How to make an icosahedron from paper? How to make an octahedron out of paper (video, photo)? Paper octahedron diagram with dimensions

One of the best shapes that are suitable for gluing paper is the octahedron. The octahedron is a regular polyhedron that has been known to mankind since ancient times. Among its main properties, it is worth noting that it is characterized by symmetry and versatility. In many cultures, this figure had a sacred meaning. In today's world, this figure is studied in all schools in geometry courses.

However, to better understand its structure, you can make an octahedron out of paper or wire. Of course, the paper version is much simpler and best suited for children. This is exactly what will be discussed in our article.

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Preparing the components

Before you start making a paper octahedron, you should stock up on the right components and tools. You will need:

1. Scissors
2. Glue
3. Pencil or pen
4. Ruler
5. Thick paper or thin cardboard.

As you can see, all of these materials are very common and you can easily find them at home. It is worth noting that a homemade toy will help the child better develop spatial thinking, and in the process of making it, fine motor skills when working with paper.

Where to begin?

First you need to prepare a sketch for the future figure. Below you can find a ready-made template that should be printed and transferred to the appropriate material. When you have already made the correct markings, you will need to cut them out and glue everything together. At the same time, try to ensure that the template comes out accurate - the more accurate it is, the better the figure itself will turn out. If a child cannot cut out a template on his own, he may need help.

Making the template yourself

If you do not want to use a ready-made workpiece, you can make it yourself, using the basic geometric provisions. It is worth noting that the octahedron has the following properties:

1. Has eight sides
2. Six Peaks
3. Twelve ribs. Four of which converge at the vertices.
4. Has a triangular base
5. The octahedron is completely symmetrical.

So, as you learned, the development of a figure consists of a certain number of regular and equal triangles, which are arranged in a certain sequence. So, for example, six of them must be placed in a row, according to the “jack” principle. The other two triangles are adjacent to the middle figures.

As you can understand, making an octahedron with your own hands from paper is actually very simple and easy. Before flattening, you should select the edge length. In addition, you should make “wings” with which you will glue the figure. After you cut out the shape, use a ruler to make sure all the lines are well bent.

In addition to the above development, there is another version of the development of such a figure - it is called a “cuboctahedron”. Externally, it consists of six squares and eight triangles. Initially, this figure was known back in the time of Leonardo da Vinci and was called the “stellated octahedron”.

Second way

Below we will present another method that will help you make an octahedron out of paper. To do this you will need:

A sheet of cardboard or very thick paper
Ruler for marking and bending lines
Pencil
Protractor for measuring angles
1. Scissors
Paper glue (for example, PVA)

How to do it?

1. The octahedron has eight faces, which are equilateral triangles. In geometry, this figure is constructed using a cube, into which it is inscribed or described around. In addition to this method, there is a simpler one - simply glue two identical tetrahedral pyramids.
2. To begin, draw a square on paper and build a regular triangle on its side. To simplify this, you should consider that each angle of such a triangle is sixty degrees. Mark these two angles using a protractor, counting from the side of the square. Extend the rays to the intersection - you will get a third angle. After this, repeat the same steps with each side of the square.
3. Thus, you will have a pyramid blank, which will subsequently need to be glued together. To do this, you need to make “allowances” on each free side of the triangles. Cut out the blank, bend all the lines and glue the pyramid. Make another one of the same kind. When gluing, make sure that everything matches as it should, otherwise the figure will not come out beautiful.
4. Once the two pyramids are ready, you will need to glue them together. Spread the contact area with PVA glue (the bottom of the pyramids) and wait until everything dries. The octahedron is ready.

In addition to using paper for such purposes, you can use wire as the main material. True, for this you will need additional tools, because working with wire is much more difficult than working with paper. First of all, you may need metal scissors and wire cutters to cut the material.

You can make an octahedron from wire using the same principle as from paper - make a rigid square base and attach the remaining triangle parts to it.

When choosing a material, you should choose one that can be soldered - this will simplify the connection of all parts. When working with wire, try to act extremely carefully - you may accidentally harm yourself. The same applies to using a soldering iron - remember the safety precautions for using electrical appliances.

Which appeared in ancient times, one of the most interesting is the octahedron. This figure is one of the five solids called Platonic solids. It is regular, symmetrical and multifaceted, and also has a sacred meaning in terms of stereometry, practiced in Ancient Greece. Today, this geometric body is studied by children at school, and in order to understand its structure better, we will look at how to make an octahedron from paper.

Let's prepare materials

All that is needed during this process is scissors, a ruler, and the paper itself, which will form the basis of the future craft. It is believed that the independent production of such stereometric figures improves one's ability to orientate oneself in space. Therefore, with the help of our short lesson, you can catch up on the geometric skills you missed at school, or invite your child to create something similar if he has problems with the perception of geometric spaces and figures.

Sketch is a faithful assistant

The first option on how to make an octahedron out of paper is a ready-made sketch. The article offers a picture that shows this figure in a scan, and all that remains for you is to print it and glue it along the outlined lines. This way your craft will have the most accurate parameters. Just remember to glue the paper onto the cardboard first so that the octahedron is more durable and lasts longer. This is especially important if it is intended for a child.

Everything with your own hands

Another option for making an octahedron out of paper is based on simple formulas and drawing. This geometric figure consists of 8 faces, 6 vertices and 12 edges, each 4 of which converge at one vertex. If you add up all the angles of the octahedron into a single number, the sum will be equal to 240 degrees. It is also worth noting that this mythical stereo figure has a triangular base and is completely symmetrical, which is why it is often called an antiprism.

Stereometry lessons

The development of the octahedron represents a certain set of triangles that are completely equal to each other. Six of them are lined up in a single row according to the “jack” principle, and the other two with their bases are adjacent to the two middle figures from different sides. Therefore, the answer to the question of how to make an octahedron from paper without layouts is simple. You just need to choose a single edge size and base it on eight. Just remember to leave allowances on the fold line along which you will glue the future craft.

Intricate mysteries of geometry

There are various varieties of such a stereo figure. Among these is the cuboctahedron. The development consists of 6 squares and 8 regular triangles, which are assembled into a three-dimensional body according to the rules of symmetry. This figure is semi-regular, and, it is worth noting, quite young. It was discovered by the creator Leonardo da Vinci, and was then called the “stellated octahedron”. You can also do it according to the scheme proposed in the article.

A dodecahedron is a regular polyhedron made up of twelve regular pentagons. This spectacular three-dimensional figure has a center of symmetry called the center of the dodecahedron. In addition, it contains fifteen planes of symmetry (in each face, any of them passes through the middle of the opposite edge and the vertex) and fifteen axes of symmetry (intersecting the midpoints of parallel opposite edges). Each of the vertices of the dodecahedron is the vertex of three pentagons of regular shape.

The design received its name from the number of faces included in it (traditionally, the ancient Greeks gave polyhedrons names reflecting the number of faces that make up the structure of the figure). Thus, the concept of “dodecahedron” is formed from the meanings of two words: “dodeca” (twelve) and “hedra” (face). The figure belongs to one of the five Platonic solids (along with tetrahedron, octahedron, hexahedron (cube) and). Interestingly, according to numerous historical documents, all of them were actively used by the inhabitants of Ancient Greece in the form of table dice and were made from a wide variety of materials.

Regular polyhedra have always attracted people with their beauty, organic nature and extraordinary perfection of forms, but the dodecahedron has a special history, which from year to year acquires new, sometimes completely mystical, facts. Representatives of many civilizations saw in it a supernatural and mysterious essence, claiming that: “Many things grow from the number twelve.” In the territories of ancient destroyed states, small figurines in the form of dodecahedrons made of bronze, stone or bone are still found. In addition, during excavations in the lands of modern England, France, Germany, Hungary, and Italy, archaeologists discovered several hundred so-called “Roman dodecahedrons” dating back to the 2nd-3rd centuries AD. The main dimensions of the figures range from four to eleven centimeters, and they are distinguished by the most incredible patterns, textures and execution techniques. The version put forward back in the time of Plato that the Universe is a huge dodecahedron was confirmed at the beginning of the 21st century. After a thorough analysis of the data obtained using WMAP (NASA's multifunctional spacecraft), scientists agreed with the assumption of ancient Greek astronomers, mathematicians and physicists, who at one time dealt with the study of the celestial sphere and its structure. Moreover, modern researchers believe that our Universe is an infinitely repeating set of dodecahedrons.

How to make a regular dodecahedron with your own hands

Today, the design of this figure is reflected in many forms of artistic creativity, architecture and construction. Folk craftsmen make unusually beautiful origami in the form of openwork dodecahedrons from colored or white paper, and they make original ones from cardboard, etc.). On sale you can purchase ready-made kits containing everything you need to make souvenirs, but the most interesting thing is to do the entire process of work with your own hands, from building individual parts to assembling the finished structure.

Materials:

In order to make a regular dodecahedron from cardboard, you need the material itself and available tools:

scissors,
pencil,
eraser,
ruler,
glue.

It’s good to have a dull knife or some kind of device for bending seam allowances, but if you don’t have them, then a metal ruler or the same scissors will do just fine.

How to make a stellated dodecahedron

Stellated dodecahedrons have a more complex structure compared to ordinary ones. These polyhedra are divided into small (of the first extension), medium (of the second extension) and large (the last stellate shape of the regular dodecahedron). Each of them has its own design features and assembly. To work, you will need the same materials and tools as for making a standard dodecahedron. If you decide to make the first option (small dodecahedron), then you need to build a drawing of the first element, which will become the basis for the entire structure (later it will be glued or parts assembled using paper clips).

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. By using 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 lines. This is necessary in order to leave free space for gluing parts of the figure to each other. Be diligent in cutting 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. Making an icosahedron from paper is an interesting process that requires 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 develop not only spatial skills and imaginative thinking, but also become more familiar with the world of geometry. For an adult, the creative process of constructing a paper icosahedron with your own hands will allow you to pass the time, as well as 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 possible to create not only them, but also complex 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. Turn 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.

Let's prepare materials

All that is needed during this process is scissors, glue, a pencil, a ruler, and the paper itself, which will form the basis of the future craft. It is believed that independent production of such stereometric figures improves abstract thinking and allows better orientation in space. Therefore, with the help of our short lesson, you can catch up on the geometric skills you missed at school, or invite your child to create something similar if he has problems with the perception of geometric spaces and figures.

Sketch is a faithful assistant

Everything with your own hands

Stereometry lessons

The development of the octahedron represents a certain set of triangles that are completely equal to each other. Six of them are lined up in a single row according to the “jack” principle, and the other two with their bases are adjacent to the two middle figures from different sides. Therefore, the answer to the question of how to make an octahedron from paper without layouts is simple. You just need to choose a single face size and base it on eight equilateral triangles. Just remember to leave allowances on the fold line along which you will glue the future craft.