How to build an octahedron out of paper step by step. Geometric figures. Octahedron. Sketch is a faithful assistant

A dodecahedron is a regular polyhedron composed 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 (crossing the midpoints of parallel opposite edges). Each of the vertices of the dodecahedron is the vertex of three regular pentagons.

The construction got its name by the number of its faces (traditionally, the ancient Greeks gave polyhedra names that reflect 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 "khedra" (face). The figure belongs to one of the five Platonic solids (along with the 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 Interestingly, according to numerous historical documents, all of them were actively used by the inhabitants of Ancient Greece in the form of table dice and were made from a wide variety of materials.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 polyhedrons have always attracted people with their beauty, organicity and extraordinary perfection of forms, but the dodecahedron has a special history, which from year to year is overgrown with new, sometimes completely mystical, facts. Representatives of many civilizations saw in him a supernatural and mysterious essence, arguing that: "Many things grow out of the number of twelve." In the territories of the ancient ruined 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, Italy, archaeologists discovered several hundred so-called "Roman dodecahedrons" dating back to the 2nd-3rd centuries AD. The main dimensions of the figurines range from four to eleven centimeters, and they differ in the most incredible patterns, textures and technique. The version put forward in the time of Plato that the Universe is a huge dodecahedron was confirmed already at the beginning of the 21st century. After a thorough analysis of the data obtained using WMAP (NASA 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 the correct dodecahedron with your own hands

Today, the design of this figure has found its reflection in many variants of artistic creativity, architecture and construction. Craftsmen make unusually beautiful origami in the form of openwork dodecahedrons from colored or white paper, and original ones are made from cardboard, etc.). On sale, you can purchase ready-made kits containing everything you need to make souvenirs, but it is most interesting to do the whole process of working with your own hands, from building individual parts to assembling the finished structure.

Materials:

In order to make the correct dodecahedron out of cardboard, you need the material itself and the tools at hand:

scissors,
pencil,
eraser,
ruler,
glue.

It is good to have a dull knife or some kind of device for bending allowances, but if they are not there, then a metal ruler or the same scissors is quite suitable.

How to make a stellated dodecahedron

Stellated dodecahedrons have a more complex structure than conventional dodecahedrons. These polyhedra are subdivided into small (of the first continuation), medium (of the second continuation) and large (the last stellate form of a regular dodecahedron). Each of them has its own features of construction and assembly. For work, you will need the same materials and tools as for the manufacture of 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 is glued or the parts are assembled using paper clips).

In this article, we will look at how to assemble a paper octahedron, what this figure is and how it can be useful for each of us. Perhaps someone will find themselves in a similar kind of creativity, or finally get the option of a homemade gift to a loved one. The instructions for making this stereometric figure are simple, and even a child can handle it, so let's go.

We start to collect the figure

Many, looking at an octahedron in a sweep, are perplexed about how to correctly connect all its faces and sides to each other so that the figure acquires the correct one. Let's try to uncover this geometric secret and understand how to assemble a paper octahedron. To begin with, we need the figure itself in the scan. You can retake it from a book or print the image that is attached to the article. We can say that this is a ready-made scheme. How to make an octahedron yourself? It's also easy. As a basis, take any value that you need. This will be the length of each face of the shape. Then draw six as you see them in the picture, and draw two more to them.

Face mounting features

Many, even after they have built a development of the figure, do not understand how to assemble a paper octahedron so that it turns out to be correct. To do this, when cutting out the figure, leave allowances for gluing. They have beveled edges, and one of the drawings shows them. Now we apply glue to these allowances and assemble a stereometric figure along the fold lines. It is important that you first make clear folds at the intersection of all the edges of the figure. So your craft will have a more beautiful and accurate, from a geometric point of view, appearance.

Required Skills

When you know how to assemble a paper octahedron, it is easier for you to work with more complex ones. Some people need this for work, others just add to their knowledge base and expand their creative range in this way. It is worth noting that you can collect this craft from ordinary thin paper if you are training. When the octahedron is needed for any other purpose, it is advisable to use whatman paper or thick cardboard.

Useful Geometric Souvenirs

An interesting veil is that a paper octahedron can be a great gift idea for a loved one. If you intend to present only a cute trinket as a present, then immediately before gluing, a drawing must be applied to each face of the octahedron. It can be made in the form of an application or drawn if you have the skills of an artist. A more "serious" undertaking would be an octahedron calendar. Instead of drawings, we mark the days on each face, except for two. Each triangle should fit two months.

Education and hobbies

In many preschools, children are taught how to draw flattened stereometric figures and then assemble and glue them together. If you see that your child does not have these skills, teach him all this at home. Also, if you have gaps in this area, you can always catch up and, by printing the attached diagram, assemble a paper octahedron using a regular stationery set.

Positive polyhedra were already famous in Ancient Greece. They are called the "Platonic" solids. Four positive polyhedra - tetrahedron, icosahedron, cube and octahedron- personify the four "essences", the elements. The octahedron symbolizes air.

You will need

- paper;
- pencil;
- ruler.

Instruction

1. The octahedron consists of eight faces that are positive triangles. In a right triangle, all sides are equal. The angles between the sides of such a triangle are 60°. Heights, medians, bisectors are the same. To build the correct octahedron and you need a cube.

2. To build a cube, draw a square. Step back some distance to the right and up, build another similar square (left and bottom lines will be dashed). Unite the corresponding paired points of both squares until the cube is rendered. From the fact that on its basis you will build octahedron, make it huge and distinct.

3. Let a cube be given. Need to build octahedron inscribed in it. Draw diagonals for the entire face of the cube. Mark the intersection points of the diagonals. Combine all received points with each other. Positive octahedron, inscribed in a cube, is ready.

4. To prove that the resulting figure is positive octahedron, you need to confirm the correctness of the triangles. In order to confirm that the face triangles are correct, draw perpendiculars from their vertices to the edges of the cube. Use the properties of right triangles and a cube.

5. It is also allowed to build octahedron, described about the given cube. Let a be the length of the edge of the cube. Find the centers of the entire face (these are the intersection points of the diagonals). Draw straight lines through the centers of opposite faces. They intersect in the center of the cube, the one that can be designated as point O.

6. It turns out that there are two lines intersecting at point O. Set aside on each of the lines on both sides a segment equal to 3a / 2. Unite the ends of the segments you have obtained. This will be the frame of the positive octahedron a, described near the cube.

The medians of a triangle are the segments that are drawn from the corresponding vertices of the triangle to opposite sides and divide them into 2 equal parts. In order to build medians in a triangle, you need to take 2 steps.

You will need

- A pre-drawn triangle, the dimensions of the sides are arbitrary;
-Ruler;
-Pencil and pen.

Instruction

1. A pencil and a ruler are taken, and after that, with their support, points are marked on the sides of the triangle so that they divide the corresponding sides of the triangle in half. One of the options for how they need to be noticed is in Figure 1.

2. Now, with the help of a red / blue or other colored pen and ruler, a segment is drawn from any vertex of the triangle, and in such a way that it connects the vertices of the triangle with their corresponding opposite lines at the points that were erected in the first step. An approximate version of how this should turn out is shown in Figure 2.

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Note!
In order to check the correctness of the construction, it must be taken into account that all the medians of the triangle intersect at one point, which is called the center point.

If you have a sauna on the site, be aware that this is not an easy sanitary and hygienic room. This is a real home hospital. In the steam room, your skin is cleansed, all unnecessary is removed from the body, irritation and fatigue leave it. And what kind of massage can be compared with a birch broom? Decided to raise steam room !

Instruction

1. Building a sauna with a steam room is not an easy task. the sauna is an object of increased fire hazard. It is for this reason that even in the old days baths in Russia were built away from housing, in vegetable gardens, to store rivers, reservoirs. Secondly, not every tree is suitable for building a steam room. Before starting construction, you need to select the right material. Thirdly, decide on the design of the furnace, which will provide the necessary heat and steam for the steam room in the required number.

2. It turns out that a place for construction has been chosen, taking into account the location of a nearby water source. Place the bath in a colorful corner of your site. For a log cabin of a steam room, a beam made of coniferous wood is better than anyone. As a vapor barrier, use aluminum foil laid on the inside of the walls of the steam room. In this case, two snags go away - the third-party smell from the sealing material disappears, the one that was used when laying the timber (tow, tourniquet, etc.); and, in addition, you will get rid of the origin of the resin on the walls. To sheathe the ceiling, walls over the vapor barrier is cooler than any board made of aspen or alder. These types of wood make the air in the steam room “lighter”.

3. The best wood for shelves is linden. On the floor, any board is suitable, in addition to birch or made from conifers. The first does not work well in conditions of high humidity (it rapidly rots), the second releases resin when heated, which causes some inconvenience. Shelves are made on different tiers. This is necessary in order for you to enter steam room, could first warm up and get used to its warmth, settling on the lower shelves. And after that, it’s great to take a steam bath on the top shelf. Their dimensions must be such that it is allowed to comfortably sit on them lying down.

4. And finally, the heart of the steam room is the oven. The easiest and far from the worst option is a standard metal furnace. Firewood is used as fuel in it. Make a firebox with access to the dressing room. This will allow, without making additional devices, to provide it with heat (heat will come from the front wall of the furnace) and each garbage (wood chips, etc.) will be outside the washing and steam compartments. Isolate the furnace outlet from the wall by lining it with bricks. Also cover the stove inside the steam room with red brick. The distance between the walls of the furnace and the brickwork should be within 2-3 cm. The gap is needed so that the furnace, expanding when heated, cannot break the brickwork. Brick lining the furnace is necessary in order to smooth out the steep temperature of the metal, and the heated bricks save the heat received from the iron longer.

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The octahedron is one of the four positive polyhedra to which people have attached magical significance since ancient times. This polyhedron symbolized air. The demonstration model of the octahedron can be made of thick paper or wire.

You will need

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

Instruction

1. The octahedron has eight faces, each of which is an equilateral triangle. In geometry, an octahedron is traditionally built, inscribed in a cube or described near it. In order 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. Construct a right triangle on one of its sides, in which all sides are equal, and each of the angles is 60 °. It is convenient to build a triangle using a protractor, setting aside from 2 corners of the square adjacent to the same side of 60 °. Draw rays through the marks. The point from the intersection will be the third corner, and in the future - the top of the pyramid. Build the same triangles on the other sides of the square.

3. You will have to glue the pyramid. This will require allowances. Four allowances are enough, one for each triangle. Cut out what you have. Make a second one of the same kind. Bend the fold lines to the wrong side.

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

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

6. In order to make a model of an octahedron out of wire, you will need a cardboard or wooden square. However, it is allowed to get by with an ordinary triangle - in order to bend the workpiece at a right angle, it is absolutely enough. 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 fastening them at 2 points together, and, if necessary, attaching them to the corners of the square. It depends on the wire. If the material is solderable, 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 rest of the blanks in the same way. Unite the ends of the ribs located on one side of the square base with each other. True triangles will turn out by themselves. Do the same operation with the ends of the ribs located on the other side of the base. The octahedron is ready.

Helpful advice
Wire for similar models must be chosen one that perfectly holds its shape.

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

You will need

- a sheet of A4 paper
- scissors

Instruction

1. To produce a three-dimensional geometric figure, an octahedron needs a square sheet of paper. It is allowed to 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 narrow side of the sheet at the mark you made. Cut off the unwanted piece of paper. Fold the square in half.

2. Attach the top right corner to the center fold. Align the upper left corner so that the fold line passes through the attached upper right corner.

3. Fold the lower left corner of the square towards the midline. Aligning the lower right corner like the upper corners, make a fold. Later, the workpiece must be overturned.

4. Fold the bottom right corner of the part 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 out the workpiece with your hand.

6. Bend the sides of the figure to the middle line of the square. Turn the part over to the opposite side.

7. Fold the piece from bottom to top in 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 center triangle.

9. Make stripes on the upper sides of the figurine. The fold point of the strips will begin at the bottom of the inner cutout of the "V".

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

11. Fold the left side down.

12. The figure shows pockets and inserts for assembling an 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. Then assemble all 4 modules together.

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

Helpful advice
Lines that are not visible must be dashed. If you are in doubt whether one or the other segment is visible, build a dotted one in a cooler way. It is easier to complete a dotted line to a solid line than to correct a solid line to a dotted line.

Creating crafts with your own hands is interesting not only for children, but also for adults. However, for adults, a sufficient number of models have been invented, which differ in the complexity of implementation and the time spent on their creation. Recently, adults and children have become interested in creating complex geometric shapes. This type of figure includes the icosahedron, which is a regular polygon and is one of the Platonic solids - regular polyhedra. This figure has 20 triangular faces (equilateral triangles), 30 edges and 12 vertices, which are the junction of 5 edges. It is quite difficult to assemble a regular paper icosahedron, but it is interesting. If you are passionate about origami, then making a paper icosahedron with your own hands will not be difficult for you. Make it from colored, corrugated paper, foil, wrapping paper for flowers. Using a variety of materials, you can give even greater beauty and showiness to your icosahedron. It all depends only on the imagination of its creator and the available material on the table.

We offer you several options for developing the icosahedron, which can be printed, transferred to thick paper and cardboard, bent along the lines and glued.

How to make an icosahedron out of paper: diagram

In order to assemble an icosahedron from a sheet of paper or cardboard, you must first prepare the following materials:

layout of the icosahedron;
PVA glue;
scissors;
ruler.

During the creation of the icosahedron, it is important to pay special attention to the process of folding all the details: in order to evenly bend the paper, you can use a regular ruler.

It is noteworthy that the icosahedron can also be found in everyday life. For example, a soccer ball is made in the form of a truncated icosahedron (a polyhedron consisting of 12 pentagons and 20 regular hexagons). This is especially evident if you color the resulting icosahedron in black and white, like the ball itself.

You can make such a soccer ball yourself by printing a preliminary scan of a truncated icosahedron in 2 copies:

Creating an icosahedron with your own hands is an interesting process that requires thoughtfulness, patience and a lot of paper. However, the result obtained in the end will please the eye for a long time. The icosahedron can be given to a child to play if he has already reached the age of three. Playing with such a complex geometric figure, he will develop not only figurative thinking, spatial skills, but also get acquainted with the world of geometry. If an adult decides to create an icosahedron on his own, then such a creative process of constructing an icosahedron will allow you to pass the time, as well as boast to your loved ones about your ability to create complex shapes.

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

Instruction

1. For building icosahedron and use the construction of a cube. Let us denote one of its faces 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, set aside 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. The segments CC1 and DD1 are the edges of the icosahedron a. Raising 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 true polyhedron inscribed in the cube - icosahedron. With the help of the hexahedron, any true polyhedron can be built.

The 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 hefty exciting. It can be made from corrugated, packaging or colored paper, foil. By using different materials, you can add even more show and beauty to your icosahedron.

You will need

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

Instruction

1. Print the layout of the icosahedron 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 with each other. Take the trouble to cut out the icosahedron as leisurely as possible, on the contrary, with the slightest shift, your craft will look ugly as a result. The need for a hefty neat cut is due to the fact that all triangles in the correct 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 the solid lines, then use glue to glue the places that are outlined by the dotted line and combine the adjacent sides of the triangles with each other. For more tight fixation, each glued side must be held in this state for 20 seconds. It is true that all other sides of the icosahedron should be glued in the same way. It is more difficult for everyone to glue the last two ribs together, because patience and dexterity are needed to connect them. Your paper icosahedron is ready.

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

4. Making an icosahedron from paper is an interesting process that requires patience, thoughtfulness and a lot of paper. But the result obtained will be pleasing to the eye for a long time. A paper icosahedron can be given as an educational toy to a child who has reached the age of 3. Playing with this geometric figure, the kid will develop not only spatial skills and imaginative thinking, but also get to know the world of geometry better. For an adult, the creative process of constructing a paper icosahedron with his own hands will allow him to pass the time, as well as impress his loved ones with the knowledge of making difficult figures.

Helpful advice
During the manufacture of a paper icosahedron, special attention must be paid to the process of folding its sides. In order 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 in ancient times. This polyhedron symbolized air. The demonstration model of the octahedron can be made of thick paper or wire.

You will need

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

Instruction

1. The octahedron has eight faces, all of which are an equilateral triangle. In geometry, an octahedron is usually built, inscribed in a cube or described near it. In order 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. Construct a positive triangle on one of its sides, in which all sides are equal, and all of the angles are 60 °. It is convenient to build a triangle using a protractor, setting aside from 2 corners of the square adjacent to the same side of 60 °. Draw rays through the marks. The point from the intersection will be the third corner, and in the future - the top of the pyramid. Build the same triangles on the other sides of the square.

4. Bend each of the triangles to the wrong side. Spread allowances with PVA glue. 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 second, aligning the sides and corners. Let the octahedron dry.

8. Find the middle of the piece, wind or solder it to the corner of the square. Attach the rest of the blanks in the same way. Unite the ends of the ribs located on one side of the square base with each other. Positive triangles will turn out by themselves. Do the same operation with the ends of the ribs located on the other side of the base. The octahedron is ready.

Helpful advice
Wire for similar models must be chosen one that keeps its shape well.

You will need

- a sheet of A4 paper
- scissors