Dodecahedron pattern for gluing. Geometric figures. Dodecahedron

ICOSAHEDRON

SCAN ICOSAHEDRON. The sweep consists of twenty regular triangles, in addition, the sweep also includes valves.

HOW TO MAKE ICOSAHEDRON BY SCAN. Bend the scan along all the necessary lines with a “mountain”. If the scan is made on thick paper, then draw along all the fold lines along the inside with the sharp edge of the scissors.

HOW TO GLUE ICOSAHEDRON? After the reamer is bent, coat the valves with glue (PVA is better), and glue the 20-sided ball.

ANOTHER WAY OF GLUING THE ICOSAHEDRON.

20 separate circles are cut out of paper, into which regular triangles are inscribed.

We bend the prepared circles along the edges of the triangle and glue them together. And - at will: edges outward or edges inward.

Icosahedron - Zmeinogorsk Altai

HERALDRY.

Icosahedron ball on the coat of arms of the city of Zmeinogorsk, Altai.

NAME. Kepler's star or double tetrahedron.

REVIEW OF THE STAR OCTAHEDRON. The sweep consists of 24 regular triangles, in addition, the sweep also includes valves.

HOW TO MAKE A STAR OCTAHEDRON BY SCAN. Bend the scan along all the necessary lines. If the scan is made on thick paper, then draw along all the fold lines along the inside with the sharp edge of the scissors. If you want to get a two-color tetrahedron, then color the triangles marked with dots in a different color.

Sweep option

photo of Nata

APPEARANCE. A star octahedron is a conglomerate of two regular tetrahedra.

DOVECAHEDRON REVEL

DODECAHEDRON - one of the five regular polyhedra, the so-called Platonic solid.

NAME. Translated "dodecahedron" means - "12 faces

IN NUMERICAL EXPRESSION. The dodecahedron has 12 faces, 20 vertices, 30 edges.

DOVECAHEDRON REVEL. The reamer consists of twelve regular pentagons, in addition, the reamer also includes valves.

HOW TO MAKE A DODECAHEDRON BY SCAN. Bend the scan along all the necessary lines with a “mountain”. If the scan is made on thick paper, then draw along all the fold lines along the inside with the sharp edge of the scissors.

SPATIAL CONSTRUCTION. Three pentagons converge at each vertex of the dodecahedron

ELEMENTS. According to some medieval scholars, the dodecahedron corresponds to Ether (that is, emptiness)

Perhaps the oldest object in the form of a dodecahedron was found in northern Italy, near Padua, at the end of the 19th century, it dates back to 500 BC. e. and was presumably used by the Etruscans as a dice.

The dodecahedron was considered in their writings by ancient Greek scientists. Plato compared various classical elements with regular polyhedra. About the dodecahedron, Plato wrote that "... his god determined for the Universe and resorted to him as a model"

In 2003, when analyzing data from the WMAP spacecraft, it was hypothesized that the universe is a dodecahedral Poincaré space

On the territory of several European countries, many objects have been found, called Roman dodecahedrons, dating back to the 2nd-3rd centuries. n. e., whose purpose is not entirely clear.

The ancient sages said: "To know the invisible, look carefully at the visible." In terms of sacred forces, the dodecahedron is the most powerful polyhedron. No wonder Salvador Dali chose this figure for his Last Supper. In it, from twelve pentagons - also a strong figure, forces are concentrated at one point - on Jesus Christ.

Now look at the dodecahedron and realize that the number 5 forms the CRYSTAL OF POWER.

The figure refers to one of the five Platonic solids (along with the tetrahedron, octahedron, hexahedron (cube) and icosahedron). 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.

DODECAHEDRON IN NATURE. A crystal of pyrite - sulfur pyrite - FeS2 - is very beautiful, and, according to legend, it was he who suggested to the Greeks the idea of ​​a "correct" dodecahedron.

If the edge length of the dodecahedron is taken as , then the area of ​​the entire surface of the dodecahedron is equal to

The radius of a sphere circumscribed around a dadecahedron is calculated as follows:

The calculation of the radius of a sphere inscribed in a dodecahedron can be done as follows:

The dodecahedron is a very unusual three-dimensional figure, consisting of 12 identical faces, each of which represents. To assemble a dodecahedron with your own hands, it is not at all necessary to have special skills, even a child can handle this task. A little skill, and you will definitely succeed!

Necessary materials and tools

• A sheet of white and colored paper. Optimum density - 220 g/m 2 . Very thin paper wrinkles too much when assembled, and very thick cardboard breaks when folded.
• Dodecahedron net (template).
• Thin or very sharp scissors.
• A simple pencil or marker.
• Protractor.
• Long line.
• liquid glue.
• Tassel.

Instruction

1. If you have a printer, you can print the template directly on the sheet, but you can draw it yourself. Pentagons are built using a protractor and a ruler, the angle between adjacent lines should be exactly 108 o, choosing the length of the face, you can make a large or small dodecahedron. The development is 2 connected "flowers", consisting of 6 figures. Be sure to leave small allowances, they are needed for gluing.
2. Carefully cut the workpiece with scissors or a knife on a special one so as not to damage the surface of the table. Next, go through the places of the folds with an acute angle of the ruler, this will greatly facilitate the assembly of the figure and make the edges more accurate.
3. Using a brush, apply a little glue to the allowances and assemble the figure by folding the edges inward. If you decide to make a dodecahedron with your own hands, and you didn’t even have adhesive tape at hand, cut out 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 simply insert the edges into the grooves and the structure will hold quite firmly.

The finished figure can be painted or decorated with stickers. A 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. Well, if they are colored and double-sided, you can choose several shades.
2. Manufacturing of modules. Mentally draw a sheet into four identical strips and fold it like an accordion. Bend the corners to one side in opposite directions, the resulting figure should resemble a parallelogram. It remains to bend the workpiece along a short diagonal. Make 30 modules and start building.
3. The dodecahedron has 10 nodes, each assembled from three elements. Prepare all the parts and put them inside each other. To prevent the modules from moving apart, fix the joints with paper clips, when you fully assemble the figure, they can be removed.

As soon as you master the technique you like, you can teach your child or friend to assemble the dodecahedron with your own hands. After all, the manufacture of three-dimensional figures not only develops finger motor skills well, but also forms spatial imagination.

The dodecahedron is a three-dimensional geometric figure that has 12 faces. This is its main characteristic, since the number of vertices and the number of edges can vary. Consider in the article the properties of this figure, its current use, as well as some interesting historical facts associated with it.

General concepts about the figure

Dodecahedron is a word taken from the language of the ancient Greeks, which literally means "a figure with 12 sides". Its faces are polygons. Given the properties of space, as well as the definition of a dodecahedron, we can say that its polygons can have 11 sides or less. If the faces of the figure are formed by regular pentagons (a polygon with 5 sides and 5 vertices), then such a dodecahedron is called regular, it is one of the 5 Platonic objects.

Geometric properties of a regular dodecahedron

Having considered the question of what a dodecahedron is, we can proceed to characterize the basic properties of a regular three-dimensional figure, that is, formed by identical pentagons.

Since the figure under consideration is voluminous, convex and consists of polygons (pentagons), then the Euler rule is valid for it, which establishes an unambiguous relationship between the number of faces, edges and vertices. It is written as: G + B = P + 2, where G is the number of faces, B - vertices, P - edges. Knowing that a regular dodecahedron is a dodecahedron, the number of vertices of which is 20, then, using the Euler rule, we get: P \u003d G + B - 2 \u003d 30 edges. The angles between adjacent faces of this Platonic figure are the same, they are equal to 116.57 o .

Mathematical formulas for regular dodecahedron

Below we give the basic formulas of the dodecahedron, which consists of regular pentagons. These formulas allow you to calculate its surface area, volume, and also determine the radii of spheres that can be inscribed in a figure or described around it:

• The surface area of ​​the dodecahedron, which is the product of 12 areas of pentagons with side "a", is expressed by the following formula: S = 3*√(25 + 10*√5)*a 2 . For approximate calculations, you can use the expression: S = 20.65 * a 2.
• The volume of a regular dodecahedron, as well as its total area of ​​faces, is uniquely determined from the knowledge of the side of the pentagon. This value is expressed by the following formula: V \u003d 1/4 * (15 + 7 * √5) * a 3, which is approximately equal to: V \u003d 7.66 * a 3.
• The radius of the inscribed circle, which touches the inner side of the faces of the figure at their center, is defined as follows: R 1 = 1/4*a*√((50 + 22*√5)/5), or approximately R 1 = 1.11*a.
• The circumscribed circle is drawn through 20 vertices of a regular dodecahedron. Its radius is determined by the formula: R 2 = √6/4*a*√(3 + √5), or approximately R 2 = 1.40*a. The figures given say that the radius of the inner sphere inscribed in the dodecahedron is 79% of that for the circumscribed sphere.

Symmetry of a regular dodecahedron

As can be seen from the figure above, the dodecahedron is a fairly symmetrical figure. To describe these properties, crystallography introduces the concepts of symmetry elements, the main of which are rotational axes and reflection planes.

The idea of ​​using these elements is simple: if you set an axis inside the crystal in question, and then rotate it around this axis by some angle, then the crystal will completely coincide with itself. The same applies to the plane, only the operation of symmetry here is not the rotation of the figure, but its reflection.

The dodecahedron has the following symmetry elements:

• 6 axes of the fifth order (that is, the figure is rotated through an angle of 360/5 = 72 o), which pass through the centers of pentagons located opposite each other;
• 15 axes of the second order (the symmetrical angle of rotation is 360/2 = 180 o), which connect the midpoints of the opposite edges of the octahedron;
• 15 planes of reflection passing through the figures located opposite the edges;
• 10 axes of the third order (the operation of symmetry is carried out when turning through an angle of 360/3 = 120 o), which pass through the opposite vertices of the dodecahedron.

Modern use of the dodecahedron

Currently, geometric objects in the form of a dodecahedron are used in some areas of human activity:

• Dice for board games. Since the dodecahedron is a Platonic figure with high symmetry, objects of this shape can be used in games where the continuation of events is probabilistic. Most dice are made in a cubic shape because they are the easiest to make, but modern games are becoming more complex and varied, which means they require dice with more options. Dodecahedron-shaped dice are used in the role-playing board game Dungeons and Dragons. A feature of these bones is that the sum of the numbers located on opposite faces is always equal to 13.

• Sound sources. Modern loudspeakers are often made in the shape of a dodecahedron because they propagate sound in all directions and shield it from ambient noise.

Historical reference

As mentioned above, the dodecahedron is one of the five Platonic solids, which are characterized by the fact that they are formed by identical regular polyhedra. The other four Platonic solids are the tetrahedron, octahedron, cube and icosahedron.

Mentions of the dodecahedron date back to the Babylonian civilization. However, the first detailed study of its geometric properties was made by ancient Greek philosophers. So, Pythagoras used a five-pointed star built on the tops of the pentagon (facets of the dodecahedron) as the emblem of his school.

Plato described in detail the correct three-dimensional figures. The philosopher believed that they represent the main elements: the tetrahedron is fire; cube - earth; octahedron - air; icosahedron - water. Since the dodecahedron did not get any element, Plato suggested that it describes the development of the entire universe.

Many may consider Plato's thoughts primitive and pseudoscientific, but here's what's curious: modern studies of the observable Universe show that cosmic radiation coming to Earth has anisotropy (direction dependence), and the symmetry of this anisotropy is in good agreement with the geometric properties of the dodecahedron.

Dodecahedron and sacred geometry

Sacred geometry is a collection of pseudo-scientific (religious) knowledge that ascribes a certain sacred meaning to various geometric shapes and symbols.

The significance of the dodecahedron polyhedron in sacred geometry lies in the perfection of its form, which is endowed with the ability to bring the surrounding bodies into harmony and evenly distribute energy between them. The dodecahedron is considered an ideal figure for the practice of meditation, as it plays the role of a conduit of consciousness to another reality. He is credited with the ability to relieve stress in humans, restore memory, improve attention and concentration abilities.

Roman dodecahedron

In the middle of the 18th century, as a result of some archaeological excavations in Europe, a strange object was found: it had the shape of a dodecahedron made of bronze, its dimensions were several centimeters, and it was empty inside. However, the following is curious: a hole was made in each of its faces, and the diameter of all the holes was different. Currently, more than 100 such objects have been found as a result of excavations in France, Italy, Germany and other European countries. All these items date back to the II-III century AD and belong to the era of the domination of the Roman Empire.

How the Romans used these objects is not known, since not a single written source has been found that would contain an accurate explanation of their purpose. Only in some works of Plutarch can one find a mention that these objects served to understand the characteristics of the 12 signs of the Zodiac. The modern explanation of the mystery of the Roman dodecahedrons has several versions:

• objects were used as candlesticks (remains of wax were found inside them);
• they were used as dice;
• dodecahedrons could serve as a calendar that indicated the time of planting crops;
• they could be used as the basis for fastening the Roman military standard.

There are other versions of the use of Roman dodecahedrons, however, none of them has exact evidence. Only one thing is known: the ancient Romans highly valued these items, because in excavations they are often found in hiding places along with gold and jewelry.

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 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 "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 buy ready-made kits containing everything you need to make souvenirs, but the most interesting thing is 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).

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;
• - compass;
• - ruler;
• - square;
• - a piece of thin wire;
• - scissors;
• - glue.

Instruction

• Start by drawing a central regular pentagon. To do this, draw a circle with a compass. Draw a 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 identical 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 dividing the ruler, or measure it with a piece of thin wire, fold it three times, then straighten it, put it on the diameter and mark the points at the bends.
• As a result of dividing the diameter into three parts, two points will be obtained on it. Draw a perpendicular through one of them to the diameter using a square. It will cross the circle in two places. From each of them, draw a ray passing through the second point on the diameter. They will intersect the circle in two more places, but the fifth place of intersection forms 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, arranging them in such a way that a figure similar to the one shown in the figure is obtained. Draw small petals to its sides on the side to facilitate gluing. 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, it is possible to make voluminous, stable table calendars in the form of this figure. To do this, first draw on each of the faces according to the calendar for one month, and only then cut and glue the figure. Also, such a calendar can be generated automatically by clicking on the link below. The year will be determined automatically by the server's built-in clock, and the language of the names of the months and days of the week will be determined by your browser settings.