Name of the figures. Geometric shapes for children

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Introduction

Geometry is one of the most important components of mathematical education, necessary for the acquisition of specific knowledge about space and practically significant skills, the formation of a language for describing objects in the surrounding world, for the development of spatial imagination and intuition, mathematical culture, as well as for aesthetic education. The study of geometry contributes to the development logical thinking, formation of proof skills.

The 7th grade geometry course systematizes knowledge about the simplest geometric figures and their properties; the concept of equality of figures is introduced; the ability to prove the equality of triangles using the studied signs is developed; a class of problems involving construction using a compass and ruler is introduced; one of the the most important concepts- the concept of parallel lines; new interesting and important properties triangles; one of the most important theorems in geometry is considered - the theorem on the sum of the angles of a triangle, which allows us to classify triangles by angles (acute, rectangular, obtuse).

During classes, especially when moving from one part of the lesson to another, changing activities, the question arises of maintaining interest in classes. Thus, relevant the question arises about using problems in geometry classes in which there is a condition problematic situation and elements of creativity. Thus, purpose This study is to systematize tasks of geometric content with elements of creativity and problem situations.

Object of study: Geometry tasks with elements of creativity, entertainment and problem situations.

Research objectives: Analyze existing geometry tasks aimed at developing logic, imagination and creative thinking. Show how you can develop interest in a subject using entertaining techniques.

Theoretical and practical significance research is that the collected material can be used in the process of additional lessons in geometry, namely at Olympiads and competitions in geometry.

Scope and structure of the study:

The study consists of an introduction, two chapters, a conclusion, bibliography, contains 14 pages of main typewritten text, 1 table, 10 figures.

Chapter 1. FLAT GEOMETRIC FIGURES. BASIC CONCEPTS AND DEFINITIONS

1.1. Basic geometric figures in the architecture of buildings and structures

There are many material objects in the world around us. different forms and sizes: residential buildings, car parts, books, jewelry, toys, etc.

In geometry, instead of the word object, they say geometric figure, while dividing geometric figures into flat and spatial. In this work, we will consider one of the most interesting sections of geometry - planimetry, in which only plane figures are considered. Planimetry(from Latin planum - “plane”, ancient Greek μετρεω - “measure”) - a section of Euclidean geometry that studies two-dimensional (single-plane) figures, that is, figures that can be located within the same plane. A flat geometric figure is one in which all points lie on the same plane. Any drawing made on a sheet of paper gives an idea of ​​such a figure.

But before considering flat figures, it is necessary to get acquainted with simple but very important figures, without which flat figures simply cannot exist.

The simplest geometric figure is dot. This is one of the main figures of geometry. It is very small, but it is always used for building various forms on surface. The point is the main figure for absolutely all constructions, even the highest complexity. From a mathematical point of view, a point is an abstract spatial object that does not have such characteristics as area or volume, but at the same time remains a fundamental concept in geometry.

Straight- one of the fundamental concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry (Euclidean). If the basis for constructing geometry is the concept of distance between two points in space, then a straight line can be defined as a line along which the path is equal to the distance between two points.

Lines in space can occupy various provisions, let's look at some of them and give examples found in the architectural appearance of buildings and structures (Table 1):

Table 1

Parallel lines	Properties of parallel lines
	If the lines are parallel, then their projections of the same name are parallel:	Essentuki, mud bath building (photo by the author)
Intersecting lines	Properties of intersecting lines	Examples in the architecture of buildings and structures
	Intersecting lines have a common point, that is, the intersection points of their projections of the same name lie on a common connection line:	"Mountain" buildings in Taiwan https://www.sro-ps.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane
Crossing lines	Properties of skew lines	Examples in the architecture of buildings and structures
	Straight lines that do not lie in the same plane and are not parallel to each other are intersecting. None is a common communication line. If intersecting and parallel lines lie in the same plane, then intersecting lines lie in two parallel planes.	Robert, Hubert - Villa Madama near Rome https://gallerix.ru/album/Hermitage-10/pic/glrx-172894287

1.2. Flat geometric shapes. Properties and Definitions

Observing the forms of plants and animals, mountains and river meanders, landscape features and distant planets, man borrowed from nature its correct forms, sizes and properties. Material needs prompted people to build houses, make tools for labor and hunting, sculpt dishes from clay, and so on. All this gradually contributed to the fact that man came to understand the basic geometric concepts.

Quadrilaterals:

Parallelogram(ancient Greek παραλληλόγραμμον from παράλληλος - parallel and γραμμή - line, line) is a quadrilateral with opposite sides are pairwise parallel, that is, they lie on parallel lines.

Signs of a parallelogram:

A quadrilateral is a parallelogram if one of the following holds true: following conditions: 1. If in a quadrilateral the opposite sides are equal in pairs, then the quadrilateral is a parallelogram. 2. If in a quadrilateral the diagonals intersect and are divided in half by the point of intersection, then this quadrilateral is a parallelogram. 3. If two sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.

A parallelogram whose angles are all right angles is called rectangle.

A parallelogram in which all sides are equal is called diamond

Trapezoid— It is a quadrilateral in which two sides are parallel and the other two sides are not parallel. Also, a trapezoid is a quadrilateral in which one pair of opposite sides is parallel, and the sides are not equal to each other.

Triangle is the simplest geometric figure formed by three segments that connect three points that do not lie on the same straight line. These three points are called vertices triangle, and the segments are sides triangle. It is precisely because of its simplicity that the triangle was the basis of many measurements. Land surveyors, when calculating land areas, and astronomers, when finding distances to planets and stars, use the properties of triangles. This is how the science of trigonometry arose - the science of measuring triangles, of expressing the sides through its angles. The area of ​​any polygon is expressed through the area of ​​a triangle: it is enough to divide this polygon into triangles, calculate their areas and add the results. True, it was not immediately possible to find the correct formula for the area of ​​a triangle.

The properties of the triangle were especially actively studied in the 15th-16th centuries. Here is one of the most beautiful theorems of that time, due to Leonhard Euler:

A huge amount of work on the geometry of the triangle, carried out in the XY-XIX centuries, created the impression that everything was already known about the triangle.

Polygon - it is a geometric figure, usually defined as a closed polyline.

Circle- the geometric locus of points in the plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If the radius is zero, then the circle degenerates into a point.

There are a large number of geometric shapes, they all differ in parameters and properties, sometimes surprising with their shapes.

In order to better remember and distinguish flat figures by properties and characteristics, I came up with a geometric fairy tale, which I would like to present to your attention in the next paragraph.

Chapter 2. PUZZLES FROM FLAT GEOMETRIC FIGURES

2.1.Puzzles for constructing a complex figure from a set of flat geometric elements.

After studying flat shapes, I wondered if there were any interesting problems with flat shapes that could be used as games or puzzles. And the first problem I found was the Tangram puzzle.

This is a Chinese puzzle. In China it is called "chi tao tu", or a seven-piece mental puzzle. In Europe, the name “Tangram” most likely arose from the word “tan”, which means “Chinese” and the root “gram” (Greek - “letter”).

First you need to draw a 10 x 10 square and divide it into seven parts: five triangles 1-5 , square 6 and parallelogram 7 . The essence of the puzzle is to use all seven pieces to put together the figures shown in Fig. 3.

Fig.3. Elements of the game "Tangram" and geometric shapes

Fig.4. Tangram tasks

It is especially interesting to make “shaped” polygons from flat figures, knowing only the outlines of objects (Fig. 4). I came up with several of these outline tasks myself and showed these tasks to my classmates, who happily began to solve the tasks and created many interesting polyhedral figures, similar to the outlines of objects in the world around us.

To develop imagination, you can also use such forms of entertaining puzzles as tasks for cutting and reproducing given figures.

Example 2. Cutting (parqueting) tasks may seem, at first glance, to be quite diverse. However, most of them use only a few basic types of cuts (usually those that can be used to create another from one parallelogram).

Let's look at some cutting techniques. In this case, we will call the cut figures polygons.

Rice. 5. Cutting techniques

Figure 5 shows geometric shapes from which you can assemble various ornamental compositions and create an ornament with your own hands.

Example 3. Another interesting task that you can come up with on your own and exchange with other students, and whoever collects the most cut pieces is declared the winner. There can be quite a lot of tasks of this type. For coding, you can take all existing geometric shapes, which are cut into three or four parts.

Fig. 6. Examples of cutting tasks:

------ - recreated square; - cut with scissors;

Basic figure

2.2. Equal-sized and equally-composed figures

Let's consider another interesting technique for cutting flat figures, where the main “heroes” of the cuts will be polygons. When calculating the areas of polygons, a simple technique called the partitioning method is used.

In general, polygons are called equiconstituted if, after cutting the polygon in a certain way F into a finite number of parts, it is possible, by arranging these parts differently, to form a polygon H from them.

This leads to the following theorem: Equilateral polygons have the same area, so they will be considered equal in area.

Using the example of equipartite polygons, we can consider such an interesting cutting as the transformation of a “Greek cross” into a square (Fig. 7).

Fig.7. Transformation of the "Greek Cross"

In the case of a mosaic (parquet) composed of Greek crosses, the parallelogram of the periods is a square. We can solve the problem by superimposing a mosaic made of squares onto a mosaic formed with the help of crosses, so that the congruent points of one mosaic coincide with the congruent points of the other (Fig. 8).

In the figure, the congruent points of the mosaic of crosses, namely the centers of the crosses, coincide with the congruent points of the “square” mosaic - the vertices of the squares. By moving the square mosaic in parallel, we will always obtain a solution to the problem. Moreover, the problem has several possible solutions if color is used when composing the parquet ornament.

Fig.8. Parquet made from a Greek cross

Another example of equally proportioned figures can be considered using the example of a parallelogram. For example, a parallelogram is equivalent to a rectangle (Fig. 9).

This example illustrates the partitioning method, which consists in calculating the area of ​​a polygon by trying to divide it into a finite number of parts in such a way that these parts can be used to create a simpler polygon whose area we already know.

For example, a triangle is equivalent to a parallelogram having the same base and half the height. From this position the formula for the area of ​​a triangle is easily derived.

Note that the above theorem also holds converse theorem: if two polygons are equal in size, then they are equivalent.

This theorem, proven in the first half of the 19th century. by the Hungarian mathematician F. Bolyai and the German officer and mathematics lover P. Gerwin, can be represented in this way: if there is a cake in the shape of a polygon and a polygonal box of a completely different shape, but the same area, then you can cut the cake into a finite number of pieces (without turning them cream side down) that they can be placed in this box.

Conclusion

In conclusion, I would like to note that there are quite a lot of problems on flat figures in various sources, but those that were of interest to me were the ones on the basis of which I had to come up with my own puzzle problems.

After all, by solving such problems, you can not only accumulate life experience, but also acquire new knowledge and skills.

In puzzles, when constructing actions-moves using rotations, shifts, translations on a plane or their compositions, I got independently created new images, for example, polyhedron figures from the game “Tangram”.

It is known that the main criterion for the mobility of a person’s thinking is the ability to recreate and creative imagination perform certain actions within a set period of time, and in our case, moves of figures on the plane. Therefore, studying mathematics and, in particular, geometry at school will give me even more knowledge to later apply in my future professional activities.

Bibliography

1. Pavlova, L.V. Non-traditional approaches to teaching drawing: tutorial/ L.V. Pavlova. - Nizhny Novgorod: Publishing house NSTU, 2002. - 73 p.

2. encyclopedic Dictionary young mathematician/ Comp. A.P. Savin. - M.: Pedagogy, 1985. - 352 p.

3.https://www.srops.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane

4.https://www.votpusk.ru/country/dostoprim_info.asp?ID=16053

Annex 1

Questionnaire for classmates

1. Do you know what a Tangram puzzle is?

2. What is a “Greek cross”?

3. Would you be interested to know what “Tangram” is?

4. Would you be interested to know what a “Greek cross” is?

22 8th grade students were surveyed. Results: 22 students do not know what “Tangram” and “Greek cross” are. 20 students would be interested in learning how to use the Tangram puzzle, consisting of seven flat figures, to obtain a more complex figure. The survey results are summarized in a diagram.

Appendix 2

Elements of the game "Tangram" and geometric shapes

Transformation of the "Greek Cross"

At the same time as learning colors, you can start showing your child cards of geometric shapes. On our website you can download them for free.

How to study figures with your child using Doman cards.

1) You need to start with simple shapes: circle, square, triangle, star, rectangle. As you master the material, begin to study more complex shapes: oval, trapezoid, parallelogram, etc.

2) You need to work with your child using Doman cards several times a day. When demonstrating a geometric figure, clearly pronounce the name of the figure. And if during classes you also use visual objects, for example, collecting inserts with figures or a toy sorter, then your child will master the material very quickly.

3) When the child remembers the name of the shapes, you can move on to more complex tasks: now showing the card, say - this is a blue square, it has 4 equal sides. Ask your child questions, ask him to describe what he sees on the card, etc.

Such activities are very useful for the development of a child’s memory and speech.

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Educational video for children (toddlers and preschoolers) made according to the Doman method “Prodigy from the cradle” - educational cards, educational pictures on various topics from part 1, part 2 of the Doman method, which can be watched for free here or on our Channel Early childhood development on youtube

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards based on Glen Doman's method with pictures of flat geometric shapes for children

Educational cards geometric shapes according to Glen Doman's method with pictures of flat geometric shapes for children

Educational cards geometric shapes according to Glen Doman's method with pictures of flat geometric shapes for children

Educational cards geometric shapes according to Glen Doman's method with pictures of flat geometric shapes for children

More of our Doman cards using the “Prodigy from the Diaper” method:

1. Domana Cards Tableware
2. Doman cards National dishes

Geometric figures represent a complex of points, lines, bodies or surfaces. These elements can be located both on the plane and in space, forming a finite number of straight lines.

The term “figure” implies several sets of points. They must be located on one or more planes and at the same time limited to a specific number of completed lines.

The main geometric figures are the point and the straight line. They are located on a plane. In addition to them, among simple figures there is a ray, a broken line and a segment.

Dot

This is one of the main figures of geometry. It is very small, but it is always used to build various shapes on a plane. The point is the main figure for absolutely all constructions, even the highest complexity. In geometry, it is usually denoted by a letter of the Latin alphabet, for example, A, B, K, L.

From a mathematical point of view, a point is an abstract spatial object that does not have such characteristics as area or volume, but at the same time remains a fundamental concept in geometry. This zero-dimensional object simply has no definition.

Straight

This figure is completely placed in one plane. A straight line does not have a specific mathematical definition, since it consists of a huge number of points located on one endless line, which has no limit or boundaries.

There is also a segment. This is also a straight line, but it starts and ends from a point, which means it has geometric limitations.

The line can also turn into a directional beam. This happens when a straight line starts from a point, but does not have a clear ending. If you put a point in the middle of the line, then it will split into two rays (additional), and oppositely directed to each other.

Several segments that are sequentially connected to each other by ends at a common point and are not located on the same straight line are usually called a broken line.

Corner

Geometric figures, the names of which we discussed above, are considered key elements used in the construction of more complex models.

An angle is a structure consisting of a vertex and two rays that extend from it. That is, the sides of this figure connect at one point.

Plane

Let's consider another primary concept. A plane is a figure that has neither end nor beginning, as well as a straight line and a point. When considering this geometric element, only its part, limited by the contours of a broken closed line, is taken into account.

Any smooth bounded surface can be considered a plane. This could be an ironing board, a piece of paper, or even a door.

Quadrilaterals

A parallelogram is a geometric figure whose opposite sides are parallel to each other in pairs. Among the particular types of this design are diamond, rectangle and square.

A rectangle is a parallelogram in which all sides touch at right angles.

A square is a quadrilateral with equal sides and angles.

A rhombus is a figure in which all sides are equal. In this case, the angles can be completely different, but in pairs. Each square is considered a diamond. But in the opposite direction this rule does not always apply. Not every rhombus is a square.

Trapezoid

Geometric shapes can be completely different and bizarre. Each of them has a unique shape and properties.

A trapezoid is a figure that is somewhat similar to a quadrilateral. It has two parallel opposite sides and is considered curved.

Circle

This geometric figure implies the location on one plane of points equidistant from its center. In this case, a given non-zero segment is usually called a radius.

Triangle

This is a simple geometric figure that is very often encountered and studied.

A triangle is considered a subtype of a polygon, located on one plane and limited by three edges and three points of contact. These elements are connected in pairs.

Polygon

The vertices of polygons are the points connecting the segments. And the latter, in turn, are considered to be parties.

Volumetric geometric shapes

• prism;
• sphere;
• cone;
• cylinder;
• pyramid;

These bodies have something in common. All of them are limited to a closed surface, inside of which there are many points.

Volumetric bodies are studied not only in geometry, but also in crystallography.

Curious facts

Surely you will be interested in reading the information provided below.

• Geometry was formed as a science back in ancient times. This phenomenon is usually associated with the development of art and various crafts. And the names of geometric figures indicate the use of the principles of determining similarity and similarity.
• Translated from ancient Greek, the term “trapezoid” means a table for a meal.
• If you take different shapes whose perimeter is the same, then the circle is guaranteed to have the largest area.
• Translated from Greek, the term “cone” means a pine cone.
• There is a famous painting by Kazemir Malevich, which, since the last century, has attracted the views of many painters. The work “Black Square” has always been mystical and mysterious. The geometric figure on the white canvas delights and amazes at the same time.

There are a large number of geometric shapes. They all differ in parameters, and sometimes even surprise in shape.

The scope of study of the science of geometry includes flat (two-dimensional) figures and three-dimensional figures (three-dimensional).

From flat:

Studies them planimetry. A point is also a flat figure.

Of the volumes known:

Studies them stereometry.

Two-dimensional figures - triangle, square, rectangle, rhombus, trapezoid, parallelogram, circle, oval, ellipse, polygons (pentagon, hexagon, heptagon, octagon and others).

The point also belongs to the figures.

Three-dimensional figures - cube, sphere, hemisphere, cone, cylinder, pyramid, parallelepiped, prism, ellipsoid, dome, tetrahedron and many others arising from the above. Next come very complex geometric figures - various polyhedra, which essentially can contain an infinite number of faces. For example, a large clinocorona - consists of 2 squares and 16 regular triangles or a clinocorona, composed of 14 faces: 2 squares and 12 regular triangles.

Speaking about geometric figures, we can distinguish two regular groups:

1) Two-dimensional figures;

2) And three-dimensional figures.

So, in more detail about two-dimensional ones, these include such figures as:

But as for three-dimensional figures, here’s what they can be:



The outlines of figures and all possible actions with them are studied by the mathematical sciences of geometry (studies of flat figures) and stereometry (the subject of study is three-dimensional figures). At school I loved both sciences.

This is how flat (2D) figures are classified:



With three sides it is a triangle. With four sides - a square, a rhombus, a rectangle, a trapezoid. There can also be a parallelogram and a circle (oval, circle, semicircle, ellipse).

Volumetric figures (3D) are classified as follows:



These are cube, parallelepiped, tetrahedron, cylinder, pyramid, icosahedron, sphere, dodecahedron, cone, octahedron, prism, sphere. In addition, there are truncated figures (pyramid, cone). Depending on the base, a pyramid or prism is divided into triangular, tetrahedral, and so on.

Children's toys (pyramids, mosaics and others) allow children to be introduced to geometric three-dimensional figures from early childhood. And flat shapes can be drawn and cut out of paper.

The two-dimensional ones include the following:

• circle;
• oval;
• square;
• rectangle;
• parallelogram;
• trapezoid;
• pentagon (hexagon, etc.);
• rhombus;
• triangle.

With three-dimensional ones it’s a little more complicated:

• cylinder;
• cone;
• prism;
• sphere or ball;
• parallelepiped;
• pyramid;
• tetrahedron;
• icosahedron;
• octahedron;
• dodecahedron.

I think many, having read the latest titles, asked themselves: What, what? For clarity, here is an illustration:



In fact, there are enough figures in mathematics. Flat figures these are rectangles, square, triangle, pentagon, hexagon, circle. Volumetric figures or 3D figures are a pyramid, a cube, a dodecahedron, and so on.

• Personally I know:

  1 From two-dimensional figures:

  circle, triangle, square, rhombus, rectangle, trapezoid, parallelogram, oval and polygon. Another star (pentagram), if it can be called a figure.

  2 From three-dimensional figures:

  Prism, pyramid, parallelepiped, prism, ball (sphere), cylinder, hemisphere (half of a sphere, that is, a ball cut in half) and cone. Pyramids are divided into triangular, quadrangular, and so on (almost ad infinitum). The more corners a pyramid has at its base, the more it resembles a cone.

Two-dimensional shapes (2D): angle; polygon (varieties of polygons: triangle, quadrangle; varieties of quadrangle: parallelogram, rectangle, rhombus, square, trapezoid, deltoid, pentagon, hexagon, etc. ad infinitum); circle, circle, circular segment, circular sector, ellipse, oval...

Three-dimensional figures (3D): dihedral angle, polyhedral angle; polyhedron (varieties of polyhedra: prism, varieties of prism: parallelepiped, cube, antiprism, pyramid, variety of tetrahedron, truncated pyramid, bipyramid, variety of octahedron, dodecahedron, icosahedron, wedge, obelisk); cylinder, truncated cylinder, cylinder segment (aka cylindrical horseshoe or hoof), cone, truncated cone, sphere, ball, spherical segment, spherical layer, spherical sector, ellipsoid, geoid...

From the very beginning, in geometry lessons we study simple figures that are flat, that is, located on the same plane.

So, the list of main figures can be studied below.





IN Lately I just had to tell my granddaughters and grandson what geometric shapes can be.

Starting with flat figures cut out of cardboard or made of plastic, children learned to distinguish between a triangle and a square, an oval and a circle, a rectangle, a rhombus and a polygon.

These special toys with holes of a certain shape also helped in remembering the names of the figures.



Later they switched to three-dimensional figures, cubes and cones, parallelepipeds, balls and rings, pyramids and cylinders.



They are not old enough to go to school yet, but when they go, they will be taught to distinguish between isosceles and equilateral triangles, learn about a ray and a point, about a circle and everything else.

Geometry – precise mathematical science, which studies spatial and other similar relationships and forms. But it is often called “dry” because it is not able to describe the shape of many natural objects, because clouds are not spheres, mountains are not cones, and lightning does not travel in straight lines. Many objects in nature have complex shapes compared to standard geometry.

However, there are a number of surprising figures that are not usually studied in school lessons geometry, but it is they that surround a person in the real world: in nature and architecture, puzzles, computer games, etc.

The main property of this complex geometric figure is self-similarity, that is, it consists of several parts, each of which is similar to the whole object. It is this property that distinguishes fractals from objects of classical (or, as they say, Euclidean) geometry.

Moreover, the term “fractal” itself is not mathematical and does not have an unambiguous definition, therefore it can be applied to objects that are self-similar or approximately self-similar. It was invented in 1975 by Benoit Mandelbrot, borrowing the Latin word “fractus” (broken, crushed).

Fractal forms are the best suited for describing the real world and are often found among natural objects: snowflakes, plant leaves, the blood vessel system of humans and animals.

This is one of the most extraordinary three-dimensional shapes in geometry, which is easy to make at home. To do this, it is enough to take a paper strip, the width of which is 5-6 times less than its length, and, twisting one of the ends 180°, glue them together.

If everything is done correctly, you can check its amazing properties yourself:

• The presence of only one side (without division into internal and external). This can be easily checked if you try to paint over one of its sides with a pencil. Regardless of where and in what direction you start painting, the end result will be that the entire tape will be painted with the same color.
• Continuity: If you draw a line along the entire surface with a pen, its end will connect to the starting point without crossing the boundaries of the surface.
• Two-dimensionality (connectedness): when cutting a Möbius strip lengthwise, it remains intact, new shapes are simply obtained (for example, when cut in half, one larger ring is obtained).
• Lack of orientation. A journey along such a Mobius strip will always be endless, it will lead to starting point paths, only in mirror image.

Mobius strips are widely used in industry and science (in conveyor belts, matrix printers, sharpening mechanisms, etc.). In addition, there is a scientific hypothesis according to which the Universe itself is also a Mobius strip of incredible size.

Polyomino

These are flat geometric shapes that are formed by connecting several squares of equal sizes along their sides.

The names of polyominoes depend on the number of squares from which they are formed:

• monomino – 1;
• domino – 2;
• trimino – 3;
• tetromino – 4, etc.

Moreover, for each variety there is a different number of types of figures: dominoes have 1 type, triminos have 3 types, hexaminos (of 6 squares) have 35 types. The number of different variations depends on the number of squares used, but no scientist has yet been able to find an amazing formula that will express this dependence. From polyomino parts you can lay out both geometric shapes and images of people, animals, and objects. Despite the fact that these will be sketchy silhouettes, the main features and shapes of the objects make them quite recognizable.

Polyamond

Along with polyominoes, there is another amazing geometric figure used to compose other shapes - polyamong. It is a polygon formed from several equilateral triangles of equal size.

The name was invented by the mathematician T. O'Beirne based on one of the names of the rhombus in English language– a diamond that can be made from 2 equilateral triangles. By analogy, O’Beirne called a figure of 3 equilateral triangles a triamond, a figure of 4 - a tetriamond, etc.

The main question of their existence remains the question of the possible number of polyamides that can be made from a certain number of triangles. The use of polyamunds in real life also similar to using polyominoes. These can be various kinds of puzzles and logical tasks.

Reuleaux triangle

As surprising as it sounds, you can drill a square hole with a drill, and the Reuleaux triangle helps with this. It represents the area formed by the intersection of 3 equal circles, whose centers are the vertices regular triangle, and the radii are equal to its side.

The Reuleaux triangle itself is named after the German scientist-engineer, who was the first to study its features in more detail and use it for his mechanisms at the turn of the 19th-20th centuries. century, although its amazing properties were already known to Leonardo da Vinci. Whoever was its discoverer, modern world This figure is widely used in the form:

• Watts drill, which allows you to drill holes of an almost perfect square shape, only with slightly rounded edges;
• a mediator necessary for playing plucked musical instruments;
• cam mechanisms used to create zigzag seams in sewing machines, as well as German watches;
• pointed arches, characteristic of the Gothic style in architecture.

Impossible figures

The so-called impossible figures deserve special attention - amazing optical illusions that at first glance seem to be a projection of a three-dimensional object, but upon closer examination unusual combinations of elements become noticeable. The most popular of them are:

Tribar created by father and son Lionel and Roger Penrose, which features an image equilateral triangle, but has strange patterns. The sides that form the top of the triangle appear perpendicular, but the right and left sides at the bottom also appear perpendicular. If we consider each part of this triangle separately, we can still recognize their existence, but in reality such a figure cannot exist, since the correct elements were incorrectly connected when it was created.

The Endless Staircase, the authorship of which also belongs to the father and son Penroses, is therefore often called by their name - the “Penrose Staircase”, as well as the “Eternal Staircase”. At first glance, it looks like an ordinary staircase leading up or down, but a person walking along it will continuously ascend (counterclockwise) or descend (clockwise). If you visually travel along such a staircase, then at the end of the “journey” your gaze stops at the starting point of the path. If such a staircase existed in reality, it would have to be climbed and descended an infinite number of times, which can be compared to an endless Sisyphean task.

The Impossible Trident is an amazing object, looking at which it is impossible to determine where the middle prong begins. It is also based on the principle of irregular connections, which can only exist in two dimensions, but not three-dimensional space. Looking at the parts of the trident separately, 3 round teeth are visible on one side, and 2 rectangular ones on the other side.

Thus, the parts of the figure enter into a kind of conflict: firstly, the foreground and background change, and secondly, the round teeth in the lower part are transformed into flat ones in the upper part.