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Volumetric dodecahedron. Geometric figures. Dodecahedron. How to make an icosahedron out of paper: diagram

A dodecahedron is a three-dimensional figure consisting of twelve pentagons. To get this figure, you must first draw its scan on thick paper, and then assemble it from this scan in space.

You will need

  • - thick paper;
  • - pencil;
  • - compasses;
  • - ruler;
  • - square;
  • - a piece of thin wire;
  • - scissors;
  • - glue.

Instructions

  • Start by drawing a central, regular pentagon. To do this, draw a circle with a compass. Draw the diameter through its center. Now it needs to be divided into three parts. There is a theorem proving that trisection (that is, dividing a segment or angle into three equal parts) using a ruler without divisions and a compass is impossible. Therefore, either measure the diameter with a ruler and divide it by three, and then mark the corresponding points on it by the divisions of the ruler, or measure it with a piece of thin wire, fold it in three, then straighten it, put it on the diameter and mark the points at the folds.
  • As a result of dividing the diameter into three parts, you get two points on it. Through one of them, draw a perpendicular to the diameter using a square. It will cross the circle in two places. Draw a ray from each of them through the second point on the diameter. They will intersect the circle in two more places, and the fifth intersection is formed by the diameter itself. It remains only to connect them together, and you get a regular pentagon inscribed in a circle.
  • Draw eleven more pentagons in the same way, positioning them so that you get a shape like the one shown in the figure. Draw small petals on the side of its edges to make gluing easier. Then cut it out and glue it. What should be the result is shown in the illustration in the title of the article.
  • Since the dodecahedron has exactly twelve faces, this figure can be used to make voluminous, stable table calendars. To do this, first make up a calendar for one month on each of the faces, and only then cut and glue the shape. Also, such a calendar can be generated automatically by clicking on the link below. The year will be determined automatically by the built-in server clock, and the language of the names of the months and days of the week will be determined by your browser settings.

You will need

  • - thick paper;
  • - pencil;
  • - compasses;
  • - ruler;
  • - square;
  • - a piece of thin wire;
  • - scissors;
  • - glue.

Instructions

Start by sketching the centerpiece. To do this, draw a circle with a compass. Draw the diameter through its center. Now it needs to be divided into three parts. There is a theorem proving that trisection (that is, dividing a segment or angle into three) using a ruler without divisions and a compass. Therefore, either measure the diameter with a ruler and divide it by three, and then mark the corresponding points on it by the divisions of the ruler, or measure it with a thin piece, fold it in three, then straighten it, put it on the diameter and mark the points at the folds.

As a result of dividing the diameter into three parts, you get two points on it. Through one of them, draw a perpendicular to the diameter using a square. It will cross the circle in two places. Draw a ray from each of them through the second point on the diameter. They will intersect the circle in two more places, and the fifth intersection is formed by the diameter itself. It remains only to connect them together, and you will get the correct one, inscribed in a circle.

Draw eleven more pentagons in the same way, positioning them so that you get a shape like the one shown in the figure. Draw small petals on the side of its edges to make gluing easier. Then cut it out and glue it. What should be the result is shown in the illustration in the title of the article.

Since the dodecahedron has exactly twelve faces, this figure can be used to make voluminous, stable table calendars. To do this, first make up a calendar for one month on each of the faces, and only then cut and glue the shape. Also, such a calendar can be generated automatically by clicking on the link below. The year will be determined automatically by the built-in server clock, and the language of the names of the months and days of the week will be determined by your browser settings.

Sources:

  • Dodecahedron calendar generator
  • how to make a regular dodecahedron

Stereometry, as a part of geometry, is much brighter and more interesting precisely because the figures here are not plane, but three-dimensional. In numerous tasks, it is required to calculate the parameters of parallelepipeds, cones, pyramids and other three-dimensional shapes. Sometimes, already at the construction stage, difficulties arise that can be easily eliminated if you follow the simple principles of stereometry.

You will need

  • - ruler;
  • - pencil;
  • - compasses;
  • - protractor.

Instructions

Decide on the number of faces, as well as the number of corners in the polygons of the faces themselves in front. If the condition says about a regular polyhedron, then build it so that it is convex (not broken), so that the faces are regular polygons, and the same number of edges converge at each vertex of the three-dimensional figure.

Remember about special polyhedra, for which there are constant characteristics:
- a tetrahedron consists of triangles, has 4 vertices, 6 edges, converging at the vertices by 3, as well as 4 faces;
- hesahedron, or cube, consists of squares, has 8 vertices, 12 edges, converging by 3 at the vertices, as well as;
- the octahedron consists of triangles, has 6 vertices, 12 edges adjoining 4 each to the vertices, as well as 8 faces;
- is a twelve-sided figure, consisting of pentagons, with 20 vertices, as well as 30 edges adjacent to the vertex by 3;
-, in turn, has 20 triangular faces, 30 edges, adjoining 5 to each of the 12 vertices.

Start with parallel lines if the edges of the polyhedron are parallel. This applies to the parallelepiped,

A dodecahedron is a very unusual three-dimensional figure, consisting of 12 identical faces, each of which represents. A little skill and you will definitely succeed!

Required materials and tools

  • A sheet of white and colored paper. The optimum density is 220 g / m 2. Very thin paper wrinkles too much during assembly, and very thick cardboard breaks at the folds.
  • Unfolding of the dodecahedron (pattern).
  • Thin or very sharp scissors.
  • A simple pencil or marker.
  • Protractor.
  • Long ruler.
  • Liquid glue.
  • Brush.

Instructions

  1. If you have a printer, you can print the template directly on the sheet, but it is quite possible to draw it yourself. Pentagons are constructed using a protractor and a ruler, the angle between adjacent lines should be exactly 108 о, choosing the length of the face, you can make a large or small dodecahedron. The unfolding represents 2 connected "flowers", consisting of 6 shapes. Be sure to leave small allowances, they are needed for gluing.
  2. Carefully cut the workpiece with scissors or a special knife so as not to damage the table surface. Next, go through the places of the folds with an acute angle of the ruler, this will noticeably facilitate the assembly of the figure and make the edges more accurate.
  3. Using a brush, apply a little glue to the seam allowances and collect the shape by folding the edges inward. If you decided to make a dodecahedron with your own hands, and you didn't even have adhesive tape at hand, cut the allowances of one half of the template in the form of elongated triangles, and make small cuts on the folds of the second part. Then just insert the edges into the grooves, and the structure will hold pretty firmly.

The finished shape can be painted or decorated with stickers. The large model can be turned into an original calendar, because the number of sides corresponds to the number of months in a year. If you are fond of Japanese, you can make a dodecahedron with your own hands using the modular origami technique.

  1. Prepare 30 sheets of plain office paper. It is good if they are colored and double-sided, you can choose several shades.
  2. Manufacturing of modules. Mentally trace the sheet into four identical strips and fold it like an accordion. Bend the corners to one side in opposite directions, the resulting shape should resemble a parallelogram. It remains to bend the workpiece along a short diagonal. Make 30 modules and start assembling.
  3. The dodecahedron has 10 nodes, each assembled from three elements. Prepare all the pieces and nest them inside each other. To prevent the modules from moving apart, fix the joints with paper clips, when you completely assemble the figure, they can be removed.

Once you have mastered the technique you like, you can teach your child or comrade how to assemble a dodecahedron with your own hands. After all, making three-dimensional figures not only develops finger motor skills well, but also forms spatial imagination.

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From the history of the dodecahedron

Everyone who went to school, studied geometry, someone loved it, and someone not very much, and someone just has to get acquainted with this science. And, of course, everyone was asked to draw or assemble various geometric shapes, and then the best work was evaluated. But, unfortunately, not all teachers talk about the origin of geometric shapes, what they are for, what value they have and where they are used. And the figures have a very rich history, they are as important as any discoveries in our world. And they are found everywhere, we just do not always notice them. Today we are going to tell you about the dodecahedron.

The word dodecahedron is of Greek origin and consists of 2 words: dodeka (twelve) and hedra (edge). The dodecahedron has 12 faces, 20 vertices, in each of which 3 edges and 30 edges converge. The sum of the flat angles is 324 °. It is a dodecahedron, which is made up of twelve regular pentagons. The dodecahedron is a regular polyhedron, it has three stellate shapes.

The dodecahedron was already known in ancient times. For example, pupils of the Pythagorean school were forbidden to pronounce this word outside the school, as for this they could lose their lives. This figure was treated as a holy figure, they were even afraid to say anything about her. Only two hundred years later, at the time of Plato, they began to speak very carefully about this figure. It was forbidden to say something unnecessary, the more offensive or dismissive. They believed that the dodecahedron is located in the energy field of people and is the highest form of human consciousness. In addition, it was believed that people live inside a huge dodecahedron, in which our universe is located, and when a person's mind reaches the very limit of the space of the Cosmos, he stumbles upon a dodecahedron, closed in a sphere.

Dodecahedron in our life

Where can you find a dodecahedron? Think hard! Probably almost everyone saw it as a random number generator, for example, on TV in a lotto game or in tabletop role-playing games. The dodecahedron can be found in the game "Pentacor", the world of which is represented by this figure. And, of course, everyone has heard of the Pentagon, this building of the US Department of Defense has the shape of a regular pentagon.

In August 2006, when mapping the distribution of dark matter in a cluster of galaxies, it was concluded that our universe looks like a set of infinitely repeating dodecahedrons.

Regular polyhedra have always attracted by the perfection of their forms, complete, seemingly impossible symmetry. Some of these bodies are found in nature, for example in the form of crystals, while others can be in the form of viruses or protozoa.

You can assemble this amazing shape using our dodecahedron sweeps.

Unfolded dodecahedron made of paper or cardboard

Scheme of a regular dodecahedron Dodecahedron diagram with formulas Diagram of a dodecahedron with the great discoveries of mankind

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

The structure got its name from the number of faces included in it (traditionally, the ancient Greeks gave polyhedrons 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 "hedra" (edge). The figure refers 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, organic nature 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, claiming that: "Out of the twelve, much grows." On the territories of ancient destroyed states, small figures in the form of dodecahedrons, made of bronze, stone or bone, are still found. In addition, during excavations on the lands of modern England, France, Germany, Hungary, Italy, archaeologists have discovered several hundred so-called "Roman dodecahedrons" dating back to the 2nd-3rd centuries AD. The main sizes of the figures are from four to eleven centimeters, and they differ in the most incredible patterns, textures and technique of execution. The version put forward in the days of Plato that the Universe is a huge dodecahedron was confirmed already at the beginning of the 21st century. After a careful 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 were engaged in 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 correct dodecahedron with your own hands

Today, the design of this figure has found its reflection in many variants of artistic creation, architecture and construction. Folk craftsmen make origami of unusual beauty in the form of openwork dodecahedrons from colored or white paper, and make original ones from cardboard, etc.). On sale you can buy ready-made kits containing everything you need to make souvenirs, but it is most interesting to do the whole process of work with your own hands, from building individual parts to assembling a 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 blunt knife or some kind of device for bending the allowances, but if they are not there, then a metal ruler or the same scissors is fine.

How to make a stellated dodecahedron

Stellate dodecahedrons have a more complex structure than ordinary dodecahedrons. These polyhedra are subdivided into small (first extension), medium (second extension), and large (last stellate form of a regular dodecahedron). Each of them has its own construction and assembly features. 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 (in the future, it is glued or parts are assembled using paper clips).