A relief polyhedron is called a positive polyhedron if all its faces are equal, positive polygons, and an identical number of edges converge at its entire vertex. There are five regular polyhedra - tetrahedron, octahedron, icosahedron, hexahedron (cube) and dodecahedron. An icosahedron is a polyhedron whose faces are twenty equal right triangles.

Instructions

1. For building icosahedron Let's use the construction of a cube. Let us denote one of its faces as SPRQ.

2. Draw two segments AA1 and BB1 so that they connect the midpoints of the edges of the cube, that is, as = AP = A1R = A1Q = BS = BQ.

3. On segments AA1 and BB1, lay equal segments CC1 and DD1 of length n so that their ends are at equal distances from the edges of the cube, i.e. BD = B1D1 = AC = A1C1.

4. Segments CC1 and DD1 are the edges of the construction icosahedron A. By constructing the segments CD and C1D, you will get one of the faces icosahedron a – CC1D.

5. Repeat constructions 2, 3 and 4 for all faces of the cube - as a result you will get a regular polyhedron inscribed in the cube - icosahedron. With the help of the hexahedron it is possible to construct any regular polyhedron.

An icosahedron is a regular polygon. Such a geometric figure has 30 edges, 20 triangular faces and 12 vertices, which are the junction of five edges. Assembling an icosahedron from paper is quite difficult, but very exciting. It can be made from corrugated, packaging or colored paper or foil. Applying different materials, you can add even greater impact and beauty to your icosahedron.

You will need

• – layout of the icosahedron;
• - paper;
• - scissors;
• - ruler;
• - PVA glue.

Instructions

1. Print out the icosahedron layout on a piece of paper, then cut it out along the dotted line. This is necessary in order to leave free space for gluing parts of the figure to each other. Try to cut out the icosahedron as leisurely as possible; on the contrary, at the slightest shift, your craft will end up looking ugly. The need for very neat cutting is due to the fact that all triangles in a regular icosahedron have identical sides. Consequently, if any side begins to differ in its length, as a result such a discrepancy in size will be invisible.

2. Fold the icosahedron along solid lines, then use glue to glue the places outlined by the dotted line and connect the adjacent sides of the triangles with each other. For a tighter fixation, each glued side must be kept in this state for 20 seconds. It is true that all other sides of the icosahedron should be glued in the same way. The last two ribs are the hardest to glue because they require patience and skill to join them together. Your paper icosahedron is ready.

3. Such a geometric figure can be seen in Everyday life. For example, a soccer ball is made in the shape of a truncated icosahedron (a polyhedron consisting of 20 hexagons and 12 pentagons). This becomes especially invisible if the resulting icosahedron is painted black and white. You can make a soccer ball out of paper yourself by printing out a scan of a truncated icosahedron in 2 copies in advance.

4. The production of an icosahedron from paper is interesting process, requiring patience, thoughtfulness and a lot of paper. But the resulting result will please the eye for a long time. A paper icosahedron can be given as a developmental toy to a child who has reached the age of 3. By playing with this geometric figure, the baby will begin to develop not only spatial skills and creative thinking, but also get to know the world of geometry better. For an adult creative process How to construct a paper icosahedron with your own hands will allow you to pass the time, and also amaze your loved ones with the knowledge of making difficult figures.

Helpful advice
When making a paper icosahedron, you need to pay special attention to the process of bending its sides. To bend the paper evenly, you can use an ordinary ruler.

The octahedron is one of the four true polyhedra, to which people attached magical significance back in ancient times. This polyhedron symbolized air. A demonstration model of the octahedron can be made from thick paper or wire.

You will need

• – thick paper or cardboard;
• - ruler;
• - pencil;
• – protractor;
• - scissors;
• - PVA glue.

Instructions

1. The octahedron has eight faces, all of which are an equilateral triangle. In geometry, an octahedron is usually constructed, inscribed in a cube or described around it. To make a model of this geometric body, difficult calculations are not required. The octahedron will consist of 2 identical tetrahedral pyramids glued together.

2. Draw a square on a piece of paper. On one of its sides, construct a positive triangle in which all sides are equal and all of the angles are 60°. It is convenient to construct a triangle using a protractor, setting aside 60° corners of a square adjacent to the same side. Draw rays through the marks. The point from the intersection will be the third angle, and in the future - the top of the pyramid. Build the same triangles on the remaining sides of the square.

3. You will have to glue the pyramid together. This will require allowances. Four allowances are enough, one for each triangle. Cut out what you have. Make a second similar piece. Fold the fold lines to the wrong side.

4. Fold each of the triangles to the wrong side. Apply PVA glue to the allowances. Glue two identical pyramids together and let them dry.

5. Now we need to glue the pyramids together. Spread the square bottom of one of them with glue, press the bottom of the 2nd one, aligning the sides and corners. Let the octahedron dry.

6. To make a wire octahedron model, you will need a cardboard or wooden square. However, you can get by with an ordinary triangle - in order to bend the workpiece at a right angle, it is absolutely sufficient. Bend the wire into a square.

7. Cut 4 identical pieces of wire the size of 2 sides of the square, plus an allowance for attaching them at 2 points to each other, and, if necessary, attaching them to the corners of the square. It depends on the wire. If the material can be soldered, the length of the edges is equal to twice the side of the square without any allowances.

8. Find the middle of the piece, wind or solder it to the corner of the square. Attach the remaining pieces in the same way. Connect the ends of the ribs on one side of the square base to each other. Positive triangles will appear by themselves. Perform the same operation with the ends of the ribs located on the other side of the base. The octahedron is ready.

Helpful advice
For similar models, you must choose the wire that holds its shape well.

The art of origami came to us from Ancient China. At the dawn of their formation, figures of animals and birds were made from paper. But today it is allowed to create not only them, but also difficult geometric figures.

You will need

• – a sheet of A4 paper
• - scissors

Instructions

1. To produce a three-dimensional geometric figure, an octahedron, you need a square sheet of paper. You can make it from an ordinary A4 sheet. To do this, bend the upper right or left corner of the sheet to the opposite side. Make a note on a piece of paper. Draw a line parallel to the tight side of the sheet along the mark you made. Cut off the unwanted piece of paper. Fold the square in half.

2. Place the top right corner on the center fold. Align the top left corner so that the fold line goes through the attached top right corner.

3. Fold the bottom left corner of the square toward the center line. Aligning the bottom right corner similar to the top corners, make a fold. After which the workpiece must be overturned.

4. Fold the bottom right corner of the piece and the top left corner to the center fold. Iron the workpiece with your hand and turn it over to the other side.

5. Align the top and bottom sides with the resulting fold line. Smooth the workpiece with your hand.

6. Bend the sides of the figure towards the middle line of the square. Flip the piece over to the opposite side.

7. Fold the piece from bottom to top along a horizontal line. The result should be a figure resembling the Latin letter “V”.

8. Fold the left side down along the left side of the center triangle. Fold the right side down along the right side of the central triangle.

9. Make stripes on the top sides of the figure. The fold point of the strips will begin at the bottom point of the inside cutout of the "V".

10. Fold the upper left corner to the fold line of the strip. Then fold the strip down. Fold the right corner and strip in the same way.

11. Fold the left side down.

12. The illustration shows the pockets and inserts for assembling the octahedron.

13. To construct an octahedron, you need to make 4 such modules. Align the two modules at an angle, tucking the protruding parts into the pockets. After this, assemble all 4 modules together.

14. The result is a geometric figure called an octahedron.

If you are interested in creating kusudama, you will probably want to collect it to see the beauty and perfection of the polyhedron shapes. If so, then you definitely need to study the Sonobe module, which has become the basis of most.

This part is a parallelogram with pockets for connecting several modules together. With their help you can make any three-dimensional figure. The name came from the inventor, Mitsunobu Sonobe.

To assemble a kusudama, you need to prepare 30 modules from squares of size 9X9 cm or 5X5 cm. In fact, the size is a matter of personal preference, as is the choice of color for the icosahedron. Experiment and create your own unique creations.

How to make a Sonobe module

1. Take a square sheet and fold it in half.
2. Direct opposite sides towards the intended center line. Iron the folds well.
3. Straighten the sheet. Fold the top left end and the bottom right.
4. Repeat again, bending the same ends.
5. Bring opposite sides together in the center.
6. Point the bottom left corner up so the sides meet. On the other side, repeat the fold in the opposite position.
7. Push the loose ends into the pockets to create an envelope-like shape.
8. Bend the workpiece diagonally.
9. Bend the ends outward on both sides so that they turn into small triangles.
10. Expand the Sonobe module.
11. Make 29 more of these using squares of different colors or the same tone.

Assembling an icosahedron from paper

Assemble a pyramid from three modules by inserting the ends into the pockets of the adjacent part. It will turn out as in the photo.

Continue assembling, focusing on the five pyramids combined together. Gradually the ball will become rounded and you will get a beautiful kusudama.

By changing the Sonobe module at your discretion, you can get new creations. This exciting activity, accessible to everyone! We wish you success and inspiration!

The icosahedron is one of the types of regular polyhedra. It has a convex shape and is characterized by the presence of 20 identical faces, which are equilateral triangles. In addition, this volumetric figure contains 12 vertices and 30 edges.
The name of the figure is translated from Greek as “twenty bases.” This name of the figure fully characterizes its structural features.
Such geometric objects are rarely found in everyday life, so they can only be observed in some playing elements, crystals of various minerals and in molecular compounds. There is also an opinion that this figure is a more accurate representation of the shapes of the Earth and some planets.

How to make an icosahedron

There are many different but simple ways, which allow you to recreate the icosahedron with your own hands. This will allow you to clearly appreciate the mystery and complexity of this figure.

Method No. 1 Icosahedron from a finished layout

The first way comes down to finding an image of the figure’s development on the Internet and submitting it for printing. After this, cut along the kennel and fold in accordance with the indicated fold lines. To make it more effective, the resulting figure can be painted and varnished. This will not only make the icosahedron brighter and more impressive, but also extend its life.

Method No. 2 How to make an icosahedron by hand

You can make a model of the icosahedron without additional materials. To do this you will need paper, a pencil and a ruler.
Using a ruler, we draw outlines - for this you need to draw a set of triangles or to simplify rectangles. You should end up with a shape that resembles a lopsided stack of blocks or dominoes. After this, we cut out a homemade masterpiece and fold the icosahedron. For clarity, you can use the following diagram. By the way, it is also suitable for the first method.

Method No. 3 Polymer clay icosahedron – flower pot

This figure can be easily used to create interesting things in terms of design, for example, a flower pot. To the list of tools required in the second method, add polymer clay and you can begin production. Cut out a triangle from a piece of paper. Its dimensions depend on the size of the desired pot. Next, we lay the clay in the shape of this triangle. You should get 15 clay triangles. Next, we fold them in the shape of an icosahedron, leaving the upper part empty. Next, in the lower part, while the clay is soft, we make holes for water to drain. You don't need too many of them. After this, we attach the legs, which we also make from clay, and send our flower pot to the oven. There it will acquire sufficient strength and will delight you and your guests. It can be painted, giving each face its own color, or choose colors that will harmoniously combine with the interior of the room.

In the same way you can make a candlestick or whatnot. In general, everything is limited only by imagination.

Many designers develop the habit of mentally changing objects and structures that come into their hands or eyes, in search of a more rational solution or simply out of curiosity: what will come of it? The example below illustrates this kind of exercise-entertainment for the designer.

In Figure 1, solid lines show a scan consisting of twenty identical equilateral triangles.

If you draw a pattern on thick paper, cut it out, cut the paper with a not very sharp knife along the lines separating the triangles from each other and from the legs, bend the pattern along these lines in one direction, glue together the ends of the strip consisting of triangles 2, 4 , 6, 8, 10, 12, 14, 16, 18, 20, and from triangles 1, 5, 9, 13, 17 and 3, 7, 11, 15, 19, glue two pentagonal pyramids, then you will be fully rewarded for your work. In your hands will be a body remarkable in perfection shape - correct twenty-hedron (icosahedron), having twenty identical faces - equilateral triangles, thirty identical edges and twelve protrusions consisting of pentahedral pyramids. Suddenly, instead of two glued pyramids, there were six pairs of them with six axes passing through these pairs. The icosahedron is symmetrical about all six axes. The top of each of the twelve pyramids and the three corners of each face touch the spherical surface. The remaining points of the faces are close to it. Compared to the faces of other regular polyhedra, the faces of the icosahedron are closest to the surface of the circumscribed sphere, the number of faces is maximum, and its shape is closest to the shape of a ball. This gives rise to the possibility of constructing, for example, a map of the planet on twenty equilateral triangles, projecting points of the sphere using its radii on the face of an inscribed icosahedron. The possibility of using this method can be clarified by a more in-depth analysis.

Now let's imagine that the icosahedron is not a shell, but a solid body. We will mentally change its shape, gradually and evenly cutting off the tops of all pyramids with planes perpendicular to their axes. Twelve new faces will appear in the form of regular pentagons, and the corners of the former triangular faces will be cut off, they will turn into hexagons with three new small sides instead of the cut corners. With further cutting of the pyramids, the pentahedrons increase, and for hexagons short sides grow, the long ones shorten, and finally a new interesting shape of a polyhedron is obtained, consisting of twelve equilateral pentagons and twenty equilateral hexagons. Soccer balls are made from this pattern.

If you cut the pyramids further, the area of ​​the pentagons continues to increase, and the hexagons become unequal, their old sides will become shorter than the new ones, and this will continue until the old sides disappear and the new ones close into triangles. We get a new interesting polyhedron shape, consisting of twelve regular pentagons and twenty equilateral triangles. With further cutting of material from the plane of the pentahedrons, they will turn into decahedrons, and the triangles will decrease in size. There will come a moment when the unequal sides of the decahedrons become equal and you get new form- twelve equilateral decagons and twenty small equilateral triangles. Continuing to remove material from the planes of the decagons, in the end we will again get twelve equilateral pentagons, and the triangles will disappear. This will be the well-known pentagon-dode-caedron dodecahedron shape. From these twelve plates, but extruded into a sphere, a Soviet pennant was made and sent to the Moon. The figure shows its development (Fig. 2).

When twenty triangular corners are cut off, we get twenty triangles instead, and the pentagonal faces turn into decagonal ones. If we continue this operation further, we will obtain the same shapes as when cutting off the corners of the icosahedron, but in the reverse order and in the end we will again obtain an icosahedron, but of much smaller dimensions.

The practical applicability of the forms discussed here is quite limited; they can only be used in cutting precious stones.

It is much more interesting to study the icosahedron not as a solid body, but as a shell. In this case, it is a closed volume, for example, a vessel for liquid and gas, made of a flat sheet. The ribs give rigidity to the shell. The ribs can be replaced by rods or threads, and then other variations arise: a hard basket or a soft mesh with large cells.

We will make further variations with a sweep (Fig. 1), modification of which will sometimes lead to unexpected results.

Let's add four more triangles to the development, as shown by the dotted line in Figure 1. The six equilateral triangles on each side of the tape will no longer be bent into pyramids, but will fit into flat regular hexagons and can be replaced by them on the development. After gluing, we obtain a drum consisting of a twelve-sided shell and two hexagonal bottoms (Fig. 3).

A similar drum can be obtained from an icosahedron if two opposite pentagonal pyramids are replaced with pentagonal bottoms.

Let us now cut off triangles 17-20 from the development. From the remaining triangles 1-16 we obtain a hexahedron with two tetrahedral pyramids and one longitudinal axis (Fig. 4).

If we cut off the tetrahedral pyramids and replace them with square faces, we get a decahedron consisting of eight triangular and two square faces (Fig. 5).

Let us now cut off four more faces from the development (Fig. 1). The remaining triangles 1-12 unexpectedly form a hexagon because each pair of triangles forms one diamond-shaped face (Fig. 6).

This is a rhombic dodecahedron, let's call it a “rhomboid,” which, like a cube, has six faces, eight trihedral angles and twelve edges. If you place it on one of the faces, then it is easy to recognize it as a cube skewed diagonally. If such a rhomboid is made of twelve rods instead of ribs, connecting them hingedly at the corners, then when stretched along the longitudinal axis, the rods will form a stick consisting of three rods at the ends and six in the middle. When this stick is longitudinally compressed, the rods will first diverge into an elongated rhomboid, then into a cube, then into a flattened rhomboid, and finally fit into one plane in the form of a regular hexagon. Here's an idea for a designer - a stool and an umbrella that fold into the shape of a stick.

The rhomboid variant, strongly elongated along its axis (Fig. 7, scan 8), is of particular interest.

Such a body with a large aspect ratio λ = 1/d (that is, with a large ratio of length 1 to thickness d), oriented during flight so that the axis is directed along the flight, and moving at a speed equal to or greater than the speed of sound, will probably have the least drag compared to other bodies of the same elongation, because the front and rear ribs of the body are directed along the flow, and the middle six ribs form sharp angles with the flow. This statement requires further proof or verification by experiment.

By cutting off both trihedral pyramids from the rhomboid (Fig. 6) (for which all rhombuses will have to be cut in half), we again unexpectedly obtain the well-known regular octahedron - the octahedron (Fig. 9). Its development consists of triangles 1, 2, 4, 6, 8, 10, 11, 12. There are “related” relationships between the octahedron and the cube, similar to the relationships between the icosahedron and the Pentagon-dodecahedron.

By cutting off the corners of the first, the second is obtained through intermediate tetragons.

From a development consisting of triangles 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, a regular decahedron is glued together, consisting of two pentahedral pyramids folded at the bases. From triangles 2, 4, 6, 8, 10, 12 we obtain the development of a regular hexagon, which is two tetrahedrons attached to each other, and the development of a tetrahedron - a regular tetrahedron - consists of triangles 2, 4, 6, 8 (Fig. 10).

It is interesting to note that the tetrahedron has four faces and four protrusions, therefore, from the tetrahedron, cutting off the trihedral corners, we again obtain a tetrahedron through intermediate octahedrons with triangular and hexagonal faces.

Finally, you can also glue a “body” from two triangles, but it will be a flat triangle, two-sided, that is, a body that has no volume.

So, it turns out that regular polyhedra can be glued together from an even number of equilateral triangles. In this case, the two result in a “body without volume.” From twelve triangles, a rhomboid is obtained, that is, a hexagon with rhombic edges or a body without volume in the form of two glued regular hexagons. From twenty-four triangles we obtain a fourteen-sided one, which has two faces - regular hexagons. Along the way, a task is proposed for readers: is it possible to glue a closed figure in another way from fourteen, eighteen and twenty-two equilateral triangles?

Let's consider another possibility of varying the scan shown in Fig. 1. If we discard the upper and lower teeth and leave only the tape consisting of even numbers of triangles, and then fold several such tapes with their side edges, we will obtain the development shown in Figure 11.

The development is given for twelve triangles in each tape. Having drawn and cut out this development, bend it along oblique lines in one direction, and along horizontal lines in the other. When glued together, we get a figure close to a round cylinder, but with a faceted side surface. This figure is rigid in torsion, bending, longitudinal compression and has local rigidity of the side wall. This variation will probably be the most valuable practical application. It can serve as a blueprint for a building structure that is light, strong, rigid and earthquake-resistant. It is not too difficult to manufacture and can be implemented both in the wall version and in the truss version, if the ribs are replaced with rods. In the second case, composed of triangles, it will be statically definable.

(in biology)

One of the most popular areas in origami is 3D modeling. Creating three-dimensional figures captures the attention of not only children, but also adults. If you have already mastered the simplest patterns and techniques and learned how to make at least a cube out of paper, you can move on to more complex models. It is best to practice creating the so-called "Platonic solids". There are only five of them: tetrahedron, icosahedron, hexahedron, dodecahedron and octahedron. All figures are based on the simplest. Today you will learn how to make an icosahedron from paper.

List of materials and tools

• One sheet of thin colored cardboard (preferred density is 220 g/m2).
• Sharp scissors or a utility knife.
• Simple NV.
• Long wooden ruler (at least 20 cm).
• Eraser.
• Liquid PVA glue or pencil.
• Brush.

Instructions

If you fully understand how to make an icosahedron from paper, you can practice assembling a more complex model - a truncated icosahedron. This figure consists of 32 faces: 12 equilateral pentagons and 20. In finished form and with proper coloring, it is very reminiscent of paper. The assembly principle is similar, the only differences are in the template. The development of a truncated icosahedron is very difficult to construct, so it is better to print it on a printer. You should choose very thick paper, otherwise the figure will not hold its shape, and sagging may form in places where pressure is applied.

Origami and 3D modeling are a great way to while away a friendly or family evening. Such activities create a good intellectual background and help develop spatial imagination.

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