Home » Master classes on handicrafts » Working with empty space: why emptiness influences design so much. Filling the space with polyhedrons Doodling paradoxically does not distract, but helps to retain the attention of the drawer

Working with empty space: why emptiness influences design so much. Filling the space with polyhedrons Doodling paradoxically does not distract, but helps to retain the attention of the drawer

Spouses who are faced with problems of emotional intimacy must take responsibility for solving them. They should begin the journey to each other's hearts again. For example, a wife who is ashamed of her own excessive emotional vulnerability must admit to herself and her husband: yes, she has fear and anxiety that her husband will find out about her shortcomings and want to break off relations with her. This will soften the husband’s heart and give the wife the opportunity to make sure that all this is just her “invention.” I know spouses who began to devote a lot of time to the church or find a lot of other activities for themselves. The wife decided that after so many years of living together she had become uninteresting to her husband and that constant participation in any activity would distract her husband and he would not be so bored with her.

She took a risk and confessed to him:

I take on all these things because it began to seem to me that you are no longer interested in me.

He was very surprised and saddened.

Nothing of the kind. I miss you terribly when you're not around.

When a husband distances himself from his wife because he cannot stand some of her qualities, he should frankly admit: he is afraid of becoming involved in his wife’s failures and therefore condemns her behavior. The wife should tell her husband how much she feels about his alienation. They must think together about what to do.

Here's another example. A wife cannot stand her husband's anger because it reminds her of her own hidden rage, and she needs to responsibly deal with her negative emotions. And the husband must let her understand that he is lonely and lacks her love when she withdraws into herself if he is angry.

Alienation of spouses often occurs due to unformed boundaries. Alienation is the only limit they set. They cannot come to terms with each other's shortcomings. If they need to become emotionally close and at the same time set some restrictions, it turns out that rapprochement does not occur or the existing problem is not solved. Spouses need to learn to be both loving and sincere.

They can agree: as soon as one of them feels that there is no more love and sincerity in the relationship, he will immediately inform the other about this. If you feel that you are afraid to tell your spouse the truth, say so: “I am afraid to tell you the truth.” First of all, deal with this fear together, and only then communicate the truth. If you notice that your spouse is talking to you in a distant manner, let him know that he is emotionally distant from you. The two of you discuss what is happening to him at the moment.

Ignorance of one's own boundaries

Dale and Margaret are my friends. Dale is an energetic optimist. He loves to take part in all church and social events; creates and trains sports teams according to the most different types the sports his children play; loves his work and can do it endlessly. Margaret is his complete opposite. Having received two higher education, she nevertheless decided that she main role- follow Dale as a shadow and clear away the rubble left by him. When he takes on too much, she helps him choose which activities to take part in and which ones to skip. When he gets into debt, she figures out how to pay it off. And although she considers this an integral part of her marriage, she is very worried about how her husband views her role in family life value too small. “Dale is a person who is interested in everything in the world, incredibly in love with life,” she told me. - For him, life is an endless adventure. But he never views the two of us as a married couple.”

Often married couples experience outside interference in their relationship because one or both spouses simply do not know how best to manage their time, energy, and abilities. It seems to them that they do not allow the fire of love that once flared up between them to fade away and maintain it at a certain level. They sincerely want to take part in the life of their spouse, talk to him about everything, maintain a romantic relationship, but not now. And very often the right moment never comes. Something outside interferes with the marriage and begins to dominate it.

Such problems usually arise among “unbridled” spouses who are unable to foresee the consequences of their actions. Throughout life, someone always ends up next to such a person and smoothes out all the unpleasant consequences of his behavior. First, parents do this, then friends, colleagues, and finally spouse. A careless attitude towards family life is a consequence of a careless attitude to everything in general. Spouses in such a marriage believe that a safety net is constantly stretched underneath them, like a circus performer. The role of this network is played by the people around them, who by their behavior inspire them that: 1) no matter how irresponsibly they treat their responsibilities, nothing bad will happen anyway; 2) and even if it happens, no one will pay attention to it; 3) and if someone still feels uncomfortable about this, there will definitely be someone who will help them out of trouble, and everything will be settled. Such people live by the principle: everything will always end well. They are unable to face the truth.

When Margaret told Dale about her experiences, he was quite surprised. It seemed to him that his wife, like him, was constantly in high spirits thanks to his “omnivorousness.” And when he heard from her: “I love you, but I will no longer encourage everything that comes between us - your participation in all types of activities,” he was terribly offended. But Margaret stopped playing the role of her husband's savior, and in the end he clearly felt the full consequences of his eternal scatteredness. Faced with the dissatisfaction of the people he had let down by being late everywhere and everywhere, he began to look at things more realistically. At the same time, he realized how much Margaret had done for him. Dale began to appreciate his wife. He began to regret the lost time when he had allowed so many unimportant things to come between them.

Dale learned to correctly perceive the surrounding reality, which is a combination of “carrot and stick”: the stick of the real state of affairs and the carrot of love for Margaret.

When decorating the interior, windows are often ignored; at most, curtains are replaced or the windows themselves are replaced.

But the space around them can also be used as practically as possible.

There really is a lot to roam around here, the range of ideas is so wide that we couldn’t stop at just one thing and decided to immediately offer 25 wonderful ideas for transforming your interior.

Ideas for a children's room

“The space near the window in the children's room.

The space near the window in the children's room.

There are never too many storage systems in a nursery; toys fill all the space and their number is only increasing. If you don’t know where to put them, then this item is for you. Make open or closed cabinets around the window, let there be a soft corner for the baby, and hidden storage systems under the window sill. This complex looks stylish and is sure to please the child.

Neat storage in the children's room. Home library storage

Home library.

Paper books take up a lot of space in any interior and, usually, their number does not decrease, but only increases. And book lovers begin to puzzle over where to put their favorite literature, where to find a place for a home library. This is where the window space comes in handy. You can use the upper part under the ceiling or organize storage of books under the windowsill, and if there are two windows in the room, then take up the entire space between them and make an impromptu bookcase out of it. Another idea is to create open shelves on either side of the window and stack books on them.

Neat storage of books.

Storing books under the windowsill.

Stylish storage of books in the apartment.

Storing books between two windows.

Stylish storage for your home library.

Reading corner

A cozy place to read.

Since we're talking about books, we can't ignore the design of a reading corner. The window is ideal for these purposes. Use soft floor pillows for this; they will serve as a seat, and their decorative counterparts will also go under your back. Create your own cozy corner where you will have a pleasant time on your day off. In case you get caught up in reading and it gets dark outside, make sure there is an additional source of artificial lighting nearby.

Reading place by the window.

Place to rest

A stylish place to relax.

When a beautiful view opens from the window, it’s simply a sin not to make a place for relaxation and contemplation on the windowsill. Let a calm color scheme prevail here, be sure to equip the seat with a soft one, do not forget to decorate the place with several decorative pillows, and place a blanket nearby. Such an interior composition will please the eye and fill the room with a special atmosphere. In addition, after a busy day at work, you can always relax and get aesthetic pleasure.

A place to relax.

The most comfortable place in the house.

A beautiful place to relax.

Neat and stylish.

Home office

Home office by the window.

A good solution is to organize a work area near the window. Use the window sill as a tabletop or place a side table on it, make convenient shelves for storing papers and stationery. Don't forget to choose a chair at a comfortable height and don't hang curtains on the window, they will block natural light from entering the room.

Dining area

Dining area by the window.

When it comes to finding a place for a dining area, turn your attention to the window and the space around it, especially if there is a free corner next to it. A corner corner or several comfortable chairs/armchairs would fit perfectly here. Choose laconic and extremely simple furniture in light shades for this area. This will make the dining room seem more cozy.

Corner dining room by the window.

Cozy mini-dining room.

Window in the kitchen

Practical use of a window in the kitchen.

A window in the kitchen is a godsend for the housewife; not only can the window sill be used as a work area, but it can also use the space by the window. Organize open shelves and place dishes, pots, cereals and spices in jars there.

In the bathroom

Storage on the bathroom window.

If the bathroom has a window, this is a great success. Make the most of it, such as storing clean bath towels here. And if you make several shelves, you can easily place personal hygiene products or toilet paper.

In the dressing room

Using the space near the window in the dressing room.

The window in the dressing room can also be put to good use. Place comfortable open niches on both sides and place your favorite bags or shoes there, which should always be within minutes’ reach. Do not forget to periodically wipe them from dust so that they do not lose their presentable appearance.

Storage systems in the bedroom

Storage systems in the bedroom.

Open storage systems will also come in handy in the bedroom. It is convenient to store home clothes, bed linen, towels, personal hygiene items, and books here. For neat storage, use auxiliary accessories - boxes, plastic boxes, wicker baskets.
For aesthetic pleasure
The space near the window can be used not only for practical purposes, but also to make a stand out of it for displaying beautiful decor. These could be souvenirs brought from travel, a collection of porcelain figurines, porcelain, vases and anything else that pleases the eye. The main thing is not to overload the shelves, leave some free space for an airy interior effect.

Stands for decoration.

When decorating the interior, windows are often ignored; at most, curtains are replaced or the windows themselves are replaced.

But the space around them can also be used as practically as possible. There really is a lot to roam around here, the range of ideas is so wide that we couldn’t stop at just one thing and decided to immediately offer 25 wonderful ideas for transforming your interior.

Ideas for a children's room

“The space near the window in the children's room.

The space near the window in the children's room.

Reading corner

A cozy place to read.

Place to rest

A stylish place to relax.

Home office

Home office by the window.

A good solution is to organize a work area near the window. Use the window sill as a tabletop or place a side table on it, make convenient shelves for storing papers and office supplies. Don't forget to choose a chair at a comfortable height and don't hang curtains on the window, they will block natural light from entering the room.

Dining area

Dining area by the window.

Window in the kitchen

Practical use of a window in the kitchen.

In the bathroom

Storage on the bathroom window.

In the dressing room

Using the space near the window in the dressing room.

Storage systems in the bedroom

Storage systems in the bedroom.

Filling space with polyhedra

What polyhedra can be used to fill space so that any two polyhedra either have a common face, or a common edge, or a common vertex, or have no common points? This filling of space with polyhedra is called spatial parquet.

It is clear that if there is a parquet on a plane consisting of polygons, then the prisms, the bases of which are these polygons, will form a spatial parquet (Fig. 1). In particular, spatial parquet can be composed of an arbitrary parallelepiped, a regular triangular prism, a regular hexagonal prism, etc.

Let's find out which regular polyhedra can be used to make a spatial parquet. Note that when filling space with polyhedra, the sum of the dihedral angles of polyhedra adjacent to one edge must be 360°. Therefore, from regular polyhedra of the same name, a spatial parquet can be made only from those whose dihedral angles have the form .

Of course, spatial parquet can be made up of equal cubes. The dihedral angles of a cube are 90°.

Let's find the dihedral angles of a regular tetrahedron. Let ABCD- regular tetrahedron with edge 1 (Fig. 2). From the tops A And D let's drop the perpendiculars A.E. And DE on the edge B.C.. Corner AED will be the linear angle j of the desired dihedral angle. In a triangle ADE we have:

. From where φ ≈ 70°30".

Rice. 2

Thus, if less than six tetrahedra converge on one edge, then the sum of their dihedral angles is less than 360°, but if we take six or more tetrahedra, then the sum of their dihedral angles will be greater than 360°. Consequently, it is impossible to make a spatial parquet from regular tetrahedra.

Let's find the dihedral angles of the octahedron. Let's consider a regular octahedron with edge 1 (Fig. 3).

Rice. 3

From the tops E And F let's drop the perpendiculars E.G. And FG on the edge B.C.. Corner EGF EGF we have:

Using the cosine theorem, we find . Hence φ ≈ 109°30". Thus, if less than four octahedra converge on one edge, then the sum of their dihedral angles is less than 360°, but if we take four or more octahedra, then the sum of their dihedral angles will be greater than 360°. Consequently, from It is impossible to form a spatial parquet of regular octahedra.

Let's find the dihedral angles of the icosahedron. Consider a regular icosahedron with edge 1 (Fig. 4).

Rice. 4

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From the tops A And C let's drop the perpendiculars A.G. And C.G. on the edge B.F.. Corner A.G.C. will be the linear angle j of the desired dihedral angle. In a triangle A.G.C. we have:

Using the cosine theorem, we find . Hence φ ≈ 138°11". Thus, if less than three icosahedra converge on one edge, then the sum of their dihedral angles is less than 360°, but if we take three or more icosahedra, then the sum of their dihedral angles will be greater than 360°. Consequently, from It is impossible to form a spatial parquet using regular icosahedrons.

Let's find the dihedral angles of the dodecahedron. Consider a regular dodecahedron with edge 1 (Fig. 5).

From the tops A And C let's drop the perpendiculars A.G. And C.G. on the edge B.F.. Corner A.G.C. will be the linear angle φ of the desired dihedral angle. In a regular pentagon ABCDE sides are equal . A.C. is the diagonal of this pentagon, and therefore . Besides, .

Using the cosine theorem, we find . Hence φ ≈ 116°34". Thus, if less than three dodecahedrons converge on one edge, then the sum of their dihedral angles is less than 360°, but if we take three or more dodecahedrons, then the sum of their dihedral angles will be greater than 360°. Consequently, from It is also impossible to form a spatial parquet of regular dodecahedrons.

Rice. 5

As a result, we find that the only regular polyhedron that can fill space, that is, create a spatial parquet, is a cube.

Using a cube, you can give examples of other polyhedra from which you can make a spatial parquet.

So, for example, a cube can be divided into regular quadrangular pyramids, the bases of which are the faces of the cube, and the top is the center of the cube (Fig. 6). One of these pyramids is the pyramid OABCD. If in a spatial parquet of cubes each cube is divided into regular quadrangular pyramids, then we obtain a spatial parquet of regular quadrangular pyramids.

Rice. 6

Regular quadrangular pyramid OABCD(Fig. 7) can be divided into two equal triangular pyramids OABC And OACD. Dividing cubes into such pyramids gives a spatial parquet consisting of triangular pyramids - tetrahedrons. For a unit cube, these tetrahedra have edges equal to . Tetrahedron OABC can be divided into two equal tetrahedrons OABP And OBCP. The edges of these tetrahedra are equal Tetrahedron OABP, in turn, can be divided into two equal tetrahedrons OARP And OBRP. The edges of these tetrahedra are equal Finally, from two tetrahedrons equal to a tetrahedron OABP, you can make one tetrahedron OABQ, from which you can also make spatial parquet. The edges of this tetrahedron are equal Note that the faces of the last tetrahedron are equal isosceles triangles with sides

Rice. 7

It turns out that there are no other tetrahedra from which a spatial parquet can be made, except for the four tetrahedra listed above (see).

Let us give other examples of polyhedra from which spatial parquets can be made.

Figure 8 shows a rhombic dodecahedron - a polyhedron whose surface consists of twelve equal rhombuses. The garnet crystal has the shape of a rhombic dodecahedron. Therefore it is also called garnetohedron.

Rice. 8

A rhombododecahedron can be obtained from two cubes as follows. Let's cut one of the cubes into six equal regular quadrangular pyramids with vertices in the center of the cube, the bases of which are the faces of the cube. Let us place each such pyramid with its base on the face of an uncut cube. We obtain a rhombic dodecahedron (Fig. 9).

Rice. 9

Let's now start creating the parquet. Let's consider a spatial parquet made of cubes painted black and white colors in a checkerboard pattern so that only black and white cubes touch on their faces. Let's divide the white cubes into regular quadrangular pyramids and attach them to the adjacent black cubes. As a result, we obtain the desired spatial parquet of rhombic dodecahedrons.

Using a rhombic dodecahedron, we give an example of another polyhedron from which a spatial parquet can be made.

Rice. 10

Let us cut the rhombic dodecahedron with a plane passing through the center of the cube inscribed in it, parallel to one of the faces of the cube. The cross section will be square ABCD with a side equal to the diagonal of the cube face (Fig. 10, A). Let us insert a regular quadrangular prism between the two halves of the rhombic dodecahedron. We obtain a polyhedron whose surface consists of twelve faces: eight rhombuses and four hexagons (Fig. 10, b).

Let us show that from such dodecahedrons it is possible to create a spatial parquet. To do this, we cut the parquet of rhombic dodecahedrons into planes passing through the centers of the black cubes and parallel to one selected face of the black cube. At the intersection of each such plane with rhombic dodecahedrons, a flat parquet of squares is formed. In each cut we insert regular quadrangular prisms, the bases of which are squares of flat parquet. As a result, we obtain the desired spatial parquet.

Let us give an example of another polyhedron from which a spatial parquet can be made. It is called a truncated octahedron and is obtained from an octahedron by cutting off regular quadrangular pyramids from its vertices, the side edges of which are equal to one third of the edge of the given octahedron (Fig. 11, A). The faces of the truncated octahedron are six squares and eight regular hexagons (Fig. 11, b).

Rice. 11

Cut the truncated octahedron into eight equal parts planes passing through pairs of opposite edges of the octahedron (Fig. 12).

Rice. 12

Each such part is a half of a cube, obtained by cutting the cube along a plane that gives a regular hexagon in the section of the cube.

If we take two equal truncated octahedrons, cut one of them into eight equal parts and attach these parts to the hexagonal faces of the uncut truncated octahedron, we get a cube.

Let us consider a spatial parquet consisting of cubes with truncated octahedra inscribed in them. These truncated octahedra do not fill all the space. There are empty spaces between them. However, these voids are located around the vertices of the cubes and represent a union of eighths of truncated octahedra and are therefore themselves truncated octahedra. Thus, the entire space turns out to be divided into truncated octahedra, and any two such truncated octahedra are obtained from each other by parallel transfer.

Note that in five of the spatial parquets discussed above, the polyhedra are located parallel to each other. These are parquets made of hexagonal prisms, cubes (parallelepipeds), rhombic dodecahedrons, dodecahedrons obtained from the rhombic dodecahedron by adding regular quadrangular prisms and truncated octahedra.

Such convex polyhedra, from which a spatial parquet can be composed so that any two polyhedra from this parquet can be obtained from each other by parallel translation, are called parallelohedra. Domestic mathematician and crystallographer E.S. Fedorov (1853–1919) proved that there are only five types of parallelohedrons: a cube (parallelepiped), a regular hexagonal prism, a truncated octahedron, a rhombic dodecahedron, and a dodecahedron derived from a rhombic dodecahedron (see).

Let us give examples of spatial parquets composed of several different polyhedra.

Figure 13 shows a polyhedron called a truncated cube. Its faces are regular triangles and octagons. It is obtained from a cube by cutting off regular triangular pyramids from its vertices. Direct calculations show that for a unit cube the lateral edges of these pyramids should be equal . If in a spatial parquet of cubes you replace the cubes with truncated cubes, then between them there will be voids in the form of octahedrons. Thus, truncated cubes and octahedra form a spatial parquet.

Rice. 13

Figure 14 shows a polyhedron called a cuboctahedron. Its faces are six squares (like a cube) and eight regular triangles (like an octahedron). It is obtained from a cube by cutting off regular triangular pyramids from its vertices, the side edges of which are equal to half the edge of the cube. If in a spatial parquet of cubes you replace the cubes with cuboctahedra, then between them there will remain voids in the form of octahedra. Thus, cuboctahedra and octahedra form a spatial parquet.

Rice. 14

Let's consider a spatial parquet consisting of cubes with regular tetrahedrons inscribed in them (Fig. 15).

Rice. 15

These tetrahedrons do not fill all the space. There are empty spaces between them. However, these voids are located around the vertices of the cubes and represent a union of eighths of octahedra and are therefore octahedra themselves. Thus, we have a spatial parquet composed of regular tetrahedra and octahedra.

Figure 16 shows a polyhedron called a rhombicuboctahedron. Its faces are squares and regular triangles. It is obtained from a unit cube as follows. Let us move the faces of the cube in the direction from its center to a distance equal to . The vertices of these faces will serve as the vertices of the desired rhombicuboctahedron. We will fill the space with rhombicuboctahedra, combining their faces obtained from the faces of the cube. We will place cubes on the remaining square faces of the rhombocuboctahedra, and we will place cuboctahedra on the triangular faces. As a result, we obtain a spatial parquet composed of rhombicuboctahedra, cubes and cuboctahedra.

Figure 17 shows a polyhedron called a truncated cuboctahedron. Its faces are regular octagons, hexagons and squares. It is obtained from a truncated cube as follows. Let us move the octagonal faces of a truncated cube, whose edges are equal to 1, in the direction from its center to a distance equal to . The vertices of these faces will serve as the vertices of the desired truncated cuboctahedron.

We will fill the space with truncated cuboctahedra, combining their faces obtained from the octagonal faces of a truncated cube, so that the hexagonal faces of one truncated cuboctahedron adjoin the square faces of another cuboctahedron. The voids between these truncated cuboctahedra will have the shape of truncated octahedra. Thus, these truncated cuboctahedra and truncated octahedra will form a spatial parquet.

In conclusion, we offer exercises for independent solution.

Exercises

1. Is it possible to create a spatial parquet from arbitrary:

a) triangular prism;

b) quadrangular prism;

c) a hexagonal prism?

2. Is it possible to make a parquet from some pentagonal prism?

3. Find the dihedral angles formed by the faces:

a) truncated octahedron;

b) rhombic dodecahedron.

4. The vertices of which polyhedron are the centers of the faces of the rhombic dodecahedron?

5. Show that equal regular quadrangular and hexagonal pyramids can be used to form a spatial parquet.

6. Find the dihedral angles of the tetrahedrons from which a spatial parquet can be made.

7. Is it possible to make a spatial parquet from a spatial cross - a polyhedron obtained by combining seven cubes (Fig. 18).

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8. Figure 19 shows a polyhedron called a stellated octahedron, resulting from a continuation of the faces of the octahedron. It was discovered by Leonardo da Vinci, then, almost a hundred years later, rediscovered by I. Kepler and named by him Stella octangula- octagonal star. What regular polyhedron needs to be added to it so that they can be used to create a spatial parquet?

Rice. 19

9. The dual to a spatial parquet consisting of polyhedra having a center of symmetry is a spatial parquet of polyhedra whose vertices are the centers of the polyhedra of this parquet. What spatial parquets are dual to parquet: a) made of cubes; b) regular triangular prisms; c) regular hexagonal prisms?

10. Find spatial parquets dual to parquets:

a) from truncated octahedra;

b) rhombic dodecahedrons;

c) dodecahedrons obtained from rhombic dodecahedrons?

Literature

1. Bonchkovsky R.N. Filling space with tetrahedrons // Mathematical education, 1935, No. 4, p. 26-40. (Available on the website www.mccme.ru)
2. Delaunay B., Zhitomirsky O. Problem book on geometry. - M.–L.: State. ed. technical-theoret. literature, 1950. (Available on the website www.mccme.ru)
3. Smirnova I.M., Smirnov V.A. Geometry. Textbook for grades 10–11 in general education institutions. - M.: Mnemosyne, 2006.

Since the times of the ancient Greeks, five Platonic solids have been known - regular polyhedra, differing highest degree symmetry. These are the tetrahedron, cube, octahedron, icosahedron and dodecahedron, they are shown in Fig. 1.

It is easy to fill the entire space with identical cubes without voids or overlaps so that any two adjacent cubes intersect either at a vertex, or along an edge, or along a face (Fig. 2).

Task

A) Prove that other Platonic solids do not allow such filling of space.

b) Come up with, how to fill the space if you can use different Platonic solids.

Hint 1

Suppose there is some filling of space with Platonic solids (not necessarily identical). Let's consider the edge of one of them. Then the sum of the dihedral angles of the polyhedra adjacent to this edge is 360°.

Hint 2

Show that the dihedral angles of a tetrahedron, octahedron, icosahedron and dodecahedron are equal to , , and accordingly.

Hint 3

Show that space can be filled with tetrahedra and octahedra.

Solution

Let's first look at filling space with cubes to understand how it is achieved. Let AB- edge of one of the cubes (Fig. 3.). Then it is the edge of three more cubes. In order for the space to be filled without voids, the sum of the dihedral angles whose edge is AB, should be 2 π . Since the dihedral angle of a cube is π /2, then the sum of four such angles is exactly what we need.

Thus, in order for any Platonic solid to pave space in the manner specified in the problem statement, it is necessary that the dihedral angle of this Platonic solid has the form 2 π /n, Where n- some natural number greater than two.

Now let's find the dihedral angles of all other Platonic solids. Making sure that none of them can be represented as 2 π /n, we will prove point a). Let's start with the tetrahedron.

We will assume that all sides of the tetrahedron ABCD are equal to 1. Let M- middle of the side B.C., D.H.- height (Fig. 4). Then point H is the center of the face ABC, which means it lies on the segment A.M. and divides it in the ratio 2: 1, counting from the point A. Considering that A.M. = DM, it follows that cos . That is, the dihedral angle of the tetrahedron is equal to .

Next, consider the octahedron ABCDEF(Fig. 5). As with the tetrahedron, we will assume that the length of each side of the octahedron is 1. Let M be the midpoint of the side B.F., A.H.- perpendicular dropped onto a plane BCF from point A, H 1 and H 2 - centers of faces BCF And ADE respectively. Then A.M. = C.M., AHH 1 H 2 is a rectangle, and . Besides, . Hence, And . Thus, the dihedral angle of the octahedron is equal to .

Before moving on to the icosahedron and dodecahedron, we should take a closer look at the regular pentagon. Let in a regular pentagon PQRST diagonals PS And QT intersect at a point K(Fig. 6). Since each angle of a regular pentagon is 3 π /5, then the angles at the bases of isosceles triangles PST And QTP equal π /5. This means that the angles at the bases of isosceles triangles KPQ And KTS equal 2 π /5; in particular, this means that any diagonal of a regular pentagon divides it into an isosceles triangle and a trapezoid.

Let's draw in a triangle KPQ bisector P.M.. Then it's easy to see that KPM = π /5 and PKM = PMK = 2π /5. From this we conclude that isosceles triangles KTS And KPM similar. This fact allows us to express all the elements of the pentagon PQRST through the length of its side.

Indeed, for simplicity we will assume that PQ= 1. Then ST = KQ= 1. Let us denote KT through x. Then PK = P.M. = MQ = x, K.M. = 1 – x. Hence, . Transforming this equality, we obtain the relation x 2 + x– 1 = 0, from where we find .

Now it's easy to find different elements. So, for us it will matter that the length of the diagonal of a regular pentagon with side 1 is . Another important point- values of trigonometric functions at points π /5 and 2 π /5. For example,

Besides, - this is the side of a regular pentagon, which is cut out if we draw in the pentagon PQRST all diagonals.

Let us finally move on to the icosahedron. In order to find its dihedral angle, it will be enough for us to consider the “cap” of the icosahedron - a regular pentagonal pyramid ABCDEF. Let M- middle of the side A.C.(Fig. 7). Then, assuming that all sides of the pyramid are equal to 1, we easily obtain , . According to the cosine theorem, BD 2 = B.M. 2 + DM 2 – 2 · B.M. · DM cos BMD. Means,

Thus, the dihedral angle of the icosahedron is equal to .

Let's move on to the dodecahedron. As with all other Platonic solids, we will assume that the length of each of its edges is equal to 1. Let us introduce the notation as indicated in Fig. 8. Let M be the midpoint of the side B.C.. Then the required angle EMG can be found by applying the cosine theorem for an isosceles triangle EMG. All that remains is to find the sides of this triangle.

The sides of the EMG triangle are easy to find. Really,

In order to find E.G., consider the section of the dodecahedron by the plane DEG(Fig. 9). This plane carves a hexagon from the dodecahedron DEKLGH, who has DE = KL = G.H.= 1 and HD = E.K. = G.L.= (as the diagonal of a regular pentagon with side 1). From symmetry considerations it is clear that the hexagon DEKLGH inscribed in a circle, and DEG = EGH = KHG = π /3. It follows that straight lines DE And H.K. are parallel, and the triangle H.G.I., Where I- point of intersection E.G. And KH, equilateral. Means, G.I. = G.H.= 1, a EI = D.H.= . Thus, we get E.G. = G.I. + EI = .

Let's return to the dihedral angle of the dodecahedron. As follows from the cosine theorem for a triangle EMG, EG 2 = EM 2 + GM 2– 2 · E.M. · GM cos EMG. Means,

Thus, the dihedral angle of the dodecahedron is equal to .

Now we can start analyzing the results obtained. As we already said at the very beginning, in order for copies of a certain Platonic solid to fill the entire space without a trace, it is necessary that the dihedral angle of this Platonic solid has the form 2 π /n. The values of the cosines of angles of this type, corresponding to the values n= 2, 3, 4, 5, 6 are:

Since on the interval the function cos x decreases monotonically, then to compare angles it is enough to compare the values of their cosines with each other. Let's do this.

The dihedral angle of a tetrahedron is . The following inequalities hold for cosines:< 1/3 < 1/2. Значит, 2π /5 > > 2π /6.

The dihedral angle of the octahedron is . The following inequalities hold for cosines: –1/2< –1/3 < 0. Значит, 2π /3 > > 2π /4.

The dihedral angle of the icosahedron is equal to . The following inequalities hold for cosines: –1< –√5/3 < –1/2. Значит, 2π /2 > > 2π /3.

The dihedral angle of a dodecahedron is . The following inequalities hold for cosines: –1/2< –1/√5 < 0. Значит, 2π /3 > > 2π /4.

Thus, the dihedral angles of no Platonic solid, except the cube, are angles of type 2 π /n. Point a) has been proven.

Let's move on to point b) of the problem. Consider equal octahedra ABCDEF And PQRCBS, which have a common edge BC. Then if the ribs B.C., DE And QR lie in the same plane, then the distance between the vertices A And P equal to the distance between the centers of the octahedra (Fig. 10). However, the latter is equal to the length of the edge of the octahedron. So, in a tetrahedron ABCP all sides are equal, and he is correct.

This consideration allows us to fill the space with tetrahedra and octahedra in the required manner. First, we fold them into a tetrahedral pipe (Fig. 11). This pipe in cross-section gives a rhombus. However, we can cover a plane with copies of any quadrilateral (and even more so a rhombus) without gaps or overlaps (see the “Tilings” problem). Therefore, the entire space is also easily filled with such pipes.

“Circles in a circle”) and balls in space. Despite the fact that almost all formulations sound very natural, such problems are quite complex, and in most cases they are just waiting to be solved.

From the point of view of other sciences, the problem under consideration is interesting, first of all, because the answer to it allows us to predict what the structure of the crystals of a particular substance is, how various atoms and molecules combine in order to form these crystals. It turns out that crystals are mostly arranged regularly, which makes it possible to describe them uniformly using discrete subgroups of movements space. The connection between crystals and subgroups of motions is explained as follows: for each discrete subgroup of motions of space, one can select the largest connected piece of space, no two points of which can be translated into each other by any motion from this subgroup. Generally speaking, there can be many such pieces; any of them is called fundamental area subgroups of movements. In the case where the fundamental domain is limited, the discrete subgroup of motions is called crystallographic. This name explains the nature of the bond: molecules and atoms of regularly arranged crystals can often be considered as the fundamental region of certain crystallographic groups of motions.

The number of flat crystallographic groups is 17. In three-dimensional space there are already 219 crystallographic groups. In a space of dimension 4, the number of groups is even greater: 4783. Each such group generates a certain partition of the plane or space into identical pieces. For example, the division of a plane into equal squares, the sides of which are equal to 1 (checkered paper), is generated by a crystallographic group consisting of parallel transfers to all possible vectors of the form ( m, n), Where m And n— integers, as well as rotations by angles π /4, π /2 and 3 π /4 relative to the centers and vertices of the squares. A similar crystallographic group generates the filling of space with cubes. The regular filling of space with tetrahedra and octahedra also corresponds to a crystallographic group - it consists of all such movements that transfer the filling into itself. However, neither the octahedron nor the tetrahedron will be its fundamental region.

Working with empty space: why emptiness influences design so much. Filling the space with polyhedrons Doodling paradoxically does not distract, but helps to retain the attention of the drawer

Ignorance of one's own boundaries

Ideas for a children's room

Reading corner

Place to rest

Home office

Dining area

Window in the kitchen

In the bathroom

In the dressing room

Storage systems in the bedroom

Ideas for a children's room

Reading corner

Place to rest

Home office

Dining area

Window in the kitchen

In the bathroom

In the dressing room

Storage systems in the bedroom

Task

Hint 1

Hint 2

Hint 3

Solution

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