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Presentation on geometry on the topic: "Symmetry in architecture and construction." Geometry presentation on symmetry in architecture Presentation on symmetry in architecture

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
PKS-15-2 groups
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.
Reveal how symmetry is used in
architectural structures.

Tasks

Describe the types of symmetry.
Consider examples.
Learn to distinguish the types of symmetry in
architecture of the native city.

Symmetry

Symmetry is what we call the same
arrangement of equal parts in relation to
planes or lines. Symmetry
structure is one of the reasons for its
active influence on perception.

Asymmetry

In terms of mathematical concepts
asymmetry is just the absence of symmetry.
In architecture, symmetry and asymmetry are two opposite methods of regularity.
spatial organization.

Asymmetry

Asymmetric compositions in progress
development of architecture arose as
the 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
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,
character-defining composition
symmetrical layout.

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
where any point Z goes to
symmetrical to it with respect to the plane α
point Z1. (fig.5)
Fig.5

Central symmetry

In addition to mirror symmetry
considered 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 normally
called the center of rotation.

Axial symmetry

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

Axis of symmetry

In orthogonal drawings, the plane
symmetry is represented by a line, so its
often referred to as 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

The highest degree of symmetry
has a ball in the center of which
intersect an infinite number
axes and planes of symmetry. (Fig.9)
Fig.9

The Importance of Symmetry in Architecture

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

Fig.10
Fig.11

Symmetry in ancient Russian buildings

In ancient Russian architecture there are many
examples of intuitive or conscious
use of symmetry, this and
bell towers (Fig. 12), watchtowers
towers (Fig. 13). A clear imprint of symmetry
bear on themselves and later buildings:
stone Russian temples (Fig. 14),
palaces (Fig. 15).

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

Architectural buildings of the city of Irkutsk

Palace of Sports "Trud" - Symmetrical (Fig.16)
Musical Theatre. N.M. Zagursky -
Symmetrical (Fig.17)
House of Music Denis Matsuev - Dissymmetric
(Fig.18)
Fig.17
Fig.18
Fig.16

Conclusion

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

Description of the presentation on individual slides:

1 slide

Description of the slide:

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) Building of the Senate of the Moscow Kremlin 7) Pashkov House 8) Asymmetry and dissymmetry 9) St. Basil's Cathedral 10) Conclusions

3 slide

Description of the slide:

Symmetry in architecture Symmetrical objects are highly functional in many 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 the symmetry is most clearly manifested in the ancient buildings of Ancient Greece. From that time to the present day, symmetry in the human mind has become an objective sign of beauty. 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, people consider them beautiful. What is it connected with?

4 slide

Description of the slide:

Geometry possesses two treasures: one of them is the Pythagorean theorem, the other is the division of a segment in the middle and extreme ratios ... The first can be compared with a measure of gold, the second is more like a precious stone. I. Kepler The golden section The golden section is the law of the proportional connection of the whole and the parts that make up this whole, when the whole relates to the larger part in the same way as the larger part to the smaller

5 slide

Description of the slide:

Parthenon Sacred hill and temple of Divine Athena, Magnificent Parthenon, Having buried the forgotten ruins, Aspires to the gods of Olympus. N. Vasyutinsky PARTHENON, the main temple of the Athenian Acropolis, dedicated to Athena Parthenos (ie 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 the decoration (mainly sculptural work) continued until 432 BC. The Parthenon is a masterpiece of ancient Greek architecture and a symbol of Greek genius.

6 slide

Description of the slide:

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 ratio 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 ones. The ledges are made entirely of squares of Pentilean marble. Sculptures The ratio of the height of a building to its length is 0.618. Parthenon

7 slide

Description of the slide:

The most perfect of all classical monuments preserved in the Italian capital is the Pantheon. For a long time it was believed that it was built in 27 AD, but excavations have shown that the Pantheon is a reconstructed building 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, as well as its height, is 43 meters. The sky is visible through the opening in the dome. This creates an atmosphere of solemnity. Between the second and third chapels is the tomb of Raphael. 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 got its name from the name of its creator - Pharaoh Cheops (about 2551 - 2528 BC). The height of the pyramid of Cheops is 138 meters (and originally 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 horizontally gives a 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.

9 slide

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This name comes from the Latin word clafssicus, which means exemplary in translation. The style itself took shape in the 17th century in France, having adopted the ancient heritage as the norm, as an ideal model in all types of arts - literature, painting, architecture, theatrical art. The architecture of Classicism is characterized by clarity and geometrism of forms, symmetry, logical planning and discreet decor. This style had a great influence on the formation of the image 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. 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

Description of the slide:

The 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 a masterpiece of architecture. Contemporaries called it the Russian Pantheon. The hall with a diameter of 24.7 m and a height of 27 m with a colonnade of the Corinthian order along the perimeter is covered with a coffered dome, at the base of which there are 24 light windows. Sculptural themed bas-reliefs adorn the walls between the columns and windows.

11 slide

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The plan of the building is an isosceles triangle with a courtyard divided by additional buildings into three parts: a central pentagonal and lateral triangular 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 simultaneously the center of one of its compositional axes. All the premises of the Senate are interconnected by a corridor running along the perimeter of the courtyard. Senate building

12 slide

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At the very end of Mokhovaya Street, on a hill, rises a magnificent palace made of white stone, known in Moscow as the Pashkov House. The combination of antique 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 who restored the house according to the drawings of the 18th century managed to recreate it. Before the war of 1812, there was a garden in the courtyard of the house, where 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 commissioned a wealthy nobleman Lieutenant Captain of the Life Guards of the Semenovsky Regiment Pyotr Pashkov to build a palace opposite. Turning it back to the Kremlin. Pashkov House

13 slide

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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 building 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 Is full of yesterday's execution; Torment fresh trail all around: Where is the corpse, chopped on a grand scale, Where is the pillar, where is the pitchfork; there the cauldrons are full of tar that has cooled; Here is an overturned chopping block: Iron prongs stick out, Heaps of ash smolder with bones, On stakes, crouching, the dead, numb, turn black... (Description of Red Square after the mass execution by order of Ivan the Terrible. A.S. Pushkin. Oprichnik) Asymmetry

14 slide

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

15 slide

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St. Basil's Cathedral This whimsical composition of ten temples, each of which has a central symmetry, as a whole has neither mirror nor rotational symmetry. The symmetrical architectural details of the cathedral whirl in their asymmetrical, disorderly dance around its central tent: they either rise or fall, or seem to run into each other, or lag behind, creating an impression of joy and celebration. Without its amazing asymmetry, St. Basil's Cathedral is simply unthinkable! Investigating it, scientists came to the conclusion about the predominance of the golden section series 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 In this division and concluded the main architectural idea of creating a cathedral, the same for all eight domes, uniting them into one composition.

17 slide

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In modern architecture, the techniques of both antisymmetry and dissymmetry are increasingly used. These searches often lead to very interesting results. A new urban aesthetic 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 hence 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.

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

Symmetry in architecture Symmetrical objects are highly functional in many 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 the symmetry is most clearly manifested in the ancient buildings of Ancient Greece. From that time to the present day, symmetry in the human mind has become an objective sign of beauty. 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, people consider them beautiful. What is it connected with?

Pyramid of Cheops Pyramid of Cheops Egyptian Pyramid of Cheops is the oldest and at the same time the only miracle of the world that has survived to this day. It got its name from the name of its creator - Pharaoh Cheops (about BC). The height of the pyramid of Cheops meters (and was originally 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 horizontally gives a 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.

The Senate building The Senate building was built according to the project of the architect M.F. Contemporaries called it the Russian Pantheon. The hall with a diameter of 24.7 m and a height of 27 m with a colonnade of the Corinthian order along the perimeter is covered with a coffered dome, at the base of which there are 24 light windows. Sculptural themed bas-reliefs adorn the walls between the columns and windows.

At the very end of Mokhovaya Street, on a hill, rises a magnificent palace made of white stone, known in Moscow as the Pashkov House. The combination of antique 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 the years. commissioned by the wealthy landowner Pashkov. In 1812 the house was destroyed by fire. However, the architects who restored the house according to the drawings of the 18th century managed to recreate it. Before the war of 1812, there was a garden in the courtyard of the house, where 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 commissioned a wealthy nobleman Lieutenant Captain of the Life Guards of the Semenovsky Regiment Pyotr Pashkov to build a palace opposite. Turning it back to the Kremlin. Pashkov House

This bizarre composition of ten temples, each of which has a central symmetry, as a whole has neither mirror nor rotational symmetry. The symmetrical architectural details of the cathedral whirl in their asymmetrical, disorderly dance around its central tent: they either rise or fall, or seem to run into each other, or lag behind, creating an impression of joy and celebration. Without its amazing asymmetry, St. Basil's Cathedral is simply unthinkable! Investigating it, scientists came to the conclusion about the predominance of the golden section series 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 In this division and concluded the main architectural idea of creating a cathedral, the same for all eight domes, uniting them into one composition.

Dissymmetry is a partial lack of symmetry, symmetry disorder, expressed in the presence of some symmetrical properties and the absence of others. An example of dissymmetry in an architectural structure is the Catherine Palace in Tsarskoye Selo near St. Petersburg. Almost all symmetry properties are fully maintained in it, with the exception of one detail. The presence of the Palace Church upsets the symmetry of the building as a whole. If this church is not taken into account, then the Palace becomes symmetrical. Dissymmetry

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 shapes. Two points lying on the same perpendicular to a given plane (or line) on different sides and at the same distance from it are called symmetrical with respect to this plane (or line). A figure is symmetrical with respect to a line (axis of symmetry) or a 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 lack of symmetry. Symmetry is axial and central. Axial example: Central example:

The presence of symmetry. The image on the plane of objects of the world around us has an axis of symmetry or a center of symmetry. Many tree leaves, flower petals are symmetrical about the middle stem.

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

Symmetry in architecture. The main techniques in creating architectural compositions are the combination of different volumes - high and low, rectilinear and curvilinear, the alternation of spaces - open and closed, the symmetry and asymmetry of the building. During construction, first of all, architects consider the symmetry of the building, since in case of an error the object may collapse.

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

Some styles of architecture. Gothic style Renaissance Classicism Next slide

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

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

Style - classicism. All buildings built in the style of classicism have clear rectilinear forms and symmetrical compositions. Against the background of smooth walls, porticos and colonnades protrude, which give the buildings a solemn monumentality and splendor. Decorative decoration of bas-reliefs and statues enliven the appearance of buildings. The masters of classicism deliberately borrowed the techniques of antiquity and the renaissance, applied 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 a central symmetry; in general, it has neither mirror nor rotational symmetry. The symmetrical architectural details of the cathedral whirl in their asymmetrical, disorderly dance around its central tent: they either rise or fall, or seem to run into each other, or lag behind, creating an 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 LV - This amazingly symmetrical world.

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Slides captions:

St. Petersburg at a geometry lesson Types of symmetry in architecture

Axial symmetry Two points are said to be symmetrical with respect to a line if this line passes through the midpoint of the segment connecting these points and is perpendicular to it. A B C D D c

Axial symmetry in the architecture of St. Petersburg

St. Isaac's Cathedral The largest Orthodox church SPb. It was built in 1818-1858 according to the project of O. Montferan. Height 101.5 m.

Catherine's Palace in Pushkin Former Imperial Palace. Located in the city of Pushkin (formerly Tsarskoye Selo). The building was founded in 1717 by order of Catherine I. Represents an example of the late Baroque. During the war, the palace was badly damaged. Its restoration took many years.

Central symmetry Two points are said to be symmetric with respect to a given point if this point is the midpoint of the segment connecting the points. A B O

In 1798-1810, work was carried out on the construction of the granite embankments of the Moika, which were decorated with a cast iron fence with a clear pattern of geometric elements. CENTRAL SYMMETRY IN THE FAMOUS GRATES OF ST.

Belinsky Bridge In the 18th century, 7 stone drawbridges of the same type were built on the Fontanka. Only two of them have survived in their original form. One of them is Simeonovsky Bridge (Belinsky Bridge)

Fragment of the lattice of the Summer Garden

MIRROR SYMMETRY Mirror symmetry (symmetry with respect to the plane) is such a mapping of space onto itself, in which any point M passes into a point N symmetrical to it with respect to the plane.

Palace and Park Ensemble of Oranienbaum

Mirror symmetry in the gardens and parks of St. Petersburg

In this picture you can see different types symmetry The Marble Palace was built in 1768-1785 according to the project of A. Rinaldi by order of Catherine II. The first building in St. Petersburg, the facades of which are lined with natural stone.

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