4 x dimensional figures. Four-dimensional cube

Bakalar Maria

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

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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 pursuing 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, a straight line is “immersed” in a plane, and a point is in a straight line. In addition, four-dimensional space plays an important role in the modern theory of relativity (the so-called space-time or Minkowski space), and can also be considered as a special casedimensional Euclidean space (for).

A four-dimensional cube (tesseract) is an object of four-dimensional space that has the maximum possible dimension (just like an ordinary cube is an object of 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 of a linear function of four variables is sought), and is also used in digital microelectronics (when programming the operation of an electronic clock display). In addition, the very process of studying a four-dimensional cube contributes to the development of spatial thinking and imagination.

Therefore, the study of the structure and specific properties of a four-dimensional cube is quite relevant. It is worth noting that in terms of structure, the four-dimensional cube has been studied quite well. Of much greater interest is the character 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 a four-dimensional space is 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.

This goal determined the objectives of the study:

1) Study the basic facts of multidimensional analytic geometry;

2) Study the features of building 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 sweeps and central projections of three-dimensional and four-dimensional cubes.

6) Using the apparatus of multidimensional analytic geometry, describe three-dimensional objects obtained by intersecting a four-dimensional cube by hyperplanes parallel to one of its three-dimensional faces, or by hyperplanes perpendicular to its main diagonal.

The information obtained in this way will make it possible to better understand the structure of the tesseract, as well as to reveal a deep analogy in the structure and properties of cubes of various dimensions.

Main part

First, we describe the mathematical apparatus that we will use in the course of this study.

1) Vector coordinates: if, then

2) The equation of a hyperplane with a normal vector has the form Here

3) Planes and are parallel if and only if

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

5) Orthogonality condition for vectors:

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

If we talk about the geometric method of assigning, then it is advisable to follow the process of constructing cubes, starting from zero dimension. A zero-dimensional cube is a point (note, by the way, that a point can also play the role of a zero-dimensional ball). Next, we introduce the first dimension (abscissa axis) and mark two points (two zero-dimensional cubes) on the corresponding axis, located at a distance of 1 from each other. The resulting segment is a one-dimensional cube. Let us immediately note a characteristic feature: The boundary (ends) of a one-dimensional cube (segment) are two zero-dimensional cubes (two points). Next, we introduce the second dimension (the ordinate axis) and on the planewe 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 an orthogonal projection of the other). Connecting the corresponding ends of the segments, we get a square - a two-dimensional cube. Again, note that the boundary of a two-dimensional cube (square) is four one-dimensional cubes (four line segments). Finally, we introduce the third dimension (the applicate axis) and plot in spacetwo squares in such a way that one of them is an orthogonal projection of the other (while the corresponding vertices of the squares are at a distance of 1 from each other). We 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 make it possible to reveal the following pattern: at each stepthe dimensional cube "moves, leaving a trace" ine measurement at distance 1, while the direction of movement is perpendicular to the cube. It is the formal continuation of this process that allows us to come to the concept of a four-dimensional cube. Namely, let's make the three-dimensional cube move in the direction of the fourth dimension (perpendicular to the cube) at a distance of 1. Acting similarly to the previous one, that is, connecting the corresponding vertices of the cubes, we will get a four-dimensional cube. it should be noted that geometrically such a construction is impossible in our space (because 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 the four-dimensional cube. It is also obtained formally, by analogy. So, the analytical specification of a zero-dimensional unit cube is as follows:

The analytical specification of a one-dimensional unit cube is as follows:

The analytical specification of a two-dimensional unit cube is as follows:

The analytical task of a three-dimensional unit cube is as follows:

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

As you can see, both in the geometric and in the analytical methods of defining 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 has. First, let's find out what elements are included in it. Here again you can use an analogy (to put forward a hypothesis). The boundaries of a one-dimensional cube are points (zero-dimensional cubes), a two-dimensional cube - segments (one-dimensional cubes), a three-dimensional cube - squares (two-dimensional faces). It can be assumed that the boundary of the tesseract is three-dimensional cubes. In order to prove this, let us clarify what is meant by vertices, edges and faces. We call its corner points the vertices of a cube. That is, the coordinates of the vertices can be zeros or ones. Thus, a connection is found between the dimension of the cube and the number of its vertices. We apply the combinatorial product rule - since the vertexdimensional cube has exactlycoordinates, each of which is equal to zero or one (regardless of all the others), then in total there ispeaks. Thus, at any vertex, all coordinates are fixed and can equal or ... If we fix all the coordinates (putting each of them equal or , regardless of the others), except for one, then we get straight lines containing the edges of the cube. Similar to the previous one, you can count that there are exactlythings. And if now we fix all the coordinates (putting each of them equal or , regardless of the others), except for some two, we obtain planes containing two-dimensional cube faces. Using the combinatorial rule, we find that there are exactlythings. Further, similarly - fixing all coordinates (putting each of them equal or , regardless of the others), except for some three, we obtain hyperplanes containing three-dimensional cube faces. Using the same rule, we calculate their number - exactlyetc. This will be sufficient for our study. Let us apply the obtained results to the structure of a four-dimensional cube, namely, in all 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 of its elements.

The vertices of the four-dimensional cube:

The edges of the 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 the four-dimensional cube and the methods of its assignment are described with sufficient completeness, we will 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 by hyperplanes parallel to the faceIt is known from analytic geometry that any such section will be given by the equationLet's set the corresponding sections analytically:

As you can see, the analytical task of the three-dimensional unit cube lying in the hyperplane has been obtained

To establish an analogy, we write down the section of a three-dimensional cube by the 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 unit three-dimensional cubes lying in hyperplanes respectively.

Now we will consider sections of a four-dimensional cube by hyperplanes perpendicular to its main diagonal. Let's first solve this problem for a three-dimensional cube. Using the above method of specifying a unit three-dimensional cube, he concludes that as the main diagonal, one can take, for example, a segment with ends and ... Hence, the vector of the main diagonal will have coordinates... Therefore, the equation of any plane perpendicular to the main diagonal will have the form:

Determine the boundaries of the parameter change... Because , then, adding these inequalities term by term, we obtain:

Or .

If, then (due to restrictions). Similarly, if, then . Hence, for and for the cutting plane and the cube have exactly one common point ( and respectively). Now, let's note the following. If(again due to variable constraints). The corresponding planes intersect three faces at once, because, otherwise, the cutting plane would be parallel to one of them, which is not the case by the condition. If, then the plane intersects all the faces of the cube. If, then the plane intersects the faces... Here are the corresponding calculations.

Let be Then the planecrosses the line in a straight line, moreover. Edge, moreover. Edge plane intersects in a straight line, moreover

Let be Then the planecrosses the line:

straight edge, and.

straight edge, and.

straight edge, and.

straight edge, and.

straight edge, and.

straight edge, and.

This time, six segments are obtained, having successively common ends:

Let be Then the planecrosses the line in a straight line, moreover. Edge plane intersects in a straight line, moreover. Edge plane intersects in a straight line, moreover ... That is, three segments are obtained that have pairwise common ends:Thus, for the specified values ​​of the parameterthe plane will intersect the cube in a regular triangle with vertices

So, here is an exhaustive 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, in what sets it intersects them, 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 an equilateral triangle (which is proved by direct calculation of the lengths of the segments), the vertices of which are these ends of the segments.

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

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

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

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

4) For and the secant hyperplane and the four-dimensional cube have one common point ( and respectively);

5) When a regular tetrahedron will be obtained in the section;

6) When an octahedron will be obtained in the section;

7) When a regular tetrahedron will be obtained in the section.

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

Let us illustrate what has been said with a specific example. Namely, we investigate the section of the four-dimensional cube by the hyperplaneDue to the limitations of the variables, this hyperplane intersects the following three-dimensional faces: Edge intersects on a planeDue to the limitations of the variables, we have:We get a triangular region with verticesFurther,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... It is easy to calculate that this tetrahedron is indeed correct.

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 described analytically and geometrically, models of sweeps and central projections of three-dimensional and four-dimensional cubes were made, three-dimensional objects obtained by intersecting a four-dimensional cube by hyperplanes parallel to one of its three-dimensional faces, or by hyperplanes perpendicular to its main diagonal.

The study made it possible to reveal a deep analogy in the structure and properties of cubes of different dimensions. The used analogy technique 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 a given point, which is called the center of the sphere. Further,dimensional simplex can be defined as partdimensional space, limited by the minimum numberdimensional hyperplanes. For example, a one-dimensional simplex is a segment (a part of a one-dimensional space bounded by two points), a two-dimensional simplex is a triangle (a part of a two-dimensional space bounded by three straight lines), a three-dimensional simplex is a tetrahedron (a part of a three-dimensional space bounded by four planes). Finally,the dimensional simplex is defined as a 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 study.

Bibliography

1) Bugrov Y.S., Nikolsky S.M.Higher mathematics, v.1 –M .: Bustard, 2005 - 284 p.

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

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

In geometry hypercube- this is n-dimensional analogy of the square ( n= 2) and cube ( n= 3). It is a closed, convex shape made up of groups of parallel lines located at opposite edges of the shape, and connected to each other at right angles.

This figure is also known as tesseract(tesseract). Tesseract refers to a cube as a cube refers to a square. More formally, a tesseract can be described as a regular convex four-dimensional polytope (polytope) whose boundary consists of eight cubic cells.

According to the Oxford English Dictionary, the word tesseract was coined in 1888 by Charles Howard Hinton and used in his book A New Era of Thought. The word was formed from the Greek "τεσσερες ακτινες" ("four rays"), there are four axes of coordinates. In addition, in some sources, the same figure was called tetracube(tetracube).

n-dimensional hypercube is also called n-cube.

A point is a hypercube of dimension 0. If you move a point by a unit of length, you get a segment of unit length - a hypercube of dimension 1. Further, if you move a segment by a unit of length in the direction perpendicular to the direction of the segment, you get a cube - a hypercube of dimension 2. Shifting a square by a unit of length in the direction perpendicular to the plane of the square, a cube is obtained - a hypercube of dimension 3. This process can be generalized to any number of dimensions. For example, if you move a cube one unit of length in the fourth dimension, you get a tesseract.

The family of hypercubes is one of the few regular polyhedra that can be represented in any dimension.

Hypercube elements

Dimension hypercube n has 2 n"sides" (one-dimensional line has 2 points; two-dimensional square - 4 sides; three-dimensional cube - 6 faces; four-dimensional tesseract - 8 cells). The number of vertices (points) of the hypercube is 2 n(for example, for a cube - 2 3 vertices).

Quantity m-dimensional hypercubes on the border n-cube equals

For example, the border of a hypercube contains 8 cubes, 24 squares, 32 edges, and 16 vertices.

Plane projection

The formation of a hypercube can be represented in the following way:

• Two points A and B can be connected to form a line segment AB.
• Two parallel line segments AB and CD can be connected to form a square ABCD.
• Two parallel squares ABCD and EFGH can be connected to form a cube ABCDEFGH.
• Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to form the hypercube ABCDEFGHIJKLMNOP.

The latter structure is not easy to visualize, but it is possible to depict its projection onto a 2D or 3D space. Moreover, projections onto a 2D plane can be more useful by being able to rearrange the positions of the projected vertices. In this case, images can be obtained that no longer reflect the spatial relationships of elements within the tesseract, but illustrate the structure of vertex connections, as in the examples below.

The first illustration shows how, in principle, a tesseract is formed by connecting two cubes. This diagram is similar to the diagram for creating a two-square cube. The second diagram shows that all the edges of the tesseract have the same length. This scheme also forces you to search for cubes connected to each other. In the third diagram, the vertices of the tesseract are located in accordance with the distances along the edges relative to the bottom point. This scheme is interesting in that it is used as a basic scheme for the network topology of connecting processors when organizing parallel computing: the distance between any two nodes does not exceed 4 edge lengths, and there are many different ways to balance the load.

Hypercube in art

The hypercube has appeared in science fiction literature since 1940, when Robert Heinlein, in his story "And He Built a Crooked House", described a house built in the shape of a tesseract sweep. In the story, this Further, this house collapses, turning into a four-dimensional tesseract. After that, the hypercube appears in many books and novels.

The movie "Cube 2: Hypercube" tells the story of eight people trapped in a network of hypercubes.

The painting by Salvador Dali "Crucifixion" ("Crucifixion (Corpus Hypercubus)", 1954) depicts Jesus crucified on a tesseract scan. This painting can be seen at the Metropolitan Museum of Art in New York.

Conclusion

The hypercube is one of the simplest four-dimensional objects, by the example of which you can see all the complexity and unusualness of the fourth dimension. And what looks impossible in three dimensions, perhaps in four, for example, impossible figures. So, for example, the bars of an impossible triangle in four dimensions will be connected at right angles. And this figure will look like this from all points of view, and will not be distorted, unlike the realizations of the impossible triangle in three-dimensional space (see.

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 a dimension of more 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 greater dimension. To understand how this analogy is used, consider a very simple four-dimensional object - a hypercube, that is, a four-dimensional cube. Let, for definiteness, suppose we want to solve a specific problem, namely, to count the number of square faces of a four-dimensional cube. The entire consideration below will be very lax, without any evidence, purely by analogy.

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

To build a cube from a subcube, you need to stretch the subcube in the direction perpendicular to the plane of the subcube in the direction of the third dimension. In this case, a subcube will grow from each side of the original subcube, which is a lateral two-dimensional face of the cube, which will limit the three-dimensional volume of the cube from 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 came at the end of the construction of the cube.

What you have just read has been set out in excessive detail and with a lot of clarifications. And not casual. Now we will do this trick, we will replace some words in the previous text formally 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 the 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 a lateral three-dimensional face of the hypercube, which will limit the four-dimensional volume of the hypercube from 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 that 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? Because exactly the same formal replacement of words we get the description of the construction of the cube from the description of the construction of the 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, a cube has 12 edges, which means that an additional 12 new faces (subcubes) will appear in those 6 cubes that limit the four-dimensional volume along the three axes of three-dimensional space. And there are still two cubes that limit this four-dimensional volume from below and from above along the fourth axis. Each of these cubes has 6 faces.

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

The next picture shows the logical structure of a hypercube. It is like a projection of a hypercube onto a three-dimensional space. This results in a three-dimensional frame made of ribs. In the figure, of course, you can see the projection of this frame onto the plane as well.

On this frame, the inner cube is, as it were, 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 up along the fourth dimension axis and it passes into the outer cube. So the outer and inner cubes from this figure limit the hypercube along the fourth dimension axis.

And between these two cubes, 6 more new cubes are visible, which are in common contact with the first two. These six cubes constrain our hypercube along three axes of three-dimensional space. As you can see, they are in contact not only with the first two cubes, which are internal and external on this three-dimensional frame, but they are still in contact with each other.

You can calculate right in the figure and make sure that the hypercube really has 24 faces. But this question arises. This hypercube skeleton in 3D space is filled with eight 3D cubes without any gaps. To make a real hypercube out of this three-dimensional projection of a hypercube, it is necessary to turn this frame inside out so that all 8 cubes limit the 4-dimensional volume.

This is how it is done. We invite a resident of the four-dimensional space to visit and ask him to help us. He grabs the inner cube of this skeleton and shifts 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 inner 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 abdomen and end up in a parallel three-dimensional space. Therefore, the four-man is politely refused.

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

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

The inhabitants of the two-dimensional space built on the plane a frame of the projection of the cube on the 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. At the same time, two-dimensional inhabitants see the complete disappearance of the entire inner frame, and they only have the frame of the outer square. With this operation, all the squares that were in contact with their edges continue to touch the same edges as before.

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

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

If you are a fan of the Avengers movies, the first thing that comes to your mind when you hear the word "Tesseract" is the transparent cube-shaped vessel of the Infinity Stone containing boundless power.

For fans of the Marvel Universe, the Tesseract is a glowing blue cube that makes people from not only Earth, but other planets also go crazy. This is why all the Avengers have banded together to protect the Earthlings from the extremely destructive forces of the Tesseract.

However, the following must be said: The Tesseract is an actual geometric concept, or rather, a form that exists in 4D. This isn't just a blue cube from the Avengers ... it's a real concept.

Tesseract is an object in 4 dimensions. But before we explain it in detail, let's start from the very beginning.

What is dimension?

Everyone has heard the terms 2D and 3D, representing respectively two-dimensional or three-dimensional objects in space. But what are these dimensions?

Measurement is simply the direction you can go. For example, if you draw a line on a piece of paper, you can go either left / right (x-axis) or up / down (y-axis). Thus, we say that the paper is two-dimensional, since you can only walk in two directions.

There is a sense of depth in 3D.

Now, in the real world, in addition to the two directions mentioned above (left / right and up / down), you can also go to / from. Hence, a sense of depth is added in 3D space. Therefore, we say that real life is 3-dimensional.

A point can represent 0 dimensions (since it does not move in any direction), a line represents 1 dimension (length), a square represents 2 dimensions (length and width), and a cube represents 3 dimensions (length, width, and height).

Take a 3D cube and replace each face (which is currently a square) with a cube. And so! The shape you get is the tesseract.

What is a tesseract?

Simply put, a tesseract is a cube in 4-dimensional space. You can also say that it is a 4D analog of a cube. It is a 4D shape where each face is a cube.

Here's a simple way to conceptualize dimensions: a square is two-dimensional; therefore, each of its corners has 2 lines extending from it at an angle of 90 degrees to each other. The cube is 3D, so each of its corners has 3 lines descending from it. Likewise, the tesseract is a 4D shape, so each corner has 4 lines extending from it.

Why is it difficult to imagine a tesseract?

Since we, as humans, have evolved to visualize objects in three dimensions, anything that goes into extra dimensions such as 4D, 5D, 6D, etc., does not make much sense to us, because we cannot have them at all. imagine. Our brain cannot understand the 4th dimension in space. We just can't think about it.

However, just because we cannot visualize the concept of multidimensional spaces does not mean that it cannot exist.

Mathematically, a tesseract is a perfectly accurate form. Likewise, all forms in the higher dimensions, i.e. 5D and 6D, are also mathematically plausible.

Just as a cube can be expanded into 6 squares in 2D space, a tesseract can be expanded into 8 cubes in 3D space.

Surprising and incomprehensible, isn't it?

So the tesseract is a "real concept" that is absolutely plausible mathematically, not just the glowing blue cube that is fought over in the Avengers movies.

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

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

I remember that my dear friends sometimes like a moment of mathematical educational program. Therefore, I will write a piece of my lecture for mathematicians here as well. And I will try without tediousness. 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're miserable because we're only three-dimensional,” said my Sunday school teacher, who was the first to tell me what a 4-dimensional cube is. Sunday school was, of course, extremely religious - mathematics. This time we studied hyper-cubes. A week before that, mathematical induction, a week after that, Hamiltonian cycles in graphs - respectively, this is the 7th grade.

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?

OK OK! I am not asking you for a clear mathematical definition. Just imagine the simplest and most common three-dimensional cube. Have you presented?

Good.
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!

The square has 2 coordinates. The cube has three. Points of a square 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 such a thing with 4 coordinates and everything from 0 to 1.

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

So, stop, how do you draw a 4-dimensional cube? After all, we cannot draw 4-dimensional space on a plane!
But we also do not draw 3-dimensional space on a plane, we draw it projection onto the 2-dimensional plane of the drawing. We place the third coordinate (z) at an angle, imagining that the axis from the plane of the drawing goes "towards us".

Now it is quite clear how to draw a 4-dimensional cube. In the same way as we placed the third axis at a certain angle, 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 a whisper from the back desks. Let me explain in more detail what this mess of lines is.
Look first at the three-dimensional cube. What have we done? We took a square and dragged it along the third axis (z). It is like many, many paper squares glued together in a pile.
It's the same with a 4-dimensional cube. Let's call the fourth axis the "time axis" for convenience and for science fiction purposes. We need to take an ordinary three-dimensional cube and drag it in time from time "now" to 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 "now cube" leaves a trace in time - a segment. Connecting her present with her future.

In short, without lyrics: we drew two identical 3-dimensional cubes and connected the corresponding vertices.
Exactly as it was done 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 the 4-dimensional cube (a 4-dimensional cube with a fifth coordinate 0 and a 4-dimensional cube with a fifth coordinate 1) and connect the corresponding vertices with edges. True, such a jumble of edges will come out on the plane that it will be almost impossible to understand anything.

When we imagined a 4-dimensional cube and even managed to draw it, we can explore it in any way. Do not forget to explore it both in the mind and in 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.
Hence, a 4-dimensional cube must be limited to eight 3-dimensional cubes. On each of the 4 coordinates - on both sides. In the picture above, we clearly see 2 faces that bound it along the "time" coordinate.

Here are two cubes (they are slightly oblique because they have 2 dimensions projected onto a plane at an angle), bounding our hyper-cube to the left and right.

It is also easy to notice the "top" and "bottom".

The most difficult thing is to understand visually where "front" and "back" are. The front one starts from the front face of the "cube now" and to the front face of the "cube of the future" - it is red. Rear, respectively, purple.

They are the hardest to spot because other cubes get tangled under your feet, which constrain the hypercube to a different projected coordinate. But note that the cubes are still different! Here's another picture, where the "now" and "future cube" are highlighted.

It is of course 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 skeletons and connect their corresponding vertices with a new edge.
I don't have such a model now. At the lecture, I show the students a slightly different 3-dimensional model of a 4-dimensional cube.

You know how a cube is projected onto a plane like this.
As if we are looking at a cube from above.

The closest line 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 bigger now, we see the cube of the future in the distance, so it looks smaller.

On the other side. From the side of the top.

Straight straight from the side of the face:

From the side of the rib:

And the last angle, asymmetrical. From the section "You also tell me that I looked between his ribs."

Well, then you can come up with anything. For example, as is the development of a 3-dimensional cube on a plane (this is how you need to cut out a sheet of paper in order to get a cube when folding), there is also an unfolding of a 4-dimensional cube into space. It's like cutting out 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 generally n-dimensional cubes. For example, is it true that the radius of a sphere circumscribed around an n-dimensional cube is 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)?