The artist Vasily Surikov said "that there is an immutable law in the composition, when there is nothing to remove anything in the picture, neither add, even the extra point is impossible, this is a real mathematics." For a long time, the artists of the investigators of this law are intuitive, but after Leonardo da Vinci, the process of creating a picturesque canvase is no longer necessary without solving geometric tasks. For example, Albrecht Dürer To determine the points of the golden section used the proportional circulation invented.

Art historian F. V. Kovalev, examining in detail the picture of Nikolai Ge "Alexander Sergeevich Pushkin in the village of Mikhailovsky," notes that every detail of the canvas whether a fireplace, shelf, chair or poet himself are strictly inscribed in golden proportions. The investigors of the golden section without tired and measured the masterpieces of architecture, claiming that they became such because they were created on gold canons: in their list, the Great Pyramids Giza, the Cathedral of the Parisian Mother of God, the Temple of Basil Blessed, Parfenon.

And today, in any art, the spatial forms are trying to follow the proportions of the golden section, since they, according to art historians, facilitate the perception of the work and form an aesthetic feeling at the viewer.

Goethe, poet, naturalist and artist (he painted and wrote watercolor), dreamed of creating a unified teaching on the form, education and transformation of organic bodies. This he introduced the term in a scientific way morphology.

Pierre Curie at the beginning of our century formulated a number of deep ideas of symmetry. He argued that it was impossible to consider the symmetry of any body without taking into account the symmetry of the environment.

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

Golden section and symmetry

The golden cross section cannot be viewed in itself, separately, without a symmetry. Great Russian crystallograph of G.V. Wulf (1863 ... 1925) considered the golden section by one of the manifestations of symmetry.

The gold division is not a manifestation of asymmetry, something opposite symmetry. According to modern ideas, the gold division is asymmetric symmetry. The science of symmetry includes such concepts as static and dynamic symmetry. Static symmetry characterizes peace, equilibrium, and dynamic - movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art characterizes peace, equilibrium and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characteristic of equal segments, equal values. The dynamic symmetry is typical of the increase in segments or their decrease, and it is expressed in the values \u200b\u200bof the golden section of an increasing or decreasing range.

Word, sound and film

Forms of temporary art in their own way demonstrate to us the principle of golden division. Literary critics, for example, noticed that the most popular number of lines in the poems of the late period of Pushkin's creativity corresponds to the Fibonacci row - 5, 8, 13, 21, 34.

There is a Rule of the Golden section and in separately taken works of Russian classics. So the climax of the "peak lady" is the dramatic scene of Herman and Countess, ending with the death of the latter. In the lead of 853 lines, and the climax accounts for 535 row (853: 535 \u003d 1.6) - this is the point of the golden section.

The Soviet musicologist E. K. Rosenov notes the striking accuracy of the ratios of the golden section in the strict and free forms of the works of Johanne Sebastian Baha, which corresponds to a thoughtful, concentrated, technically verified style of the master. This is true in relation to the outstanding creations of other composers, where the most bright or unexpected musical solution is usually accounted for on the golden section.

Film director Sergei Eisenstein Scenario of his film "Bramenos Potemkin" consciously agreed with the Rule of Golden Section, dividing the tape for five parts. In the first three sections, the action unfolds on the ship, and in the last two - in Odessa. Go to the scene in the city and there is a golden middle of the film.

We invite you to discuss the topic in our group -

Each person coming across the geometry of objects in space is well familiar with the method of golden section. It is used in art, interior design and architecture. Last century, the golden cross section turned out to be so popular that now many supporters of mystical vision of the world gave him another name - a universal harmonious rule. The features of this method should be considered more. This will help you to find out why he enjoys interest in several spheres of activity - art, architecture, design.

The essence of the universal proportion

The principle of the golden section is just the dependence of the numbers. However, many treat him biased, attributing to this phenomenon some kind of mystical forces. The reason lies in the unusual properties of the rule:

Many living facilities have the proportions of the body and limbs approximate to the testimony of the golden section.
The dependences of 1.62 or 0.63 define the ratios of the size only for living beings. Objects relating to inanimate nature, very rarely correspond to the value of the harmonic rule.
The gold proportions of the structure of the body of living beings are an integral condition for the survival of many biological species.

The golden section can be found in the structure of the bodies of various animals, tree trunks and shrubs roots. Supporters of the universality of this principle are trying to prove that its meanings are vital for representatives of the living world.

You can explain the gold section method using the image of a chicken egg. The ratio of segments from the shell points, equally deleted from the center of gravity, is equal to the golden cross section. The most important for the survival of the birds with an egg indicator is its form, and not the strength of the shell.

Important! The gold cross section is designed based on measurements of many living objects.

The origin of the golden section

About universal rule was known to mathematic Ancient Greece. She was used by Pythagoras and Euclide. In the famous architectural masterpiece - Hope's pyramid, the ratio of the size of the main part and the length of the parties, as well as bas-reliefs and decorative parts correspond to the harmonic rule.

The golden section method was adopted not only architects, but also artists. The mystery of the harmonic proportion was considered one of the greatest mysteries.

The first, documented universal geometric proportion, was the Franciscan monk of Luka Pachet. His ability to mathematics were brilliantly. A wide recognition of the golden cross section received after the publication of the results of studies of the golden section of the procession. He studied human body proportions, ancient monuments of sculpture, plants.

How to calculate the golden cross section

To figure out what a golden cross section will help an explanation based on the lengths of segments. For example, there are several small inside. Then the lengths of small segments belong to the total length of a large segment, like 0.62. Such a definition helps to figure out how many parts can be divided by a certain line so that it corresponds to the harmonic rule. Another plus use of this method - you can find out what should be the ratio of the largest segment to the length of the entire object. This ratio is 1.62.

Such data can be represented as the proportions of the measured objects. At first they were looking for, selecting the experienced way. However, now accurate ratios are known, so build an object in accordance with them will not be difficult. The golden cross section finds so ways:

Build a rectangular triangle. Split one of his sides, and then hold perpendiculars with securable arcs. When calculating, it follows from one end of the segment to construct a perpendicular equal to ½ of its length. Then the rectangular triangle is then completed. If you note the point on hypotenuse, which will show the length of the perpendicular segment, then the radius equal to the remaining part of the line will dissect the base for two halves. The resulting lines will relate to each other according to the golden cross section.
Universal geometric values \u200b\u200bare also obtained by another way - building Durera's 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 cross section.
In architecture, harmonic proportion is applied in a modified form. To do this, the rectangular triangle should be broken down by hypotenuse.

Important! If we compare with the classic concept of the gold section method, the architect 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 for architectural structures, 44:56 were used more often. This is due to the need to build high-rise buildings.

The secret of the golden section

If, in the case of alive objects, a golden cross section, manifested in the proportions of the body of people and animals, can be explained by the need to adapt to the environment, then in the use of the rule of optimal proportions in the 12th century for the construction of houses was new.

Parthenon, preserved since the times of ancient Greece, was erected by the method of the Golden section. Many castles of the melief of the middle ages were created with parameters corresponding to the harmonic rule.

Golden section in architecture

A variety of antiquity buildings that have been preserved still serve as a confirmation of the fact that architects from the Middle Ages were familiar with the harmonious rule. A very well noticeable desire to keep the harmonic proportion in the construction of churches, significant public buildings, the Residence of the Royal Persians.

For example, the Cathedral of the Paris's Mother of God was erected in such a way that many of his sites correlate with the Rule of Golden Section. You can find a lot of works of the 18th century architecture, which were built in harmony with this rule. Many Russian architects also applied the rule. Among them was M. Cossacks, who created projects of the estates and residential buildings. He designed the building of the Senate and the Golitsyn hospital.

Naturally, at home with such respects, parts were erected before the opening of the Rules of the Golden section. For example, such buildings include the church of the Intercession to Nerli. The beauty of the building acquires even greater mystery, considering that the building of the Pokrovskaya Church was erected in the XVIII century. but modern view Building acquired after restoration.

In the works about the gold section, it is mentioned that in the architecture, the perception of objects depends on who observes. The proportions formed with the help of a gold section give the most calm ratio of the parts of the structure relative to each other.

A bright representative of a number of buildings corresponding to the universal rule is the architectural monument to Parthenon, erected back in the fifth century BC. e. Parthenon is arranged with eight columns on smaller facades and seventeen - along large. The temple is erected from noble marble. Due to this, the use of coloring is limited. The height of the structure refers to its length of 0.618. If separating the parfenon according to the proportions of the golden section, certain protrusions of the facade will be obtained.

All these facilities have one similarity - the harmonicity of the combination of forms and the excellent quality of construction. This is explained by the use of the harmonic rule.

The importance of the golden section for man

The architecture of ancient buildings and medieval houses is quite interesting for modern designers. This is explained by such reasons:

Thanks to the original design of houses, you can prevent the boring stamps. Each such building is an architectural masterpiece.
Mass application Rules for decorating sculptures and statues.
Due to the observance of harmonic proportions, the look is attracted to more important details.

Important! When creating a project for the construction and creating an appearance, the architects of the Middle Ages were used by universal proportions, based on the patterns of human perception.

Today, psychologists concluded that the principle of the golden section is nothing but a human reaction to a certain ratio of dimensions and forms. In one experiment, the test group was offered to bend the paper sheet in such a way that the parties are turned out to be optimal proportions. In 85 results from 100 people flex a sheet in almost exactly according to the harmonic rule.

According to modern scientists, the indicators of the golden section are more likely to the sphere of psychology, rather than characterize the patterns of the physical world. This explains why such interest on the side of the mystorts is manifested. However, when building objects according to this rule, a person perceives them more comfortably.

Using a golden section 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 design. A lot of attention is paid to the correct allocation of attention within the house.

The modern interpretation of the golden section is no longer applies only to the rules of geometry and forms. Today, the principle of harmonic proportions is subject not only to the size of the details of the facade, the area of \u200b\u200bthe rooms or the length of the frontones, but also the color palette used when creating the interior.

Making a harmonious structure on a modular base is much simpler. Many branches and premises in this case are performed as separate blocks. They are designed in strict accordance with the harmonic rule. Building a building as a set of individual modules is much easier than creating a single box.

Many firms involved in the construction of country houses, when creating a project, keep the harmonious rule. This allows you to create customers an impression that the building design has been worked out in detail. Such houses usually describe how the most harmonious and comfortable in use. With the optimal choice of the area of \u200b\u200bthe rooms, the tenants psychologically feel soothing.

If the house is erected without the number of harmonic proportions, you can create a layout that will be approximated by the ratio of the walls 1: 1.61. For this, additional partitions are installed in the rooms, or the furniture items are rearranged.

Similarly, the dimensions of the doors and windows change in such a way that the opening is to have a width, the indicator of which is less than the height value of 1.61 times.

It is more difficult to select color solutions. In this case, you can observe the simplified value of the golden section - 2/3. The main color background should take 60% of the room space. The extensive shade takes 30% of the room. The remaining surface area is painted close to each other with tones that enhances the perception of the selected color.

The inner walls of the rooms divide the horizontal strip. It is 70 cm from the floor. The height of the furniture should be in a harmonic ratio with the height of the walls. This rule refers to the length distribution. For example, the sofa must have dimensions that would be no less than 2/3 of the length of the simplest. The area of \u200b\u200bthe room, which is occupied by furniture objects, should also have a certain meaning. It refers to K. common Square Total rooms as 1: 1.61.

Gold proportion is difficult applicable in practice due to the presence of only one number. That's why. I design harmonious buildings, use a number of Fibonacci numbers. This ensures a variety of options for the forms and proportions of the details of the structure. A number of Fibonacci numbers are also called gold. All values \u200b\u200bstrictly correspond to a certain mathematical dependence.

Except for a row of Fibonacci, in modern architecture Another design method is used - the principle laid down by the French architect Le Corbusier. When this method is selected, the starting unit of measurement is the growth of the owner of the house. Based on this indicator, the size of the building and interior is calculated. Thanks to this approach, the house is not only harmonious, but also acquires individuality.

Any interior will acquire a more completed look if you use eaves. When using universal proportions, it is possible to calculate its size. The optimal indicators are 22.5, 14 and 8.5 cm. Install the corzeal follows according to the rules of the golden section. The small side of the decorative element should relate to the greater as it belongs to the folded values \u200b\u200bof both sides. If the big party is equal to 14 cm, then the small one should take 8.5 cm.

It is possible to give the cozy room by dividing wall surfaces using plaster mirrors. If the wall is divided by a border, from the remaining majority of the wall should be taken away the height of the cornisa plank. To create a mirror of the optimal length from the border and the eaves, you should retreat the same distance.

Conclusion

Houses built on the principle of the golden section are really obtained very convenient. However, the price of building such buildings is quite high, since the cost of building materials due to atypical sizes is increased by 70%. This approach is not completely new, as most of the houses of the last century created based on the parameters of the hosts.

Through the use of the Golden section method in the construction and design of the building, not only comfortable, but also durable. They look harmonious and attractive. The interior is also drawn up by universal proportion. This allows you to competently use space.

In such rooms, a person feels as comfortable as possible. Build a house using the principle of the golden section can be independently. The main thing is to calculate the load on the elements of the structure, and choose the correct materials.

The golden section method is used in the interior design, placing decorative elements of certain sizes in the room. This allows you to give the room to the coziness. Color solutions are also chosen in accordance with universal harmonic proportions.

A person distinguishes the surrounding items in shape. Interest in the form of a subject can be dictated by the vital necessity, and may be caused by the beauty of the form. The form, based on the construction of which is a combination of symmetry and the golden section, contributes to the best visual perception and the appearance of the feeling of beauty and harmony. The integer always consists of parts, part of different values \u200b\u200bare in a certain respect 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 (Lat. Proportio) call the equality of two relationships: a. : b. = c. : d..

Cut straight AU You can divide into two parts in the following ways:

into two equal parts - AU : AC = AU : Sun;

on two unequal parts in any respect (such parts of the proportion are not formed);

so when AU : AC = AC : Sun.

Last and there is a gold division or division of a segment in extreme and middle.

The gold cross section is such a proportional division of the segment to unequal parts, in which the entire segment belongs to the most part, as most of the most relate to the smaller; or in other words, a smaller cut is so relate to more as bigger to everything

a. : b. = b. : c. or from : b. = b. : but.

Fig. one. Geometric image of golden proportion

A practical acquaintance with the golden section starts from the division of a straight line in a gold proportion with a circulation and a ruler.

Fig. 2. The division of the segment is straight along the golden section. BC. = 1/2 AB; CD = BC.

From the point IN Regenerated perpendicular equal to half AU. Point received FROM connected line with a point BUT. A segment is postponed on the resulting line Sunending in point D.. Section AD It is transferred to straight AU. The point obtained E. divide cut AU In the ratio of the golden proportion.

Segments of gold proportions are expressed by an endless irrational fraction AE \u003d 0.618 ... if AU take a unit Ve \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If cut AU To adopt for 100 parts, most of the segment is 62, and the smaller - 38 parts.

The properties of the gold cross section are described by the equation:

x. 2 - x. - 1 = 0.

Solution of this equation:

The properties of the golden section created around this number a romantic halo of mysteriousness and almost a mystical worship.

Second golden cross section

The Bulgarian Magazine "Fatherland" (№10, 1983) published the article Color Tsekova-pencil "On the second gold section", which follows from the main cross section and gives another ratio of 44: 56.

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

Fig. 3. Building a second golden section

The division is as follows (see Fig.3). Section AU It is divided into the proportion of the golden section. From the point FROM Regenerated perpendicular Cd.. Radius AU There is a point D.which connects the line with a point BUT. Right angle ACD shares in half. From the point FROM Line is held before crossing the line AD. Point E. divide cut AD In terms of 56: 44.

Fig. four. The division of the rectangle of the line of the second golden section

In fig. 4 shows the position of the line of the second golden section. It is in the middle between the line of the golden section and the middle line of the rectangle.

Golden Triangle

To find segments of the gold proportion of the ascending and downward rows, you can use pentagram.

Fig. five. Building the right pentagon and pentagram

To build a pentagram, it is necessary to build the right pentagon. The way to build it developed a German painter and chart of Albrecht Dürer (1471 ... 1528). Let be O. - center of the circle, A. - point on circle and E. - Middle Cut OA. Perpendicular to radius OAdot ABOUTintersects with a circle at point D.. Using a circulation, postpone on the diameter of the segment CE = ED. The length of the side inscribed in the circumference of the correct pentagon is equal to DC. Put on the circumference of the segment DC And we get five points to draw the right pentagon. We connect the corners of the pentagon through one diagonals and get a pentagram. All diagonals of the pentagon shall form each other for segments related to the golden proportion.

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

Fig. 6. Building a golden triangle

We carry out straight AU. From the point BUT lay on it three times the segment ABOUT arbitrary magnitude through the resulting point R We carry out perpendicular to the line AU, perpendicular to the right and left from the point R lay off the segments ABOUT. Points received d. and d. 1 Connect straight with a point BUT. Section dD. 1 postpone on line AD 1, getting a point FROM. She divided the line AD 1 in the proportion of the golden section. Lines AD 1 I. dD. 1 Use to build a "golden" rectangle.

History of golden section

It is believed that the concept of the golden division introduced the scientific use of Pythagoras, an ancient Greek philosopher and mathematician (VI century. BC). There is an assumption that Pythagoras his knowledge of the golden division borrowed from Egyptians and Babylonians. And indeed, the proportions of the pyramid of Heops, temples, bas-reliefs, the objects of life and jewelry from the tombs of Tutankhamon show that the Egyptian masters used the ratios of gold division when creating them. The French architect Le Corbusier found that in the relief from the Temple of the pharaoh of Network I in ABIDOS and in the relief, depicting the Pharaoh Ramses, the proportions of the figures correspond to the values \u200b\u200bof the golden division. The architect Hesier, depicted on the relief of a wooden board from the tomb of his name, keeps in the hands of measuring instruments in which the proportions of the gold division are fixed.

The Greeks were skilled geometers. Even arithmetic trained their children with the help of geometric shapes. The square of Pythagora and the diagonal of this square were the basis for building dynamic rectangles.

Fig. 7. Dynamic rectangles

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

Fig. eight. Antique Circle Golden Section

In the antique literature, the gold division was first mentioned in the "beginning of" Euclide. In the 2nd book "Beginning", the geometric construction of gold division is given after Euclidean study of the golden division were engaged in the gypsy (II century BC), PAPP (III century. N.E.) and others. In medieval Europe with golden division I got acquainted on the Arab translation "beginning" Euclida. Translator J. Campano from Navarre (III century) Made to transfer comments. The secrets of the golden division were jealously worried, stored in a strict mystery. They were known only dedicated.

In the era of the Renaissance, interest in the gold division among scientists and artists in connection with its use both in geometry and in art, especially in the architecture of Leonardo da Vinci, an artist and a scientist, saw that Italian artists have a big empirical experience, and there are few knowledge . He conceived and began writing a book on geometry, but at that time the book of the monk Luke Pacheti appeared, and Leonardo left his venture. According to contemporaries and historians of science, Luka Pacheti was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileem. Luka Pacheli was a student of the artist Piero della Franni, who wrote two books, one of which was called "about the future in painting." He is considered the creator of the descriptive geometry.

Luka Pacheti perfectly understood the importance of science for art. In 1496, at the invitation of the Duke Moro, he comes to Milan, where he lectures in mathematics. In Milan, at the courtyard of Mora, Leonardo da Vinci worked at that time. In 1509, in Venice, the book of Luke Pachet "Divine Proportion" was published with brilliantly fulfilled illustrations, due to what Leonardo da Vinci did them. The book was an enthusiastic anthem of the golden proportion. Among the most advantages of the golden proportion to the monk of Luka Pacholi did not fail to call her "Divine Essence" as the expression of the divine trinity God Son, God Father and God's Holy Spirit (I understood that the small segment is the personification of God's Son, a bigger segment - God Father, and the whole cut - God of the Holy Spirit).

Leonardo da Vinci also paid a lot of attention to studying the golden division. He produced a sequence of a stereometric body formed by the right pentagons, and each time he received rectangles with the relations of the parties in the gold division. So he gave this division name golden cross section. So it still keeps the most popular.

At the same time in the north of Europe, in Germany, Albrecht Durer worked on the same problems. It makes an outline of introduction to the first variant of the treatise on the proportions. Durer writes. "It is necessary that one who knows anything, taught this other who need it. That I am removed to do. "

Judging by one of the letters of Durera, he met with Luke Pachet during his stay in Italy. Albrecht Dürer develops in detail the theory of the proportions of the human body. An important place in its system relations Durer took the golden cross section. The growth of a person is divided into the golden proportions of the belt line, as well as a line spent through the tips of the middle fingers of lowered hands, the lower part of the face - mouth, etc. The proportional durer circuit is known.

Great astronomer XVI century. Johan Kepler called the Golden section by one of the treasures of geometry. It first draws attention to the value of the golden proportion for botany (plant growth and their structure).

Kepler called the gold proportion continuing himself "it works so," he wrote, - that two younger members of this endless proportion in the amount give the third dick, and any two last member, if they are folded, give the following member, and the same proportion remains to Infinity. "

The construction of a series of segments of the golden proportion can be made both towards the increase (increasing row) and in the direction of the reduction (downward row).

If on a straight arbitrary length, postpone the segment m., Next lay off the segment M.. Based on these two segments, build the scale of segments of the gold proportion of the ascending and downward rows

Fig. nine. Building a gold proportion segment scale

In the next centuries, the golden proportion rule turned into an academic canon and, when in the art, the struggle against academic routine began in art, in the heat of struggle "and the child was splashing with water." The golden cross section was "openly" again in the middle of the XIX century. In 1855, the German investigator of the Golden section, Professor Zeasing, published his work "aesthetic research." It was exactly what was happening with Cainling, which was to inevitably happen to occur with the researcher, who considers the phenomenon as such, without communication with other phenomena. He absolutized the proportion of the Golden section, declaring it universal for all the phenomena of nature and art. Casesing had numerous followers, but there were opponents who declared his doctrine on the proportions of "mathematical aesthetics".

Fig. 10. Golden proportions in human body parts

Ceasing has done a colossal job. He measured about two thousand human bodies and came to the conclusion that the golden cross section expresses the average statistical law. Body division Pup point is the most important indicator of the golden section. The proportions of the male body fluctuate within the middle ratio of 13: 8 \u003d 1.625 and are somewhat closer to the golden section than the proportions of the female body, in respect of which the average value of the proportion is expressed in the ratio of 8: 5 \u003d 1.6. In a newborn, the proportion is ratio of 1: 1, by 13 years it is equal to 1.6, and by 21 years equals male. The proportions of the golden section appear in relation to other parts of the body - the length of the shoulder, forearm and brushes, brushes and fingers, etc.

Fig. eleven. Golden proportions in a person figure

The justice of its theory Ceasing was checked in Greek statues. In the most detailed, he developed the proportions of Apollo Belvedere. We have been studied by Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, poetic sizes. Cainling gave the definition of the golden section, showed how it is expressed in the segments of direct and in numbers. When the numbers expressing the lengths of the segments were obtained, Cainsing saw that they constitute a number of fibonacci, which could be continued to infinity in one and the other side. The next book was the name "Golden division as a major morphological law in nature and art." In 1876, a small book was published in Russia, almost a brochure, with the statement of this work of Ceasing. The author covered under the initials Yu.F.V. In this edition, no product of painting is not mentioned.

IN late XIX. - early XX centuries. A lot of purely formalistic theories appeared on the use of a gold section in works of art and architecture. With the development of design and technical aesthetics, the law of the Golden section has spread to the design of machines, furniture, etc.

Fibonacci row

With the history of the Golden section indirectly, the name of the Italian mathematics of the monk Leonardo from Pisa, more famous under the name Fibonacci (son Bonachci). He traveled a lot in the east, introduced Europe with Indian (Arabic) figures. In 1202, his mathematical work "Book On Abak" (counting board) was published, in which all the tasks known at that time were collected. One of the tasks read "How many rabbit pairs in one year from one couple will be born." Reflecting on this topic, Fibonacci lined up such a series of numbers:

A number of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Known as a number of fibonacci. The peculiarity of the sequence of numbers is that each member, starting from the third, is equal to the sum of the two previous 2 + 3 \u003d 5; 3 + 5 \u003d 8; 5 + 8 \u003d 13, 8 + 13 \u003d 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers of the series is approaching the ratio of gold division. So, 21: 34 \u003d 0.617, and 34: 55 \u003d 0.618. This ratio is indicated by the symbol. F.. Only this ratio is 0.618: 0.382 - it gives continuous division of a straight line in a gold proportion, an increase in it or a decrease to infinity, when a smaller cut does so relate to more as greater than everything.

Fibonacci also dealt with the decision of the practical needs of the trade: With which the smallest number of Giri, you can weigh the goods? Fibonacci proves that the optimal is such a system of Irrows: 1, 2, 4, 8, 16 ...

Generalized golden cross section

Fibonacci series could remain only a mathematical incident, if not the fact that all researchers in the golden division in the plant and in the animal world, not to mention the art, invariably came to this row as arithmetic expression Law of golden division.

Scientists continued to actively develop the theory of Fibonacci numbers and a golden section. Yu. Matyatsevich using Fibonacci numbers solves the 10th problem of Hilbert. Elegant methods of solving a number of cybernetic tasks (search theory, games, programming) using Fibonacci and Golden Section are arising. Even Mathematical Fibonachchi-Association is created in the USA, which since 1963 produces a special magazine.

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

Fibonacci row (1, 1, 2, 3, 5, 8) and the Outlook the same "binary" range of Girric 1, 2, 4, 8, 16 ... At first glance, completely different. But their construct algorithms are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 \u003d 1 + 1; 4 \u003d 2 + 2 ..., in the second - this is the sum of two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... whether it is impossible to find the general mathematical formula from which it turns out and " binary "row, and a number of fibonacci? Or maybe this formula will give us new numeric sets with some new unique properties?

Indeed, set the numeric parameter S.which can take any values: 0, 1, 2, 3, 4, 5 ... Consider a numeric row, S. + 1 of the first members of which are units, and each of the following is equal to the sum of the two members of the previous and separating from the previous one S. Steps. If a n.-y member of this series we denote by φ s ( n.), then get general formula φ s ( n.) \u003d φ s ( n. - 1) + φ s ( n. - S. - 1).

Obviously, when S. \u003d 0 From this formula, we get a "binary" series, when S. \u003d 1 - Fibonacci row, with S. \u003d 2, 3, 4. New rows of numbers that were called S.- Located Fibonacci.

IN general Golden S.-Pages there is a positive root of the gold equation S.-sectation x s + 1 - x s - 1 \u003d 0.

It is easy to show that when S. \u003d 0 It turns out the division of the segment in half, and when S. \u003d 1 -Baby classic gold cross section.

Relationships of neighboring S.- Located fibonacci with absolute mathematical accuracy coincide in the limit with gold S.-Porporations! Mathematics in such cases say gold S.-Schens are numeric invariants S.- Located Fibonacci.

Facts confirming the existence of gold S.-sens in nature, leads the Belarusian scientist E.M. Soroka in the book "Structural Harmony Systems" (Minsk, "Science and Technology", 1984). It turns out that, for example, that well-studied double alloys have special, pronounced functional properties (thermally resistant, solid, wear-resistant, resistant to oxidation, etc.) only if the specific gravity of the source components are connected to each other of one of Golden S.-Pages. This allowed the author to put forward the hypothesis that gold S.-Engthening there are numeric invariants of self-organizing systems. Being confirmed experimentally, this hypothesis may have a fundamental importance for the development of synergetics - a new field of science learning the processes in self-organizing systems.

Using gold codes S.-Pages can be expressed by any real number in the form of the amount of the Golden degrees S.-Pages with whole coefficients.

The fundamental difference of this method of encoding the numbers is that the foundations of new codes that are gold S.-Proids, S. \u003e 0 turn out to be irrational numbers. Thus, new surgery systems with irrational bases as it were "from head to feet" historically established hierarchy of relations between the numbers rational and irrational. The fact is that at first there were "open" numbers of natural; Then their relationships are rational numbers. And only later - after opening with Pythagorians, irrational numbers appeared, irrational numbers appeared. For example, in decimal, five-hour, binary and other classical positional systems, natural numbers were chosen as a kind of primary source - 10, 5, 2, from which all other natural, as well as rational and irrational numbers were constructed.

A kind of alternative to existing methods of note is a new, irrational system, as a primary source, the start of the number of which is chosen an irrational number (which is reminding the root of the golden section equation); Other valid numbers are already expressed through it.

In such a number system any natural number Always represent the ultimate - and not infinite, as they thought before! - the amount of degrees of any of the gold S.-Pages. This is one of the reasons why the "irrational" arithmetic, possessing an amazing mathematical simplicity and grace, as if absorbed the best qualities of the classic binary and "fibonaccia" arithmetic.

Forming principles in nature

Everything that has acquired some kind of form has been formed, grew, sought to take a place in space and keep themselves. This desire finds implementation mainly in two versions - growth up or pouring on the surface of the earth and spiral spiral.

The shell is spinled on the helix. If it is deployed, it turns out the length, a little inferior to the length of the snake. A small ten-centimeter shell has a spiral of 35 cm long. The spirals are very common in nature. The idea of \u200b\u200bthe golden section will be incomplete if not to say about the spiral.

Fig. 12. Spiral Archimedes

The shape of spirally curled shell attracted the attention of Archimedes. He studied it and brought the spiral equation. The spiral drawn through this equation is called his name. The increase in its step is always evenly. Currently, the Archimeph Spiral is widely used in the technique.

Gethete also emphasized the trend of nature to spiral. The screw and spiral arrangement of the leaves on the branches of the trees was noticed for a long time. The spiral saw in the location of sunflower seeds, in pine cones, pineapples, cactus, etc. The joint work of nerds and mathematicians shed light on these amazing phenomena of nature. It turned out that in the location of the leaves on the branch (phyotaxis), the seeds of sunflower, the pine cones show itself a number of fibonacci, and therefore, the law of the golden section is manifested. Spider rods spiral spiral. Hurricane spiral spiral. A frightened flock of reindeer is running around the spiral. DNA molecule is twisted with a double helix. Goethe called the spiral of the "curve of life."

Among the roadside herbs there is no remarkable plant - chicory. I look at it carefully. From the main stem, the process was formed. Immediately the first sheet is located.

Fig. 13. Chicory

The process makes a strong release into space, stops, produces a sheet, but already shorter than the first, again makes a release into space, but already less power, releases a leaflet of even smaller size and emissions again. If the first emission is taken for 100 units, the second is 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subordinate to the golden proportion. In growth, the conquest of space, the plant retained certain proportions. The impulses of its growth gradually decreased in the proportion of the Golden Section.

Fig. fourteen. Lizard niphelistic

In a lizard at first glance, pleasant for our eye proportion - the length of her tail is as follows to the length of the rest of the body, like 62 to 38.

Both in the plant, and in the animal world persistently breaks through the formative tendency of nature - symmetry relative to the direction of growth and movement. Here, the golden cross section is manifested in the proportions of parts perpendicular to the direction of growth.

Nature made division into symmetric parts and gold proportions. In parts manifests the repetition of the structure of the whole.

Fig. fifteen. Egg bird

Great Goethe, poet, naturalist and artist (he painted and wrote watercolor), dreamed of creating a unified teaching on the form, education and transformation of organic bodies. It he introduced the term morphology into scientific use.

Golden section and symmetry

The gold division is not a manifestation of asymmetry, something opposite symmetry according to modern ideas the gold division is an asymmetric symmetry. The science of symmetry includes such concepts as static and dynamic symmetry. Static symmetry characterizes peace, equilibrium, and dynamic - movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art characterizes peace, equilibrium and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characteristic of equal segments, equal values. The dynamic symmetry is typical of the increase in segments or their decrease, and it is expressed in the values \u200b\u200bof the golden section of an increasing or decreasing range.