Examples of the golden ratio in architecture can be found everywhere when you know how to see it. Even a schoolboy can figure it out. In 2013, a 10th grade student Elena Sivakova conducted her own study of buildings of the 19th and 20th centuries. Let's trace how she did it, and learn to see and define it in architectural structures in 5 minutes. After reading the article, there will be no questions about what it is, and whether its unusual properties can be used in your life.
The buildings of the historical center of St. Petersburg are built in different, such as baroque, empire, eclecticism, neo-baroque, neo-gothic. Do they obey the golden rule?
Auguste Montferrand, the court architect of Alexander I, built this cathedral from 1819 to 1858. The style of the late, in which the features of neo-Renaissance and eclecticism are already manifested. Elena asked herself the question: "What is the reason for the harmony of a rather bulky building?"
The first row is determined by the width of the building, which is taken as 400 units. and represents such numbers 400, 247, 153, 94, 58 ...
If 400 is divided by the number ≈1.618, then we get approximately 247; repeat the action with the following number: 247: 1.618≈153.
And so we find all the numbers. Now we look at the picture. The main part with columns fits into a rectangle with sides 400 and 247. Since the sides are in the ratio Ф≈1.618, they form the Golden Rectangle.
The next row is represented by the height of the building: 370, 228, 140, 87, 53, 33, 20, 12. These dimensions are incorporated into smaller details. Vertically, St. Isaac's Cathedral is divided by the Golden Section at the base of the dome, which makes the ratio of the main part and the dome harmonious.
The third row of sizes starts from 113, and is the width of the base of the main dome: 113, 69, 42, 26, 16. The numbers of this row are found in the sizes of windows, in the heights of columns and other details of the cathedral.
Golden rectangular and isosceles triangles take place in the building of St. Isaac's Cathedral, as can be seen from the figure.
On the Universitetskaya embankment of Vasilievsky Island there is the Kunstkamera building, founded in 1718 under the direction of the German architect Georg Mattarnovi: the Petrovskoe Baroque, two 3-storey buildings and a complex multi-tiered dome tower.
The study begins with the main values: the height and length of the building, from which the golden row is built. Length - 450 units, then 277, 170, 105, 65, 40, 24. Such dimensions can be seen in the height and width of different levels of the tower, the length of the hulls. The tower itself is inscribed in a golden isosceles triangle from base to top. The golden ratio can be seen to a greater extent in this main element, which is correct from the point of view of architecture. Conclusion: the basis of the Kunstkamera obeys the golden rule and preserves compositional harmony.
The height of the building starts a new golden row: 211, 130, 80, 49, 30. Looking at the dimensions of the drawing, it becomes clear that the choice of a three-story type of buildings is due to its proportionality with the tower.
Built in 1907 according to the project of Vladimir Alexandrovich Lipsky and Konstantin Nikolaevich de Rochefort (Rochefort). In 1905 the Belgian S. Esders and the Dutchman N. Scheufals applied for permission to build a five-story building with a dome and a spire on the corner tower for their trading house instead of the old one.
From a building length of 671 units. begins the series of the Golden Section, observed in sizes: 671, 414, 256, 158, 98, 60, 37, 23. Pay attention to the main element - the spire. We make sure that the compositional solution is completed by a harmonious combination of altitude values.
Built in 1941 according to the design of Noah Abramovich Trotsky. The building of the Soviet period is seen as a creative interpretation. The central portico with fourteen columns completes the sculptural ensemble on the theme of building socialism and the coat of arms of the Russian Soviet Federative Socialist Republic.
Five-story buildings are symmetrically located on the sides. The length of the House reaches 1472 units, from which, by dividing by the number F, a number of sizes of building elements are obtained: 1472, 909, 562, 34, 214, 132, 81, 50 (Appendix 21): the height of the structure, the height of the entrance, etc.
The top of the Golden Isosceles Triangle coincides with the top of the building, and its sides pass through the apex points of the main entrance. The rectangular golden triangle is formed by the vertices at the top of the building and at the end of the inner part of the side wing. Proportionality is obvious, although it does not have much compositional significance.
A team headed by B.M. Iofan worked on his project, who was later removed from the post of chief architect. An example of post-war Soviet architecture was built from 1949 to 1953.
BM Iofan proposed a composition of five components with a central tower. During the construction years, it was the tallest building in Europe.
The length of the building is 1472 units. and starts the row: 909, 562, 347, 214, 132, 81, 50. The heights are mainly subject to the Golden Ratio. Another row follows from the width of the tower: 538, 332, 205, 126, which is visible in latitudinal dimensions.
A golden right-angled triangle with a hypotenuse goes through the corner of the building and captures the annexes.
Thus, in all the buildings studied, the student discovered the Golden Ratio, which preserves harmony.
To simplify the task of searching for the ES, one can take rational fractions 3/2; 5/3; 8/5; 13/8; 21/13; 34/21; 55/34; 89/55; and so on. The pattern is clear: 3 + 2 = 5; 5 + 3 = 8; 8 + 5 = 13 ... Or even simpler. Make yourself a compass to determine the proportion according to the instructions in the video. Time will take about 10 minutes. How to use this compass to determine the proportionality of elements will also be told and shown.
Using this method, we find the golden proportion of the Russian architect Matvey Kazakov in the Kremlin building of the Senate, and in all other works: the Prechistensky Palace in Moscow, the Noble Assembly, the Golitsyn Hospital (named after Pirogov) ...
The Pashkov house in Moscow, created by another great architect Vasily Ivanovich Bazhenov (Russian state library) are ranked among the samples of perfect architectural monuments, in which it is easy to determine the AP.
When the metal Eiffel Tower was being assembled in Paris, many Frenchmen were outraged. Critics wrote about her as "the ugliness of the city", "the shame of Paris", "a skinny pyramid of metal stairs." Among them were Emile Zola, Dumas Jr., Guy de Maupassant. Now this most visited monument is the pride of Parisians. Maybe the “divine” proportion is to blame?
It is also observed in the most famous French cathedral, Notre Dame de Paris.
Intuitively or deliberately, did the great architects construct buildings with these proportions in mind? Ancient mathematicians have known about the golden ratio since the time of Pythagoras. There are more and more confirmations of its use in architectural proportions. However, there is not a single ancient record with a direct recommendation to use "divine proportion". Neither does Vitruvius (1st century BC), who wrote "Ten Books on Architecture", in which he considered proportionality, as well. It's a strange fact, isn't it?
Maybe all of the above studies are a fit for a known result? It is not so difficult to choose from a variety of architectural elements those that confirm the hypothesis, since no one demands absolute accuracy. It is logical to think about the question: "What if the Greeks did NOT use the golden ratio?"
Strictly speaking, for Luca Pacioli, who wrote the work "Divine Proportion" in 1509, its applied significance was not so important. It was important to substantiate its mystical nature. And they began to use it consciously only from the moment the book was published.
Beautiful and harmonious objects always meet the ZS rule, and this proportionality is determined when analyzing the values. Art critics have carefully studied the Greek Parthenon, erected in honor of the victory over the Persians - the temple of the goddess Athena. The ratio of the length of the temple to the width gives the golden number with a small margin of error. If you subtract 14 cm from the length of the structure and add to the width, you get a complete coincidence with the mathematical value. The facade of the building tapers slightly upward, deviates from the rectangular shape. Considering the visual perception, this was done deliberately by the builders. Therefore, it is not entirely correct to consider it a rectangle of the golden ratio. But the proportions are respected, so it is logical to assume that the architects Iktin and Kallikrates deliberately laid the rule in the project?
The pyramid of Cheops is also built taking into account this condition. Without going into the mathematical proof of the existence of the golden formula, we will only say that it contains a right-angled golden triangle, the sides of which are the height and half of the side of the base of the structure. No wonder?
But then the question arises about the level of ancient Egyptian mathematics. It turns out that the Pythagorean theorem was known to them two millennia before the birth of the scientist himself. Attention is drawn to the fact that the heirs of Cheops built their pyramids with different proportions. Why?
It has been established that pyramidal structures with GSs have a phenomenal effect on those located in them: plants grow better, metals become stronger, water remains fresh for a long time. Scientists have been working with these riddles for many years, but the mystery remains.
It is noticed that the pyramid brings the structure of space into a harmonious state. Everything that falls into the zone of action is also organized in a similar way: the psychoemotional state of people improves, radiation harmful to humans decreases, and geopathogenic zones disappear. The Internet claims that if the size of a figure is doubled, then the influence of the pyramid is amplified a hundredfold.
Correct distribution of energy inside the house, harmonious designs combined with ecology and safety building materials encourage contemporary architects and designers to use the principles and concepts of the Golden Ratio. This increases the estimate and gives the impression of a deep study of the project. The cost increases by 60-80%.
For talented artists and architects, the rule is kept intuitively during the creative process. However, some of them consciously implement this provision.
In nature, this proportionality is found everywhere. Anyone who feels the harmony of space will create a proportional building without special efforts for this.
For example, our ancestors built mansions commensurate with humans. They measured the height and length in fathoms, elbows, arshins, spans. Nobody objects that the human body has a golden proportion? The length of the arm from the fingertips to the armpit refers to the distance from the same point to the elbow as it refers to the size of the palm.
The famous French architect Le Corbusier used the growth of the owner as a starting unit to calculate the parameters of the future home and interior. All his works are truly individual and harmonious.
The application of this norm in, as an architecturally designed space, is combined with the concepts of self-organization, recursion, asymmetry, and beauty.
What is it? The segments of the golden ratio are expressed by an infinite irrational fraction, decimal value which is approximately equal to the number Ф≈1.618 or Ф≈1.62. In other words: if we take the whole and divide it into two parts so that one of them is 62%, and the other is 38%, we get the Golden Ratio.
Golden rectangle: when the length of the larger side is divided by the length of the smaller one, we obtain the number F. When dividing the smaller by the larger, the inverse value is φ ≈ 0.618.
Golden isosceles triangle: if the ratio of the size of one side and the size of the base is the golden number Ф; the angle between equal sides is 36 °.
Kepler's golden right-angled triangle combines the Pythagorean theorem and the ZS: the ratio of the squares of its sides is 1.618.
Watch an informative video on the topic
Any person who, at least indirectly, had to deal with the geometry of spatial objects in interior design and architecture, is probably well aware of the principle of the golden ratio. Until recently, several decades ago, the popularity of the golden section was so high that numerous supporters of mystical theories and the structure of the world call it the universal harmonic rule.
Another thing is surprising. The reason for the biased, almost mystical attitude towards such a simple numerical dependence was several unusual properties:
Today, the golden ratio is found in the structure of the body of animals, shells and shells of mollusks, the proportions of leaves, branches, trunks and root systems in a fairly large number of shrubs and grasses.
Many followers of the theory of universality of the golden section have repeatedly attempted to prove the fact that its proportions are the most optimal for biological organisms in the conditions of their existence.
Typically, the structure of the shell of Astreae Heliotropium, one of the marine molluscs, is given as an example. The carapace is a coiled calcite shell with a geometry that almost coincides with the proportions of the golden ratio.
A clearer and more obvious example is a regular chicken egg.
The ratio of the main parameters, namely, large and small focus, or distances from equidistant points of the surface to the center of gravity, will also correspond to the golden ratio. At the same time, the shape of the bird's egg shell is the most optimal for the survival of the bird as a biological species. In this case, the strength of the shell does not play a major role.
For your information! The golden ratio, also called the universal proportion of geometry, was obtained as a result of a huge number of practical measurements and comparisons of the sizes of real plants, birds, animals.
The ancient Greek mathematicians Euclid and Pythagoras knew about the golden ratio of the section. In one of the monuments of ancient architecture - the Cheops pyramid, the aspect-to-base ratio, individual elements and wall bas-reliefs are made in accordance with the universal proportion.
The method of the golden section was widely used in the Middle Ages by artists and architects, while the essence of the universal proportion was considered one of the secrets of the universe and was carefully hidden from the common man in the street. The composition of many paintings, sculptures and buildings was built strictly in accordance with the proportions of the golden section.
For the first time, the essence of the universal proportion was documented in 1509 by the Franciscan monk Luca Pacioli, who had brilliant mathematical abilities. But the real recognition took place after the German scientist Zeising carried out a comprehensive study of the proportions and geometry of the human body, ancient sculptures, works of art, animals and plants.
In most living objects, some body sizes obey the same proportions. In 1855, the scientist concluded that the proportions of the golden section are a kind of standard for the harmony of body and form. We are talking, first of all, about living things, for a dead nature, the golden ratio is much less common.
It is easiest to imagine the proportion of the golden ratio as the ratio of two parts of the same object of different lengths, separated by a point.
Simply put, how many lengths of a small segment will fit inside a large one, or the ratio of the largest segment to the entire length of a linear object. In the first case, the golden ratio is 0.63; in the second, the aspect ratio is 1.618034.
In practice, the golden ratio is just a proportion, the ratio of segments of a certain length, sides of a rectangle or other geometric shapes, related or conjugate dimensional characteristics of real objects.
Initially, the golden proportions were deduced empirically using geometric constructions. There are several ways to build or derive harmonic proportions:
For your information! Unlike the classic golden ratio, the architectural version assumes a 44:56 aspect ratio.
If the standard version of the golden ratio for living creatures, painting, graphics, sculptures and antique buildings was calculated as 37:63, then the golden ratio in architecture from the end of the 17th century was increasingly used 44:56. Most experts consider the change in favor of more "squared" proportions to be widespread in high-rise construction.
If the natural manifestations of the universal section in the proportions of the bodies of animals and humans, the stem base of plants can still be explained by evolution and adaptability to the influence of the external environment, then the discovery of the golden section in the construction of houses in the XII-XIX centuries was a certain surprise. Moreover, the famous ancient Greek Parthenon was built in compliance with the universal proportion, many houses and castles of wealthy nobles and wealthy people in the Middle Ages were deliberately built with parameters very close to the golden ratio.
Many of the buildings that have survived to this day indicate that the architects of the Middle Ages knew about the existence of the golden ratio, and, of course, when building the house, they were guided by their primitive calculations and dependencies, with the help of which they tried to achieve maximum strength. The desire to build the most beautiful and harmonious houses in the buildings of royal residences, churches, town halls and buildings of particular social importance in society.
For example, the famous Notre Dame Cathedral in its proportions has many sections and size chains corresponding to the golden ratio.
Even before the publication of his research in 1855 by Professor Zeising, at the end of the 18th century, the famous architectural complexes of the Golitsyn hospital and the building of the Senate in St. Petersburg, the Pashkov House and the Petrovsky Palace in Moscow were built using the golden ratio proportions.
Of course, houses with strict observance of the rule of the golden ratio were built earlier. It is worth mentioning the monument of ancient architecture of the Church of the Intercession on the Nerl, shown in the diagram.
All of them are united not only harmonious combination forms and high quality of construction, but also, first of all, the presence of the golden ratio 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 Church of the Intercession dates back to the 13th century, but the building received its modern architectural appearance at the turn of the 17th century as a result of restoration and restructuring.
The ancient architecture of buildings and houses of the Middle Ages remains attractive and interesting for modern man for many reasons:
Important! When designing a house and developing the appearance, medieval architects applied the rule of the golden ratio, unconsciously using the peculiarities of the perception of the human subconscious.
Modern psychologists have experimentally proved that the golden ratio is a manifestation of a person's unconscious desire or reaction to a harmonious combination or proportion in sizes, shapes and even colors. An experiment was conducted, during which a group of people who were unfamiliar with each other, did not have common interests, different professions and age categories, were offered a number of tests, among which was the task of bending a sheet of paper in the most optimal proportion of the sides. According to the test results, it was found that in 85 cases out of 100, the subjects were bent almost exactly along the golden ratio.
Therefore, modern science believes that the phenomenon of universal proportion is a psychological phenomenon, and not the action of any metaphysical forces.
The principles of applying the golden ratio have become extremely popular in the construction of private houses in the past few years. The ecology and safety of building materials have been replaced by the harmonious design and the correct distribution of energy inside the house.
The modern interpretation of the rule of universal harmony has long spread beyond the usual geometry and shape of an object. Today, the rule is obeyed not only by the dimensional chains of the length of the portico and pediment, individual elements of the facade and the height of the building, but also the area of rooms, window and doorways, and even the color scheme of the interior of the room.
The easiest way to build harmonious house on a modular basis. In this case, most departments and rooms are made in the form of independent blocks or modules, designed in compliance with the rule of the golden ratio. It is much easier to construct a building in the form of a set of harmonious modules than to build a single box, in which most of the facade and interior spaces must be within the rigid framework of the proportions of the golden ratio.
Many construction firms that design private households use the principles and concepts of the golden ratio to increase estimates and create an impression of deep study of the structure of the house for clients. As a rule, such a house is declared as very comfortable and harmonious to use. Correctly selected ratio of room areas guarantees mental comfort and excellent health of the owners.
If the house was built without taking into account the optimal ratios of the golden ratio, you can re-plan the rooms so that the proportions of the room correspond to the ratio of the walls in a ratio of 1: 1.61. For this, furniture can be moved or additional partitions can be installed inside the rooms. Similarly, the dimensions of window and door openings are changed so that the width of the opening is less than the height door leaf 1.61 times. Furniture planning is carried out in the same way, household appliances, wall and floor decoration.
Harder to choose color scheme... In this case, instead of the usual ratio of 63:37, the followers of the golden rule adopted a simplified interpretation - 2/3. That is, the main color background should occupy 60% of the room space, no more than 30% is given to the shading color, and the rest is assigned to various related tones, designed to enhance the perception of the color scheme.
The interior walls of the room are divided by a horizontal belt or border at a height of 70 cm, the installed furniture should be commensurate with the height of the ceilings according to the ratio of the golden ratio. The same rule applies to the distribution of lengths, for example, the size of a sofa should not exceed 2/3 of the length of the wall, and the total area occupied by furniture refers to the area of the room, as 1: 1.61.
The golden ratio is difficult to en masse in practice due to only one cross-section value, therefore, when designing harmonious buildings, they often resort to a number of Fibonacci numbers. This allows you to expand the number possible options proportions and geometric shapes of the main elements of the house. In this case, a series of Fibonacci numbers, linked by a clear mathematical relationship, is called harmonic or golden.
In the modern method of housing design based on the principle of the golden ratio, in addition to the Fibonacci series, the principle proposed by the famous French architect Le Corbusier is widely used. In this case, the height of the future owner or the average height of a person is chosen as the starting unit of measurement by which all parameters of the building and the interior are calculated. This approach allows you to design a house that is not only harmonious, but also truly individual.
In practice, according to the reviews of those who decided to build a house according to the rule of the golden ratio, a well-built building is indeed quite convenient for living. But the cost of the building is due to individual design and the use of building materials of non-standard sizes increases by 60-70%. And this approach is nothing new, since most of the buildings of the last century were built specifically for the individual characteristics of the future owners.
Another tool in the architect's arsenal is scale and proportion. They refer to how the individual parts of a building relate to each other and how the whole unusual house in harmony with the surrounding landscape.
Note that scale does not necessarily mean size. Apartments can be quite large, but have a comfortable and intimate environment for a person. And vice versa. V small house you can live perfectly fine using small elements and other design features.
Some designers and architects intuitively design buildings with great proportions, while others use systems such as the golden ratio. Today we'll show you how to use these tools to design your own creative masterpiece.
This mansion was divided into separate areas, each with own roof, giving it a visually smaller appearance. Materials, palette and proportions tie the different parts together, as well as the residence with the surrounding hills and green spaces.
Project by Mahoney Architects & Interiors
A house in a field or in another large area should have the effect of presence. To do this, use bold colors or architectural details that accentuate its appearance.
Decor by Eck | MacNeely Architects inc.
Reduce the size of the architectural details, as there is a direct relationship between the human body and the dimensions of the structure.
Project from Bud Dietrich, AIA
Determine the height and width of the room using the golden ratio, a technique that was developed a thousand years ago.
Sketch by Bud Dietrich, AIA
In this room, the height of the ceiling and the arrangement of decorative panels on brickwork the wall was determined by a specialist on the basis of dimensional rules.
Bedroom by Bud Dietrich, AIA
Pick up stylish furniture , accessories and finishes to keep the human scale alive. A fireplace, a seating area and a carpet create an intimate atmosphere in a fairly large room with a high ceiling.
Living Room by Christopher A Rose AIA, ASID
In this tiny room, a vaulted ceiling, fireplace and balustrades create a sense of spaciousness and volume.
Living room by Eck | MacNeely Architects inc.
Moreover, this architectural element allows you to create a more comfortable atmosphere in the room.
Design by Lisa Henderson - Harvest Architecture
Pay attention to the following photo, the ledge from the roof is harmoniously combined with the overall structure thanks to the use of similar tiles and window blocks.
Facade by One Room at a Time, Inc.
Categories:Today, when the technical revolution is over, modern construction possibilities allow you to make almost any imagination of an architect. In individual construction, we see many different architectural projects, structures and materials. Do we like every house? There are houses that are just good, and there are those that are pleasing to the eye. The latter are somewhat similar to old buildings, although they look completely different. Each of you was once in old houses, they have something fascinating, something special. What is there in them that is not in others? And why is not every modern building so pleasing to the eye and the feeling of beauty in your heart?
Earlier, at the dawn of architecture, an architect was called the "Architect". A good architect created and embodied his buildings using the golden proportion. It is the buildings created according to the golden ratio that look the most beautiful and harmonious for people.
The Golden Ratio is a division of some value in the ratio of 62% and 38% ( = 1: 1.618). For example: The ancient Greek Parthenon is striking in its grandeur and proportionality. (Fig 1)
Fig 1 Parthenon
Ancient architects found the golden proportion in nature. A shell, a flower, waves, trees, the universe are built according to the golden ratio ... (Fig. 2)
Fig 2. The golden proportion in nature
Man is also created according to the golden ratio. (Fig 3) For example, from the 2nd to the 4th month of pregnancy, when the baby's body is being actively formed, the mother's tummy grows in accordance with the golden ratio.
Rice. 3. Vitruvian Man (Drawing by Leonardo da Vinci)
Not only my mother's tummy, but all parts of our body are in tune with each other in accordance with the golden ratio. Architect Le Corbusier in 1948 displayed the proportional system of the human body. There are other examples, such as the old Russian measure "Sazheni". The only difference is that Le Corbusier's initial value is human height - 1.82 m. And the folk fathom is equal to height - 1.76 m.
It is very convenient to use the golden proportion to create houses - in order to maintain harmony in nature and create the most comfortable space inside. To build quality home it is necessary to take into account 3 fundamental rules that were formulated by the architect Vitruvius in the 1st century BC. - "Benefit - Strength - Beauty". And today, these rules are indisputably the key to quality architecture.
The domed house we built has the following benefits for the owner's family:
There are designs in which it is extremely difficult to take into account the benefits, strength, and beauty. For example, modern "Steklyashki" - huge glass buildings reflecting clouds - are useful, durable, but not always beautiful. The item "Beauty" most often creates additional expenses. For example, buildings in the style 
Baroque, the decoration of the facade of which sometimes took more money than the construction of the load-bearing part. And there are buildings that in themselves embody the naturalharmony, which leads to minimal costs.
One of the geometric shapes that has all three qualities and has its prototypes in the nature and architecture of antiquity is the dome. Domes are different.
For example, the Cathedral of St. Perth in the Vatican is one of the oldest buildings (1626). Several generations of great masters worked on its creation:
Bramante, Raphael, Michelangelo, Bernini. The dome of the cathedral rises to a height of 136.57 meters. (Fig 4) This is the tallest dome in the world. Michelangelo designed a hemispherical dome. However, later the structure was considered not strong enough, and the dome acquired an elongated egg-shaped silhouette.
From the time of Michelangelo, architecture and construction have advanced very strongly. New technologies and materials have been created, which have allowed significantly more people to build a house for themselves, combining the best architectural and technological solutions.
Dome houses allow you to combine the golden proportion and the three rules of the ancient architect. For private houses it looks like this (Figure 5):
Benefit (convenience)
Strength (safety):
Beauty (harmony)
Dome houses designed according to the golden ratio are the harmony of nature. Please note that everything in nature is in the best balance, living in a house based on the golden ratio, as on a "solid foundation", you will feel the harmony of life and inner balance in all areas: at work, family, leisure, and inner soul comfort.
Architect Olga Voron
Gileva Anastasia
XIV municipal competition
educational research work of students
"Golden section" in the architecture of a traditional peasant house
Work completed:
Gileva Anastasia Vasilievna,
student of grade 8A MOU SOSH №8
Supervisor:
Gileva Irina Ivanovna,
teacher informatics MOU secondary school №8
Zoya Egorovna Golubleva,
teacher of mathematics MOU secondary school №8
Krasnovishersk - 2010
Introduction | |
Chapter 1 "Golden Ratio" | |
Chapter 2 Features of building peasant houses | |
Bychina, Gileva, Paleva, Semina | |
Bychina, Gileva, Paleva, Semina for the presence of a "golden ratio" relationship | |
Conclusion | |
Literature | |
Application |
Introduction
There are things that cannot be explained. Here 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 not both. You will sit so that the ratio of one part of the bench to the other, relative to your body, is approximately 1.62. A simple thing, absolutely instinctive ... Sitting on the bench, you made the "golden ratio".
They knew about the golden ratio even in ancient egypt and Babylon, India and China. The great Pythagoras created a secret school where the mystical essence of the "golden section" was studied. Euclid applied it, creating his geometry, and Phidias - his immortal sculptures. Plato said that the universe is arranged according to the "golden ratio". And Aristotle found the correspondence of the "golden section" to 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.
You will certainly see this proportion in the curves of sea shells, and in the form of flowers, and in the form of beetles, and in a beautiful human body. Inanimate nature does not know what the “golden ratio” is, but it is used in architecture and sculpture, in painting and mathematics, in music and poetry ...
Egyptian pyramids, buildings of 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 peasant, not knowing the basics of architecture, "lift" such proportional buildings?
Looking at the abandoned huts of the villages of Bychina, Gilev, Paleva, Semina, ... we asked ourselves: is there a golden ratio in the architecture of these ancient houses?
The purpose of our work: to investigate the architecture of peasant huts in the villages of Bychina, Gilev, Paleva, Semin for the presence of the golden ratio.
To achieve this goal, it is necessary to solve the following tasks:
Chapter 1 "Golden Ratio"
1.1. The "golden proportion" and related ratios
The question of the mathematical prerequisites of beauty, of the role of mathematics in art, worried the ancient Greeks, and they inherited their interest from previous civilizations. In our time, geometry - necessary element general education and culture - is of great historical interest, has serious practical application and has inner beauty.
Johannes Kepler wrote: “Geometry owns 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 with the value of gold, the second can be called a precious stone. "
There are many ratios of the "golden ratio", but in his work mWe will consider only two relationships: the “golden ratio” of the segment and the “golden rectangle”. This is no coincidence, since we will investigate the linear dimensions of houses (height, length and width).
Let's follow the example of L.S. Sagatelova. and determine the ratio of the segments at the "golden ratio" and the aspect ratio of the "golden rectangle".
The division of a segment in the middle and extreme ratio is called the "golden section". Another name has been established in history - the "golden proportion".
Let C AB and produces, as they say, the "golden section" of the segment.
(1)
CB: AB = AC: CB
The golden ratio is a division of a segment in which a large part refers to the whole, as a smaller part to a larger one.
If the length of the segment AB is denoted by a, and the length of the AC - through x, then a-x - the length of the segment CB, and the proportion (1) will take the form:
(2)
In proportion, as you know, the product of the extreme terms is equal to the product of the means and the proportion (2) is rewritten as:
x 2 = a (a-x)
We get a quadratic equation:
x 2 + ax-a 2 = 0.
The length of the segment is expressed as a positive number, therefore, of the two roots
x 1,2 = 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 creations it is found many times. The number is irrational, it is written like this: = 0.61803398 ...
But in practice, they use a number taken to the nearest thousandths of 0.618, or to hundredths of 0.62, or to tenths of 0.6.
If, then, and a-x = 0.38a.
Thus, parts of the "golden ratio" are approximately 62% and 38% of the entire segment.
During the Renaissance, the golden ratio was very popular among artists, sculptors and architects. So, choosing the size of the picture, the artists tried to make the relationship of its sides equal. This rectangle was called "golden".
The algorithm for constructing the "golden" rectangle has come down to us since the time of Euclid:
Let's find the exact aspect ratio of the constructed rectangle.
Let us denote the side of the original square by a ; express through a the length of the diagonal AB is the hypotenuse of a right-angled 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 is. 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 section" of the segment - 62% and 38%, as well as for 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 one.
1.2. "Golden proportion" in architecture
The golden proportion is a mathematical concept. But it is a criterion for harmony and beauty, and these are already categories of art.
In books about the golden ratio, you can find a remark that in architecture, as in painting, everything depends on the position of the observer, and that if some proportions in a building on the one hand 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 (V century BC) - the temple of Athena.
The dimensions of the Parthenon are well understood. 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 a rectangle has a size, therefore, the rectangle of the facade is the starting point in the construction of the Parthenon's geometry.
Many researchers, seeking to reveal the secret of the harmony of the Parthenon, searched for and found the "golden" proportion in the ratios of its parts.
A regular series of a regular series of golden proportions has been established. Taking the width of the front facade of the temple as a unit, the researchers obtained a progression consisting of 8 members of the row:
1; where = 0.618.
Careful measurements of the Parthenon have shown 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 a distance appear to be bent in the middle.
Another example from the architecture of antiquity is the Pantheon.
The famous Russian architect M. Kazakov widely used the “golden ratio” in his work. His talent was multifaceted, but to a greater extent he was revealed in numerous implemented projects 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 now called the First Clinical Hospital named after N.I. Pirogov (Leninsky prospect, 5). Another architectural masterpiece of Moscow - Pashkov's house - is one of the most perfect pieces of architecture by V. Bazhenov (Appendix 1).
The construction of village houses was carried out by peasants who did not possess knowledge of the basics of architecture in general and the concept of the "golden section" in particular. However, inproportional relationships can be distinguished in the structure of traditional rural houses. 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 a proportional structure of a peasant dwelling house 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.
A special place among different systems proportioning takes the "golden ratio". However, the use of the proportions of the "golden ratio" 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 the national dwelling, relations that exactly correspond to the golden ratio are quite rare, and much more often - very close to it. ...
We have not found scientific papers dedicated to the direct study of the use of the "golden ratio" in the architecture of a traditional peasant house. The more interesting the topic we are researching.
Chapter 2 Features of the construction of peasant houses.
2.1. Technology of building a peasant house in villagesBychina, Gileva, Paleva, Semina.
According to Mark Yakovlevich Gilev, a resident of the village of Bychin, the technology of building a peasant house included several stages:
The first stage is logging. For building a house, they choose spruce, pine, less often fir. The forest is harvested in late autumn, for the old month. The forest lies all winter.
The second stage is timber processing. In the spring, logs are scraped from the bark and the blockhouse is chopped. Prepare the material for the floor and roof, for this "dissolve" the logs on the boards. At the same time, moss is being harvested. Sphagnum is usually used.
The third stage is drying. In summer, cooked log cabins, moss and boards dry naturally. Drying boards are not laid tightly, so that the "air moves".
The fourth stage is raising the log house. In the old days, at the base of the house were placed racks made of larch or cedar - the most resistant to decay of coniferous species. Currently, the prepared log house is being laid on the foundation. The logs are shifted with moss.
The fifth stage is the final one. A year later, when the blockhouse is settled, they carry out carpentry work: they close gable roof, build a ceiling, put windows, doors, layinsulated double floors with earthen backfill etc.
Usually, 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 log house are 6x7, 7x7 or 7x8 - closer to the square. The larger the house, the higher the frame is raised (the number of crowns - horizontal rows of logs - increases). There are no definite norms, the builder does everything "by eye" as he likes.Logs were usually not spliced along the length, the dimensions of the building were increased by cutting another frame to the existing one or by installing a new frame close to the old one.
Observations show that village houses, although they are based on a frame close to a square, are more reminiscent of elongated parallelepipeds in shape. This is achieved by attaching outbuildings to the main frame. Both the living quarters and the outbuildings are under the same roof.
The technology described above, as we can 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 villagesBychina, Gileva, Paleva, Semin for the presence of the "golden ratio" relationship.
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 uppermost crown 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 erected under one roof, that is, the internal division of the house into zones was not taken into account.
The data obtained are presented in Table 1.
P / p No. | House name | Linear dimensions of the whole house | ||
Height | Width | Length |
||
D. Semina Gilev Arkady Semenovich (year of construction - ...) | ||||
D.N-Bychina Primary school building (built in 1916) | ||||
D.N-Bychina Mitrakov Andrey Egorovich (built in 1930) | ||||
D.V-Bychina Gilev Mark Yakovlevich (built -1930) | ||||
D.V-Bychina Bychin Egor Vasilievich (year of construction - ...) | 6.8 (2nd floor) | |||
D.N-Bychina (year of construction - late 19th century) | 8 (2nd floor) | |||
7 | D. Paleva Gilev Nikolay Konstantinovich (built in 1950) (built in 1978) | 4,2 | 6,8 | 8,5 |
D. Bychina Bychin Fedor Andreevich (built ~ 1820) | 10,5 |
|||
D.Ivachina Bychina Natalia Yakovlevna (built in 1924) | ||||
11 | D. Paleva Sobyanina Antonina Yakovlevna (built in 1931) new house | 2,9 | 4,9 | 8,5 |
D. Paleva Mitrakov Alexander Egorovich (built in 1910) | 3,45 | 12,4 |
||
D. Semina Mitrakova Lyudmila Alexandrovna (built in 1963) | 10,9 |
The obtained data were processed using the Ms Excel spreadsheet processor (Table 2). Correlation coefficients were found to determine the presence of a relationship between the values 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 coefficients 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 considered arrays.
The calculation of the values of the ratios of width to height, length to height and length to width of the house confirmed the above.
table 2
House no. | 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 |
The analysis of the obtained results 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 front part of the building faces the street and its appearance great attention was paid during the construction. The builder strove to give the facade a harmonious form, based on his intuition.
The rest of the dimensions received less attention and, as studies show, 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 striven for beauty and harmony. Mathematics asserts that the basis of beauty is the harmonious ratio of the parts of the whole - the "golden proportion". A person notices this proportion in all living things and strives to take into account and use it when creating his works.
In our work, we set out to find the ratio of the "golden ratio" in the architecture of a peasant house.
The study of 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 shown that in the construction of a traditional peasant house, the use of proportions of the "golden ratio" is based more on intuition than on deliberate and accurate calculation. Quite rarely, there are relationships that exactly correspond to the "golden ratio", and much more often - very close to it.
We examined the basic rectangles: the front part, the base of the house, the end part. The data obtained with the help of correlation analysis prove the presence of the "golden section" in the facade of the building and its absence in the rest of the base rectangles.And this is no coincidence, since the front part of the building faces the street and great attention was paid to its appearance during construction. The builder strove to give the facade a harmonious form, based on his intuition. Less attention was paid to the rest of the sizes and, as studies show, their value depended on the size of the outbuildings, that is, it was directly related to the practical needs of the owners of the house.
Literature
Annex 1
Pashkov House in Moscow
Senate in the Kremlin
Golitsyn hospital in Moscow
