Municipal budgetary educational institution

“Secondary school No. 27”, Perm

The world of geometry in city architecture

Vedernikova Ekaterina, 16 years old,

student of class 10A

MBOU "Secondary School No. 27"

Perm city, Russia

Supervisor:

Kustova Tatyana Semenovna,

mathematic teacher

MBOU "Secondary School No. 27"

Perm, 2013

Introduction…….….…………………………………………………………..page 3

Theoretical part………………………………………………………….page 4

Practical part…..……………………………………………………...page 7

Conclusion…………………………………………………………………………………page 10

References…………………………………………………………………page 11

Introduction

deepen knowledge of geometry, in particular, stereometry, since in architecture, both before and now, various geometric figures and bodies are used,

identify the dependence of architecture and geometry on each other,

consider examples of the variety of buildings in our city, present a photo report on the work done with interesting architectural solutions.

Project implementation:

During the search activity, find in our city buildings that have an unusual architectural structure, using geometric bodies and figures in their composition. Provide photographs and descriptions of the buildings (indicating the geometric shapes and bodies) included in the structure.

Theoretical part

“Centuries have passed, but the role of geometry
has not changed. She's still
remains the architect's grammar"
Le Corbusier

Geometry is the science of the properties of geometric figures

Architecture is an art form that is a system of buildings and structures that form the spatial environment for human life.

Some people may think that various intricate lines, shapes, and surfaces can only be found in books or textbooks. However, it is worth looking around, and we will see that many buildings and structures have the shape of geometric figures known to us.

The origin of architecture dates back to the time of the primitive communal system, when the first artificially constructed dwellings and settlements arose. With the emergence of states, a new form of settlement arose - the city as a center of administration, craft production and trade. In ancient centuries, large states of Egypt, Greece, Japan, the Roman Empire, and China emerged where unique architecture was created. Already at that time, the abstract concept of a geometric body (figure) arose and the connection between geometry and the real world was noted. Geometry, as a practical science, was used by the Egyptians to restore land after each flood of the Nile, during various economic works, during the construction of irrigation canals, grandiose temples and pyramids, and when carving the famous sphinxes from granite. The breadth of urban planning distinguishes Roman architecture. The Romans perfected and implemented strict organized planning in large-scale cities.

Of course, we can only talk approximately about the correspondence of architectural forms to geometric figures, ignoring small details. Almost all geometric shapes are used in architecture. The choice of using one or another figure in an architectural structure depends on many factors: the aesthetic appearance of the building, its strength, and ease of use. The aesthetic features of architectural structures changed during the historical process and were embodied in architectural styles. Style is usually called the totality of the main features and characteristics of the architecture of a certain time and place. Geometric shapes characteristic of architectural structures as a whole and their individual elements are also signs of architectural styles.

Modern architecture

Architecture these days has an increasingly unusual character. Buildings come in all different shapes. Many buildings are decorated with columns and stucco moldings. Geometric figures of various shapes can be seen in the construction of bridge structures. The “youngest” buildings are skyscrapers and underground structures with a modernized design. Such buildings are designed using architectural proportions.

Practical part

There are many interesting buildings in our city. I would like to present photographs here and consider what geometric bodies and figures Perm sights are made of. And thereby prove that the geometry of our city is very diverse and interesting.

Basically, the building is constructed in the form of a rectangular parallelepiped; I consider the main feature to be the hemisphere included in the building, which adorns the second building of the “Family” hypermarket.

The facade of the shopping mall "Capital" is made in the shape of an arc.

From this viewing angle we can see that the wall above the entrance is also in the shape of an arc, it turns out to be “1/4 cylinder”

f The front of the house, the Victoria residential complex, has an interesting shape, the corner of the house is not straight, the cross-section is in the shape of a rhombus.

Conclusion

In the course of my work, I examined the dependence of architecture on geometry, I was convinced of this in practice and presented photos and drawings of individual geometric bodies. The purpose of my work was to study geometry outside the school curriculum. I tried to reveal the use of geometry in the practical activities of man, in the construction of famous buildings.

Internet resources:

1. /library/material/140875/

2. /slide/40354/

3. /view.aspx?id=555977

4. /mathematics/00077208_0.html

Sources of illustrations.

Slide 1

Slide 2



Slide 3

Cylinder A cylinder is a body obtained by rotating a rectangle around a straight line containing its side.

Slide 4

built in the 2nd century. BC. in Rome at the Forum Boarium. It is the earliest surviving marble building in Rome. Twenty columns of Pentelic marble, resting on a tuff pedestal, were installed under the direction of a Hellenic architect, possibly Hermodorus of Salamis. ` Temple of Hercules

Slide 5

“temple of all gods” in Rome, a monument of centric-dome architecture of Ancient Rome, built in the 2nd century AD. e. under Emperor Hadrian on the site of the previous Pantheon, built two centuries earlier by Marcus Vipsanias Agrippa. Pantheon

Slide 6

a separate rotunda chapel, built by Donato Bramante by order of the Spanish monarchs Ferdinand and Isabella on the Roman Janiculum hill in 1502 Tempietto

Slide 7

in the city of Kasimov, Ryazan region, built in 16 by the first ruler of the Kasimov Khanate, Tsarevich Kasim. A two-story building with a dome in the classicist style with a two-tier minaret in the form of a low cylinder under a pointed dome placed on a massive base. Khan Mosque

Slide 8

located in the Belgorod region, it was built in 1790. The brick structure consists of two cylinders - a large one, with a diameter of 26 m, and a small one inside it, with a diameter of approx. 10 m. The inner cylinder rises 4 m above the outer one in the form of a drum and ends with a dome. round building

Slide 9

Chaev's mansion is located in St. Petersburg. Built in 1906-1907. for railway engineer S. N. Chaev. Architects: Apyshkov V.P., Lidval F.I., Roslavlev M.I. The plan has a diagonal axis on which three cylindrical volumes are placed: a vestibule, a hall and a winter garden.

Slide 10

built in 1927-1929. in Moscow according to the project of Konstantin Melnikov. The volumetric composition of the house consists of two vertical cylinders of different heights of the same diameter, cut into each other by a third of the radius, thereby forming an unusual plan shape in the form of the number “8”. House-workshop of architect K.S. Melnikova

Slide 11

built in 1927-1929. in Moscow according to the project of Ilya Golosov. The compositional center of the building is a vertical glass cylinder, on which the entire building with unusually large window surfaces seems to be “put on.” The staircase was designed in such an impressive way. House of Culture named after Zuev

Slide 12

built in 1930-1935. Architects: Fomin I.I., Daugul V.G., Serebrovsky B.M. A round tower is adjacent to the horizontal building. The building is emphasized asymmetrically. The southern wing has a round shape. Inside it there is a huge hall covered with a dome. Moscow district council

Slide 13

is located in Munich. Its construction was completed in time for the 1972 Olympics. In 2004 it was closed for reconstruction (part of the exhibition was exhibited near the museum). On June 21, 2008, the museum was reopened - a new pavilion was added to the museum premises, which expanded the total area of ​​the museum to 5000 m². BMW Museum

Slide 14

The administrative complex Porta Fira Towers was built in Barcelona (Spain) in 2004-2008. Architect: Toyo Ito. The design of the hotel building is made in the form of a distorted cylinder, expanded towards the top. The second tower - an office building - is a rectangle. Porta Fira Towers

Slide 15

Parallelepiped A parallelepiped is a prism with a parallelogram at its base. All faces of a parallelepiped are parallelograms. The opposite faces of a parallelepiped are equal and parallel.

Slide 16

Muslim shrine in the form of a cubic building in the courtyard of the Holy Mosque (Mecca, Saudi Arabia). The Kaaba serves as the qibla - the landmark to which Muslims around the world turn their faces during prayer. Kaaba

Slide 17

church in Palermo (Italy), located in Piazza Bellini, next to the Temple of Martorana. The church in the name of Saint Cataldo was founded by Mayo from Bari in 1161. The building of San Cataldo is an almost regular parallelepiped, on which is erected a smaller parallelepiped, decorated with three hemispherical domes. San Cataldo

Slide 18

Museum of Modern Art in New York. Built in 1977. Japanese designers Kazuo Sejima and Ryue Nishizawa were invited to construct the building of the New Museum of Contemporary Art. This is how this unusual structure appeared in a minimalist style, similar to six shoe boxes stacked on top of each other. Modern Art Museum

Slide 19

built in Rotterdam and Helmond by architect Piet Blom in 1984. In Rotterdam, houses are located on Overblaak Street, not far from the metro station of the same name. Blom's radical solution was that he rotated the parallelepiped of the house by 45 degrees and placed it at an angle on a hexagonal pylon. Cube Houses or Cube Houses

Slide 20

a complex of three skyscrapers in the center of Tel Aviv (Israel). Construction1996-2007 The square tower has 42 floors and 154m, it is the lowest of the three towers of the Azrieli complex. Aeriel Tower

Slide 21

built for the 2008 Olympics in Beijing. The structure was built by the Australian company PTW. The total area of ​​the complex is about 70 thousand square meters. m. The design used elements that externally resemble a crystal lattice of water bubbles. water cube

Slide 22

located in Lumino (Switzerland). This house with a total area of ​​220 square meters was built according to the design of the architectural bureau Davide Macullo Architects in 2007-2009. The plan of the villa has the form of two offset parallelepipeds, the deformation of which is due to the natural topography of the area. House in Lumino

Slide 23

Cube Tube is a massive cube-shaped building, it seems very light. The house was designed by architecture and design studio Sako Architects and built in the Jinhua Economic Development Zone in China. Built in 2010 Cube Tube

Mazlova Ekaterina, Mishkevich Albina 6th grade MOBU Secondary School No. 5, Meleuz RB

Scientific research work on the topic: in architecture Meleuz city »

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Slide captions:

Scientific research work on the topic: “Geometric shapes and figures in the architecture of the city of Meleuz”
Completed by: students of grade 6 a MOBU secondary school No. 5. Meleuz Mishkevich Albina and Mazlova Ekaterina

The purpose of our work: to find out how geometry decorates the city of Meleuz; explore what geometric shapes, bodies and figures are found on the streets of our city. Tasks: 1. Study the variety of geometric shapes and figures. 2. Consider options for using geometric shapes and bodies in individual architectural objects of our city. 3. Find out which geometric shapes are more common and Why.

Objects of research: architectural buildings and structures, streets of the city of Meleuz. Subject of research: geometric shapes and figures in the architecture of the city of Meleuz. Research hypothesis: geometric figures, being ideal objects, find their visual embodiment in a variety of architectural structures.

Research methods: 1. Analyze the literature on the topic under study. 2. Consider the diversity of architectural structures in the city of Meleuz. 3. Show what shape or set of geometric shapes the selected structures have.4. Questionnaire.5. Experiment 6. Presentation of research results. Relevance of the work Architectural objects are an integral part of our life. Our mood and attitude depend on what buildings surround us. There is a need to study the diversity of objects that have appeared in our world.
1) Polygons, types of polygons
BASIC GEOMETRIC FIGURES AND FORMS
2) Round shapes
BASIC GEOMETRIC FIGURES AND FORMS
3) Polyhedra
BASIC GEOMETRIC FIGURES AND FORMS
4) Bodies of revolution
IS EVERYTHING SATISFIED WITH EVERYTHING ABOUT THE ARCHITECTURE OF OUR CITY?
WHAT ARCHITECTURAL STRUCTURES WOULD YOU LIKE TO SEE IN OUR CITY?
WHAT GEOMETRICAL FIGURES AND FORMS ARE USED IN THE ARCHITECTURE OF OUR CITY?
WHAT GEOMETRICAL FIGURES AND FORMS MAKE BUILDINGS MORE CLEAR AND EXPRESSIVE?
EXPERIMENTAL "IF I WERE AN ARCHITECT"
The work was completed by 30 students (1st grade) 28 students - used polygons (rectangle, square, rhombus) 2 students - used a circle and an oval.
FIRST DIRECTION
SECOND DIRECTION
SLIDESHOW “GEOMETRY OF OUR CITY”
CONCLUSION:
All architectural structures of the city of Meleuz consist of geometric figures and their combinations (mostly polyhedra).
LE CORBUSIER:
“……The world around us is a world of geometry, pure, true, impeccable in our eyes. Everything around is geometry. Never have we seen so clearly such shapes as a circle, a rectangle, an angle, a cylinder, a sphere, executed so clearly, with such care and so confidently.”


Preview:

Municipal educational budgetary institution

Secondary school No. 5

municipal district Meleuzovsky district

Republic of Bashkortostan

Research work

on the topic of:

"Geometric shapes and figures

in architecture

Meleuz cities »

Completed:

6th grade students

MOBU secondary school No. 5

municipal district

Meleuzovsky district

Republic of Bashkortostan

Mishkevich Albina and

Mazlova Ekaterina

Head: mathematics teacher

MOBU secondary school No. 5

Melkova Anzhelika Nikolaevna

Meleuz 2014

Introduction…………………………………………………………………………………... 3

1. Basic geometric figures and shapes………………………….. 5

a) polygons, types of polygons…………………………………. 6

b) rounded shapes……………………………………………………….... 8

c) polyhedra………………………………………………………………………………… 8

d) bodies of revolution………………………………………………………………………………. 10

II. Review of the architectural structures of the city…………………………….... 11

a) questionnaire…………….………………………………………………… 12

b) experiment………………………………………………………………………………… 13

c) review of architectural structures…………………………….……… 13

Conclusion. ………………………………………………………………....... 17

Literature..……………………………………………………………………………… 19

Applications……………………………………………………………….. 20

Introduction

We live in the city of Meleuz in the Republic of Bashkortostan. The city of Meleuz is the regional center. It is located at the confluence of the Meleuz River and the Belaya River.

Cities are like people... sometimes they suffer from their own imperfections, make mistakes, rejoice - there is a holiday on their streets. Sometimes it seems that the city is sad or even crying.

Modern residential complexes, stylish shopping centers and beautiful shops - the architectural appearance of Meleuz is changing every year, the city is becoming prettier before our eyes.

We love our city and proudly say to all nonresidents: “I am a Meleuzian.” We assure you, we have something to be proud of - our city has flourished and become a real beauty. Clean asphalt streets, beautiful flower beds, fountains and various shaped buildings.

Observing the architectural structures of our city, we were interested in the following: is it possible to determine the relationship between geometric shapes and architectural structures.

We decided to consider the question of the geometry of the city and whether it affects its image, because each city has its own structure and each city has its own aura.

The purpose of our work: find out how geometry decorates the city of Meleuz; explore what geometric shapes, bodies and figures are found on the streets of our city.

Tasks:

1. Study the variety of geometric shapes and figures;

2. Consider options for using geometric shapes and bodies in individual architectural objects of our city;

3. Find out which geometric shapes are more common and why.

Objects of study:architectural buildings and structures, streets of Meleuz.

Subject of study:geometric shapes and figures in the architecture of the city of Meleuz.

Research hypothesis:geometric figures, being ideal objects, find their visual embodiment in a variety of architectural structures.

Place and timing of the study:Republic of Bashkortostan, Meleuz, September 2013 – February 2014

Research methods:

1. Analyze the literature on the topic under study.

2. Consider the variety of architectural structures of the city of Meleuz.

3. Show what form or set of geometric figures they have

Selected structures.

4. Questionnaire.

5. Experiment.

6. Registration of research results.

Relevance of our workis that architectural objects are an integral part of our lives. Our mood and attitude depend on what buildings surround us. There is a need to study the diversity of objects that have appeared in our world.

The structure of the sections is related to the general idea of ​​the work.

The main part consists of two chapters. The first examines the basic geometric figures and shapes. The second section provides an overview of the notable architectural structures of the city of Meleuz with comments regarding their forms.

Main expected result of the study– collecting material for use in geometry lessons in high school, designing a slide show “Geometry of our city.”

I. Basic geometric figures and shapes

Amazing country - Geometry!

Figures and lines live in it,

They measure, draw and find out:

Perimeter, area, length, width,

Diameter, radius and height.

Gather your knowledge quickly!

Prepare a simple pencil as soon as possible!

Triangles, squares, rhombuses, circles... every student encounters them in geometry lessons at school.

Geometric figures occupy a central place in the school curriculum.

The first geometric concepts arose in prehistoric times.

For primitive people, the shape of the objects around them played an important role. By shape and color, they distinguished edible mushrooms from inedible ones, trees suitable for buildings from trees that could only be used for firewood. Sometimes they found crystals of minerals, from which they made tools for hunting and home use. Thus, mastering the world around them, people became acquainted with the simplest geometric figures.

And when people began to build houses, they had to understand more deeply what shape to give to the walls and roof. It became clear that it was better to trim the logs and make the roof sloping so that water would drain from it. And, without knowing it, people were studying geometry all the time. Women were engaged in geometry, making clothes, hunters, making spears and boomerangs of complex shapes. Only the word “geometry” itself did not exist then, and the shape of bodies was not considered separately from their other properties.

When they began to build houses from stone, they had to drag heavy stone blocks. For this purpose, rollers have been used since ancient times. This is how people became acquainted with one of the most important figures - the cylinder. It was difficult to transport loads on rollers due to the large weight of the logs themselves. To make the work easier, people began to cut thin flat round plates from the trunks. This is how the first wheel appeared. The unknown inventor of the first wheel made a great discovery! Just imagine for a moment that all the wheels on earth have disappeared. It will be a real disaster. Because every car, from pocket watches to spaceships, has dozens and hundreds of different wheels.

But it was not only in the process of work that people became acquainted with geometric figures. For a long time they loved to decorate themselves, their homes and their clothes. Ancient craftsmen learned to give beautiful shapes to bronze and gold, silver and precious stones. And artists, painting palaces, found new geometric shapes. The potter needed to know what shape to make a vessel so that it would contain this or that amount of liquid, and the ancient Egyptians learned to find the volumes of quite complex figures. Astronomers who observed the sky and, based on their observations, gave instructions when to begin field work, had to learn to determine the position of the stars in the sky. To do this, it was necessary to measure the angles.

The shape of peasant fields was also different. The fields were separated from each other by boundaries, and the flood of the Nile washed away these boundaries every spring. Therefore, there were special officials who were engaged in land surveying, in Russian, land surveyors. Thus, from the practical problem of land surveying, the science of land surveying arose. In Greek, the earth was called “geos”, I measure - “metrio”, and therefore the science of measuring fields was called “geometry”. Just don’t think of calling a modern geometer a land surveyor. For many thousands of years since its inception, it has only been involved in land surveying to a small extent.

Geometric figures interested our ancestors not only because they helped solve practical problems. Some of the figures had magical meaning for people. Thus, the triangle was considered a symbol of life, death and rebirth; the square is a symbol of stability. The universe and infinity were designated by a regular pentagon - a pentagon, a regular hexagon - a hexagon, was a symbol of beauty and harmony. The circle is a sign of perfection.

There are a variety of geometric shapes created by nature and human hands; in geometry they are considered as flat forms (figures) and volumetric forms (bodies).

Geometry is divided into two sections: planimetry and stereometry.

It is with planimetry that the study of geometry in schools begins.

Planimetry comes from the Latin "planum" - plane, and the Greek "metreo" - measure.

This section of geometry studies figures that are located on a plane: point, straight line,square, rectangle, triangle, rhombus, pentagon and other polygons, circle, oval. Geometric figures on a plane have two dimensions: length and width.

Stereometry is a branch of geometry that studies figures in space.In addition to length and width, they have height.

Volumetric ones include: cube, parallelepiped, prism, pyramid, cylinder, cone, ball.

So, what geometric figures and shapes have we studied?

1) Polygons, types of polygons

Polygon is a geometric figure bounded on all sides by a closed broken line, consisting of three or more segments (links).

If a closed broken line consists of three segments, then such a polygon is called triangle , from four segments -quadrilateral, from five segments - pentagon, etc.

a) Triangles

Triangle is a flat geometric figure consisting of three points that do not lie on the same straight line and three segments connecting these points.

A triangle is the simplest closed rectilinear figure, one of the first whose properties man recognized in ancient times, since this figure has always been widely used in practical life.

b) Quadrilaterals

Quadrangleis a flat geometric figure consisting of four points (vertices of a quadrilateral) and four consecutively connecting them segments (sides of a quadrilateral). They have four corners and four sides. A quadrilateral never has three vertices on the same line.

Quadrilaterals are divided into:

1. If opposite sides are parallel in pairs

Parallelogram is a quadrilateral whose opposite sides are parallel in pairs, that is, they lie on parallel lines.

The square and rectangle we knew from childhood turned out to be a special case of a parallelogram.

Square - a regular quadrilateral or rhombus, in which all angles are right, or a parallelogram, in which all sides and angles are equal.

A square, by definition, has equal sides and angles, and, as it turns out, has all the properties of a parallelogram, rectangle and rhombus.

Rectangle is a parallelogram in which all angles are right.


Rhombus is a parallelogram in which all sides are equal.

A rhombus also has all the properties of a parallelogram, but its diagonals are mutually perpendicular and are bisectors of the angles. The heights of the rhombus are equal.

2) if only two sides are parallel

Trapezoid - a quadrilateral with exactly one pair of opposite sides parallel.


The trapezoid is calledisosceles (or isosceles),if its sides are equal.

A trapezoid, one of whose angles is right, is called rectangular.

Rectangular trapezoid Equilateral trapezoid


2) Round shapes

Circle - the geometric locus of points of the plane equidistant from a given point, called the center, at a given non-zero distance, called its radius.

Circle is a part of the plane bounded by a circle.

The circle is only part of the circle, its boundary, while -

Me as a circle is a more extensive and full-fledged figure.

Oval - it is a flat geometric figure.

It is a circle slightly elongated horizontally or vertically. Unlike a circle, an oval does not have an even shape. At some points the oval shape is most curved.

1. Polyhedra

a) Prism

A prism is a polyhedron that consists of two flat polygons lying in different planes and combined by parallel translation, and all the segments connecting the corresponding points of these polygons.

Based on: triangular prism, quadrangular prism, pentagonal prism, etc.

According to the location of the lateral ribs:

Oblique prism– the side edge is inclined to the base at an angle other than 90º.

Straight prism – the side edge is located perpendicular to the base.

Pentagonal, inclined Triangular, inclined Pentagonal, straight

b) Parallelepiped

A parallelepiped is a prism with a parallelogram at its base.

Parallelepipeds, like any prisms, can be straight or inclined.

Inclined parallelepiped- this is an inclined prism, at the base of which is a parallelogram (Fig.a).

Right parallelepiped- this is a straight prism, at the base of which is a parallelogram (Fig.b) or a parallelepiped, the side edge of which is perpendicular to the plane of the base.

Rectangular is a right parallelepiped, at the base of which is a rectangle (ora straight prism with a rectangle at its base).

Cube is a right parallelepiped, all of whose faces are squares.

c) Pyramid

A pyramid is a polyhedron that consists of a flat polygon - the base of the pyramid, a point not lying in the plane of the base - the top of the pyramid and all the segments connecting the top of the pyramid with the points of the base.

The segments connecting the top of the pyramid with the vertices of the base are called lateral edges.

1. Bodies of rotation

A new group of geometric bodies is bodies of revolution, because obtained by rotating flat figures.

A) Cylinder.

A cylinder is a body that consists of two circles, combined by parallel translation, and all the segments connecting the corresponding points of these circles. The circles are called the base of the cylinder, and the segments are called the generators of the cylinder. The bases of the cylinder are equal and lie in parallel planes, the generators are parallel and equal. The cylinder is obtained by rotating a rectangle around one of its sides.

b) Cone

A cone is a body that consists of a circle - the base of the cone, a point not lying in the plane of this circle - the vertex of the cone and all segments connecting the vertex of the cone with the points of the base.

Cone - formed by a right triangle rotating around one of the legs.

B) Sphere and ball.

A sphere is the set of all points in space located at a positive distance R from a given point O, called the center of the sphere.

The word sphere is the Latin form of the Greek word (sphaira) - ball.

A ball is the set of all points in space whose distance from a given point does not exceed a given positive number R. A ball is obtained by rotating a semicircle relative to its diameter.

The beauty of geometry has repeatedly fascinated the human eye. It would seem that you are building the most ordinary and rather ordinary constructions, and then, if you look at them from a different point of view and try to change the picture a little, you end up with something different, unusual, very beautiful. Thus, from geometric shapes, you can get unusual and fascinating constructions.

II. Overview of the city's architectural structures

Some people may believe that various intricate lines, figures, and surfaces can only be found in the books of mathematicians. However, it’s worth looking around and we will see that many objects have a shape similar to the geometric shapes already familiar to us. It turns out there are a lot of them. We just don't always notice them.

Architectural structures consist of individual parts, each of which is built on the basis of certain geometric shapes or their combination. In addition, the shape of any architectural structure has as its model a certain geometric figure. A mathematician would say that this structure “fits” into a geometric figure.

So, when constructing both modern buildings and buildings of past centuries, knowledge of geometry is required. Architectural shaping using geometric structures is preserved in all cases. This problem faced the architects of past centuries, and it has not disappeared today.

Of course, we can only talk approximately about the correspondence of architectural forms to geometric figures, ignoring small details. Each geometric figure has a unique, from an architectural point of view, set of properties.

Modern architecture boldly uses a variety of geometric shapes. The house approximately has the shape of a rectangular parallelepiped. At the same time, many residential buildings and public buildings are decorated with columns.

The circle as a geometric figure has always attracted the attention of artists and architects. Solemnity and upward aspiration - this effect in the architecture of buildings is achieved by using arches representing arcs of circles. The architecture of Orthodox churches includes, as mandatory elements, domes, arches, and rounded vaults, which visually enlarge the space and create the effect of flight and lightness.

And how many geometric shapes can be found in bridge designs. Lifebuoys are often attached to the bridge parapet. They are very close in shape to a torus.

In our work, we explored what geometric shapes and bodies surround us, and we were convinced of how many different geometric lines and surfaces people use in their activities - in the construction of various buildings, bridges, fences and fences. They are used not out of simple love for interesting geometric shapes, but because the properties of these geometric lines and surfaces make it possible to solve various problems with the greatest simplicity.

A) survey results

Before starting to work on the topic, we conducted a sociological survey among students at our school. 54 6th grade students took part in the survey.

During the survey, students were asked to answer the following questions:

Questionnaire

1. Are you satisfied with everything in the architecture of our city?

and all -

b) partially -

c) I would like changes -

2. What architectural structures would you like to see in our city?

a) arrange these -

b) more modern -

c) radically change the architecture of the city -

3.What geometric shapes and shapes are used in the architecture of our city?

b) pyramid -

c) triangle -

d) circle -

e) polygons -

4. What geometric figures and shapes make buildings more clear and expressive?

a) rectangular parallelepiped -

b) pyramid -

c) triangle-

d) circle -

e) polygons -

5. The most beautiful building in our city?

The results of the survey are given in Appendix 1.

Many of the children surveyed would like to see the city as a modern metropolis, and many would like to radically change its architecture.

The guys believe that the use of various geometric shapes will make the city more attractive not only for residents, but also for guests.

To the question, Which building in the city do they consider the most beautiful? 38 students answered that they consider the City Palace of Culture to be the most beautiful building in our city.

A city, urban space can be reduced to a certain set of elements. In fact, everything that surrounds us in the city is a set of geometric shapes. This “geometricism” is practically not perceived on an everyday level from the point of view of a city resident, passerby, or tourist.

Almost all geometric shapes are used in architecture. The choice of using a particular figure in an architectural structure depends on many factors: the aesthetic appearance of the building, its strength, ease of use, etc. The basic requirements for architectural structures, formulated by the ancient Roman architectural theorist Vitruvius, sound like this: “strength, usefulness, beauty".

b) experiment.

Each of us played the game “Cubes” as a child, came up with building designs and built them, considering ourselves a builder or architect. Most often we used cube, parallelepiped, cone and cylinder in construction. In the shape of the first two, bricks and concrete blocks were made, from which buildings were erected, cones - roofs, cylinders - columns.

One of the questions in the survey was: What geometric shapes and shapes are used in the architecture of our city? Most of the guys answered that this is a rectangular parallelepiped and various types of polygons.

To test the assumption that cube, parallelepiped, cone and cylinder are most often used in construction, an experiment was conducted.

Pupils of grade 1b were asked to make a paper application on the topic: “If I were an architect”(Appendix 2) .

The guys were offered a set of geometric shapes (rectangle, square, pyramid, cone, circle, cylinder). It turned out that the majority (28 out of 30 students) used only triangles, rectangles and squares. Only 2 guys additionally used a circle and an oval.

This experiment confirmed the hypothesis that geometric figures, being ideal objects, find their visual embodiment in a variety of architectural structures.

c) overview of the architectural structures of the city

In the modern world, we are surrounded by many buildings consisting of complex geometric shapes, most of which are polyhedra. There are a lot of examples of this, just look around and we will notice that the buildings in which we live, the shops we go to, schools and kindergartens, etc. presented in the form of polyhedra.

Let's turn to the modern urban landscape. There are two directions here.

1) public and cultural buildings

These buildings are created to attract people's attention and create positive emotions in them. When designing them, architects used combinations of various geometric shapes and bodies. And our gaze most often stops at buildings that combine various geometric shapes.

For example, in Meleuz these are the buildings of the city Palace of Culture, an Orthodox church and a mosque(Appendix 3 photo 1, photo 2, photo 3).

The word TEMPLE is of Russian origin (from the word mansion - a festive building). The temple is God's house on earth. Every detail of the temple has deep meaning and significance.

The construction of a new Orthodox church for the city of Meleuz began during the era of Perestroika - in 1990 on the site of the large Trinity-St. Nicholas Church, demolished shortly before (the building existed since ~1898, and before that the old Trinity Church stood here).

In 1994, the construction of a new brick church, named “Kazansko-Bogorodsky” in memory of the last functioning church of Meleuz, was completed.

The temple is 7-domed with a three-tier bell tower.

The architectural style is modern eclecticism with distant Eastern Romanesque motifs.

The “onion” shape of the dome was not chosen by chance. It resembles a flame pointing upward, a burning candle that is lit during prayer. This dome shape symbolizes spiritual ascent and the pursuit of perfection.

The onion is a part of a sphere, smoothly transitioning and ending in a cone.

Domes are a hemisphere or simply a part of a sphere bounded by a plane. The figure lying at the base of the dome is a regular hexagonal, octagonal prism.

The spiers are either pyramids or cones.

The architecture of the church includes arches and rounded vaults as mandatory elements, which visually enlarge the space and create the effect of flight and lightness. A round dormer window at the end of the wall in the shape of a circle.

The mosque is a very unusual structure in appearance.

It is decorated with an octagonal minaret (regular octagonal straight prism), which ends with a high pyramidal (regular octagonal pyramid) tent. The minaret spire is usually crowned with a crescent.

2) residential buildings

High-rise buildings are structures made of rectangular parallelepipeds. The predominant geometric shapes are squares and rectangles (cubes and parallelepipeds). And upon closer examination, you can notice such geometric shapes as cylinders and cones, with which the facades of houses are decorated. Modern architecture boldly uses a variety of geometric shapes. Many residential buildings and public buildings are decorated with columns.(Appendix 3 photo 4, photo 5, photo 6).

One of the most “durable”, “stable” and “confident” geometric figures is the well-known square, in other words, an absolutely regular rectangle. The shape of a rectangle is a brick, board, slab, glass - that is, everything we need to build a building is rectangular in shape.

For example, a rectangle is the basic part of the building, and cylinders and cones are components of the porch and railings.

Without geometry there would be nothing, because all the buildings that surround us are geometric figures. First - simpler ones, such as a square, rectangle, ball. Then - more complex ones: prisms, tetrahedrons, pyramids, etc. But we don't always pay attention to the buildings around us.

3) fences, front gardens

Geometric figures of various shapes can be recognized in other remarkable structures erected by builders and architects(Appendix 4).

The circle as a geometric figure has always attracted the attention of artists and architects. “Cast iron lace”—garden fences, bridge railings, balcony grilles and lanterns—causes delight and surprise. Clearly visible against the background of the facades of buildings in summer, in the frost in winter, it gives a special charm to the city.

Cones are not used as independent structures in construction. They almost always make up some part of the building, such as roofs and architectural decorative details. Conical piles are also used in construction.

The expressive contrast of triangle and rectangle on the façade attracts attention. Round, rectangular, square - all these shapes coexist perfectly in the building.

Unfortunately, Meleuz is a young city; there are practically no historical buildings in it that would have their own individual personality. But it should be noted that construction is currently actively developing in our city. In recent years, architects have been attracted to more modern designs in development. Buildings with unusual shapes attract much more attention than buildings with standard shapes.

The “youngest” buildings are the Arkaim shopping center, the Sweet Dream shopping center, and the Solnechny market. These structures have modern, non-standard forms, fundamentally different from the already familiar “structures - parallelepipeds. These new objects will be a kind of “calling card not only of Meleuz and the Republic of Bashkortostan, but also of the time in which we live.

More and more objects being built have regular geometric shapes, and glass predominates in facade solutions (stained glass, panoramic, frameless, continuous and structural glazing of facades)

Widespread use of steel and glass, metal and plastic, many floors, extreme functionality and laconism - these are the features of the city of Meleuz in the 21st century.

It should be noted that by using different geometric shapes in architecture, you can create a variety of architectural structures that are different from each other. Analyzing some architectural structures of cities and comparing the geometric shapes included in their designs, one can notice that, despite the similarity of the buildings, the architecture of each has geometric shapes that make them different.

CONCLUSION

Geometry decorates the city, gives it severity, individuality and beauty.

By studying the literature used to prepare this work, a lot of interesting knowledge was acquired from the history of architecture and geometry, which once again convinces of the versatility of the application of this science (geometry) and the need to study it.

Thus, the following conclusions can be drawn:

The use of various geometric shapes in architectural structures makes it possible to change the traditional architecture of the city.

Building a city with abstract, modern designs makes it more attractive.

So, we plunged into the world of architecture, studied some of its forms, designs, and compositions. Having examined many of its objects, we were convinced that geometry plays an important, if not the main role in architecture.

Conclusion: All architectural structures of the city of Meleuz consist of geometric figures and their combinations (mostly polyhedra).

We believe that our work corresponds to the goals and objectives stated earlier.

Results of our workcan be used as a teaching aid in geometry lessons or elective classes to study this subject.

How main result of our researchThis was the creation of the slide show “Geometry of our city”.

What did we succeed in our work?

First, we got acquainted with the teaching material on geometry.

Secondly, we did painstaking work and collected working material to study the relationship between architecture and geometry.

Thirdly, we collected a lot of interesting material about the architectural structures of our city, which allowed us to come to certain conclusions regarding its architecture:

1) city architects have recently, when creating projects for new buildings, included various geometric shapes in their designs;

2) the beauty of buildings in our city lies in their symmetry and dissymmetry;

3) the use of various geometric shapes in architectural structures makes it possible to change the traditional architecture of the city;

4) building up the city with abstract, modern structures makes it more attractive to residents and guests.

The purpose of this work was to highlight the main geometric shapes (using the example of modern architecture).

To achieve this goal:

The main geometric shapes were identified.

An experiment was conducted to study the most frequently used geometric shapes in applications.

The main features of the use of various geometric shapes in architecture are analyzed.

The features of modern architecture of Meleuz are characterized.

Man is gradually reducing the number of geometric forms used, particularly in architecture, in favor of rectilinear ones (cubes and parallelepipeds), thereby impoverishing the world around him.

This raises several questions that could be topics for future research. How will the reduction of geometric shapes surrounding a person in favor of rectilinear ones affect people’s health, in particular their vision? Who invented polygons and polyhedra, where are they used?

And we would like to finish our work with the words of the great French architect, creator of international style architecture, artist and designer of the 20th century, Le Corbusier (1887-1965): “I think that never before have we lived in such a geometric period. It is worth reflecting on the past, remembering what happened before, and we will be stunned to see that the world around us is a world of geometry, pure, true, impeccable in our eyes. Everything around is geometry. Never have we seen so clearly such shapes as a circle, a rectangle, an angle, a cylinder, a sphere, executed so clearly, with such care and so confidently.”

One can only agree with Le Corbusier’s thought. Years and centuries pass, but the role of geometry does not change.

LITERATURE

1. A.V. Voloshinov. "Mathematics and Art".

M.: Enlightenment. 2000.

2. Magazine “Mathematics at School”. – 2005. - No. 4.

3. A.V. Ikonnikov. "The artistic language of architecture."

M: Stroyizdat. 1992.

4. A.V. Pogorelov. "

Geometry grades 10-11."

M.: Enlightenment. 2009.

5. L.S. Atanasyan. "Geometry 7-9 grades"

M.: Enlightenment. 2011.

6. Internet resources: http://ru.wikipedia.org

Fifth Lyceum Scientific and Practical Conference “Cognition and Creativity”

Physics mathematics

Topic: “Architecture in geometric shapes”

Research project

Student 9 "A" class MAOU

"Lyceum No. 21"

Supervisor:

Krotova Irina Leonidovna,

mathematic teacher

Table of contents

Relevance Nowadays, cities and countries are increasingly built up. New structures are appearing. New architects are appearing, new directions in architecture are emerging. As Louis Henry Sullivan said: “Architecture is the art that affects people most slowly, but most lastingly.” Our worldview and mood depend on what is happening in the city and what it looks like. And it seems to me that any building or structure is built on the basis of geometric figures and combinations of geometric bodies.

And no other form of art is so closely related to geometry as architecture. Everyone should understand architecture, because it surrounds and accompanies us all our lives.

Hypothesis

All the buildings that surround us are geometric figures; on the one hand, they are abstractions from real objects, and, on the other, they are prototypes, models of the shape of those objects that the architect creates.

Goals and objectives:

Target:

Consider what types of buildings there are and what geometric shapes they consist of.

Tasks:

Study the history of the emergence of geometry and architecture

Find geometric shapes in buildings:
In my city

Find modern Russian architects

Create your own building in geometric shapes

Theoretical part

“The world around us is a world of pure, true, impeccable geometry in our eyes. Everything around is geometry.” Le Corbusier

Geometry is a branch of mathematics that studies spatial relationships and forms, as well as other relationships and forms that are similar to spatial ones in their structure.

Architecture is the art of modeling the human environment and designing people's behavior in this environment, through a special functional and artistic organization of space and form, artistic work with the plasticity of elements and colors.

Story

It is traditionally believed that the founders of geometry as a systematic science are the ancient Greeks, who adopted the craft of surveying and measuring the volume of bodies from the Egyptians, and turned it into a strict scientific discipline. Based on the discovery of many geometric properties, Greek scientists were able to create a coherent system of knowledge in geometry. Geometric science was based on the simplest geometric properties taken from experience. The remaining provisions of science were derived from the simplest geometric properties using reasoning. This entire system was published in its completed form in Euclid's Elements around 300 BC. The first proofs of geometric statements appeared in the works of Thales and apparently used the principle of superposition, when figures whose equality needed to be proven were superimposed on each other.

Thanks to the great Archimedes, who was able to calculate the number Pi, and was also able to determine ways to calculate the surface of a ball, a problem that no one could solve before him. Archimedes asked to knock out a ball inscribed in a cylinder on his grave. Archimedes was able to establish that the volumes of a cone and a sphere inscribed in a cylinder and the cylinder itself are in a ratio of 1:2:3. The system developed by Euclid was considered immutable for more than two thousand years. However, later the history of the development of geometry took an unexpected turn, when in 1826 the brilliant Russian mathematician N.I. Lobachevsky was able to create a completely new geometric system named after him. Lobachevsky's axiom states that through a point that does not lie on a line, more than one line parallel to the given one can be drawn. In fact, the main provisions of his system differ from the provisions of Euclid’s geometry in only one point, but it is from this point that the main features of Lobachevsky’s system flow. This is the position that the sum of the angles of a triangle in Lobachevsky geometry is always less than 180 degrees. At first glance, it may seem that this statement is incorrect, but with small triangles, modern measuring instruments do not allow one to correctly measure the sum of its angles. The further history of the development of geometry proved the correctness of Lobachevsky's brilliant ideas and showed that Euclid's system is simply incapable of solving many issues.

Thus, geometry from its inception has studied some properties of the real world.

The first architectural structures had a religious purpose. The ancient pagan tribes used obelisks for rituals. The main problem was vertical instability; science was not yet well developed at that time. Then it is believed that the construction of the Egyptian pyramids began.

The Greeks made architecture itself, or more precisely, the story of the work of its structures, the theme of architecture as an art. From this point on, the supports of the post-and-beam system not only decorate the building, but also show that they support something and that they are heavy. They ask for the sympathy of the audience and, to be convincing, imitate the structure and proportions of the human figure - male, female or female.

The Romans began to widely use arches and arched structures (vaults and domes). The horizontal beam may crack if it is too long; the wedge-shaped parts in the arched arc do not break under load, but are compressed, and it is not easy to destroy a stone with pressure. Therefore, arched structures can cover much larger spaces and load them much more boldly.

A technological breakthrough in Byzantine architecture is the placement of the dome, invented in Ancient Rome, not on round walls enclosing the internal space, but on four arches - respectively, with only four points of support. Between the arches and the dome ring, biconcave triangles were formed - sails.

By the beginning of the second millennium AD, powerful empires began to emerge in Europe, and each considered itself the heir of Rome. The traditions of Roman architecture were also revived. The majestic Romanesque cathedrals were again covered with arched structures similar to ancient ones - stone and brick vaults.

The Renaissance gave the world the greatest domes, but from that moment on, great styles arose not so much as a result of construction innovations, but as a result of changes in the very picture of the world. The Renaissance, Mannerism, Baroque, Rococo, Classicism and Empire style were born more thanks to philosophers, theologians, mathematicians and historians (and to some extent to those who brought gallant manners into fashion) than to the inventors of new ceiling designs. Until the era of the Industrial Revolution, innovations in construction technologies ceased to be a determining factor in changing styles.

In 1850, the factory production of large-sized window glass made it possible to develop technologies for the construction of first large greenhouses, and then grandiose buildings for other purposes, in which either all the walls or roofs were made of glass. Fairy-tale “crystal palaces” began to become reality.

The history of architecture is a science of both historical and theoretical profile. This feature is due to the specifics of the subject - the history of the emergence and development of architecture, theoretical knowledge about architecture, architectural language, architectural composition, as well as the observation of such common features and signs of architecture of a certain time and place that allow us to distinguish architectural styles.

Architecture as a method of artistic creativity arises from the fact that the human mind has an innate need from God, while cognizing the world, to express itself, its feelings, thoughts, ideas about Infinity, composed of finite forms. Therefore, a building structure is a functional type of structure, and an architectural composition is an artistic and figurative integrity.

Architecture in geometric shapes

There are many fountains in Russia that consist of different geometric shapes. Consider the fountain in Moscow “Stone Flower”. If you look at it, you can see circles from above. There are also parts that consist of a sphere and cubes. Along the perimeter there are figures that also consist of geometric shapes.


Let's look at another fountain "Doves" in Kazan. Here we can also see circles, we can see cylinders and truncated cones.


New shopping and entertainment centers also began to appear. There are such centers in Yekaterinburg, for example, Alatyr. We can see the cube, but it is in a slice. In this cut we can see part of the cylinder.


There is also such a center “Fan-Fan”, also in Yekaterinburg. It is in the shape of a cube, but its edges are cylindrical and therefore its edges are not sharp, but rounded.


We can also find architecture in geometric shapes. These are innovation and cultural centers: in Skolkovo, in Pervouralsk - “Shaiba” (such as in Pervouralsk are planned in Vladivostok and Kaluga)





In 2015, a building, a business center, was built in Moscow, it was created by a wonderful woman, Zaha Hadid. This was her last building. She died on March 31, 2016, but left behind a lot of interesting, diverse buildings.


For example, this building is located in Baku and was built in 2012.

Hadid created a lot of things: she created the Expo Center project in Moscow; created the design of furniture, shoes, etc. for different companies, including Russian ones. But the most unusual thing is the business center in Moscow. From the outside, this building is made of several cubes of different thicknesses and sizes. They are all located differently. But inside this building looks even more unusual and creates an illusion. It looks different from different sides and angles.




Of course, Hadid has more buildings, but they all also consist of different geometric shapes.

Contemporary Russian architects

Architectural studio "MEL"

Fyodor Dubinnikov and Pavel Chaunin. Founded in 2009. The Checkers affordable housing project in 2009 brought an award from the International Architectural Biennale in Rotterdam (IABR), the Avangard Prize and the title of “Best Young Architect of Russia” as part of the curatorial program “ARCH Moscow NEXT!”

“We are creating a new typology of architecture with simple and functional solutions. The stylistic basis of our projects is minimalism and contrast. We are looking for unusual uses for ordinary materials and trying to emphasize the architectural originality of the context,” say the founders of Mel.



Architectural studio ZA BOR

Arseny Borisenko and Petr Zaitsev. Founded in 2003. Their clients are large companies and businessmen, interesting and extraordinary people. Today ZA BOR successfully conducts international practice. The bureau's portfolio is diverse and includes both private houses and interiors, as well as offices, office complexes, territory development projects, and urban planning concepts. The original projects and concepts of the workshop have been awarded dozens of awards, are included in trend books of international advertising agencies and are presented in furniture catalogs of leading office furniture manufacturers.


Architectural bureau FORM

Olga Treivas and Vera Odyn. Founded in 2011. Among the objects are the exhibition space in the new pavilion of the Garage CSK, the Russian pavilion at the International Book Fair in Turin, the Onexim Hall conference hall, converted from an old Stalinist cinema. FORM knows how to work with space, making it where it is needed, moderately unnoticeable and, on the contrary, forcing it to delicately emphasize itself when the situation requires it. It’s as if architecture suddenly “takes the form” of the art it is meant to demonstrate, rather than simply containing it within itself like a foreign body.


Of course, these are not all companies, but when looking at the projects, these guys were the ones I liked the most. There is a zest in their projects that is hidden somewhere, but you admire it. Looking through some of the projects, I wondered why they were so simple, but looking at them, I liked them more and more.

Practical part

I decided to try to draw a building from geometric shapes myself. I drew a building consisting of cubes, pyramids, cylinders and spheres. Buildings can be divided into parts. The first part is a cube-shaped entrance and the cube itself with rectangular and oval windows. The second part is also a cube, but it is very thin and a cut is made in it. It has large rectangular panoramic windows. Between these parts there is another cube with rectangular windows, but various figures are already coming out of it. There is a pyramid with a triangular window and trapezoidal windows. A ball with square windows is attached to the pyramid. In addition to the pyramids, there is another figure emerging from the cube - this is a hexagonal cylinder with windows in the shape of a circle.

I think this building can be both a shopping and entertainment center and a business center, and maybe even an innovative, cultural, scientific, etc.

When constructing such a building, you need to carefully consider the choice of material and correctly calculate it so that it does not fall from gusts of wind. You also need to choose a suitable area for construction.

Conclusion from the practical part: creating buildings and structures is very difficult, because you need to calculate practicality, choose the right material and color.

Conclusion

We looked at where geometric shapes are found in the buildings of Yekaterinburg and Pervouralsk. We examined several projects of architectural workshops. We got to know their goals and plans for the future. We also proved our hypothesis that all structures and buildings begin with the design and alignment of geometric bodies, and then calculations begin. Having seen these buildings today, we realized that the importance of choosing the use of geometric shapes and their placement, as well as the correct choice of material and color, greatly influences a person’s mood and thoughts. 12/14/2016 21.17 http://zabor.net/ 14.12.2016 22.09

Pervouralsk

2017

“Properties of a rectangular parallelepiped” - Formulate the properties of a parallelepiped. 1. All faces are parallelograms. Not cubes. 2. Opposite sides are equal and parallel. New topic. Direct. Not rectangular. Inclined. Parallelepipeds. Define a prism. Properties of a rectangular parallelepiped. A rectangular parallelepiped with equal edges is called a cube.

“Tetrahedron and parallelepiped” - Sections. Tetrahedron. The diagonals intersect and bisect at the point of intersection. Properties of a parallelepiped. Elements of a tetrahedron. Tetrahedron Parallelepiped. Section. Construction of a section. Completed by Kotlovskaya I.Yu.G.N.Novgorod. 1.Opposite faces are parallel and equal.

“Architectural monuments” - Gorno-Altaiskaya, City Palace of Culture. Practical work: Biysk tobacco factory. The city where you live. Architectural monuments of local significance: Group captains make up the overall composition. Krasnooktyabrskaya 200, Zarechny fire station. Compare the photo with the proposed template. Based on the photograph, decorate the building using the applique technique.

“Sections of a parallelepiped” - Independent work of students. MPKN - section of a parallelepiped. Task: construct a section passing through points M, N, K. M ? (ABB’A’) N ? (ABCD) K ? CC'. Task: construct a section passing through points M, N, K. Sections of a parallelepiped. 1. Teacher’s introductory speech – 3 min 2. Activation of students’ knowledge.

“Architecture of the 14th-16th centuries” - The main building material is wood. Painting. Remained predominantly ecclesiastical. Icon painting and fresco painting of churches developed. Architecture in the 14th century. Russian painting in the XIV – XVI centuries. Compare the features of the churches of Novgorod, Pskov, and Moscow. Architecture in the XIV–XV centuries. Dionysius. Construction was completed within a year.

“Calculating the volume of a parallelepiped” - Test yourself: Task 1: Calculate the volumes of the figures. Volume of a rectangular parallelepiped. Task 3: Calculate the volume of a rectangular parallelepiped. Find the volume of a cube: Mathematics 5th grade. Task 2: Which of the pictures have rectangular parallelepipeds?