Build a house according to the golden ratio. Educational and research work "golden ratio in the architecture of a traditional peasant house." House, Family Estate as an artifact

The attractiveness of a small residential building depends on many reasons and, above all, on the plan, on the proportion of the entire building and its parts, on the character building materials, quality of work, landscaping of the site.

With a “blank” plan closed in a rectangle, the result can be a box house. At the same time, a skillful layout of the plan allows you to create a cozy, sunny terrace, closed on both sides, and the placement of windows on all sides of the house makes it possible to avoid ugly blank walls.

This lays the foundation for the future attractiveness of the new home. Its appearance, further, is largely influenced by the good proportions of the structure, that is, the harmonious relationship general sizes buildings and its parts. Narrow, upturned windows or an awkward roof can ruin the appearance of any home. It is especially important not to make its top visually “heavy”. Therefore, it is better to build a pointed roof, straight, and not with a fracture.

A fracture makes the roof visually heavy, and the whole house looks ugly, like a mushroom. In addition, a roof with a fracture is structurally more complex than a straight one: it has composite rafters with notches, and the load from the roof and snow is transferred through vertical posts to the ceiling beams, which have to be made too strong and span the entire width of the house. But it is often more profitable to make a light panel ceiling supported by the middle wall - the partition. This can be done with a straight roof, and then the entire load is transferred by the rafters to the outer walls. A roof with a fracture does not provide any additional amenities.

The desire of many people to make attic rooms with sheer walls and a flat ceiling is unreasonable. It is more comfortable to live in a room with a sloping ceiling; armchairs and a bed are more comfortable under the sloping parts of the ceiling. Small veranda with pitched roof, stuck to the side of the house, is convenient, but does not decorate the building. If you cover it gable roof with a ridge (under which you can arrange sleeping places or a storage room), then the appearance of the house will noticeably change for the better, it will become more elegant and will look equally good from different sides.

The design of the veranda itself is also of great importance for the appearance of the house. To this day, verandas with frequent posts, thick, small “diamond”, “herringbone” or even more complex patterns are widespread. The window sill of such verandas is usually arranged high, so the glazed strip turns out to be narrow, and the paneling underneath is absurdly wide. On such a veranda it is always gloomy and uncomfortable. The veranda is a transitional room from the house to the site, and the more “open” it is, the better it will be. To do this, first of all, boldly lower its floor two steps below the floor in the rooms. Then the veranda will become higher. Place the window sill board at a height of 45 cm from the floor, that is, at the level of the sofa and armchairs. This will allow you to see the garden while sitting in a chair, and you will seem to be closer to the flowers and greenery. It is very important to make thin bindings and hang the frames directly on the posts in which the quarters are selected. Horizontal slabs should be thin (25-30 mm), cut into the frame so that the distance between them is slightly less than the distance between the vertical frame frames. In practice, the frame sash on the veranda is 170-180 cm high and 50-55 cm wide, and the distance between the slabs is 40-45 cm.

The porch also greatly influences the appearance of the house. It should not only protect the door from rain, but also be good place for relax. Sometimes the open part of the porch is combined with the closed part - the canopy. It's convenient and beautiful.

PRESERVE THE NATURAL BEAUTY OF THE MATERIAL

It is very important to skillfully use the natural properties of the materials you have - their texture and color. The natural appearance of each material - brick, “wild” stone, tiles, wood or plaster - is beautiful in itself, and this beauty must be protected. You should always remember the contrast of color and texture of materials used for construction.

What does it mean?
For example, you laid out a plinth made of rubble stone. It is enough to “embroider”, scratch or cut the seams in the damp mortar, clean the stones from cement - and the base will sparkle with its natural beauty. And don’t even think about plastering it! Under a layer of plaster, the natural charm and beauty of the material will perish.

If the walls of your house are light - whitewashed or plastered, then a red tiled roof will be a good decoration for it. And for red brick walls, it is better to make the roof from light tiles or white slate. Try laying red and white slate tiles on the roof in a checker or lattice pattern, it will turn out very elegant.

Smooth red brick pillars near the glazed veranda next to a white wall, plastered without grout, with bumps “like a fur coat” will create a variety that is pleasing to the eye, which cannot be achieved by any decorations. And if, over time, the wild grapes you planted cover these pillars, and if they also cover the lattice of the porch with a green carpet, then your house will become very beautiful.

A bright house always looks welcoming. And its individual parts - doors, window frames, blinds or boards under the roof overhangs - can be painted in bright colors. This will enhance the cheerful and attractive appearance of the home.

If the house is wooden - chopped, there is no need to paint it. It is best to cover the wood with drying oil with the addition of umber. The golden transparent layer will protect the wood from destruction, and at the same time the entire natural pattern of this material will be visible. The picture at the top right shows a schematic representation of a roof with a fracture. Such a roof is structurally complex and visually perceived as “heavy”. A straight, pointed roof is easier to construct and more beautiful. Below is a fragment of a pergola - a through lattice roof over a terrace made of boards and slats. Next to the pergola is a drawing of a trellis for climbing greenery. Such gratings create coziness on the site. Below are metal, welded caps for brick and asbestos-cement chimneys.

Much depends on the quality of work. Smooth masonry, cleanly planed thin frames, straight rows of slate or tiles, neat fillies under the roof overhang, smooth even painting - all this will give the house a finished and elegant look.

THE TYPE OF THE HOUSE DEPENDS ON THE IMPROVEMENT OF THE SITE

The small house is closely connected to the site. Any building looks ugly in a bare, dirty place. And if your plot is lovingly cultivated, a flower garden is arranged in front of the house, bushes and trees are planted, all the land is dug up and sown with grass, then on such a landscaped plot even the simplest and most unpretentious house will look elegant and cheerful.

On the plot of the house great importance have so-called “small forms”. This is a pergola - open terrace, which has only a through lattice roof made of poles or slats attached to poles. Ivy or wild grapes will climb along it. You can also make so-called trellis grilles that protect quiet corners near the house from prying eyes, where it is good to sunbathe or just relax. Bindweed or decorative beans planted near them will soon create a green barrier impenetrable to view. Such gratings block the latrine and compost heap on the site. It is good to hang flower boxes on the wall under the windows.

Paths and areas near the house look elegant, paved with brick in a herringbone pattern, laid with flat stones or artificial concrete slabs, split into irregularly shaped pieces. Sow grass in the cracks between slabs or bricks. Under the pergola, where rain gets through the slats, the ground should be paved with brick or stone. This is done like this: brick cells are laid out on a sandy base, and the squares between them are hammered with white cobblestones and secured with mortar. On such a grape-covered terrace with a stone floor it will be pleasant to work, relax, have lunch or drink tea. The finishing touch to the construction of your home can be a metal cap on the pipe. It protects the chimney from rain and snow and increases draft. At the same time, a welded or forged iron head with a simple ornament, an arrow weathervane or a figurine on the top will give the house a cheerful and complete look.

The golden ratio method in the construction of a harmonious country house

When arranging your home, undoubtedly, one of the main points is Harmony and Coherence in the use of housing space. However, this is not possible without a clear understanding of the basic principles in this difficult matter. For centuries, people have accumulated experience in using these principles both in the construction of individual houses and buildings, and in the construction of large-scale settlements. After all, not only the person himself and the arrangement of his life, but also the arrangement of everything in the Universe is an example of harmony, perfection and coherence. It is not without reason that many scientific minds call such impeccable coherence a truly “divine sign.” The principle of the “Golden Proportion”, which will be discussed below, is precisely based on the use of such harmony and its transfer to the sphere of arranging a human home.

Golden Ratio is the division of any value in the ratio of 62% and 38% (F=1:1.618).

Man as the standard of the “Golden Proportion”

No matter how surprising it may sound, in those days when there were no instruments for spatial measurements, the measure for the ancestors of the modern Slavs was the man himself. To be convinced of this, it is enough to recall many of the names in the Slavic measuring system: elbow, span, flywheel and oblique fathom, metacarpus, foot. Thus, the use of such measures of length already laid the foundation for the “golden” correspondence of measured objects to the proportions of the human body. And it is not surprising that buildings erected according to such natural principles were examples of harmony with the outside world and the surrounding nature.

Some of the features of Old Russian fathoms

The most common in architectural planning in Ancient Rus' there was a system of measurements using the so-called “fathoms”, of which there were a great variety. Different localities used their own fathoms, which was reflected in their names: Vladimir, Moscow, Novgorod. How can this difference be explained? Most likely, the fact that people from different areas and regions often differed in their height, size and body proportions. Moreover, many craftsmen could invent and use various personal fathoms in their work, which is quite natural - after all, any construction should be carried out according to the needs of a specific owner. If a person selects clothes taking into account the height, size and shape of the body, it would be logical to adhere to the same principles in the construction and arrangement of the home. A low house is clearly not suitable for a giant, and a short person does not need high ceilings at all. A skinny man doesn't need one that's too wide doorway, while a person with large dimensions simply needs it. Matching the size to the needs of the owner ensures coherence, harmony and comfort.

However, as various studies confirm, Old Russian fathoms were not commensurate and multiple values ​​of each other. That is why many experts consider their use irrational and devoid of convenience, preferring to resort to classical reference units such as the meter.

However, how can we explain such a widespread practice of using irrational standards among our ancestors? Unfortunately, a strictly material perception of the surrounding reality has taken root in modern official science, and as a result, many of these questions remain without an intelligible answer.

The world around us is full of numerous movements and processes, not all of which can be seen by the human eye. Many waves, vibrations, microscopic vibrations permeate the outer space everywhere every moment. This is a kind of “pulsation of nature” - not only living, but also inanimate. And what has been said fully applies to various elements of a human home, be it walls, floors or ceilings. Microscopic wave movements, elusive even for many sensitive instruments, continuously affect the human body, which cannot remain without consequences for it. As researchers in this field note, in those rooms that are built on the basis of the standard metric system, the waves take on a monotonous, “standing” character, adversely affecting human health. The body resists the constant and same type of wave action, which weakens and tires it, contributing to exhaustion.

Secrets of harmony in the home

Not being commensurate and multiple values, Old Russian fathoms are devoid of strict physical rationality. The lack of multiplicity in distances leads to an imbalance of “standing” wave oscillations. At the same time, the coherence of the proportions of a home with the proportions of its inhabitants is accompanied by the emergence of other waves that vibrate in unison with microscopic vibrations in the human body. It is this kind of room that is the best for people to live in, and therefore in many old houses people feel comfortable and relaxed, without understanding what is the reason for this.

Of course, accurate measurement systems are of utmost importance and have a wide range of applications, including in construction, but planning symmetry and proportions based on them is not a good option.

If the dwelling has already been built, then its improvement can be achieved through a visual breakdown into parts and rooms that meet the conditions of the “golden proportion”.

Putting these principles into practice will bring life to any space while promoting well-being and a more comfortable and enjoyable experience. appearance dwellings.

We will be glad to see you among our clients!

Construction according to the Golden Proportion from Center Suburban Construction"Asgard"- this is a reliable long-term cooperation on mutually beneficial terms in compliance with all terms of the contract. Join the number of grateful clients already enjoying comfortable living in their country house.

Still have questions? Get a free consultation.

Gileva Anastasia

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XIV municipal competition

educational and research works of students

"Golden ratio" in the architecture of a traditional peasant house

Work completed:

Gileva Anastasia Vasilievna,

student of class 8A, Municipal Educational Institution Secondary School No. 8

Supervisor:

Gileva Irina Ivanovna,

computer science teacher, municipal educational institution secondary school No. 8

Golubleva Zoya Egorovna,

mathematics teacher of municipal educational institution secondary school No. 8

Krasnovishersk - 2010

Introduction
Chapter 1 “The Golden Proportion”
Chapter 2 Features of building peasant houses
Bychina, Gileva, Paleva, Semina
Bychina, Gileva, Paleva, Semina for the presence of relationships of the “golden proportion”
Conclusion
Literature
Application

Introduction

There are things that cannot be explained. So you come to an empty bench and sit down on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive... Sitting on a bench, you produced the “golden ratio”.

The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing.

You will certainly see this proportion in the curves. sea ​​shells, and in the form of flowers, and in the form of beetles, and in a beautiful human body. Inanimate nature does not know what the “golden ratio” is, but it is used in architecture and sculpture, painting and mathematics, music and poetry...

Egyptian pyramids, buildings of the ancient Greeks, divine temples of great architects amaze with their beauty and harmony. We see the same beauty and harmony in a simple peasant hut. How could a simple Russian man, not knowing the basics of architecture, “raise” such proportional buildings?

Looking at the abandoned huts in the villages of Bychina, Gileva, Paleva, Semina, ... we asked ourselves: is there a golden ratio in the architecture of these ancient houses?

The purpose of our work: to study the architecture of peasant huts in the villages of Bychina, Gileva, Paleva, Semina for the presence of the golden proportion.

To achieve this goal, it is necessary to solve the following tasks:

1. study the literature on the issue of the golden proportion and related ratios used in architecture (golden section of a segment, golden rectangle);
2. carry out measurements of peasant huts in the villages of Bychina, Gileva, Paleva, Semina;
3. process the received data using computer systems;
4. analyze the results obtained.

Chapter 1 “The Golden Proportion”

1.1. "Golden proportion" and related ratios

The question of the mathematical prerequisites for beauty and the role of mathematics in art worried the ancient Greeks, and they inherited their interest from previous civilizations. Nowadays, geometry is a necessary element general education and culture - is of great historical interest, has serious practical use and has inner beauty.

Johannes Kepler said: “Geometry has two treasures: one of them is the Pythagorean theorem, the other is the division of a segment in the mean and extreme ratio.The first can be compared to the value of gold, the second can be called a precious stone."

There are many ratios of the “golden section”, but in my work IWe will consider only two ratios: the “golden ratio” of the segment and the “golden rectangle”. This is not accidental, since we will study the linear dimensions of houses (height, length and width).

Let's follow the example of L.S. Sagatelova. and determine the ratio of segments at the “golden section” and the aspect ratio of the “golden rectangle”.

The division of a segment in the mean and extreme ratio is called the “golden ratio”. Another name has become established in history - the “golden proportion”.

Let C AB produce, as they say, the “golden ratio” of the segment.

(1)

SV:AB=AS:SV

The golden ratio is a division of a segment in which the larger part is related to the whole as the smaller part is to the larger one.

If the length of segment AB is denoted by A, and the length of AC is through x, then a-x - the length of the segment CB, and proportion (1) will take the form:

(2)

In a proportion, as is known, the product of the extreme terms is equal to the product of the middle terms and we rewrite proportion (2) in the form:

x 2 =a(a-x)

We get a quadratic equation:

x 2 +ax-a 2 =0.

The length of a segment is expressed as a positive number, so from two roots

x 1.2= should choose positive or .

Number denoted by the letterin honor of the ancient Greek sculptor Phidias (born at the beginning of the 5th century BC), in whose works it appears many times. The number is irrational, it is written like this: =0.61803398…

But in practice they use a number taken with an accuracy of thousandths of 0.618, or hundredths of 0.62, or tenths of 0.6.

If, then, and a-x=0.38a.

Thus, the parts of the “golden ratio” make up approximately 62% and 38% of the entire segment.

During the Renaissance, the golden ratio was very popular among artists, sculptors and architects. So, when choosing the size of the painting, the artists tried to ensure that the ratio of its sides was equal. Such a rectangle began to be called “golden”.

The algorithm for constructing the “golden” rectangle has come down to us since the time of Euclid:

1. Draw a square and divide it into two equal rectangles.
2. Draw a diagonal AB in one of the rectangles.
3. Using a compass, draw a circle of radius AB with center at point A.
4. Continue the base of the square until it intersects with the arc at point P and draw the second side of the desired rectangle at a right angle.

Let's find the exact ratio of the sides of the constructed rectangle.

Let us denote the side of the original square by A ; let's express it through a the length of the diagonal AB is the hypotenuse of a right triangle with leg a and; i.e. AB=.

Let's find the lengths of the sides of the constructed rectangle, one of them is equal to A , and the other - . Finally, we find the ratio of the larger side of the rectangle to the smaller one, we get.

Thus, in the architecture of peasant houses we will look for parts of the “golden ratio” of the segment - 62% and 38%, as well as the “golden rectangle”, the sign of which is the number 1.62 as the ratio of the larger side of the rectangle to the smaller.

1.2. "Golden proportion" in architecture

The golden proportion is a mathematical concept. But it is a criterion of harmony and beauty, and these are already categories of art.

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

One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC) - the temple of Athena.

The dimensions of the Parthenon are well studied. It is known that the facade of the Parthenon is inscribed in a rectangle with sides 1:2, and the plan forms a rectangle with sides 1 and.

It is known that the diagonal of the rectangle has the size, therefore, the rectangle of the facade and is the initial one in the construction of the geometry of the Parthenon.

Many researchers who sought to uncover the secret of the Parthenon’s harmony sought and found the “golden” proportion in the relationships of its parts.

A regular series of golden proportions has been established. Taking the width of the end façade of the temple as one, the researchers obtained a progression consisting of 8 members of the series:

1; where =0.618.

Careful measurements of the Parthenon showed that there are no straight lines and the surfaces are not flat, but slightly curved. The architects of Greece knew that a strictly horizontal line and a flat surface to an observer from afar appear to be bent in the middle.

Another example from ancient architecture is the Pantheon.

The famous Russian architect M. Kazakov widely used the “golden ratio” in his work. His talent was multifaceted, but it was revealed to a greater extent in the numerous completed projects of residential buildings and estates. For example, the “golden ratio” can be found in the architecture of the Senate building in the Kremlin. According to the project of M. Kazakov, the Golitsyn Hospital was built in Moscow, which is currently called the First Clinical Hospital named after N.I. Pirogov (Leninsky Prospekt, 5). Another architectural masterpiece of Moscow - the Pashkov House - is one of the most perfect works of architecture by V. Bazhenov (Appendix 1).

The construction of village houses was carried out by peasants who did not have knowledge of the basics of architecture in general and the concept of the “golden ratio” in particular. However, inIn the structure of traditional rural houses, proportional relationships can be distinguished. Research has shown that proportional relationships are based on the properties of the square and its derivatives. The main compositional principle for the formation of the proportional structure of a peasant residential building was the principle of similarity, which found its expression both in the layout of the building and in the structural organization of its most important elements and details.

The “golden ratio” occupies a special place among various proportioning systems. However, the use of the proportions of the “golden section” in the formation of the architectural and artistic structure of a traditional peasant house is based more on intuition than on deliberate and accurate calculation - in the proportional structure of a people's home, it is quite rare to find relations that exactly correspond to the golden section, and much more often - very close to it .

We have not found scientific works, devoted to a direct study of the issue of using the “golden proportion” ratios in the architecture of a traditional peasant house. The more interesting the topic we are researching is.

Chapter 2 Features of building peasant houses.

2.1. Technology for building a peasant house in villagesBychina, Gileva, Paleva, Semina.

According to Mark Yakovlevich Gilev, a resident of the village of Bychin, the technology for building a peasant house included several stages:

The first stage is logging. To build a house, they choose spruce, pine, and less often fir. Timber is harvested in late autumn, in the old month. The forest lies all winter.

The second stage is wood processing. In the spring, the logs are stripped of their bark and the frame is cut down. The material for the floor and roof is prepared; for this, the logs are “unraveled” onto boards. At the same time, moss is being harvested. Sphagnum moss is usually used.

The third stage is drying. In summer, the prepared log house, moss and boards dry naturally. The drying boards are not laid tightly so that “air can move.”

The fourth stage is raising the frame. In the old days, racks made of larch or cedar, the most rot-resistant conifers, were placed at the base of the house. Currently, the prepared frame is being laid on the foundation. The logs are covered with moss.

The fifth stage is the final one. A year later, when the log house has settled down, carpentry work is carried out: they cover it with a gable roof, build a ceiling, install windows, doors, flooringinsulated double floors with earthen backfill And so on.

Typically, when building houses, logs with a length of 5 to 10 m and a diameter of 30 to 40 cm were used.The dimensions of the main frame are 6x7, 7x7 or 7x8 - closer to a square. How bigger house, the higher the frame is raised (the number of crowns - horizontal rows of logs - increases). There are no specific standards; the builder does everything “by eye”, as he likes.The logs were usually not joined lengthwise; the size of the building was increased by adding another log to the existing one or by installing a new log close to the old one.

Observations show that village houses, although they have a frame structure that is close to square, are more shaped like elongated parallelepipeds. This is achieved by attaching outbuildings to the main frame. Both the living space and outbuildings are under one roof.

The technology described above, as we see, does not provide mechanisms for calculating the basic dimensions of a house. Moreover, we received confirmation that all construction is being carried out “by eye”, without observing any proportions.

2.2. Study of the linear dimensions of houses in villagesBychin, Gileva, Paleva, Semina for the presence of “golden proportion” relationships.

We measured several houses. The measurement was carried out using a ten-meter tape measure. The height (H) of the house was taken from the ground to the very top of the main frame. Width (C) of the house - along the front of the house (without protruding parts). The length (L) of the house was measured taking into account all extensions built under one roof, that is, the internal division of the house into zones was not taken into account.

The obtained data are presented in Table 1.

No.	House name	Linear dimensions of the whole house
		Height	Width	Length
	D.Semina Gilev Arkady Semenovich (year of construction - ...)
	D.N-Bychina Building primary school (year of construction - 1916)
	D.N-Bychina Mitrakov Andrey Egorovich (year of construction - 1930)
	D.V-Bychina Gilev Mark Yakovlevich (year of construction -1930)
	D.V-Bychina Bychin Egor Vasilievich (year of construction - ...)		6.8(2 floors)
	D.N-Bychina (year of construction - late 19th century)		8 (2 floor)
7	D.Paleva Gilev Nikolai Konstantinovich (year of construction - 1950) (year of construction - 1978)	4,2	6,8	8,5
	D. Bychina Bychin Fedor Andreevich (year of construction ~1820)			10,5
	D.Ivacina Bychina Natalya Yakovlevna (year of construction - 1924)
11	D.Paleva Sobyanina Antonina Yakovlevna (year of construction - 1931) new house	2,9	4,9	8,5
	D.Paleva Mitrakov Alexander Egorovich (year of construction - 1910)	3,45		12,4
	D. Semina Mitrakova Lyudmila Aleksandrovna (built 1963)			10,9

Processing of the obtained data was carried out using the Ms Excel spreadsheet processor (Table 2). Correlation coefficients were found to determine the presence of a relationship between quantities and the nature of this relationship. Correlation coefficient for the height and width of the house0.835904279 - close to +1.This means that there is a strong dependence between the arrays of values ​​and it is directly proportional. The correlation coefficient for the width and length of the house, as well as for the height and length of the house, are close to 0. This means that, as such, there is no dependence between the arrays under consideration.

Calculation of the ratios of width to height, length to height and length to width of the house confirmed the above.

table 2

House number	Height	Width (C)	Length (L)	Relationship
				1,606061	2,242424	1,396226
					2,705882	1,352941
				1,612903	2,580645
				1,666667	3,030303	1,818182
				1,942857	2,857143	1,470588
				1,666667	1,875	1,125
				1,619048	2,02381	1,25
				1,738095	2,02381	1,164384
			10,5	1,775	2,625	1,478873
				1,689655	2,931034	1,734694
				1,848485	2,606061	1,409836
	3,45		12,4	1,768116	3,594203	2,032787
			10,9	2,137931	3,758621	1,758065
	0,835904279
	0,203090205
	0,05084057

Analysis of the results obtained showed that for the front part of the house the ratio of width to height in 9 cases out of 14 is close to the proportion of the “golden rectangle”. And this is no coincidence, since the façade of the building faces the street and much attention was paid to its appearance during construction. The builder sought to give the facade a harmonious shape, based on his intuition.

The remaining dimensions received less attention and, as research shows, their size depended on the size of the outbuildings, that is, it was directly related to the practical needs of the owners of the house.

Conclusion

At all times, man has strived for beauty and harmony. Mathematics claims that the basis of beauty is the harmonious relationship of the parts of the whole - the “golden proportion”. Man notices this proportion in all living things and strives to take it into account and use it when creating his works.

In our work, we set out to find the relationships of the “golden proportion” in the architecture of a peasant house.

Studying the literature on this topic did not give us an exact answer to the question: is there a “golden ratio” in the proportions of a village hut?

Our research has proven that when building a traditional peasant house, the application of the “golden section” proportions is based more on intuition than on deliberate and precise calculation. It is quite rare to find relationships that exactly correspond to the “golden ratio”, and much more often - very close to it.

We looked at the basic rectangles: the front part, the base of the house, the end part. The data obtained using correlation analysis proves the presence of the “golden ratio” in the façade of the building and its absence in the remaining basic rectangles.And this is no coincidence, since the façade of the building faces the street and much attention was paid to its appearance during construction. The builder sought to give the facade a harmonious shape, based on his intuition. The remaining dimensions received less attention and, as research shows, their size depended on the size of the outbuildings, that is, it was directly related to the practical needs of the owners of the house.

Literature

1. Geometry: beauty and harmony. The simplest problems of analytical geometry on the plane. Golden ratio. Symmetry is all around us. 8-9 grades: elective courses / author-comp. L.S. Sagatelova, V.N. Studenetskaya. - Volgograd: Teacher, 2007. - 158 p.
2. Gutnov A.E. World of architecture: The language of architecture. - M.: Mol. Guard, 1985. - 351 p.
3. Prokhorenko A.I. Architecture of a rural house. Past and present. - M.: Mol. Guard, 1984. - 67 p.
4. Stakhov A.P. Harmony of the Universe and the Golden Section: the oldest scientific paradigm and its role in modern science, mathematics and education.//http://www.trinitas.ru/rus/002/a0232001.htm

Annex 1

Pashkov House in Moscow

Senate in the Kremlin

Golitsyn Hospital in Moscow

Golden ratio - harmonic proportion

In mathematics, proportion (lat. proportio) is the equality of two ratios: a: b = c: d.

A straight line segment AB can be divided into two parts in the following ways:
into two equal parts – AB: AC = AB: BC;
into two unequal parts in any respect (such parts do not form proportions);
thus, when AB: AC = AC: BC.

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

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

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

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

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

Segments of the golden proportion are expressed by the infinite irrational fraction AE = 0.618..., if AB is taken as one, BE = 0.382... For practical purposes, approximate values ​​of 0.62 and 0.38 are often used. If segment AB is taken to be 100 parts, then the larger part of the segment is 62, and the smaller part is 38 parts.

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

x2 – x – 1 = 0.

Solution to this equation:

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

Second golden ratio

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

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

The figure shows the position of the line of the second golden 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 the pentagram.

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht 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.

We draw straight AB. From point A we lay out 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, 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. 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 the relief wooden board from the tomb named after him, holding in his hands measuring instruments, in which the proportions of the golden division are fixed.

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

Plato(427...347 BC) also knew about the golden division. His dialogue " Timaeus"is dedicated 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.

In the ancient literature that has come down to us, the golden division was first mentioned in “ Beginnings» Euclid. 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 have a lot of empirical experience, but little knowledge. He conceived and began to write a book on geometry, but at that time a monk’s book appeared Luca Pacioli, 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, he was working on the same problems Albrecht Durer. 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. Johann 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 remains 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.

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.”

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. Proportions male body fluctuate within the average ratio of 13:8 = 1.625 and come somewhat closer to the golden ratio than the proportions 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.

IN late XIX- early 20th century 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.

Fibonacci series

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected with the history of the golden ratio. He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read “How many pairs of rabbits will be born from one pair in one year.” Reflecting on this topic, Fibonacci built the following series of numbers:

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

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

Generalized golden ratio

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

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

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

The Fibonacci series (1, 1, 2, 3, 5, 8) and the “binary” series of weights discovered by him 1, 2, 4, 8, 16... at first glance are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 = 1 + 1; 4 = 2 + 2..., in the second - this is the sum of the two previous numbers 2 = 1 + 1, 3 = 2 + 1, 5 = 3 + 2.... Is it possible to find a general mathematical formula from which we obtain and “ binary series and Fibonacci series? Or maybe this formula will give us new numerical sets that have some new unique properties?

Indeed, let us define a numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Consider a number series, S + 1 of the first terms of which are ones, and each of the subsequent ones is equal to the sum of two terms of the previous and separated from the previous one by S steps. If nth term We denote this series by φS (n), then we obtain the general formula φS (n) = φS (n – 1) + φS (n – S – 1).

It is obvious that at S = 0 from this formula we will obtain a “binary” series, at S = 1 – the Fibonacci series, at S = 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

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

It is easy to show that at S = 0 the segment is divided in half, and at S = 1 the familiar classical golden ratio results.

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

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

Using golden S-proportion codes, you can express any real number as a sum of powers of golden S-proportions with integer coefficients.

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

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

In such a number system, any natural number can always be represented as finite - and not infinite, as previously thought! – the sum of powers of any of the golden S-proportions. This is one of the reasons why “irrational” arithmetic, possessing amazing mathematical simplicity and elegance, seems to have absorbed best qualities classical binary and Fibonacci arithmetic.

Principles of formation in nature

Everything that took on some form was formed, grew, strived to take a place in space and preserve itself. This desire is realized mainly in two options - growing upward or spreading over the surface of the earth and twisting in a spiral.

The shell is twisted in a spiral. If you unfold it, you get a length slightly shorter than the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The idea of ​​the golden ratio will be incomplete without talking about the spiral.

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and came up with an equation for the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

Goethe also emphasized the tendency of nature towards spirality. The helical and spiral arrangement of leaves on tree branches was noticed a long time ago. The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that the Fibonacci series manifests itself in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, and pine cones, and therefore, the law of the golden ratio manifests itself. The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral the “curve of life.”

Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there.

The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

Rice. 13. Chicory

Rice. 14. Viviparous lizard

At first glance, the lizard has proportions that are pleasing to our eyes - the length of its tail is related to the length of the rest of the body as 62 to 38.

In both the plant and animal worlds, the formative tendency of nature persistently breaks through - symmetry regarding the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.

Nature has carried out division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.

Rice. 15. Bird's egg

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

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

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

The golden ratio cannot be considered on its own, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulf (1863...1925) considered the golden ratio to be one of the manifestations of symmetry.

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

Information sources: Kovalev F.V. Golden ratio in painting. K.: Vyshcha School, 1989. Kepler I. About hexagonal snowflakes. – M., 1982. Durer A. Diaries, letters, treatises - L., M., 1957. Tsekov-Pencil Ts. About the second golden ratio. – Sofia, 1983. Stakhov A. Codes of the golden proportion.	see also: Ernst Neufert. Construction design. Measurement system