Good day, friends!

For a long time I was going to tell you about this project of ours, but somehow the hands did not reach. And here is a miracle! The hands have arrived! So, the project is called "Polygons around us". As you may have guessed, this is the math work we did in 4th grade with my daughter Alexandra.

We approached the work creatively and we are sure that our mathematical creativity can be useful for you to prepare your abstracts, projects or research papers.

We titled the work as follows: “Mathematical Thriller. Polygon Hunter »

And now I bring you the full text, along with all the photographs. The story is told in the first person, the author of this scientific work.

The purpose of the work: the practical application of polygons in the world around us.

Problematic question: what place do polygons occupy in our life?

Since childhood, we have been familiar with various types of polygons, but how often we meet them in the world around us, we somehow do not think.

I decided to take a closer look at things familiar in everyday life and find polygons studied in mathematics lessons in the objects around us.

One day, armed to the teeth with a long heavy ruler, I went hunting for polygons.

Didn't have to go far. I looked for them at home.

I went to the door to the kitchen and, gathering my will into a fist, turned on the light! And… Oh horror!!! I felt hundreds of polygonal, sharp and blunt, as well as absolutely direct views. They were everywhere! They were staring at me without hesitation! They were not afraid of my ruler! They didn't even try to hide! This is not a kitchen! This is a real polygonal kingdom! Hundreds of polygons sat on the walls (rectangles in the wallpaper pattern). I didn't even dare to count them.

The most cunning stuck to the ceiling (ceiling tiles are in the shape of rectangles). They looked at me suspiciously from above.

And the most arrogant ones got into the dishes ... and even turned into them (the ornament on the dishes and the shape of the dishes are represented by different types of polygons).

Now I know that polygons love to mold dumplings (hexagons are visible in the dumpling mold).

They watch what I eat. And even for what my cat eats (the edges of food boxes are in the form of rectangles).

Terrified, I jumped out of the kitchen and headed into the hall. And suddenly I saw ... that one of the polygons captured my parrots (the cage consists of elements of a rectangular, triangular and quadrangular shape).

These impudent figurines did not spare even a child (constructor elements). My younger brother enthusiastically played with them, unaware of the danger.

My beloved grandmother, without stopping, looked into another polygon, which showed her what was happening in the world (the TV screen is a rectangle).

And suddenly there was a sharp squeaky sound! “What is this?” I thought in shock. And it was another representative of this polygonal kingdom (a cell phone has the shape of a rectangular parallelepiped) that gave a voice from the shelf.

I ran to the nursery, hoping to hide at least there ... But I did not succeed.

Bright, cheerful polygons, laughing happily, swayed on our curtains (geometric fabric pattern). “May you fall!”, I thought, and looked at my table…

I shouldn't have done it ... On my table, two complex polygons were talking about something. One is blue, the other is red… (plafonds of lamps can be considered as a combination of triangles and quadrilaterals).

And beside them, little polygonal cubs giggled softly (the edges of the pencils are rectangles, and the base is a hexagon).

This is not an apartment! This is a lair of polygons!!! They have a nest here!

They even celebrated the New Year with us (the shape of many Christmas decorations is a combination of different polygons)! And we didn't even know...

I realized that you can't hide from them anywhere. Even in Egypt (the faces of the pyramids are triangles, the bases are rectangles)!

Conclusion. This world belongs to polygons! And we have to come to terms with it. And learn to live in harmony with these polygonal creatures.

Here we have such an unusual project. Thanks to which, in the diary, Sasha got another five.

It was made in the Power Point program in the form of slides and presented not only at a mathematics lesson, but also at the school competition "Science and Creativity", where he was also awarded a diploma.

On our blog you will find other mathematical projects:

That's all for today!

We wish you interesting creative tasks!

What would happen if there was only one type of shape in the world, for example, a shape such as a rectangle? Some things wouldn't change at all: doors, cargo trailers, football fields - they all look the same. But what about door handles? They would be a little weird. What about car wheels? This would be inefficient. What about football? It's hard to even imagine. Fortunately, the world is full of many different forms. Do they exist in nature? Yes, and there are a lot of them.

What is a polygon?

In order for a figure to be a polygon, certain conditions are necessary. First, there must be many sides and corners. Also, it must be a closed form. is a figure with all equal sides and angles. Accordingly, for the wrong one, they may be slightly deformed.

Types of regular polygons

What is the minimum number of sides a regular polygon can have? One line cannot have many sides. The two sides also cannot meet and form a closed shape. And three sides can - so you get a triangle. And since we are talking about regular polygons, where all sides and angles are equal, we mean

If you add one more side, you get a square. Can a rectangle where the sides are not equal be a regular polygon? No, this figure will be called a rectangle. If you add the fifth side, you get a pentagon. Accordingly, there are hexagons, heptagons, octagons, and so on ad infinitum.

Elementary geometry

Polygons come in different types: open, closed, and self-intersecting. In elementary geometry, a polygon is a flat figure, which is bounded by a finite chain of straight line segments in the form of a closed polyline or contour. These segments are its edges or sides, and the points where two edges meet are its vertices and corners. The interior of a polygon is sometimes called its body.

Polyhedra in nature and human life

While many living forms abound in pentagonal patterns, the mineral world favors double, triple, quadruple, and sixfold symmetry. The hexagon is a dense shape that provides maximum structural efficiency. It is very common in the field of molecules and crystals, in which pentagonal shapes are almost never found. Steroids, cholesterol, benzene, vitamins C and D, aspirin, sugar, graphite are all manifestations of sixfold symmetry. Where are regular polyhedra found in nature? The most famous hexagonal architecture is created by bees, wasps and hornets.

Six water molecules form the core of each snow crystal. This is how a snowflake is made. The facets of the fly's eye form a densely packed hexagonal arrangement. What other regular polyhedra are there in nature? These are water and diamond crystals, basalt columns, epithelial cells in the eye, some plant cells and much more. Thus, the polyhedrons created by nature, both animate and inanimate, are present in human life in a huge number and variety.

What is the reason for the popularity of hexagons?

Snowflakes, organic molecules, quartz crystals and columnar basalts are hexagons. The reason for this is their inherent symmetry. The most striking example is the honeycomb, whose hexagonal structure minimizes the lack of space, since the entire surface is used very rationally. Why divide into identical cells? Bees create regular polyhedrons in nature in order to use them for their needs, including storing honey and laying eggs. Why does nature prefer hexagons? The answer to this question can be given by elementary mathematics.

• Triangles. Take 428 equilateral triangles with a side of about 7.35 mm. Their total length is 3 * 7.35 mm * 428/2 = 47.2 cm.
• Rectangles. Take 428 squares with a side of about 4.84 mm, their total length is 4 * 4.84 m * 428/2 = 41.4 cm.
• Hexagons. And finally, take 428 hexagons with a side of 3 mm, their total length is 6 * 3 mm * 428/2 = 38.5 cm.

The obvious is the victory of the hexagons. It is this form that helps to minimize the space to the maximum and allows you to place as many figures as possible in a smaller area. The honeycomb in which the bees store their amber nectar is a marvel of precision engineering, an array of prism-shaped cells with a perfectly hexagonal cross section. The wax walls are made to a very precise thickness, the cells are carefully tilted to prevent the gooey honey from falling out, and the entire structure is aligned with the earth's magnetic field. Surprisingly, the bees work simultaneously, coordinating their efforts.

Why hexagons? It's simple geometry

If you want to put cells of the same shape and size together so that they fill the entire plane, then only three regular shapes (with all sides and with the same angles) will work: equilateral triangles, squares and hexagons. Of these, hexagonal cells require the smallest overall wall length compared to triangles or squares of the same area.

So the bees' choice of hexagons makes sense. Back in the 18th century, scientist Charles Darwin declared that hexagonal honeycombs were “absolutely ideal in saving labor and wax.” He believed that natural selection gave the bees the instinct to create these wax chambers, which had the advantage of requiring less energy and time than other forms.

Examples of polyhedra in nature

The compound eyes of some insects are packed into a hexagon, where each face is a lens connected to a long thin retinal cell. The structures that are formed by clusters of biological cells often have shapes governed by the same rules as bubbles in soapy water. The microscopic structure of the facet of the eye is one of the best examples. Each facet contains a cluster of four photosensitive cells, which are the same shape as the cluster of four regular vesicles.

What determines these soap film rules and bubble shapes? Nature is even more concerned about economy than the bees. Bubbles and soap films are made of water (with soap added), and surface tension pulls the surface of the liquid in such a way as to give it as little area as possible. This is why drops are spherical (more or less) when they fall: a sphere has less surface area than any other shape with the same volume. On a wax sheet, water droplets are drawn into small beads for the same reason.

This surface tension explains the bubble raft and foam models. The foam will look for the structure that has the lowest total surface tension, which will provide the smallest wall area. Although the geometry of soap films is dictated by the interaction of mechanical forces, it does not tell us what the shape of the foam will be. A typical foam contains polyhedral cells of various shapes and sizes. If you look closely, then the regular polyhedra in nature are not so correct. Their edges are rarely perfectly straight.

Correct bubbles

Let's assume that you can make a "perfect" foam in which all the bubbles are the same size. What is the perfect cell shape that makes the total area of ​​the bubble wall as small as possible. This has been discussed for many years, and for a long time it was believed that the ideal cell shape is a 14-sided polyhedron with square and hexagonal sides.

In 1993, a more economical, though less ordered, structure was discovered, consisting of a repeating group of eight different cell shapes. This more complex model was used as inspiration for the foam design of the swimming stadium during the 2008 Beijing Olympics.

The rules of cell formation in foam also control some of the patterns observed in living cells. It is not only the compound eye of the fly that shows the same hexagonal packing of facets as the flat bubble. The light-sensitive cells inside each of the individual lenses also cluster into clusters that look just like soap bubbles.

The world of polyhedra in nature

The cells of many different types of organisms, from plants to rats, contain membranes with these microscopic structures. No one knows what they are for, but they are so widespread that it is fair to assume that they have some useful role. Perhaps they isolate one biochemical process from another, avoiding cross interventions.

Or maybe it's just an efficient way to create a large working plane, since many biochemical processes take place on the surface of membranes, where enzymes and other active molecules can be embedded. Whatever the function of polyhedra in nature, don't bother creating complex genetic instructions, because the laws of physics will do it for you.

Some butterflies have winged scales containing an ordered labyrinth of a tough material called chitin. Exposure to light waves bouncing off normal ridges and other structures on the surface of a wing causes some wavelengths (i.e. some colors) to fade out while others reinforce each other. Thus, the polygonal structure offers an excellent vehicle for producing animal color.

To make ordered networks out of a rigid mineral, some organisms appear to form a mold out of soft, flexible membranes and then crystallize the hard material within one of the interpenetrating networks. The honeycomb structure of hollow microscopic channels within the chitinous spines of the unusual known as the sea mouse turns these hair-like structures into natural optical fibers that can guide light, changing it from red to bluish-green depending on the direction of the light. This color change may serve to deter predators.

Nature knows better

The flora and fauna are replete with examples of polyhedrons in wildlife, as well as the inanimate world of stones and minerals. From a purely evolutionary point of view, the hexagonal structure is the leader in energy optimization. In addition to the obvious advantages (saving space), polyhedral meshes provide a large number of faces, therefore, the number of neighbors increases, which has a beneficial effect on the entire structure. The end result of this is that information spreads much faster. Why are regular hexagonal and irregular star polyhedra so common in nature? Probably so necessary. Nature knows better, she knows better.

A person shows interest in polyhedra throughout his conscious activity - from a two-year-old child playing with wooden cubes to a mature mathematician. Some of the regular and semi-regular bodies occur in nature in the form of crystals, others in the form of viruses that can only be seen with an electron microscope. What is a polyhedron? To answer this question, let us recall that geometry itself is sometimes defined as the science of space and spatial figures - two-dimensional and three-dimensional. A two-dimensional figure can be defined as a set of line segments bounding a part of a plane. Such a flat figure is called a polygon. It follows that a polyhedron can be defined as a set of polygons bounding a portion of three-dimensional space. The polygons that form a polyhedron are called its faces.

Since ancient times, scientists have been interested in "ideal" or regular polygons, that is, polygons that have equal sides and equal angles. An equilateral triangle can be considered the simplest regular polygon, since it has the smallest number of sides that can limit a part of a plane. The general picture of regular polygons of interest to us, along with an equilateral triangle, is made up of: a square (four sides), a pentagon (five sides), a hexagon (six sides), an octagon (eight sides), a decagon (ten sides), etc. Obviously, theoretically there are no restrictions on the number of sides of a regular polygon, that is, the number of regular polygons is infinite.

What is a regular polyhedron? Such a polyhedron is called regular if all its faces are equal (or congruent, as is customary in mathematics) to each other and, at the same time, are regular polygons. How many regular polyhedra are there? At first glance, the answer to this question is very simple - as many as there are regular polygons, that is, at first glance it seems that you can create a regular polyhedron, the sides of which can be any regular polygon. However, it is not. Already in the Elements of Euclid it was rigorously proved that the number of regular polyhedra is very limited and that there are only five regular polyhedra whose faces can be only three types of regular polygons: triangles, squares and pentagons. These regular polyhedra are called the Platonic solids. The first one is the tetrahedron. Its faces are four equilateral triangles. The tetrahedron has the fewest number of faces among the Platonic solids and is the three-dimensional analog of a flat regular triangle, which has the fewest number of sides among regular polygons. The word "tetrahedron" comes from the Greek "tetra" - four and "edra" - base. It is a triangular pyramid. The next body is a hexahedron, also called a cube. The hexahedron has six faces, which are squares. The faces of the octahedron are regular triangles and their number in the octahedron is eight. The next largest number of faces is the dodecahedron. Its faces are pentagons and their number in the dodecahedron is twelve. The icosahedron closes the five Platonic solids. Its faces are regular triangles and their number is twenty.

In my work, the main definitions and properties of convex polyhedra are considered. The existence of only five regular polyhedra has been proven. The relations for the regular n-gonal pyramid and the regular tetrahedron, which are the most common in stereometry problems, are considered in detail. The paper presents a large amount of analytical and illustrative material that can be used in the study of some sections of stereometry.

Plato's studies

Plato created a very interesting theory. He suggested that the atoms of the four "basic elements" (earth, water, air and fire), from which all things are built, have the form of regular polyhedra: a tetrahedron - fire, a hexahedron (cube) - earth, an octahedron - air, an icosahedron - water. The fifth polyhedron - the dodecahedron - symbolized the "Great Mind" or "Harmony of the Universe". Particles of three elements that easily turn into each other, namely fire, air and water, turned out to be made up of identical figures - regular triangles. And the earth, which is significantly different from them, consists of particles of a different type - cubes, or rather squares. Plato very clearly explained all the transformations with the help of triangles. In the restless chaos, two particles of air meet a particle of fire, that is, two octahedrons meet a tetrahedron. Two octahedrons have a total of sixteen triangular faces, a tetrahedron has four. Altogether twenty. Out of twenty, one icosahedron is easily formed, and this is a particle of water.

Plato's cosmology became the basis of the so-called icosahedral-dodecahedral doctrine, which has since run like a red thread through all human science. The essence of this doctrine is that the dodecahedron and icosahedron are typical forms of nature in all its manifestations, from the cosmos to the microworld.

Regular polyhedra

Regular polyhedra have attracted the attention of scientists, builders, architects and many others since ancient times. They were struck by the beauty, perfection, harmony of these polyhedrons. The Pythagoreans considered these polyhedra to be divine and used them in their philosophical writings about the essence of the world. The last, 13th book of the famous "Beginnings" of Euclid is devoted to regular polyhedra.

We repeat that a convex polyhedron is called regular if its faces are equal regular polygons and the same number of faces converge at each vertex.

The simplest such regular polyhedron "is a triangular pyramid, the faces of which are regular triangles. Three faces converge at each of its vertices. Having all four faces, this polyhedron is also called a tetrahedron, which means "four-hedron" in Greek.

Sometimes a tetrahedron is also called an arbitrary pyramid. Therefore, in the case when we are talking about a regular polyhedron, we will say - a regular tetrahedron.

A polyhedron whose faces are regular triangles, and at each vertex four faces converge, the surface of which consists of eight regular triangles, is called an octahedron.

A polyhedron, at each vertex of which five regular triangles converge, the surface of which consists of twenty regular triangles, is called an icosahedron.

Note that since more than five regular triangles cannot converge at the vertices of a convex polyhedron, there are no other regular polyhedra whose faces are regular triangles.

Similarly, since only three squares can converge at the vertices of a convex polyhedron, there are no other regular polyhedra with squares as faces besides the cube. A cube has six sides and is therefore called a hexahedron.

A polyhedron whose faces are regular pentagons and three faces converge at each vertex. Its surface consists of twelve regular pentagons, it is called a dodecahedron.

Since regular polygons with more than five sides cannot converge at the vertices of a convex Polyhedron, there are no other regular polyhedra, and thus there are only five regular polyhedra: tetrahedron, hexahedron (cube), octahedron, dodecahedron, icosahedron.

The names of regular polyhedra come from Greece. In literal translation from Greek "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "icosahedron" mean: "tetrahedron", "octahedron", "hexahedron". dodecahedron, dodecahedron. The 13th book of Euclid's Elements is dedicated to these beautiful bodies. They are also called Plato's bodies, because they occupied an important place in Plato's philosophical concept of the structure of the universe.

And now let's look at how many properties, lemmas and theorems associated with these figures.

Let us consider a polyhedral angle with vertex S, where all flat and all dihedral angles are equal. We choose points A1, A2, An on its edges so that SA1 = SA2 = SAn. Then the points A1, A2, An lie in the same plane and are vertices of a regular n-gon.

Proof.

Let us prove that any consecutive points lie in the same plane. Consider four consecutive points A1, A2, A3 and A4. The pyramids SA1 A2 A3 and SA2 A3 A4 are equal, since they can be combined by combining the edges SA2 and SA3 (of course, the edges of different pyramids are taken) and the dihedral angles at these edges. Similarly, it can be shown that the pyramids SA1 A3A4 and SA1 A2 A4 are equal, since all their edges are equal. This implies the equality

It follows from the last equality that the volume of the pyramid A1A2A3A4 is equal to zero, that is, these four points lie in the same plane. Hence, all n points lie in the same plane, and in the n-gon A1 A2 An all sides and angles are equal. Hence, it is correct, and the lemma is proved.

Let us prove that there are at most five different types of regular polyhedra.

Proof.

From the definition of a regular polyhedron it follows that only triangles, quadrangles and pentagons can be its faces. Indeed, let us prove, for example, that faces cannot be regular hexagons. According to the definition of a regular polyhedron, at least three faces must converge at each of its vertices. However, in a regular hexagon, the angles are 120°. It turns out that the sum of three plane angles of a convex polyhedral angle is 360°, which is impossible, since this sum is always less than 360°. Moreover, the faces of a regular polyhedron cannot be polygons with a large number of sides.

Let us find out how many faces can converge at a vertex of a regular polyhedron. If all its faces are regular triangles, then no more than five triangles can adjoin each vertex, since otherwise the sum of plane angles at this vertex will be at least 360°, which, as we have seen, is impossible. So, if all the faces of a regular polyhedron are regular triangles, then three, four or five triangles adjoin each vertex. By analogous reasoning, we make sure that at each vertex of a regular polyhedron, whose faces are regular quadrangles and pentagons, exactly three edges converge.

Let us now prove that there is only one polyhedron of a given type with a fixed edge length. Consider, for example, the case when all faces are regular pentagons. Assume the opposite: let there be two polyhedra, all of whose faces are regular pentagons with side a, and all dihedral angles in each polyhedron are equal to each other. Note that not all dihedral angles of one polyhedron are necessarily equal to the dihedral angles of another polyhedron: this is what we will now prove.

As we have shown, three edges emerge from each vertex of each polyhedron. Let the edges AB, AC and AD come out of the vertex A of one polyhedron, and the edges A1B1, A1C1 and A1D1 come out of the vertex A1 of the other. ABCD and A1B1C1D1 are regular triangular pyramids, since they have equal edges coming out of vertices A and A1 and flat angles at these vertices.

It follows that the dihedral angles of one polyhedron are equal to the dihedral angles of the other. Hence, if we combine the pyramids ABCD and A1B1C1D1, then the polyhedra themselves will also be compatible. Hence, if there exists a regular polyhedron all of whose faces are regular pentagons with side a, then such a polyhedron is unique.

Other polyhedra are considered similarly. In the case when all faces are triangles and four or five triangles adjoin each vertex, one should use Lemma 2. and a pentagon. The theorem has been proven.

Note that this theorem does not imply that there are exactly five types of regular polyhedra. The theorem only states that there are at most five such types, and now it remains for us to prove that there are indeed five of these types by presenting all five types of polyhedra.

Regular n-gonal pyramid

Consider a regular n-gonal pyramid. This polyhedron is often encountered in stereometric problems, and therefore a more detailed and thorough study of its properties is of great interest. Moreover, one of our regular polyhedra - the tetrahedron - is it.

Let SA1A2 An be a regular n-gonal pyramid. Let us introduce the following notation:

α is the angle of inclination of the side rib to the plane of the base;

β is the dihedral angle at the base;

γ is the flat angle at the top;

δ is the dihedral angle at the lateral edge.

Let O be the center of the base of the pyramid, B the middle of the edge A1A2, D the intersection point of the segments A1A3 and OA2, C the point on the side edge SA2 such that A1CSA2, E the intersection point of the segments SB and A1C, K the intersection point of the segments A1A3 and OV. Let A1OA2=. It's easy to show

We also denote the height of the pyramid through H, the apothem - through m, the side edge - through l, the side of the base - through a, and through r and R - the radii of the circles inscribed in the base and described around it.

Below are the relations between the angles α, β, γ, δ of a regular n-gonal pyramid, formulated in the form of theorems.

regular tetrahedron

Its properties

Applying the relations obtained in the previous section to a regular tetrahedron allows us to obtain a number of interesting relations for the latter. In this section, we will present the obtained formulas for this specific case and, in addition, we will find expressions for some characteristics of a regular tetrahedron, such as, for example, volume, total surface area, and the like.

Following the notation of the previous section, consider the regular tetrahedron SA1A2A3 with edge length a. We leave the notation for its angles the same and calculate them.

In a regular triangle, the length of the height is equal. Since this triangle is regular, its height is both a bisector and a median. Medians, as you know, are divided by the point of their intersection in a ratio of 2: 1, counting from the top. It is easy to find the point of intersection of the medians. Since the tetrahedron is regular, this point will be the point O - the center of the regular triangle A1A2A3. The base of the height of a regular tetrahedron, dropped from the point S, also projects to the point O. Hence,. In a regular triangle SA1A2, the length of the apothem of the tetrahedron is equal. Let's apply the Pythagorean theorem for Δ SBO:. From here.

Thus, the height of a regular tetrahedron is equal to.

The area of ​​the base of a tetrahedron - a regular triangle:

So the volume of a regular tetrahedron is:

The total surface area of ​​a tetrahedron is four times the area of ​​its base.

The dihedral angle at the side face for a regular tetrahedron is obviously equal to the angle of inclination of the side face to the base plane:

The plane angle at the vertex of a regular tetrahedron is equal to.

The angle of inclination of the side rib to the plane of the base can be found from:

The radius of an inscribed sphere for a regular tetrahedron can be found by a well-known formula relating it to the volume and total surface area of ​​the tetrahedron (note that the latter formula is valid for any polyhedron in which a sphere can be inscribed). In our case, we have

Find the radius of the circumscribed sphere. The center of the sphere circumscribed about a regular tetrahedron lies at its height, since it is the line SO that is perpendicular to the plane of the base and passes through its center, and this line must contain a point equidistant from all vertices of the base of the tetrahedron. Let this be point O1, then O1S=O1A2=R. We have. Let's apply the Pythagorean theorem to triangles BA2O1 and BO1O:

Note that R = 3r, r + R = H.

It is interesting to calculate, that is, the angle at which the edge of a regular tetrahedron is visible from the center of the described sphere. Let's find it:

This is a value familiar to us from the course of chemistry: this is the angle between the C–H bonds in the methane molecule, which can be very accurately measured in the experiment, and since not a single hydrogen atom in the CH4 molecule is obviously isolated by anything, it is reasonable to assume that this molecule has the shape of a regular tetrahedron. This fact is confirmed by photographs of a methane molecule obtained using an electron microscope.

Regular hexahedron (Cube)

Face type Square

Number of faces 6

Number of ribs 12

Number of peaks 8

Flat angle 90 o

Sum of flat angles 270 o

Is there a center of symmetry Yes (the point of intersection of the diagonals)

Number of axes of symmetry 9

Number of planes of symmetry 9

Regular octahedron

Number of faces 8

Number of ribs 12

Number of peaks 6

Flat angle 60o

Number of flat corners at the vertex 4

Sum of flat angles 240o

Is there an axis of symmetry Yes

Existence of a regular octahedron

Consider the square ABCD and build on it, as on the base, on both sides of its plane quadrangular pyramids, the side edges of which are equal to the sides of the square. The resulting polyhedron will be an octahedron.

To prove this, it remains for us to check that all dihedral angles are equal. Indeed, let O be the center of square ABCD. Connecting the point O with all the vertices of our polyhedron, we get eight triangular pyramids with a common vertex O. Consider one of them, for example, ABEO. AO = BO = EO and, moreover, these edges are pairwise perpendicular. Pyramid ABEO is regular, since its base is a regular triangle ABE. Hence, all dihedral angles at the base are equal. Similarly, all eight pyramids with apex at point O and bases - the faces of the octahedron ABCDEG - are regular and, moreover, are equal to each other. This means that all the dihedral angles of this octahedron are equal, since each of them is twice the dihedral angle at the base of each of the pyramids.

*Note an interesting fact related to the hexahedron (cube) and octahedron. A cube has 6 faces, 12 edges and 8 vertices, while an octahedron has 8 faces, 12 edges and 6 vertices. That is, the number of faces of one polyhedron is equal to the number of vertices of the other and vice versa. The cube and hexahedron are said to be dual to each other. This is also manifested in the fact that if you take a cube and build a polyhedron with vertices in the centers of its faces, then, as you can easily see, you get an octahedron. The reverse is also true - the centers of the faces of the octahedron serve as the vertices of the cube. This is the duality of the octahedron and the cube.

It is easy to figure out that if we take the centers of the faces of a regular tetrahedron, then we again get a regular tetrahedron. Thus the tetrahedron is dual to itself. *

Regular icosahedron

Face view Right triangle

Number of faces 20

Number of ribs 30

Number of peaks 12

Flat angle 60 o

Number of flat corners at the vertex 5

Sum of flat angles 300 o

Is there a center of symmetry Yes

Number of axes of symmetry Several

Number of planes of symmetry Several

Existence of a regular icosahedron

There is a regular polyhedron in which all faces are regular triangles, and 5 edges emerge from each vertex. This polyhedron has 20 faces, 30 edges, 12 vertices and is called an icosahedron (icosi - twenty).

Proof

Consider the octahedron ABCDEG with edge 1. Choose points M, K, N, Q, L, and P on its edges AE, BE, CE, DE, AB, and BC, respectively, so that AM = EK = CN = EQ = BL = BP = x. We choose x such that all the segments connecting these points are equal to each other.

It is obvious that for this it suffices to fulfill the equality KM = KQ. However, since KEQ is an isosceles right triangle with legs KE and EQ, then. We write the cosine theorem for the triangle MEK, in which:

From here. The second root, which is greater than 1, does not fit. Choosing x in this way, we construct the required polyhedron. We choose six more points that are symmetrical to the points K, L, P, N, Q, and M with respect to the center of the tetrahedron and denote them as K1, L1, P1, N1, Q1, and M1, respectively. The resulting polyhedron with vertices K, L, P, N, Q, M, K1, L1, P1, N1, Q1, and M1 is the desired one. All its faces are regular triangles, five edges emerge from each vertex. Let us now prove that all its dihedral angles are equal to each other.

To do this, we note that all the vertices of the constructed twenty-hedron are equidistant from the point O, the center of the octahedron, that is, they are located on the surface of the sphere with center O. Further, we proceed in the same way as in the proof of the existence of a regular octahedron. Let us connect all the vertices of the twenty-hedron with the point O. In exactly the same way, we prove the equality of triangular pyramids, the bases of which are the faces of the constructed polyhedron, and make sure that all the dihedral angles of the twenty-hedron are twice as large as the angles at the base of these equal triangular pyramids. Therefore, all dihedral angles are equal, which means that the resulting polyhedron is regular. It's called the icosahedron.

Regular dodecahedron

View of the Pentagon face (regular pentagon)

Number of faces 12

Number of ribs 30

Number of peaks 20

Flat angle 108 o

Number of flat corners at the vertex 3

Sum of flat angles 324 o

Is there a center of symmetry yes

Number of axes of symmetry Several

Number of planes of symmetry Several

Existence of a regular dodecahedron

There is a regular polyhedron in which all faces are regular pentagons and 3 edges emerge from each vertex. This polyhedron has 12 faces, 30 edges and 20 vertices and is called a dodecahedron (dodeka - twelve).

Proof.

As you can see, the number of faces and vertices of the polyhedron, the existence of which we are now trying to prove, is equal to the number of vertices and faces of the icosahedron. Thus, if we prove the existence of the polyhedron referred to in this theorem, then it will certainly turn out to be dual to the icosahedron. On the example of a cube and an octahedron, we have seen that dual figures have the property that the vertices of one of them lie at the centers of the faces of the other. This leads to the idea of ​​proving this theorem.

Take an icosahedron and consider a polyhedron with vertices at the centers of its faces. It is obvious that the centers of the five faces of the icosahedron that have a common vertex lie in the same plane and serve as the vertices of a regular pentagon (this can be verified in a manner similar to that used in the proof of the lemma). So, each vertex of the icosahedron corresponds to a face of a new polyhedron, the faces of which are regular pentagons, and all dihedral angles are equal. This follows from the fact that any three edges coming out of the same vertex of the new polyhedron can be considered as side edges of a regular triangular pyramid, and all resulting pyramids are equal (they have equal side edges and flat angles between them, which are the angles of a regular triangular pyramid). pentagon). From the foregoing, it follows that the resulting polyhedron is regular and has 12 faces, 30 edges and 20 vertices. Such a polyhedron is called a dodecahedron.

So, in three-dimensional space, there are only five types of regular polyhedra. We determined their form and established that all polyhedra have duals to them. The cube is dual to the octahedron and vice versa. Icosahedron to dodecahedron and vice versa. The tetrahedron is dual to itself.

Euler's formula for regular polyhedra

So, it was found out that there are exactly five regular polyhedra. And how to determine the number of edges, faces, vertices in them? This is not difficult to do for polyhedra with a small number of edges, but how, for example, to obtain such information for an icosahedron? The famous mathematician L. Euler obtained the formula В+Г-Р=2, which relates the number of vertices /В/, faces /Г/ and edges /Р/ of any polyhedron. The simplicity of this formula is that it has nothing to do with distance or angles. In order to determine the number of edges, vertices and faces of a regular polyhedron, we first find the number k \u003d 2y - xy + 2x, where x is the number of edges belonging to one face, y is the number of faces converging at one vertex. To find the number of faces, vertices and edges of a regular polyhedron, we use formulas. After that, it is easy to fill out a table that provides information about the elements of regular polyhedra:

Name Vertices (V) Edges (P) Faces (D) Formula

Tetrahedron 4 6 4 4-6+4=2

Hexahedron (Cube) 8 12 6 8-12+6=2

Octahedron 6 12 8 6-12+8=2

Icosahedron 12 30 20 12-30+20=2

Dodecahedron 20 30 12 20-30+12=2

Chapter II: Regular polyhedra in life

Space and Earth

There are many hypotheses and theories related to polyhedrons about the structure of the Universe, including our planet. Below are some of them.

An important place was occupied by regular polyhedra in the system of the harmonious structure of the world by I. Kepler. All the same faith in harmony, beauty and the mathematically regular structure of the universe led I. Kepler to the idea that since there are five regular polyhedra, only six planets correspond to them. In his opinion, the spheres of the planets are interconnected by the Platonic solids inscribed in them. Since for each regular polyhedron the centers of the inscribed and circumscribed spheres coincide, the whole model will have a single center, in which the Sun will be located.

Having done a huge computational work, in 1596 I. Kepler published the results of his discovery in the book "The Secret of the Universe". He inscribes a cube into the sphere of Saturn's orbit, into a cube - the sphere of Jupiter, into the sphere of Jupiter - a tetrahedron, and so on successively fit into each other the sphere of Mars - a dodecahedron, the sphere of the Earth - an icosahedron, the sphere of Venus - an octahedron, the sphere of Mercury. The secret of the universe seems open.

Today it is safe to say that the distances between the planets are not related to any polyhedra. However, it is possible that without the "Secrets of the Universe", "Harmony of the World" by I. Kepler, regular polyhedra there would not have been three famous laws of I. Kepler, which play an important role in describing the motion of the planets.

Where else can you see these amazing bodies? In a very beautiful book by the German biologist of the beginning of our century, E. Haeckel, "The Beauty of Forms in Nature," one can read the following lines: "Nature nourishes in its bosom an inexhaustible number of amazing creatures that far surpass all forms created by human art in beauty and diversity." The creations of nature in this book are beautiful and symmetrical. This is an inseparable property of natural harmony. But here you can also see unicellular organisms - feodarii, the shape of which accurately conveys the icosahedron. What caused such a natural geometrization? Maybe because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume and the smallest surface area. This geometric property helps the marine microorganism overcome the pressure of the water column.

It is also interesting that it was the icosahedron that turned out to be the focus of attention of biologists in their disputes regarding the shape of viruses. The virus cannot be perfectly round, as previously thought. To establish its shape, they took various polyhedrons, directed light at them at the same angles as the flow of atoms to the virus. It turned out that only one polyhedron gives exactly the same shadow - the icosahedron. Its geometric properties, mentioned above, allow saving genetic information. Regular polyhedra are the most advantageous figures. And nature takes advantage of this. The crystals of some substances familiar to us are in the form of regular polyhedra. So, the cube conveys the shape of sodium chloride crystals NaCl, the single crystal of aluminum-potassium alum (KAlSO4) 2 12H2O has the shape of an octahedron, the crystal of sulfur pyrite FeS has the shape of a dodecahedron, antimony sodium sulfate is a tetrahedron, boron is an icosahedron. Regular polyhedra determine the shape of the crystal lattices of some chemicals. We illustrate this idea with the following problem.

Task. The model of the CH4 methane molecule has the shape of a regular tetrahedron, with hydrogen atoms at four vertices and a carbon atom in the center. Determine the bond angle between two CH bonds.

Solution. Since a regular tetrahedron has six equal edges, it is possible to choose a cube such that the diagonals of its faces are the edges of a regular tetrahedron. The center of the cube is also the center of the tetrahedron, because the four vertices of the tetrahedron are also the vertices of the cube, and the sphere described around them is uniquely determined by four points that do not lie in the same plane. The desired angle j between two CH bonds is equal to the angle AOC. Triangle AOC is isosceles. Hence, where a is the side of the cube, d is the length of the diagonal of the side face or edge of the tetrahedron. So, from where = 54.73561O and j = 109.47O

The question of the shape of the Earth constantly occupied the minds of scientists of ancient times. And when the hypothesis about the spherical shape of the Earth was confirmed, the idea arose that the shape of the Earth is a dodecahedron. So, already Plato wrote: “The earth, if you look at it from above, looks like a ball sewn from 12 pieces of skin.” This hypothesis of Plato found further scientific development in the works of physicists, mathematicians and geologists. So, the French geologist de Beamont and the famous mathematician Poincaré believed that the shape of the Earth is a deformed dodecahedron.

There is another hypothesis. Its meaning is that the Earth has the shape of an icosahedron. Two parallels are taken on the globe - 30o north and south latitude. The distance from each of them to the pole of its hemisphere is 60o, between them is also 60o. On the northern of these parallels, points are marked through 1/5 of a full circle, or 72o: at the intersection with meridians 32o, 104o and 176o in. d. and 40o and 112o z. e. On the southern parallel, the points are marked at the intersections with the meridians, passing exactly in the middle between the named: 68o and 140o in. and 4o, 76o and 148o z. e. Five points on the parallel 30o s. sh. , five - on the parallel of 30o S. sh. and two poles of the Earth and will make up 12 vertices of the polyhedron.

The Russian geologist S. Kislitsin also shared the opinion about the dodecahedral shape of the Earth. He hypothesized that 400-500 million years ago the dodecahedral geosphere turned into a geo-icosahedron. However, such a transition turned out to be incomplete and incomplete, as a result of which the geo-dodecahedron turned out to be inscribed in the structure of the icosahedron. In recent years, the hypothesis of the icosahedral-dodecahedral shape of the Earth has been tested. To do this, scientists aligned the axis of the dodecahedron with the axis of the globe and, rotating this polyhedron around it, drew attention to the fact that its edges coincide with giant disturbances in the earth's crust (for example, with the Mid-Atlantic submarine ridge). Then taking the icosahedron as a polyhedron, they found that its edges coincide with smaller divisions of the earth's crust (ridges, faults, etc.). These observations confirm the hypothesis that the tectonic structure of the earth's crust is similar to the dodecahedron and icosahedron shapes.

The nodes of a hypothetical geo-crystal are, as it were, the centers of certain anomalies on the planet: they contain all the world's centers of extreme atmospheric pressure, areas where hurricanes originate; in one of the nodes of the icosahedron (in Gabon) a "natural atomic reactor" was discovered that was still operating 1.7 billion years ago. Giant mineral deposits (for example, the Tyumen oil field), anomalies of the animal world (Lake Baikal), centers for the development of human cultures (Ancient Egypt, the proto-Indian civilization Mohenjo-Daro, Northern Mongolian, etc.) are confined to many nodes of polyhedrons.

There is one more assumption. The ideas of Pythagoras, Plato, I. Kepler about the connection of regular polyhedra with the harmonious structure of the world have already found their continuation in our time in an interesting scientific hypothesis, the authors of which (in the early 80s) were Moscow engineers V. Makarov and V. Morozov. They believe that the core of the Earth has the shape and properties of a growing crystal that affects the development of all natural processes taking place on the planet. The rays of this crystal, or rather, its force field, determine the icosahedral-dodecahedral structure of the Earth, which manifests itself in the fact that in the earth's crust, as it were, projections of regular polyhedra inscribed in the globe appear: the icosahedron and the dodecahedron. Their 62 vertices and midpoints of the edges, called nodes by the authors, have a number of specific properties that make it possible to explain some incomprehensible phenomena.

Further studies of the Earth, perhaps, will determine the attitude towards this beautiful scientific hypothesis, in which, apparently, regular polyhedra occupy an important place.

And one more question arises in connection with regular polyhedra: is it possible to fill the space with them so that there are no gaps between them? It arises by analogy with regular polygons, some of which can fill the plane. It turns out that you can fill the space only with the help of one regular polyhedron-cube. Space can also be filled with rhombic dodecahedrons. To understand this, you need to solve the problem.

Task. With the help of seven cubes forming a spatial "cross", build a rhombic dodecahedron and show that they can fill space.

Solution. Cubes can fill space. Consider a part of a cubic lattice. We leave the middle cube untouched, and in each of the "bounding" cubes we draw planes through all six pairs of opposite edges. In this case, the "surrounding" cubes will be divided into six equal pyramids with square bases and side edges equal to half the diagonal of the cube. The pyramids adjacent to the untouched cube form together with the latter a rhombic dodecahedron. From this it is clear that the whole space can be filled with rhombic dodecahedrons. As a consequence, we obtain that the volume of a rhombic dodecahedron is equal to twice the volume of a cube whose edge coincides with the smaller diagonal of the dodecahedron face.

Solving this problem, we came to rhombic dodecahedrons. Interestingly, the bee cells, which also fill the space without gaps, are also ideally geometric shapes. The upper part of the bee cell is a part of the rhombic dodecahedron.

In 1525, Dürer wrote a treatise in which he presented five regular polyhedra whose surfaces serve as good perspective models.

So, regular polyhedra revealed to us the attempts of scientists to approach the secret of world harmony and showed the irresistible attractiveness of geometry.

Regular polyhedra and the golden ratio

During the Renaissance, sculptors, architects, and artists showed great interest in the forms of regular polyhedra. Leonardo da Vinci, for example, was fond of the theory of polyhedra and often depicted them on his canvases. He illustrated the book of his friend the monk Luca Pacioli (1445 - 1514) "On the Divine Proportion" with images of regular and semi-regular polyhedra.

In 1509, in Venice, Luca Pacioli published On the Divine Proportion. Pacioli found in the five Platonic solids - regular polygons (tetrahedron, cube, octahedron, icosahedron and dodecahedron) thirteen manifestations of the "divine proportion". In the chapter "On the twelfth, almost supernatural property," he considers the regular icosahedron. At each vertex of the icosahedron, five triangles converge to form a regular pentagon. If you connect any two opposite edges of an icosahedron to each other, you get a rectangle in which the larger side is related to the smaller one as the sum of the sides is to the larger one.

Thus, the golden ratio is manifested in the geometry of five regular polyhedra, which, according to the ancient scientists, underlie the universe.

The geometry of Plato's solids in the paintings of great artists

A famous Renaissance artist, also fond of geometry, was A. Dürer. In his well-known engraving "Melancholia", a dodecahedron was depicted in the foreground.

Consider the image of the painting by the artist Salvador Dali "The Last Supper". In the foreground of the painting is depicted Christ with his disciples against the background of a huge transparent dodecahedron.

Crystals are natural polyhedra

Many forms of polyhedrons were not invented by man himself, but by nature in the form of crystals.

Often people, looking at the wonderful, iridescent polyhedrons of crystals, cannot believe that they were created by nature, and not by man. That is why so many amazing folk tales about crystals were born.

Interesting written materials have survived, for example, the so-called "Ebers Papyrus", which contains a description of stone treatment methods with special rituals and spells, where mysterious powers are attributed to precious stones.

It was believed that the pomegranate crystal brings happiness. It has the form of a rhombic dodecahedron (sometimes called a rhomboidal or rhombic dodecahedron) - a dodecahedron, the faces of which are twelve equal rhombuses.

For garnet, dodecahedral crystals are so typical that the shape of such a polyhedron was even called a garnetohedron.

Garnet is one of the main rock-forming minerals. There are huge rocks that are composed of garnet rocks called skarns. However, precious, beautifully colored and transparent stones are far from common. Despite this, it is precisely the garnet - blood-red pyrope - that archaeologists consider the most ancient decoration, since it was discovered in Europe in the ancient Neolithic on the territory of modern Czech Republic and Slovakia, where it is currently very popular.

The fact that the garnet, i.e., the rhombododecahedron polyhedron, has been known since ancient times can be judged by the history of the origin of its name, which in ancient Greek meant “red paint”. At the same time, the name was associated with red - the most common color of garnets.

Garnet is highly valued by connoisseurs of precious stones. It is used to make first-class jewelry, garnet has the ability to communicate the gift of foresight to the women wearing it and drives away heavy thoughts from them, while it protects men from violent death.

Grenades emphasize the unusualness of the situation, the eccentricity of people's actions, emphasize the purity and sublimity of their feelings.

This is a talisman stone for people born in JANUARY.

Consider stones whose shape is well studied and represents regular, semi-regular and star-shaped polyhedra.

Pyrite takes its name from the Greek word pyros, meaning fire. A blow to it gives rise to a spark; in ancient times, pieces of pyrite served as flint. The specular sheen on the faces distinguishes pyrite from other sulfides. Polished pyrite shines even brighter. Mirrors made of polished pyrite were found by archaeologists in the graves of the Incas. Therefore, pyrite also has such a rare name - the stone of the Incas. During the epidemics of the gold rush, pyrite spangles in a quartz vein, in wet sand on a washing pan, turned more than one hot head. Even now, novice stone lovers mistake pyrite for gold.

But let's take a closer look at it, listen to the proverb: "Not all that glitters is gold!" the color of pyrite is brass yellow. The edges of pyrite crystals are cast with a strong metallic sheen. ? here in the break, the brilliance is dimmer.

The hardness of pyrite is 6-6.5, it easily scratches glass. It is the hardest mineral in the sulfide class.

And yet the most characteristic in the appearance of pyrite is the shape of the crystals. Most often it is a cube. From the smallest "cubes nesting along cracks, to cubes with a rib height of 5 cm, 15 cm and even 30 cm! But pyrite crystals are not only cut into cubes, in the arsenal of this mineral there are octahedrons already known to us from magnetite. For pyrite, they are quite rare.But pyrite allows you to personally admire the form with the same name - pentagondodecahedron. "Penta" is five, all the faces of this form are five-sided, and "dodeca" - a dozen - there are twelve in total. This form for pyrite is so typical that in the old days even received the name “pyritohedron.” There may also be specimens that combine faces of different shapes: a cube and a pentagondodecahedron.

cassetite

Cassiterite is a shiny, brittle brown mineral that is the main ore of tin. The shape is very memorable - high tetrahedral, sharp pyramids above and below, and in the middle - a short column, also faceted. Quite different in appearance, cassiterite crystals grow in quartz veins. On the Chukchi Peninsula there is the Iultin deposit, where veins with excellent cassiterite crystals have long been famous.

Galena looks like a metal and it is simply impossible not to notice it in the ore. It immediately gives out a strong metallic luster and heaviness. Galena is almost always silvery cubes (or parallelepipeds). And these are not necessarily whole crystals. Galena has perfect cleavage in a cube. This means that it does not break into shapeless fragments, but into neat silvery shiny cubes. Its natural crystals are shaped like an octahedron or cuboctahedron. Galena is also distinguished by such a property: this mineral is soft and chemically not very resistant.

ZIRCONIUM

"Zircon" - from the Persian words "king" and "gun" - golden color.

Zirconium was discovered in 1789/0 in precious Ceylon zircon. The discoverer of this element is M. Claport. Magnificent transparent and brightly sparkling zircons were famous in antiquity. This stone was highly valued in Asia.

Chemists and metallurgists had to work hard before zirconium rod shells and other structural details appeared in nuclear reactors.

So, zircon is an effective gem - orange, straw yellow, blue - blue, green - glitters and plays like a diamond.

Zircons are often represented by small regular crystals of a characteristic elegant shape. The motif of their crystal lattice, and, accordingly, the shape of crystals is subject to the fourth axis of symmetry. Zircon crystals belong to the tetragonal syngony. They are square in cross section. And the crystal itself consists of a tetragonal prism (sometimes it is blunted along the edges by a second similar prism) and a tetragonal bipyramid that completes the prism at both ends.

Crystals with two dipyramids at the ends are even more spectacular: one at the tops, and the other only dulls the edges between the prism and the upper pyramid.

Salt crystals have the shape of a cube, crystals of ice and rock crystal (quartz) resemble a pencil honed on both sides, that is, they have the shape of a hexagonal prism, on the base of which hexagonal pyramids are placed.

Diamond is most often found in the form of an octahedron, sometimes a cube and even a cuboctahedron.

Icelandic spar, which bifurcates the image, has the shape of an oblique parallelepiped.

Interesting

All other regular polyhedra can be obtained from the cube by transformations.

In the process of division of the egg, first a tetrahedron of four cells is formed, then an octahedron, a cube, and finally a dodecahedral-icosahedral structure of the gastrula.

And finally, perhaps most importantly, the DNA structure of the genetic code of life is a four-dimensional sweep (along the time axis) of a rotating dodecahedron!

It was believed that regular polyhedrons bring good luck. Therefore, there were bones not only in the form of a cube, but in all other forms. For example, a dodecahedron-shaped bone was called d12.

The German mathematician August Ferdinand Möbius, in his work “On the Volume of Polyhedra”, he described a geometric surface that has an incredible property: it has only one side! If you glue the ends of a strip of paper, first turning one of them 180 degrees, we get a Mobius strip or strip. Try painting the twisted ribbon 2 colors - one on the outside and one on the inside. You won't succeed! But on the other hand, an ant crawling on a Möbius strip does not need to crawl over its edge in order to get to the opposite side.

“Regular convex polyhedra are defiantly few,” Lewis Carroll once remarked, “but this detachment, very modest in number, the magnificent five, managed to penetrate deeply into the very depths of science. »

All these examples confirm the amazing insight of Plato's intuition.

Conclusion

The presented work considers:

Definition of convex polyhedra;

Basic properties of convex polyhedra, including Euler's theorem relating the number of vertices, edges and faces of a given polyhedron;

Definition of a regular polyhedron, the existence of only five regular polyhedra has been proved;

Relations between the characteristic angles of a regular n-gonal pyramid, which is an integral part of a regular polyhedron, are considered in detail;

Some characteristics of a regular tetrahedron, such as volume, surface area, and the like, are considered in detail.

The appendices contain proofs of the main properties of convex polyhedra and other theorems contained in this paper. The above theorems and relations can be useful in solving many problems in stereometry. The work can be used in the study of certain topics of stereometry as a reference and illustrative material.

Polyhedrons surround us everywhere: children's cubes, furniture, architectural structures, etc. In everyday life, we almost stopped noticing them, but it's very interesting to know the history of objects familiar to all, especially if it is so exciting.

Russkikh Egor, Tarasov Dmitry

The world around us is a world of forms, it is very diverse and amazing. We are surrounded by household items of various kinds. After studying this topic, we really saw that polygons surround us everywhere and are found in various areas of life.

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Regular polygons

Amazing polygon

Star polygons

Polygons in nature

Polygons in nature

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Regular polygons in science and some other spheres of life Authors of the project: students of the 8th grade of Russians Egor Tarasov Dmitry. Scientific adviser: teacher of mathematics Rakhmankulova E.R.

Problem question. What is the role of polygons in our life? Object of study: polygons. Subject of research: practical application of polygons in the world around us.

Purpose: systematization of knowledge on this topic and obtaining new information about polygons and their practical application. Tasks: 1. To study the literature on the topic. 2. Show the practical application of regular polygons in the world around us.

Research methods: 1. Scientific (literature study); 2. Research. Hypothesis: Polygons create beauty in human surroundings.

Regular polygons

Magic square 4 9 2 3 5 7 8 1 6

Amazing polygon

Star polygons

Polygons in nature P3: P4: P6 = 1: 0.877: 0.816

Polygons in nature

Polygons in nature

Polygons around us Parquet

Conclusion Without geometry, there would be nothing, everything that surrounds us is geometric shapes. But we forget to pay attention to it.

Conclusion The world around us is a world of forms, it is very diverse and amazing. We are surrounded by household items of various kinds. After studying this topic, we really saw that polygons surround us everywhere and are found in various areas of life.

Thank you for your attention!

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