Presentation on the topic of central symmetry in architecture. Symmetry in architecture “Architecture - the most important thing is three things: beauty, tranquility and strength of the building

Ministry of Education of the Irkutsk Region
"Irkutsk Aviation College"
(GBPOUIO "IAT")
Project
in the discipline "Mathematics"
on the topic “Symmetry in architecture”
Completed:
1st year students
Groups PKS-15-2
Sharipov D.A.
Tolmachev M.V.
Kryuchkov V.V.
Supervisor:
Maksimova R.P.
Irkutsk 2015

Goal of the work

Get to know the main types
symmetry.
Identify how symmetry is used in
architectural structures.

Tasks

Describe the types of symmetry.
Consider examples.
Learn to distinguish types of symmetry in
architecture of my hometown.

Symmetry

We call the same thing symmetry
arrangement of equal parts in relation to
planes or lines. Symmetry
structure is one of the reasons for it
active influence on perception.

Asymmetry

From the point of view of mathematical concepts
asymmetry is just the absence of symmetry.
In architecture - symmetry and asymmetry are two opposing methods of natural
organization of spatial form.

Asymmetry

Asymmetrical compositions in progress
architectural developments arose as
embodiment of complex combinations of life
processes and environmental conditions.
Specific forms of such compositions
grow as a result of unique
combinations of factors. (Fig.1)
Fig.1

Dissymmetry

This is a partial lack of symmetry, its
a disorder expressed in the presence of some
symmetrical properties and the absence of others. (Fig.2)
Fig.2

Dissymmetry

Absolute symmetry in large and complex
structures, strictly speaking, is impossible.
Complexity of functional systems
causes partial deviations from the main one,
defining the character of the composition
symmetrical scheme.

The simplest types of symmetry

Mirror symmetry (Fig. 3), symmetry
left and right (Fig. 4)
Fig.3
Fig.4

Mirror symmetry

This is a mapping of space onto itself, with
in which any point Z goes to
symmetrical to it relative to the plane α
point Z1. (Fig.5)
Fig.5

Central symmetry

In addition to mirror symmetry
central or
rotational symmetry. In this case
transition of parts to a new position and
the formation of the original figure occurs
when this figure is rotated by a certain
angle around a point that is usually
called the center of rotation.

Axial symmetry

Central axial symmetry is
symmetry about the vertical axis,
line of intersection of two (or more
number) of vertical planes of symmetry.
(Fig.6)
Fig.6

Axis of symmetry

In orthogonal drawings, the plane
symmetry is depicted by a line, so it
often called the axis of symmetry. (Fig.7)
Fig.7

Axis of symmetry

An imaginary line dividing the body into
two equal halves.(Fig. 8)
Fig.8

Highest degree of symmetry
has a ball in the center of which
an infinite set intersects
axes and planes of symmetry. (Fig.9)
Fig.9

The meaning of symmetry in architecture

Symmetrical objects have high
degree of expediency, because
symmetrical objects have more
stability and equal functionality
in different directions.
Symmetry was used to create
religious and domestic buildings since ancient times
times From then until now
symmetry in human consciousness has become
an objective sign of beauty.

Fig.10
Fig.11

Symmetry in Old Russian buildings

There are many things in ancient Russian architecture
examples of intuitive or conscious
the use of symmetry, this and
bell towers (Fig. 12), guard towers
towers (Fig. 13). A clear imprint of symmetry
later buildings also bear on them:
stone Russian churches (Fig. 14),
palaces (Fig. 15).

Fig.12
Fig.14
Fig.13
Fig.15

Architectural structures of the city of Irkutsk

Sports Palace “Trud” - Symmetrical (Fig. 16)
Musical Theater named after. N.M. Zagursky -
Symmetrical (Fig. 17)
House of Music of Denis Matsuev – Disymmetrical
(Fig.18)
Fig.17
Fig.18
Fig.16

Conclusion

Architecture is an amazing field
human activity. It's cramped in there
intertwined and strictly balanced science,
technique art. Only proportionate
the harmonious unity of these principles makes
man-made structure
architectural monument, not subject to
time, like literary monuments,
sculpture, music.

Description of the presentation by individual slides:

1 slide

Slide description:

Symmetry in architecture “Architecture has three main things: beauty, tranquility and strength of the building. To achieve this, knowledge of proportion, perspective, mechanics or physics in general serves as a guide, and the common leader of all of them is reason.” V. Bazhenov

2 slide

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Plan: 1) Symmetry in architecture 2) Golden ratio 3) Parthenon and Pantheon 4) Pyramid of Cheops 5) Classicism 6) Senate building of the Moscow Kremlin 7) Pashkov House 8) Asymmetry and dissymmetry 9) St. Basil's Cathedral 10) Conclusions

3 slide

Slide description:

Symmetry in architecture Symmetrical objects have high functionality in different directions. All this led a person to think about the degree of expediency: greater stability and equal, that in order for a structure to be beautiful it must be symmetrical. Symmetry was used in the construction of religious and domestic buildings in Ancient Egypt. But symmetry is most clearly manifested in the ancient buildings of Ancient Greece. From then to the present day, symmetry in the human mind has become an objective sign of beauty. Maintaining symmetry is the first rule of an architect when designing any structure. Architectural structures created by man are for the most part symmetrical. They are pleasing to the eye and people consider them beautiful. What is this connected with?

4 slide

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Geometry has two treasures: one of them is the Pythagorean theorem, the other is the division of a segment in the average and extreme ratio... The first can be compared to a measure of gold, the second is more reminiscent of a precious stone. I. Kepler Golden ratio The golden ratio is the law of proportional connection between the whole and the parts that make up the whole, when the whole relates to the larger part as the larger part relates to the smaller

5 slide

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Parthenon The sacred hill and temple of Divine Athena, the Magnificent Parthenon, Having buried the forgotten ruins, aims to the gods of Olympus. N. Vasyutinsky PARTHENON, the main temple of the Athenian Acropolis, dedicated to Athena Parthenos (i.e. the Virgin), the patron goddess of the city. Construction began in 447 BC, the consecration of the temple took place at the Panathenaic festival in 438 BC, but decoration (mainly sculptural work) continued until 432 BC. The Parthenon is a masterpiece of ancient Greek architecture and a symbol of the Greek genius.

6 slide

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The figure shows a number of patterns associated with the golden ratio. When looking at the Parthenon at the location of the monumental gate at the entrance to the city (propylaea), the ratio of the rock mass at the temple also corresponds to the golden ratio. Thus, the golden proportion was already used when creating the composition of the temples on the sacred hill. The Parthenon has 8 columns on the short sides and 17 on the long sides. The projections are made entirely of squares of Pentilean marble. Sculptures The ratio of the building's height to its length is 0.618. Parthenon

7 slide

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The Pantheon is considered the most perfect of all classical monuments preserved in the Italian capital. For a long time it was believed that it was built in 27 AD, but excavations have shown that the Pantheon is a reconstructed structure from the time of Hadrian (1st century AD). In 609, the pagan Pantheon became a Christian temple of the Holy Virgin Mary. The internal diameter of the Pantheon and its height are 43 meters. The sky is visible through the hole in the dome. This creates an atmosphere of solemnity. Between the second and third chapels is Raphael's tomb. On the tombstone there is an inscription: “Here lies Raphael, who competed with Mother Nature herself, and she feared that he would surpass her in creativity.” Pantheon

8 slide

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Pyramid of Cheops The Egyptian Pyramid of Cheops is the oldest and at the same time the only wonder of the world that has survived to this day. It received its name from the name of its creator - Pharaoh Cheops (about 2551 - 2528 BC). The height of the Cheops pyramid is 138 meters (and initially it was even 147 meters). It is built from 2.3 million limestone blocks, which weigh an average of two and a half tons. The pyramid stands on a specially prepared plane, which gives a horizontal deviation of less than two centimeters. The base of the pyramid is square, and the length of one side is 227.5 meters. The faces of the pyramid are oriented to the cardinal points, and their angle of inclination to the base is 52 degrees.

Slide 9

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This name comes from the Latin word clafssicus, which means exemplary. The style itself developed in the 17th century in France, accepting the ancient heritage as the norm, as an ideal example in all types of arts - literature, painting, architecture, theatrical art. The architecture of Classicism is characterized by clarity and geometricism of forms, symmetry, logical layout and restrained decor. This style had a great influence on the formation of the appearance of our northern capital. Many palaces, squares and parks created by famous architects, founders of Russian Classicism - V.I. Bazhenov, K.I. Rossi, M.F. Kazakov, gave St. Petersburg the appearance of a majestic and completely European city. 1. Palace Square with the arch of the General Staff building - architect Rossi K.I. 2. Mikhailovsky Palace in St. Petersburg - architect Rossi K.I. Classicism

10 slide

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Senate Building The Senate building was built according to the design of the architect M.F. Kazakov in 1776-1787, interior decoration continued until 1790. The Round Hall of the Senate is rightfully considered an architectural masterpiece. Contemporaries called it the Russian Pantheon. The hall with a diameter of 24.7 m and a height of 27 m with a Corinthian order colonnade along the perimeter is covered with a coffered dome, at the base of which there are 24 skylights. Sculptural thematic bas-reliefs decorate the walls between the columns and windows.

11 slide

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The building's plan is an isosceles triangle with a courtyard divided by additional buildings into three parts: a central pentagonal one and triangular side ones. The main axis of the building runs along the axis of the triangle, on which the Dome of the hall is located, oriented towards Red Square. It is at the same time the center of one of its compositional axes. All rooms of the Senate are connected to each other by a corridor running along the perimeter of the courtyard. Senate building

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At the very end of Mokhovaya Street, on a hill rises a luxurious palace made of white stone, known in Moscow as the Pashkov House. The combination of ancient severity and solemnity with purely Moscow patterning makes it a masterpiece of original Russian classicism. The palace was built by one of the best Moscow architects Vasily Bazhenov in 1784-1786. commissioned by the wealthy landowner Pashkov. In 1812 the house was destroyed by fire. However, the architects, restoring the house according to drawings from the 18th century, managed to recreate it. Before the War of 1812, in the courtyard of the house there was a garden in which parrots, peacocks and other birds unknown to Muscovites lived. On Sundays this garden was open to everyone. One of the legends about the creation of the Pashkov House says that when Vasily Bazhenov was not allowed to build a palace comparable to the Louvre for Catherine II on the territory of the Kremlin, he, by order of a wealthy nobleman, captain-lieutenant of the Semenovsky Life Guards regiment Peter Pashkov, erected a palace opposite. Turning it back to the Kremlin. Pashkov House

Slide 13

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In addition to symmetry in architecture, one can consider antisymmetry and dissymmetry. Antisymmetry is the opposite of symmetry, its absence. An example of antisymmetry in architecture is St. Basil's Cathedral in Moscow, where symmetry is completely absent in the structure as a whole. However, it is surprising that the individual parts of this cathedral are symmetrical and this creates its harmony. And the square in the twilight of the night stands full of yesterday's execution; There are traces of fresh torment all around: Where is the corpse, chopped up on a grand scale, Where is the pillar, where is the pitchfork; There are cooled cauldrons full of resin; Here is an overturned scaffold: Iron teeth stick out, With bones, piles of ashes smolder, On stakes, writhing, the dead are numb and black... (description of Red Square after a mass execution by order of Ivan the Terrible. A.S. Pushkin. Oprichnik) Asymmetry

Slide 14

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The grandiose structure was planned to reflect the course of the war, the significance of the victory and the protection provided by heavenly forces to the king and the state. Ivan the Terrible wanted to embody his plan in architecture in extraordinary detail and vividly. Therefore, a temple appeared with a more complex structure than any other religious building in Rus'. It consists of nine churches, designed like towers and united by galleries that run along the lower tier. In accordance with the king's plan, each church is dedicated to one or another holy or Christian holiday. St. Basil's Cathedral

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St. Basil's Cathedral This bizarre composition of ten churches, each of which has central symmetry, has neither mirror nor rotational symmetry as a whole. The symmetrical architectural details of the cathedral swirl in their asymmetrical, chaotic dance around its central tent: they either rise, or fall, or seem to run into each other, or lag behind, creating the impression of joy and celebration. Without its amazing asymmetry, St. Basil's Cathedral is simply unthinkable! Studying it, scientists came to the conclusion that the golden ratio predominates in it. If we take the height of the cathedral as one, then the basic proportions that determine the division of the whole into parts form the golden ratio series: 1: j: j 2: j 3: j 4: j 5: j 6: j 7, where j =0.618 In this division and contains the main architectural idea of ​​​​creating the cathedral, common for all eight domes, uniting them into one composition.

Slide 17

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In modern architecture, techniques of both antisymmetry and dissymmetry are increasingly used. These searches often lead to very interesting results. A new aesthetics of urban planning is emerging. Concluding our conversation, we can state that beauty is the unity of symmetry and dissymmetry. So, the “sphere of influence” of symmetry (and therefore its antipode, asymmetry) is truly limitless. Nature - science - art. Everywhere we see the confrontation, and often the unity of two great principles - symmetry and asymmetry, which largely determine the harmony of nature, the wisdom of science and the beauty of art. Conclusions.

Kazan Cathedral

Maintaining symmetry is the first rule of an architect when designing any structure. One has only to look at the magnificent work of A.N. Voronikhin, Kazan Cathedral in St. Petersburg, to be convinced of this. If we mentally draw a vertical line through the spire on the dome and the top of the pediment, we will see that on both sides of it there are absolutely identical parts of the structure (colonnades and cathedral buildings. But it is possible that you do not know that there is another one in the Kazan Cathedral, if one might say "failed" symmetry. The fact is that, according to the canons of the Orthodox Church, the entrance to the cathedral should be from the east, i.e. it should be from the street, which is located to the right of the cathedral and runs perpendicular to Nevsky Prospekt. But, on the other hand sides Voronikhin understood that the cathedral should be facing the main thoroughfare of the city. And then he made an entrance to the cathedral from the east, but conceived another entrance, which he decorated with a beautiful colonnade. To make the building perfect, and therefore symmetrical, the same colonnade had to be located on the other side of the cathedral. Then, if we looked at the cathedral from above, its plan would have not one, but two axes of symmetry. But the architect’s plans were not destined to come true.

Presentation on geometry on the topic: “Symmetry in architecture and construction.” Completed by students of grades 9-1: Chekalkina Ekaterina and Sokolova Ksenia Teacher Pochetukhina E.A.

Definition of symmetry. SYMMETRY is a property of geometric figures. Two points lying on the same perpendicular to a given plane (or line) on opposite sides and at the same distance from it are called symmetrical with respect to this plane (or line). A figure is symmetrical relative to a straight line (axis of symmetry) or plane if its points in pairs have the specified property. A figure is symmetrical with respect to a point (center of symmetry) if its points lie in pairs on straight lines passing through the center of symmetry, on opposite sides and at equal distances from it.

Examples of symmetry in geometric shapes. Asymmetry is the absence of symmetry. Symmetry can be axial or central. Axial example: Central example:

The presence of symmetry. The image on a plane of objects in the world around us has an axis of symmetry or a center of symmetry. Many tree leaves and flower petals are symmetrical relative to the average stem.

We often encounter symmetry in art, technology, everyday life, furniture, chemistry, helical symmetry in nature... symmetry in architecture and construction. The presence of symmetry.

Symmetry in architecture. The main techniques when creating architectural compositions are the combination of different volumes - high and low, rectilinear and curvilinear, alternation of spaces - open and closed, symmetry and asymmetry of the building. During construction, architects first consider the symmetry of the building, since if there is a mistake, the object may collapse.

An example of symmetry in architecture. In the buildings, all structures are located strictly symmetrically.

Some styles of architecture. Gothic style Renaissance Classicism Next slide

Gothic style of architecture. The Gothic style emerged in the Middle Ages. Gothic buildings are distinguished by an abundance of openwork, like lace, decorations, sculptures, and ornaments, so both outside and inside they give the impression of lightness and airiness. Windows, portals, and vaults have a characteristic lancet shape. The facades of the buildings had mirror (axial) symmetry. back

Style – Renaissance. The architects of the Renaissance created a style - the Renaissance, in which they used the heritage of ancient art and Greek architectural orders. True, they used them in a new way, more freely, with a deviation from ancient canons, in other proportions and sizes, in combination with other architectural elements. Renaissance buildings were strict in form, with clear, straight lines. The symmetry of the facades is preserved. back

Style – classicism. All buildings built in the classicist style have clear rectilinear forms and symmetrical compositions. Against the background of smooth walls, porticos and colonnades protrude, giving the buildings a solemn monumentality and splendor. Decorative decorations of bas-reliefs and statues enliven the appearance of the buildings. The masters of classicism deliberately borrowed techniques from antiquity and the Renaissance and used orders with antique proportions and details. back

An example of an amazing combination of symmetry and asymmetry is the Intercession Cathedral (St. Basil's Cathedral) on Red Square in Moscow. This composition consists of ten temples, each of which has central symmetry, but in general has neither mirror nor rotational symmetry. The symmetrical architectural details of the cathedral swirl in their asymmetrical, chaotic dance around its central tent: they either rise, or fall, or seem to run into each other, or lag behind, creating the impression of joy and celebration. Without its amazing asymmetry, St. Basil's Cathedral would be simply unthinkable.

Symmetry can be called the key to success in construction. Almost all buildings in the world, in order to avoid destruction, are built exclusively symmetrically. This is why symmetry is so important in construction.

Sources of information: www.letopisi.ru www.google.ru www.simmetr-geom.com Tarasov L.V. – This amazingly symmetrical world.

The work was carried out by a student of 8 “A” class Khabibullina Maria, Supervisor: Zykova Olga Evgenievna Municipal Educational Institution GPLI Balashov, Saratov Region Symmetry is the idea through which man for centuries has tried to comprehend and create order, beauty and perfection. G. Weil Fundamental question How often is symmetry used in the creation of architectural structures? Can we consider the use of symmetry as a technique that harmonizes the perception of architectural structures? . PURPOSE: to identify how widely symmetry is used in architectural structures. HYPOTHESIS: symmetry is widely used in the design of architectural structures and the design of building facades. OBJECTIVES: Define what is called symmetry. Consider some types of symmetry. Define what is called architecture. Consider architectural styles. Explore some architectural structures in the design of which symmetry was used. Historical background “The study of archaeological monuments shows that humanity, at the dawn of its culture, already had an idea of ​​symmetry and implemented it in drawings and in everyday objects. It must be assumed that the use of symmetry in primitive production was determined not only by aesthetic motives, but to a certain extent by human confidence in the greater suitability for the practice of correct forms,” these are the words of our wonderful compatriot, who devoted his entire long life to the study of symmetry, Academician A.V. Shubnikov (1887-1970). Definition of symmetry “Concise Oxford Dictionary” beauty, proportion, harmony, consistency “Modern Dictionary of Foreign Words” proportionality, complete correspondence in the arrangement of parts of the whole “Great Encyclopedia of Cyril and Methodius” property of geometric figures The general concept of symmetry characterizes the special structure of the organization of any systems in which Certain characteristics are preserved when certain transformations are performed. The features that will be preserved can be geometric, physical, biological, informational, etc. Definition of symmetry (Concise Oxford Dictionary) Symmetry is “beauty due to the proportionality of the parts of the body or any whole, balance, likeness, harmony, consistency.” Definition of symmetry (Modern Dictionary of Foreign Words) Symmetry is “proportionality, complete correspondence in the arrangement of parts of the whole relative to the midline, center... such an arrangement of points relative to a point (center of symmetry), straight line (axis of symmetry) or plane (plane of symmetry), with in which every two corresponding points lying on the same straight line passing through the center of symmetry, on the same perpendicular to the axis or plane of symmetry, are at the same distance from them. .." Definition of symmetry (Big Encyclopedia of Cyril and Methodius) Symmetry in geometry is a property of geometric figures. Types of symmetry Axial symmetry Central symmetry Mirror symmetry Transfer symmetry Axial symmetry Two points lying on the same perpendicular to a given plane (or straight line) on opposite sides and at the same distance from it are called symmetrical with respect to this plane (or straight line). A figure (flat spatial) with respect to symmetry) is either symmetrical with respect to a straight line or (axis of a plane (plane of symmetry), if its points in pairs have the following property. Central symmetry A figure is symmetrical with respect to a point (center of symmetry), if its points lie in pairs on straight lines passing through the center symmetry, on different sides and at equal distances from the point. Mirror symmetry In space, an analogue of the axis of symmetry is the plane of symmetry. The mapping of space onto itself relative to the plane is called mirror symmetry. This name is justified by the fact that both parts of the figure are located on opposite sides of the plane of symmetry , similar to some object and its reflection in a mirror. Transferable symmetry This type of symmetry consists in the fact that parts of the whole form are organized in such a way that each next one repeats the previous one and is spaced from it by a certain interval in a certain direction. This interval is called the step of symmetry. Portable symmetry is usually used when constructing borders. Many types of symmetry are used when constructing borders as an element of design of architectural structures. Border - edge, border, frame (French bordure, from bord - edge). To create borders of linear ornaments, the following transformations are used: a) parallel transfer; b) mirror symmetry with a vertical axis; c) mirror symmetry with a horizontal axis; d) rotational (central symmetry). Architecture Architecture is the art of construction, the ability to design and create cities, residential buildings, public and industrial buildings, squares, streets, parks. Architecture is an amazing field of human activity. Science, technology, and art are closely intertwined and strictly balanced in it. The impression that an architectural structure makes primarily depends on the composition of the building. The combination of various volumes - high and low, rectilinear and curvilinear, alternation of spaces - open and closed - these are the main techniques that the architect uses when creating architectural compositions. Only a proportionate, harmonious combination of these principles makes a structure erected by man an architectural monument. Architecture styles Style is a historically established set of artistic means and techniques. Gothic style Renaissance style Baroque style Classicism style Russian-Byzantine style Art Nouveau style Gothic style This style arose in the Middle Ages. Gothic buildings are distinguished by an abundance of openwork, like lace, decorations, sculptures, and ornaments, so both outside and inside they give the impression of lightness and airiness. Windows, portals, and vaults have a characteristic lancet shape. The facades of the buildings had mirror (axial) symmetry. Gothic Duomo cathedral in Milan. Architects A. and F. degli Organi, G.A. Amadeo and others Milan Cathedral (Italian: Duomo di Milano) is a cathedral in Milan. Built in Flaming Gothic style from white marble. Construction began in 1386, but it was completed only at the beginning of the 19th century, when, by order of Napoleon, the design of the facade was completed. Some details, however, were completed later: until 1965. It is one of the largest cathedrals in the world. The total length of the temple is 158 meters, the width of the transverse nave is 92 m, the height of the spire is 106.5 m. The cathedral can accommodate up to 40,000 people. The statistics of building gigantomania could be supplemented with some more figures: in total, the cathedral is decorated with 3,000 statues, and it took almost 600 years to build. Renaissance style This style was created by the architects of the Renaissance, in which they used the heritage of ancient art and Greek architectural orders. Renaissance buildings were strict in form, with clear, straight lines. The symmetry of the facades is preserved. The Louvre Museum in Paris is an architectural monument and a rich museum that has no equal in the variety and completeness of its collections, their artistic and historical value. Louvre. 1546-1574 Architect P. Lesko The construction of the current Louvre building lasted almost a millennium and is inseparable from the history of the city of Paris itself. Under Francis I, from 1527, a radical restructuring of the building inside and out began. The medieval towers were demolished, and the castle took on the appearance of a Renaissance palace. The main work was carried out according to the project and under the leadership of P. Lesko. In the second half of the 17th century. Large-scale work was undertaken in the Louvre to bring the appearance of the palace closer to the architecture of the Baroque era. One of the main creators of this style, L. Bernini, was invited to Paris from Rome for this purpose. However, the project he proposed was considered too pompous. The work was entrusted to French architects. C. Perrault (1613–1688) built the famous eastern colonnade in the classicism style, which was preferred in France. Baroque style This style, which replaced the Renaissance, is distinguished by an abundance of curvilinear forms. Grandiose architectural ensembles (a group of buildings united by a common design) of palaces and villas built in the Baroque style amaze the imagination with the abundance of decorations on the facades and inside the buildings. There are almost no straight lines. Architectural forms bend, pile on top of one another and intertwine with the sculpture. This creates the impression of constant mobility of forms. The most brilliant embodiment of new ideas was the Zwinger palace complex, created according to the design of the outstanding German architect Matthaus Daniel Peppelmann. Dresden. Palace ensemble Zwinger.17111732. Architect D. Peppelman. Kuskovo. The Kuskovo ensemble was built by serf architects F. Argunov, A. Mironov and G. Dikushin with the participation of K. Blank. The central place in the ensemble is occupied by the palace, built in 1769-1775 by C. Blank according to the design of C. de Wailly. The wooden building of the palace with a stone ground floor was made in the style of early classicism with baroque elements. The palace has preserved the layout and original interiors of the 18th century, as well as a unique collection of paintings. One of the most interesting buildings in Kuskovo is the Grotto, built in 1755-1761 under the leadership of F. Argunov. The Baroque stone pavilion on a three-stage plinth is lavishly decorated with sculpture in the niches, decorations on the pediments, and lion masks above the windows. The walls of the Grotto are decorated with colored glass and limestone tuff. According to the architect's plan, the Grotto was supposed to personify the elements of stone and water. Classicism style All buildings built in this style have clear rectilinear forms and symmetrical compositions. Against the background of smooth walls, porticos and colonnades protrude, giving the buildings a solemn monumentality and splendor. Decorative decorations of bas-reliefs and statues enliven the appearance of the buildings. The masters of classicism deliberately borrowed techniques from antiquity and the Renaissance and used orders with antique proportions and details. Church of the Epiphany in Elokhov. Epiphany Cathedral In Moscow, Balashov, Church of the Archangel Michael Kazan Cathedral in St. Petersburg (architectural style - classicism, architect - A.N. Voronikhin) If we mentally draw a vertical line through the spire on the dome and the top of the pediment, we will see that with On both sides of it there are absolutely identical parts of the structure (colonnades and cathedral buildings). But it is possible that you do not know that in the Kazan Cathedral there is another, so to speak, “failed” symmetry. The fact is that, according to the canons of the Orthodox Church, the entrance to the cathedral should be from the east, i.e. it should be from the street, which is located to the right of the cathedral and runs perpendicular to Nevsky Prospekt. But, on the other hand, Voronikhin understood that the cathedral should be facing the main highway of the city. And then he made an entrance to the cathedral from the east, but planned another entrance, which he decorated with a beautiful colonnade. To make the building perfect, and therefore symmetrical, the same colonnade had to be located on the other side of the cathedral. Then, if we looked at the cathedral from above, its plan would have not one, but two axes of symmetry. But the architect's plans were not destined to come true. Admiralty. The Main Admiralty building is a masterpiece of Russian national architecture, one of the highest achievements of Russian classicism architecture. The appearance of the building, as well as its role in the ensemble of the city center, was determined as a result of long-term construction activities throughout the 18th and first quarter of the 19th centuries. The Admiralty consists of two parallel shaped buildings connected by a pavilion with arches cut in the center. Spaso-Preobrazhensky Cathedral, a bright representative of the Russian-Byzantine style. The temple, founded in 1880 according to the design of B.V. Freidenberg, built by A.S. Kaminsky and was completed in 1889. Russian-Byzantine architecture is characterized by the borrowing of a number of compositional techniques and motifs of Byzantine architecture, most clearly embodied in the “model projects” of churches by Konstantin Thon in the 1840s. Ton built the Cathedral of Christ the Savior, the Grand Kremlin Palace and the Armory Chamber in Moscow, as well as cathedrals in Sveaborg, Yelets (Ascension Cathedral), Tomsk, Rostov-on-Don. Notre-Dame-de-Paris LEUVEN: City Hall building. Leuven university town; the figurine in the foreground represents the acquisition of knowledge from books. Eiffel Tower Balashov Temple of the Transfiguration Paris LISBON, Belém district: magnificent carved architecture of the Jerónimos Monastery (Mosteiro dos Jerónimos) Conclusions In our work, we examined architectural structures of various styles, built in different eras, and revealed that in the architecture of each of them one can see symmetry. Architectural monuments, widely known as examples of proportionality and harmony, are literally imbued with mathematics, numerical calculations and geometry. Conclusions: Research has shown that all types of symmetry are used in the design and construction of architectural structures and the design of building facades. Symmetry is opposed to chaos, disorder. It is present in literally everything in our lives, but we are so accustomed to it that we do not notice it. But no matter how we treat it, it is in our lives, adding to the peace, tranquility and state of something alien to the eye. We believe that no matter how art develops in the future, elements of symmetry will still prevail in it. Wishes to future architects It is important that when restoring old buildings, the architectural style of that time is not violated. We would really like modern architects to use various types of symmetry when designing buildings and other architectural structures, not only to decorate facades, but also when planning. Stages of the project Collecting material (studying popular science literature, photographs, Internet resources) Processing the information received Creating a research project in Microsoft Word Creating a presentation in Power Point Participation in the school competition “I am a researcher” Completed by: student 8 “A” class of the boarding lyceum Khabibullina Maria Head: mathematics teacher of the boarding lyceum Zykova Olga Evgenievna Information resources of the project 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. I explore the world. Moscow monasteries and temples: encyclopedia. Astrel Publishing House LLC 2006 I am exploring the world. Museums and estates of Moscow: encyclopedia. Astrel Publishing House LLC 2006 I explore the world: Children's encyclopedia.Architecture.1990 What is Who is. Volume 1, 2. Publishing house "Pedagogy", 1990. L.S. Atanasyan Geometry 7-9, 10-11. Moscow "Enlightenment" 2005 Great Encyclopedia of Cyril and Methodius. Electronic edition. http://www.elohovo.narod.ru. http://www.ru.wikipedia.org. http://www.hist-singhts.ru http://www.museum.ru