The evolution of the human brain took place in three-dimensional space. Therefore, it is difficult for us to imagine spaces with dimensions greater than three. In fact, the human brain cannot imagine geometric objects with dimensions greater than three. And at the same time, we can easily imagine geometric objects with dimensions not only three, but also with dimensions two and one.

The difference and analogy between one-dimensional and two-dimensional spaces, as well as the difference and analogy between two-dimensional and three-dimensional spaces allow us to slightly open the screen of mystery that fences us off from spaces of higher dimensions. To understand how this analogy is used, consider a very simple four-dimensional object - a hypercube, that is, a four-dimensional cube. To be specific, let’s say we want to solve a specific problem, namely, count the number of square faces of a four-dimensional cube. All further consideration will be very lax, without any evidence, purely by analogy.

To understand how a hypercube is built from a regular cube, you must first look at how a regular cube is built from a regular square. For the sake of originality in the presentation of this material, we will here call an ordinary square a SubCube (and will not confuse it with a succubus).

To build a cube from a subcube, you need to extend the subcube in the direction perpendicular to the plane subcube in the direction of the third dimension. In this case, from each side of the initial subcube a subcube will grow, which is the side two-dimensional face of the cube, which will limit the three-dimensional volume of the cube on four sides, two perpendicular to each direction in the plane of the subcube. And along the new third axis there are also two subcubes that limit the three-dimensional volume of the cube. This is the two-dimensional face where our subcube was originally located and that two-dimensional face of the cube where the subcube ended up at the end of the cube's construction.

What you have just read is presented in excessive detail and with a lot of clarifications. And for good reason. Now we will do such a trick, we will formally replace some words in the previous text in this way:
cube -> hypercube
subcube -> cube
plane -> volume
third -> fourth
two-dimensional -> three-dimensional
four -> six
three-dimensional -> four-dimensional
two -> three
plane -> space

As a result, we get the following meaningful text, which no longer seems overly detailed.

To build a hypercube from a cube, you need to stretch the cube in a direction perpendicular to the volume of the cube in the direction of the fourth dimension. In this case, a cube will grow from each side of the original cube, which is the lateral three-dimensional face of the hypercube, which will limit the four-dimensional volume of the hypercube on six sides, three perpendicular to each direction in the space of the cube. And along the new fourth axis there are also two cubes that limit the four-dimensional volume of the hypercube. This is the three-dimensional face where our cube was originally located and the three-dimensional face of the hypercube where the cube came at the end of the construction of the hypercube.

Why are we so confident that we have received the correct description of the construction of a hypercube? Yes, because by exactly the same formal substitution of words we get a description of the construction of a cube from a description of the construction of a square. (Check it out for yourself.)

Now it is clear that if another three-dimensional cube should grow from each side of the cube, then a face should grow from each edge of the initial cube. In total, the cube has 12 edges, which means that an additional 12 new faces (subcubes) will appear on those 6 cubes that limit the four-dimensional volume along the three axes of three-dimensional space. And there are two more cubes left that limit this four-dimensional volume from below and above along the fourth axis. Each of these cubes has 6 faces.

In total, we find that the hypercube has 12+6+6=24 square faces.

The following picture shows the logical structure of a hypercube. This is like a projection of a hypercube onto three-dimensional space. This produces a three-dimensional frame of ribs. In the figure, naturally, you see the projection of this frame onto a plane.

On this frame, the inner cube is like the initial cube from which the construction began and which limits the four-dimensional volume of the hypercube along the fourth axis from the bottom. We stretch this initial cube upward along the fourth axis of measurement and it goes into the outer cube. So the outer and inner cubes from this figure limit the hypercube along the fourth axis of measurement.

And between these two cubes you can see 6 more new cubes, which touch common faces with the first two. These six cubes bound our hypercube along the three axes of three-dimensional space. As you can see, they are not only in contact with the first two cubes, which are the inner and outer cubes on this three-dimensional frame, but they are also in contact with each other.

You can count directly in the figure and make sure that the hypercube really has 24 faces. But this question arises. This hypercube frame in three-dimensional space is filled with eight three-dimensional cubes without any gaps. To make a real hypercube from this three-dimensional projection of a hypercube, you need to turn this frame inside out so that all 8 cubes bound a 4-dimensional volume.

It's done like this. We invite a resident of four-dimensional space to visit us and ask him to help us. It grabs the inner cube of this frame and moves it in the direction of the fourth dimension, which is perpendicular to our three-dimensional space. In our three-dimensional space, we perceive it as if the entire internal frame had disappeared and only the frame of the outer cube remained.

Further, our four-dimensional assistant offers his assistance in maternity hospitals for painless childbirth, but our pregnant women are frightened by the prospect that the baby will simply disappear from the stomach and end up in parallel three-dimensional space. Therefore, the four-dimensional person is politely refused.

And we are puzzled by the question of whether some of our cubes came apart when we turned the hypercube frame inside out. After all, if some three-dimensional cubes surrounding a hypercube touch their neighbors on the frame with their faces, will they also touch with the same faces if the four-dimensional cube turns the frame inside out?

Let us again turn to the analogy with spaces of lower dimensions. Compare the image of the hypercube frame with the projection of a three-dimensional cube onto a plane shown in the following picture.

The inhabitants of two-dimensional space built on a plane a frame for the projection of a cube onto a plane and invited us, three-dimensional inhabitants, to turn this frame inside out. We take the four vertices of the inner square and move them perpendicular to the plane. Two-dimensional residents see the complete disappearance of the entire internal frame, and they are left with only the frame of the outer square. With such an operation, all the squares that were in contact with their edges continue to touch with the same edges.

Therefore, we hope that the logical scheme of the hypercube will also not be violated when turning the frame of the hypercube inside out, and the number of square faces of the hypercube will not increase and will still be equal to 24. This, of course, is not proof at all, but purely a guess by analogy .

After everything you've read here, you can easily draw the logical framework of a five-dimensional cube and calculate the number of vertices, edges, faces, cubes and hypercubes it has. It's not difficult at all.

As soon as I was able to give lectures after the operation, the first question the students asked was:

When will you draw us a 4-dimensional cube? Ilyas Abdulkhaevich promised us!

I remember that my dear friends sometimes like a moment of mathematical educational activities. Therefore, I will write a part of my lecture for mathematicians here. And I will try without being boring. At some points I read the lecture more strictly, of course.

Let's agree first. 4-dimensional, and even more so 5-6-7- and generally k-dimensional space is not given to us in sensory sensations.
“We are wretched because we are only three-dimensional,” as my Sunday school teacher, who first told me what a 4-dimensional cube is, said. Sunday school was, naturally, extremely religious - mathematical. That time we were studying hyper-cubes. A week before this, mathematical induction, a week after that, Hamiltonian cycles in graphs - accordingly, this is grade 7.

We cannot touch, smell, hear or see a 4-dimensional cube. What can we do with it? We can imagine it! Because our brain is much more complex than our eyes and hands.

So, in order to understand what a 4-dimensional cube is, let's first understand what is available to us. What is a 3-dimensional cube?

Okay, okay! I'm not asking you for a clear mathematical definition. Just imagine the simplest and most ordinary three-dimensional cube. Introduced?

Fine.
In order to understand how to generalize a 3-dimensional cube into a 4-dimensional space, let's figure out what a 2-dimensional cube is. It's so simple - it's a square!

A square has 2 coordinates. The cube has three. Square points are points with two coordinates. The first is from 0 to 1. And the second is from 0 to 1. The points of the cube have three coordinates. And each is any number from 0 to 1.

It is logical to imagine that a 4-dimensional cube is a thing that has 4 coordinates and everything is from 0 to 1.

/* It’s immediately logical to imagine a 1-dimensional cube, which is nothing more than a simple segment from 0 to 1. */

So, wait, how do you draw a 4-dimensional cube? After all, we cannot draw 4-dimensional space on a plane!
But we don’t draw 3-dimensional space on a plane either, we draw it projection onto a 2-dimensional drawing plane. We place the third coordinate (z) at an angle, imagining that the axis from the drawing plane goes “towards us”.

Now it is completely clear how to draw a 4-dimensional cube. In the same way that we positioned the third axis at a certain angle, let’s take the fourth axis and also position it at a certain angle.
And - voila! -- projection of a 4-dimensional cube onto a plane.

What? What is this anyway? I always hear whispers from the back desks. Let me explain in more detail what this jumble of lines is.
Look first at the three-dimensional cube. What have we done? We took the square and dragged it along the third axis (z). It's like many, many paper squares glued together in a stack.
It's the same with a 4-dimensional cube. Let's call the fourth axis, for convenience and for science fiction, the “time axis.” We need to take an ordinary three-dimensional cube and drag it through time from the time “now” to the time “in an hour.”

We have a "now" cube. In the picture it is pink.

And now we drag it along the fourth axis - along the time axis (I showed it in green). And we get the cube of the future - blue.

Each vertex of the “cube now” leaves a trace in time - a segment. Connecting her present with her future.

In short, without any lyrics: we drew two identical 3-dimensional cubes and connected the corresponding vertices.
Exactly as they did with a 3-dimensional cube (draw 2 identical 2-dimensional cubes and connect the vertices).

To draw a 5-dimensional cube, you will have to draw two copies of a 4-dimensional cube (a 4-dimensional cube with fifth coordinate 0 and a 4-dimensional cube with fifth coordinate 1) and connect the corresponding vertices with edges. True, there will be such a jumble of edges on the plane that it will be almost impossible to understand anything.

Once we have imagined a 4-dimensional cube and even been able to draw it, we can explore it in different ways. Remembering to explore it both in your mind and from the picture.
For example. A 2-dimensional cube is bounded on 4 sides by 1-dimensional cubes. This is logical: for each of the 2 coordinates it has both a beginning and an end.
A 3-dimensional cube is bounded on 6 sides by 2-dimensional cubes. For each of the three coordinates it has a beginning and an end.
This means that a 4-dimensional cube must be limited by eight 3-dimensional cubes. For each of the 4 coordinates - on both sides. In the figure above we clearly see 2 faces that limit it along the “time” coordinate.

Here are two cubes (they are slightly oblique because they have 2 dimensions projected onto the plane at an angle), limiting our hypercube on the left and right.

It is also easy to notice “upper” and “lower”.

The most difficult thing is to understand visually where “front” and “rear” are. The front one starts from the front edge of the “cube now” and to the front edge of the “cube of the future” - it is red. The rear one is purple.

They are the most difficult to notice because other cubes are tangled underfoot, which limit the hypercube at a different projected coordinate. But note that the cubes are still different! Here is the picture again, where the “cube of now” and the “cube of the future” are highlighted.

Of course, it is possible to project a 4-dimensional cube into 3-dimensional space.
The first possible spatial model is clear what it looks like: you need to take 2 cube frames and connect their corresponding vertices with a new edge.
I don't have this model in stock right now. At the lecture, I show students a slightly different 3-dimensional model of a 4-dimensional cube.

You know how a cube is projected onto a plane like this.
It's like we're looking at a cube from above.

The near edge is, of course, large. And the far edge looks smaller, we see it through the near one.

This is how you can project a 4-dimensional cube. The cube is larger now, we see the cube of the future in the distance, so it looks smaller.

On the other side. From the top side.

Directly exactly from the side of the edge:

From the rib side:

And the last angle, asymmetrical. From the section “tell me that I looked between his ribs.”

Well, then you can come up with anything. For example, just as there is a development of a 3-dimensional cube onto a plane (it’s like cutting out a sheet of paper so that when folded you get a cube), the same happens with the development of a 4-dimensional cube into space. It's like cutting a piece of wood so that by folding it in 4-dimensional space we get a tesseract.

You can study not just a 4-dimensional cube, but n-dimensional cubes in general. For example, is it true that the radius of a sphere circumscribed around n-dimensional cube less than the length of the edge of this cube? Or here’s a simpler question: how many vertices does an n-dimensional cube have? How many edges (1-dimensional faces)?

Bakalyar Maria

Methods for introducing the concept of a four-dimensional cube (tesseract), its structure and some properties are studied. The question of what three-dimensional objects are obtained when a four-dimensional cube is intersected by hyperplanes parallel to its three-dimensional faces, as well as hyperplanes perpendicular to its main diagonal is addressed. The multidimensional apparatus used for research is considered. analytical geometry.

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Introduction……………………………………………………………………………….2

Main part……………………………………………………………..4

Conclusions………….. ………………………………………………………..12

References………………………………………………………..13

Introduction

Four-dimensional space has long attracted the attention of both professional mathematicians and people far from studying this science. Interest in the fourth dimension may be due to the assumption that our three-dimensional world is “immersed” in four-dimensional space, just as a plane is “immersed” in three-dimensional space, the straight line is “immersed” in the plane, and the point is in the straight line. In addition, four-dimensional space plays an important role in modern theory relativity (the so-called space-time or Minkowski space), and can also be considered as a special casedimensional Euclidean space (with).

A four-dimensional cube (tesseract) is an object in four-dimensional space that has the maximum possible dimension (just as an ordinary cube is an object in three-dimensional space). Note that it is also of direct interest, namely, it can appear in optimization problems of linear programming (as an area in which the minimum or maximum is sought linear function four variables), and is also used in digital microelectronics (when programming the operation of an electronic watch display). In addition, the process of studying a four-dimensional cube itself contributes to the development spatial thinking and imagination.

Consequently, the study of the structure and specific properties four-dimensional cube is quite relevant. It is worth noting that in terms of structure, the four-dimensional cube has been studied quite well. Much more interesting is the nature of its sections by various hyperplanes. Thus, the main goal of this work is to study the structure of the tesseract, as well as to clarify the question of what three-dimensional objects will be obtained if a four-dimensional cube is dissected by hyperplanes parallel to one of its three-dimensional faces, or by hyperplanes perpendicular to its main diagonal. A hyperplane in four-dimensional space will be called a three-dimensional subspace. We can say that a straight line on a plane is a one-dimensional hyperplane, a plane in three-dimensional space is a two-dimensional hyperplane.

The goal determined the objectives of the study:

1) Study the basic facts of multidimensional analytical geometry;

2) Study the features of constructing cubes of dimensions from 0 to 3;

3) Study the structure of a four-dimensional cube;

4) Analytically and geometrically describe a four-dimensional cube;

5) Make models of developments and central projections of three-dimensional and four-dimensional cubes.

6) Using the apparatus of multidimensional analytical geometry, describe three-dimensional objects resulting from the intersection of a four-dimensional cube with hyperplanes parallel to one of its three-dimensional faces, or hyperplanes perpendicular to its main diagonal.

The information obtained in this way will allow us to better understand the structure of the tesseract, as well as to identify deep analogies in the structure and properties of cubes of different dimensions.

Main part

First we will describe mathematical apparatus, which we will use during this study.

1) Vector coordinates: if, That

2) Equation of a hyperplane with a normal vector looks like Here

3) Planes and are parallel if and only if

4) The distance between two points is determined as follows: if, That

5) Condition for orthogonality of vectors:

First of all, let's find out how to describe a four-dimensional cube. This can be done in two ways - geometric and analytical.

If we talk about the geometric method of specifying, then it is advisable to trace the process of constructing cubes, starting from zero dimension. A cube of zero dimension is a point (note, by the way, that a point can also play the role of a ball of zero dimension). Next, we introduce the first dimension (the x-axis) and on the corresponding axis we mark two points (two zero-dimensional cubes) located at a distance of 1 from each other. The result is a segment - a one-dimensional cube. Let us immediately note characteristic feature: The boundary (ends) of a one-dimensional cube (segment) are two zero-dimensional cubes (two points). Next, we introduce the second dimension (ordinate axis) and on the planelet's construct two one-dimensional cubes (two segments), the ends of which are at a distance of 1 from each other (in fact, one of the segments is orthogonal projection another). By connecting the corresponding ends of the segments, we obtain a square - a two-dimensional cube. Again, note that the boundary of a two-dimensional cube (square) is four one-dimensional cubes (four segments). Finally, we introduce the third dimension (applicate axis) and construct in spacetwo squares in such a way that one of them is an orthogonal projection of the other (the corresponding vertices of the squares are at a distance of 1 from each other). Let's connect the corresponding vertices with segments - we get a three-dimensional cube. We see that the boundary of a three-dimensional cube is six two-dimensional cubes (six squares). The described constructions allow us to identify the following pattern: at each stepthe dimensional cube “moves, leaving a trace” ine measurement at a distance of 1, while the direction of movement is perpendicular to the cube. It is the formal continuation of this process that allows us to arrive at the concept of a four-dimensional cube. Namely, we will force the three-dimensional cube to move in the direction of the fourth dimension (perpendicular to the cube) by a distance of 1. Acting similarly to the previous one, that is, by connecting the corresponding vertices of the cubes, we will obtain a four-dimensional cube. It should be noted that geometrically such a construction in our space is impossible (since it is three-dimensional), but here we do not encounter any contradictions from a logical point of view. Now let's move on to the analytical description of a four-dimensional cube. It is also obtained formally, using analogy. So, the analytical specification of a zero-dimensional unit cube has the form:

The analytical task of a one-dimensional unit cube has the form:

The analytical task of a two-dimensional unit cube has the form:

The analytical task of a three-dimensional unit cube has the form:

Now it is very easy to give an analytical representation of a four-dimensional cube, namely:

As we see, both for the geometric and for analytical methods to assign a four-dimensional cube, the method of analogies was used.

Now, using the apparatus of analytical geometry, we will find out what the structure of a four-dimensional cube is. First, let's find out what elements it includes. Here again we can use an analogy (to put forward a hypothesis). The boundaries of a one-dimensional cube are points (zero-dimensional cubes), of a two-dimensional cube - segments (one-dimensional cubes), of a three-dimensional cube - squares (two-dimensional faces). It can be assumed that the boundaries of the tesseract are three-dimensional cubes. In order to prove this, let us clarify what is meant by vertices, edges and faces. The vertices of a cube are its corner points. That is, the coordinates of the vertices can be zeros or ones. Thus, a connection is revealed between the dimension of the cube and the number of its vertices. Let us apply the combinatorial product rule - since the vertexmeasured cube has exactlycoordinates, each of which is equal to zero or one (independent of all others), then in total there ispeaks Thus, for any vertex all coordinates are fixed and can be equal to or . If we fix all the coordinates (putting each of them equal or , regardless of the others), except for one, we obtain straight lines containing the edges of the cube. Similar to the previous one, you can count that there are exactlythings. And if we now fix all the coordinates (putting each of them equal or , independently of the others), except for some two, we obtain planes containing two-dimensional faces of the cube. Using the rule of combinatorics, we find that there are exactlythings. Next, similarly - fixing all the coordinates (putting each of them equal or , regardless of the others), except for some three, we obtain hyperplanes containing three-dimensional faces of the cube. Using the same rule, we calculate their number - exactlyetc. This will be sufficient for our research. Let us apply the results obtained to the structure of a four-dimensional cube, namely, in all the derived formulas we put. Therefore, a four-dimensional cube has: 16 vertices, 32 edges, 24 two-dimensional faces, and 8 three-dimensional faces. For clarity, let us define analytically all its elements.

Vertices of a four-dimensional cube:

Edges of a four-dimensional cube ():

Two-dimensional faces of a four-dimensional cube (similar restrictions):

Three-dimensional faces of a four-dimensional cube (similar restrictions):

Now that the structure of a four-dimensional cube and the methods for defining it have been described in sufficient detail, let us proceed to the implementation of the main goal - to clarify the nature of the various sections of the cube. Let's start with the elementary case when the sections of a cube are parallel to one of its three-dimensional faces. For example, consider its sections with hyperplanes parallel to the faceFrom analytical geometry it is known that any such section will be given by the equationLet us define the corresponding sections analytically:

As we can see, we have obtained an analytical specification for a three-dimensional unit cube lying in a hyperplane

To establish an analogy, let us write the section of a three-dimensional cube by a plane We get:

This is a square lying in a plane. The analogy is obvious.

Sections of a four-dimensional cube by hyperplanesgive completely similar results. These will also be single three-dimensional cubes lying in hyperplanes respectively.

Now let's consider sections of a four-dimensional cube with hyperplanes perpendicular to its main diagonal. First, let's solve this problem for a three-dimensional cube. Using the above-described method of defining a unit three-dimensional cube, he concludes that as the main diagonal one can take, for example, a segment with ends And . This means that the vector of the main diagonal will have coordinates. Therefore, the equation of any plane perpendicular to the main diagonal will be:

Let's determine the limits of parameter change. Because , then, adding these inequalities term by term, we get:

Or .

If , then (due to restrictions). Likewise - if, That . So, when and when cutting plane and cube have exactly one common point ( And respectively). Now let's note the following. If(again due to variable limitations). The corresponding planes intersect three faces at once, because, otherwise, the cutting plane would be parallel to one of them, which does not take place according to the condition. If, then the plane intersects all faces of the cube. If, then the plane intersects the faces. Let us present the corresponding calculations.

Let Then the planecrosses the line in a straight line, and . The edge, moreover. Edge the plane intersects in a straight line, and

Let Then the planecrosses the line:

edge in a straight line, and .

edge in a straight line, and .

edge in a straight line, and .

edge in a straight line, and .

edge in a straight line, and .

edge in a straight line, and .

This time we get six segments that have sequentially common ends:

Let Then the planecrosses the line in a straight line, and . Edge the plane intersects in a straight line, and . Edge the plane intersects in a straight line, and . That is, we get three segments that have pairwise common ends:Thus, for the specified parameter valuesthe plane will intersect the cube along regular triangle with peaks

So, here is a comprehensive description of the plane figures obtained when a cube is intersected by a plane perpendicular to its main diagonal. The main idea was as follows. It is necessary to understand which faces the plane intersects, along which sets it intersects them, and how these sets are related to each other. For example, if it turned out that the plane intersects exactly three faces along segments that have pairwise common ends, then the section was equilateral triangle(which is proven by directly counting the lengths of the segments), the vertices of which are these ends of the segments.

Using the same apparatus and the same idea of ​​studying sections, the following facts can be deduced in a completely analogous way:

1) The vector of one of the main diagonals of a four-dimensional unit cube has the coordinates

2) Any hyperplane perpendicular to the main diagonal of a four-dimensional cube can be written in the form.

3) In the equation of a secant hyperplane, the parametercan vary from 0 to 4;

4) When and a secant hyperplane and a four-dimensional cube have one common point ( And respectively);

5) When the cross section will produce a regular tetrahedron;

6) When in cross-section the result will be an octahedron;

7) When the cross section will produce a regular tetrahedron.

Accordingly, here the hyperplane intersects the tesseract along a plane on which, due to the limitations of the variables, a triangular region is allocated (an analogy - the plane intersected the cube along a straight line, on which, due to the constraints of the variables, a segment was allocated). In case 5) the hyperplane intersects exactly four three-dimensional faces of the tesseract, that is, four triangles are obtained that have pairwise common sides, in other words, forming a tetrahedron (how this can be calculated is correct). In case 6), the hyperplane intersects exactly eight three-dimensional faces of the tesseract, that is, eight triangles are obtained that have sequentially common sides, in other words, forming an octahedron. Case 7) is completely similar to case 5).

Let us illustrate this with a specific example. Namely, we study the section of a four-dimensional cube by a hyperplaneDue to variable restrictions, this hyperplane intersects the following three-dimensional faces: Edge intersects along a planeDue to the limitations of the variables, we have:We get a triangular area with verticesNext,we get a triangleWhen a hyperplane intersects a facewe get a triangleWhen a hyperplane intersects a facewe get a triangleThus, the vertices of the tetrahedron have the following coordinates. As is easy to calculate, this tetrahedron is indeed regular.

Conclusions

So, in the process of this research, the basic facts of multidimensional analytical geometry were studied, the features of constructing cubes of dimensions from 0 to 3 were studied, the structure of a four-dimensional cube was studied, a four-dimensional cube was analytically and geometrically described, models of developments and central projections of three-dimensional and four-dimensional cubes were made, three-dimensional cubes were analytically described objects resulting from the intersection of a four-dimensional cube with hyperplanes parallel to one of its three-dimensional faces, or with hyperplanes perpendicular to its main diagonal.

The conducted research made it possible to identify deep analogies in the structure and properties of cubes of different dimensions. The analogy technique used can be applied in research, for example,dimensional sphere ordimensional simplex. Namely,a dimensional sphere can be defined as a set of pointsdimensional space equidistant from given point, which is called the center of the sphere. Next,a dimensional simplex can be defined as a partdimensional space limited by the minimum numberdimensional hyperplanes. For example, a one-dimensional simplex is a segment (a part of one-dimensional space, limited by two points), a two-dimensional simplex is a triangle (a part of two-dimensional space, limited by three lines), a three-dimensional simplex is a tetrahedron (a part of three-dimensional space, limited by four planes). Finally,we define the dimensional simplex as the partdimensional space, limitedhyperplane of dimension.

Note that, despite the numerous applications of the tesseract in some areas of science, this study is still largely a mathematical investigation.

References

1) Bugrov Ya.S., Nikolsky S.M.Higher mathematics, vol. 1 – M.: Bustard, 2005 – 284 p.

2) Quantum. Four-dimensional cube / Duzhin S., Rubtsov V., No. 6, 1986.

3) Quantum. How to draw dimensional cube / Demidovich N.B., No. 8, 1974.

Hypercube and Platonic solids

Model a truncated icosahedron (“soccer ball”) in the “Vector” system
in which each pentagon is bounded by hexagons

Truncated icosahedron can be obtained by cutting off 12 vertices to form faces in the form regular pentagons. In this case, the number of vertices of the new polyhedron increases 5 times (12×5=60), 20 triangular faces turn into regular hexagons (in total faces become 20+12=32), A the number of edges increases to 30+12×5=90.

Steps for constructing a truncated icosahedron in the Vector system

Figures in 4-dimensional space.

--à

--à ?

For example, given a cube and a hypercube. A hypercube has 24 faces. This means that a 4-dimensional octahedron will have 24 vertices. Although no, a hypercube has 8 faces of cubes - each has a center at its vertex. This means that a 4-dimensional octahedron will have 8 vertices, which is even lighter.

4-dimensional octahedron. It consists of eight equilateral and equal tetrahedra,
connected by four at each vertex.

Rice. An attempt to simulate
hyperball-hypersphere in the “Vector” system

Front - back faces - balls without distortion. Another six balls can be defined through ellipsoids or quadratic surfaces (through 4 contour lines as generators) or through faces (first defined through generators).

More techniques to “build” a hypersphere
- the same “soccer ball” in 4-dimensional space

Appendix 2

For convex polyhedra, there is a property that relates the number of its vertices, edges and faces, proved in 1752 by Leonhard Euler, and called Euler's theorem.

Before formulating it, consider the polyhedra known to us and fill out the following table, in which B is the number of vertices, P - edges and G - faces of a given polyhedron:

From this table it is immediately clear that for all selected polyhedra the equality B - P + G = 2 holds. It turns out that this equality is true not only for these polyhedra, but also for an arbitrary convex polyhedron.

Euler's theorem. For any convex polyhedron the equality holds

B - P + G = 2,

where B is the number of vertices, P is the number of edges and G is the number of faces of a given polyhedron.

Proof. To prove this equality, let us imagine the surface of this polyhedron made of an elastic material. Let's remove (cut out) one of its faces and stretch the remaining surface onto a plane. We obtain a polygon (formed by the edges of the removed face of the polyhedron), divided into smaller polygons (formed by the remaining faces of the polyhedron).

Note that polygons can be deformed, enlarged, reduced, or even curved their sides, as long as there are no gaps in the sides. The number of vertices, edges and faces will not change.

Let us prove that the resulting partition of the polygon into smaller polygons satisfies the equality

(*)B - P + G " = 1,

where B – total number vertices, P is the total number of edges and Г " is the number of polygons included in the partition. It is clear that Г " = Г - 1, where Г is the number of faces of a given polyhedron.

Let us prove that equality (*) does not change if a diagonal is drawn in some polygon of a given partition (Fig. 5, a). Indeed, after drawing such a diagonal, the new partition will have B vertices, P+1 edges and the number of polygons will increase by one. Therefore, we have

B - (P + 1) + (G "+1) = B – P + G " .

Using this property, we draw diagonals that split the incoming polygons into triangles, and for the resulting partition we show the feasibility of equality (*) (Fig. 5, b). To do this, we will sequentially remove external edges, reducing the number of triangles. In this case, two cases are possible:

a) to remove a triangle ABC it is necessary to remove two ribs, in our case AB And B.C.;

b) to remove a triangleMKNit is necessary to remove one edge, in our caseMN.

In both cases, equality (*) will not change. For example, in the first case, after removing the triangle, the graph will consist of B - 1 vertices, P - 2 edges and G " - 1 polygon:

(B - 1) - (P + 2) + (G " – 1) = B – P + G ".

Consider the second case yourself.

Thus, removing one triangle does not change the equality (*). Continuing this process of removing triangles, we will eventually arrive at a partition consisting of a single triangle. For such a partition, B = 3, P = 3, Г " = 1 and, therefore, B – Р + Г " = 1. This means that equality (*) also holds for the original partition, from which we finally obtain that for this partition of the polygon equality (*) is true. Thus, for the original convex polyhedron the equality B - P + G = 2 is true.

An example of a polyhedron for which Euler's relation does not hold, shown in Figure 6. This polyhedron has 16 vertices, 32 edges and 16 faces. Thus, for this polyhedron the equality B – P + G = 0 holds.

Appendix 3.

Film Cube 2: Hypercube is a science fiction film, a sequel to the film Cube.

Eight strangers wake up in cube-shaped rooms. The rooms are located inside a four-dimensional hypercube. Rooms are constantly moving through “quantum teleportation”, and if you climb into the next room, it is unlikely to return to the previous one. Intersect in a hypercube parallel worlds, time flows differently in some rooms, and some rooms are death traps.

The plot of the film largely repeats the story of the first part, which is also reflected in the images of some of the characters. Dies in the rooms of the hypercube Nobel laureate Rosenzweig, who calculated exact time destruction of the hypercube.

Criticism

If in the first part people imprisoned in a labyrinth tried to help each other, in this film it’s every man for himself. There are a lot of unnecessary special effects (aka traps) that in no way logically connect this part of the film with the previous one. That is, it turns out that the film Cube 2 is a kind of labyrinth of the future 2020-2030, but not 2000. In the first part, all types of traps can theoretically be created by a person. In the second part, these traps are some kind of computer program, the so-called “Virtual Reality”.

The Tesseract is a four-dimensional hypercube - a cube in four-dimensional space.
According to the Oxford Dictionary, the word tesseract was coined and used in 1888 by Charles Howard Hinton (1853-1907) in his book A New Age of Thought. Later, some people called the same figure a tetracube (Greek τετρα - four) - a four-dimensional cube.
An ordinary tesseract in Euclidean four-dimensional space is defined as a convex hull of points (±1, ±1, ±1, ±1). In other words, it can be represented as the following set:
[-1, 1]^4 = ((x_1,x_2,x_3,x_4) : -1 = The tesseract is limited by eight hyperplanes x_i= +- 1, i=1,2,3,4 , the intersection of which with the tesseract itself defines it three-dimensional faces (which are ordinary cubes) Each pair of non-parallel three-dimensional faces intersect to form two-dimensional faces (squares), and so on. Finally, the tesseract has 8 three-dimensional faces, 24 two-dimensional faces, 32 edges and 16 vertices.
Popular description
Let's try to imagine what a hypercube will look like without leaving three-dimensional space.
In a one-dimensional “space” - on a line - we select a segment AB of length L. On a two-dimensional plane at a distance L from AB, we draw a segment DC parallel to it and connect their ends. The result is a square CDBA. Repeating this operation with the plane, we obtain a three-dimensional cube CDBAGHFE. And by shifting the cube in the fourth dimension (perpendicular to the first three) by a distance L, we get the hypercube CDBAGHFEKLJIOPNM.
The one-dimensional segment AB serves as the side of the two-dimensional square CDBA, the square - as the side of the cube CDBAGHFE, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, a cube has eight. In a four-dimensional hypercube, there will thus be 16 vertices: 8 vertices of the original cube and 8 of the one shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and another 8 edges “draw” its eight vertices, which have moved to the fourth dimension. The same reasoning can be done for the faces of a hypercube. In two-dimensional space there is only one (the square itself), a cube has 6 of them (two faces from the moved square and four more that describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from its twelve edges.
Just as the sides of a square are 4 one-dimensional segments, and the sides (faces) of a cube are 6 two-dimensional squares, so for a “four-dimensional cube” (tesseract) the sides are 8 three-dimensional cubes. The spaces of opposite pairs of tesseract cubes (that is, the three-dimensional spaces to which these cubes belong) are parallel. In the figure these are the cubes: CDBAGHFE and KLJIOPNM, CDBAKLJI and GHFEOPNM, EFBAMNJI and GHDCOPLK, CKIAGOME and DLJBHPNF.
In a similar way, we can continue our reasoning for hypercubes of a larger number of dimensions, but it is much more interesting to see how a four-dimensional hypercube will look for us, residents of three-dimensional space. For this we will use the already familiar method of analogies.
Let's take the wire cube ABCDHEFG and look at it with one eye from the side of the edge. We will see and can draw two squares on the plane (its near and far edges), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic “boxes” inserted into each other and connected by eight edges. In this case, the “boxes” themselves - three-dimensional faces - will be projected onto “our” space, and the lines connecting them will stretch in the direction of the fourth axis. You can also try to imagine the cube not in projection, but in a spatial image.
Just as a three-dimensional cube is formed by a square shifted by the length of its face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in perspective will look like some rather complex figure. The four-dimensional hypercube itself consists of an infinite number of cubes, just as a three-dimensional cube can be “cut” into an infinite number of flat squares.
By cutting the six faces of a three-dimensional cube, you can decompose it into flat figure- scan. It will have a square on each side of the original face, plus one more - the face opposite to it. And the three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes “growing” from it, plus one more - the final “hyperface”.
The properties of the tesseract are an extension of the properties geometric shapes smaller dimension into four-dimensional space.