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"BEAUTY SHOULD MEET A STRICT NUMBER" B. PASCAL. For many centuries, artists have used the concept of the golden ratio to build harmonic compositions.
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THE GOLDEN SECTION IN ARCHITECTURE Sculptural structures, monuments are erected to perpetuate significant events, to preserve in the memory of descendants the names of famous people, their exploits and deeds. It is known that even in ancient times the basis of sculpture was the theory of proportions. Relationships of parts human body associated with the golden section formula. The proportions of the "golden section" create the impression of harmony of beauty, so the sculptors used them in their works.
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GOLDEN SECTION IN PAINTING During the Renaissance, the golden ratio was very popular among artists. For example, in most picturesque landscapes, the horizon line divides the canvas in height in a ratio close to the golden ratio. And, choosing the size of the picture itself, they tried to ensure that its sides were in a golden ratio.
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The canvas on which is written " The Last Supper"Salvador Dali has the shape of a golden rectangle. Smaller golden rectangles were used by the artist when placing the figures of the twelve apostles.
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The golden rectangle has many interesting properties. If you cut off a square from it, you get a golden rectangle again. And so you can continue ad infinitum. If we connect the vertices of the squares with a smooth line, we will get a curve called the GOLDEN SPIRAL.
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If the golden rectangle was used by artists to create a sense of balance and peace in the viewer, then the golden spiral was used to express disturbing, rapidly developing events. The sketch of the engraving "Massacre of the Innocents", made by Raphael, is distinguished by the dynamism and drama of the plot. The figure shows a golden spiral along which the main figures of the exposition are located.
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Many galaxies are also twisted along the golden spiral, in particular, the Galaxy of the Solar System. golden ratio, the golden rectangle, and the golden spiral are mathematical symbols for the ideal balance of form and growth. The great German poet Goethe considered them a mathematical symbol of life and spiritual development.
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Proportionality is the most striking, visible, objective and mathematically regular expression of architectural harmony. Proportion is a mathematical pattern that has passed through the soul of an architect. This is the poetry of number and geometry in architectural language. The language of proportions was spoken by architects of all times and architectural trends: ancient Egyptians and Greeks, medieval masons and ancient Russian carpenters, representatives of baroque and classicism, constructivists and modernists. website
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Architecture is triune: it eternally combines the logic of the scientist, the craft of the master and the inspiration of the artist. "Strength - usefulness - beauty" - this is the famous formula of a single architectural whole, derived by the ancient Roman theorist of architecture Marco Vitruvius. People have always sought to achieve harmony in architecture. Thanks to this desire, new inventions, designs and styles were born. "Strength - benefit - beauty"
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Harmony in nature and harmony in architecture find the same mathematical expression in the law of the golden ratio. Why is the law of the golden section so often manifested in architecture? To achieve harmony in works of art, the principle of Heraclitus must be fulfilled: "out of everything - one, from one - everything." Harmony in an architectural structure depends not so much on its size, but on the ratio between the sizes of its constituent parts.
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Ancient Egyptian pyramids The design of the ancient Egyptian pyramid is the simplest, strongest and most stable, its mass decreases as the height above the ground increases. The shape of the pyramid, emphasized by its huge size, gives it a special beauty and grandeur, evokes a feeling of eternity, immortality, wisdom and peace.
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Pyramid of Cheops, Egypt The architect Khesira is the builder of the first pyramid in Ancient Egypt. In his hands are two sticks - two standards of measure, their ratio is 1 / √ 5 = 0447!
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Secrets of ancient proportions. Parthenon
The pinnacle of Greek architecture is the temple of the goddess Athena Parthenos (Virgin), built in 447-438 BC. architects Iktin and Kallikrat in Athens
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Many researchers who sought to uncover the secret of the harmony of the Parthenon searched for and found the golden section in the ratios of its parts. If we take the end facade of the temple as a unit of width, then we get a progression consisting of eight members of the series: 1: j: j 2: j 3: j 4: j 5: j 6: j 7, where j = 1.618
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The Parthenon was and remains the most perfect of architectural structures, architectural sculpture, a marble code of laws of ancient architecture. The Parthenon is the most striking example of the use of the golden ratio in architecture.
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Cathedral of Notre Dame de Paris
Notre Dame Cathedral is the most majestic monument of early Gothic. In the proud regularity of the western facade of the cathedral horizontal lines still compete with vertical ones. The facade wall has not yet disappeared, but it has already acquired lightness and even transparency.
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Notre-Dame de Paris Cathedral The proportional basis of the western facade of the Notre-Dame Cathedral is a square, and the height of the towers of the facade is equal to half the side of this square ...
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Church of the Intercession of the Virgin on the Nerl
The cross-domed scheme underlies the Church of the Intercession on the Nerl. It is characterized by a calm balance based on symmetry. The temple seems surprisingly light, looking up.
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At the core architectural plan church lies a rectangle with sides 1 and √2 and a diagonal √5, these numbers easily guess all the components with which the golden ratio is expressed. Church of the Intercession of the Virgin on the Nerl
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Church of the Ascension in Kolomenskoye
The Temple of the Ascension is not only the anthem of Russia spreading its wings, but also the architectural anthem of geometry
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The geometry of domes - the geometry of a burning candle
In Russian church art, the desire to combine the aesthetics of feelings with the aesthetics of numbers, the beauty of a freely flowing rhythm with the beauty of a regular geometric body, manifested itself. M.V. Alpatov
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St Basil's Church
It is difficult to find a person who would not know St. Basil's Cathedral on Red Square. This temple is special, it is distinguished by an amazing variety of shapes and details, colorful coatings, it has no equal in our country. The architectural decoration of the entire cathedral is dictated by a certain logic and sequence in the development of forms.
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Exploring the temple, we came to the conclusion about the predominance of the golden ratio in it. If we take the height of the cathedral as a unit, then the main proportions that determine the division of the whole into parts form a series of the golden section: 1: j: j 2: j 3: j 4: j 5: j 6: j 7, where j = 0.618 Basil's Cathedral Blessed
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Modulor Le Corbusier
The idea of building a modulor is ingeniously simple. Modulor is a series of the golden section. “Modulor is a gamma of proportions that makes bad things difficult and good things easy” A. Einstein “Modulor is a gamma. The musician has a scale and creates music according to his abilities - banal or beautiful" Le Corbusier
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Radiant house in Marseille - embodiment common sense, clear, straightforward and rational. The chapel in Ronchamp is something irrational, plastic, sculptural, fabulous. The only thing that unites these two monuments of architecture is the modulor, the architectural range of proportions common to both works. Radiant house in Marseille Chapel in Ronchamp
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What unites all systems of proportionality?
Any proportional system is the basis, the skeleton of an architectural structure, this is the scale, or rather the mode in which architectural music will sound. Pskov Kremlin Australia Sydney Belgium Brussels Russia Tsarskoye Selo Kizhi
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Homework
Themes of reports and communications. Proportions and measures in the architecture of Ancient Rus'. Proportions of modern architectural ensembles of Russia.
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Contents The concept of the "golden section" "Golden section" of the segment "Golden" rectangle "Golden" triangle Five-pointed star "Golden section" in anatomy "Golden section" in sculpture "Golden section" in modern architecture "Golden section" in ancient architecture
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The golden section The golden section is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller segment is related to the larger one as the larger one is to the entire segment. This ratio is approximately equal to 0.618. a: b = b: c or c: b = b: a. Formula
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The “golden section” of the segment From point B, a perpendicular equal to half AB is restored. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is plotted, ending with point D. The segment AD is transferred to the straight line AB. The resulting point E divides the segment AB in the ratio of the golden ratio. The properties of the golden section are described by the equation: x * x - x - 1 \u003d 0. The solution to this equation:
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"Golden" Rectangle If a square is cut off from a rectangle, the "golden" rectangle remains again, and this process can be continued indefinitely. And the diagonals of the first and second rectangles will intersect at point O, which will belong to all the resulting “golden” rectangles.
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"Golden" Triangle The lengths of the bisectors of the angles at its base are equal to the length of the base itself.
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Five-pointed star Each end of the pentagonal star is a "golden" triangle. Its sides form an angle of 36 ° at the apex, and the base laid on the side divides it in proportion to the golden ratio
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"Golden Section" in anatomy Human height is divided in golden proportions by the belt line, as well as the line drawn through the tips of the middle fingers of the lowered hands, and the lower part of the face is divided by the mouth.
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The "Golden Section" in sculpture The golden ratio of the statue of Apollo: the height of the depicted person is divided by the umbilical line in the golden section.
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The "Golden Section" in Modern Architecture The proportions of the Pokrovsky Cathedral on Red Square in Moscow are determined by eight members of the golden section series. Many members of this series are repeated many times in the intricate elements of the temple.
GOLDEN SECTION - a proportion to which the ancient magicians attributed special properties. If you divide an object into two unequal parts so that the smaller one will relate to the larger one, as the larger one to the entire object, the so-called golden ratio will arise. Simplified, this ratio can be represented as 2/3 or 3/5. It has been noticed that objects containing the "golden section" are perceived by people as the most harmonious. The "golden ratio" is found in the Egyptian pyramids, many works of art - sculptures, paintings, and even movies. Most artists used the "golden ratio" proportions intuitively. But some did it deliberately. So S. Eisenstein artificially built the film "Battleship Potemkin" according to the rules of the "golden section". He broke the tape into five parts. In the first three, the action takes place on a ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city takes place exactly at the point of the golden ratio. Yes, and in each part there is a turning point, which occurs according to the law of the golden section. In the frame, scene, episode, there is a certain leap in the development of the theme: the plot, the mood. Since such a transition is close to the point of the golden ratio, it is perceived as the most regular and natural.
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 otherwise. The golden ratio gives the most relaxed ratio of the sizes of certain lengths. One of the most beautiful works of ancient Greek architecture is the Parthenon (V century BC). The Parthenon has 8 columns on the short sides and 17 on the long ones, the ledges are made entirely of squares of Pentile marble. The nobility of the material from which the temple was built made it possible to limit the use of coloring, common in Greek architecture, it only emphasizes the details and forms a colored background (blue and red) for the sculpture. The ratio of the height of the building to its length is 0.618. If we divide the Parthenon according to the golden section, we will get certain protrusions of the facade.
Another example from ancient architecture is the Pantheon. Also, the golden ratio can be seen in the architecture of the Notre Dame de Paris cathedral in France. The famous Russian architect M. Kazakov widely used the golden section in his work. His talent was multifaceted, but in more it was revealed in 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. The wonderful creation of V. Bazhenov has firmly entered the ensemble of the center of modern Moscow, enriched it. The external view of the house has remained almost unchanged to this day, despite the fact that it was badly burned in 1812. During the restoration, the building acquired more massive forms. The internal layout of the building has not been preserved either, which only the drawing of the lower floor gives an idea of. Many statements of the architect deserve attention today. About his favorite art, V. Bazhenov said: Architecture has three main subjects: beauty, calmness and strength of the building ... To achieve this, the knowledge of proportion, perspective, mechanics or physics in general serves as a guide, and all of them have a common leader is reason.
The length of the face of the pyramid in Giza is equal to a foot (238.7 m), the height of the pyramid is a foot (147.6 m). The length of the edge, divided by the height, leads to the ratio Ф \u003d Height of a foot corresponds to 5813 inches () - these are numbers from the Fibonacci sequence. These interesting observations suggest that the construction of the pyramid is based on the proportion Ф = 1.618. Mexican pyramids also obey such proportions. Only in the cross section of the pyramid is a staircase-like shape visible. The first tier has 16 steps, the second has 42 steps and the third has 68 steps.
The "golden ratio" is found in the Egyptian pyramids, many works of art - sculptures, paintings, and even movies. Most artists used the "golden ratio" proportions intuitively. But some did it deliberately. So S. Eisenstein artificially built the film "Battleship Potemkin" according to the rules of the "golden section". He broke the tape into five parts. In the first three, the action takes place on the ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city takes place exactly at the point of the golden ratio. Yes, and in each part there is a turning point, which occurs according to the law of the golden section. In the frame, scene, episode, there is a certain leap in the development of the theme: the plot, the mood. Since such a transition is close to the point of the golden ratio, it is perceived as the most regular and natural.
For many millennia, the shape of a tetrahedral pyramid has been the subject of reflection for an inquisitive mind. Sections of the Space of the Universe with sufficiently dense material objects (for example, the Solar System) are subject to changes (curvature) of their structure under the influence, including the mental activity of the Mind, inadequate to its Habitat. Disharmonious events in the near Cosmos and in the far Cosmos aggravate the situation. The main working hypothesis, with which specialists have been working for many years, sounds something like this: imagine the Space around us. For clarity, let's break it into cubes. We will see smooth planes, clear, slender lines - complete harmony around. Now let's put a curved mirror next to it and look into it. We will see how these even, slender lines and planes twisted and floated. Here is the curved space model. A man in a curved Space, whose structure has deviated from the state of Harmony, loses his bearings, he lives in a fog, becomes inadequate to his human essence. The consequence of the curvature of Space, the deviation of its structure from the state of Harmony are all earthly troubles: diseases, epidemics, crime, earthquakes, wars, regional conflicts, social tensions, economic cataclysms, lack of spirituality, a decline in morality.
The Pyramid in the zone of its activity directly or indirectly corrects the structure of the Space, brings it closer to the state of Harmony. Everything that is or falls into this Space begins to develop in the direction of Harmony. In this case, the probability of occurrence of all these troubles decreases. The dynamics of mitigation and elimination of all negative manifestations significantly depends on the size of the Pyramid, its orientation in space and compliance with all geometric relationships. With the doubling of the height of the Pyramid, its active influence intensifies ~ times.
Many have tried to unravel the secrets of the Giza pyramid. Unlike other Egyptian pyramids, this is not a tomb, but rather an unsolvable puzzle of numerical combinations. The key to the geometric and mathematical secret of the Giza pyramid, which had been a mystery to mankind for so long, was in fact transferred to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of each of its faces was equal to the square of its height. Area of a triangle = Area of a square =
