4 x dimensional cube. Cybercube - the first step into the fourth dimension

An image is a projection (perspective) of a four-dimensional cube onto three-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 "tetracubus".

Geometry

An ordinary tesseract in Euclidean four-dimensional space is defined as the convex hull of points (± 1, ± 1, ± 1, ± 1). In other words, it can be represented as the following set:

The tesseract is bounded by eight hyperplanes, the intersection of which with the tesseract itself defines its three-dimensional faces (which are ordinary cubes). Each pair of non-parallel 3D faces intersect to form 2D faces (squares), and so on. Finally, a tesseract has 8 3D faces, 24 2D, 32 edges, and 16 vertices.

Popular Description

Let's try to imagine what the hypercube will look like without leaving three-dimensional space.

In one-dimensional "space" - on a line - select a segment AB of length L. On a two-dimensional plane at a distance L from AB, draw a segment DC parallel to it and connect their ends. The result is a square ABCD. Repeating this operation with the plane, we get a three-dimensional cube ABCDHEFG. And by shifting the cube in the fourth dimension (perpendicular to the first three) by a distance L, we get a hypercube ABCDEFGHIJKLMNOP.
http://upload.wikimedia.org/wikipedia/ru/1/13/Construction_tesseract.PNG

The one-dimensional segment AB is the side of the two-dimensional square ABCD, the square is the side of the cube ABCDHEFG, 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, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and 8 more edges will "draw" its eight vertices, which have moved into the fourth dimension. The same reasoning can be done for the faces of the hypercube. In two-dimensional space, it is one (the square itself), the cube has 6 of them (two faces from the moved square and four more will 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.

In a similar way, we can continue the reasoning for hypercubes of a larger number of dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, inhabitants of three-dimensional space. Let's use the familiar analogy method for this.

Unfolding the tesseract

Take a wire cube ABCDHEFG and look at it with one eye from the side of the face. We will see and can draw two squares on the plane (its near and far faces), 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 fourth dimension. You can also try to imagine a 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 a face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in perspective will look like a rather complex figure. The part of it, which remained in "our" space, is drawn with solid lines, and that which has gone into hyperspace, with dotted lines. The very same four-dimensional hypercube consists of an infinite number of cubes, just as a three-dimensional cube can be "cut" into an infinite number of flat squares.

Having cut six faces of a three-dimensional cube, you can expand it into a flat shape - a sweep. It will have a square on each side of the original face plus one more - the face opposite to it. A three-dimensional unfolding of a four-dimensional hypercube will consist of the original cube, six cubes "growing" from it, plus one more - the final "hyperface".

Tesseract properties are the continuation of the properties of geometric figures of lower dimensions into four-dimensional space.

Projection

Into two-dimensional space

This structure is difficult for the imagination, but it is possible to project a tesseract into 2D or 3D spaces. In addition, projection to plane makes it easy to understand the location of the vertices of the hypercube. In this way, images can be obtained that no longer reflect spatial relationships within the tesseract, but which illustrate the structure of vertex connections, as in the following examples:


Into three-dimensional space

The projection of a tesseract onto a three-dimensional space is represented by two nested three-dimensional cubes, the corresponding vertices of which are connected by segments. The inner and outer cubes have different sizes in three-dimensional space, but in four-dimensional space they are equal cubes. To understand the equality of all the cubes of the tesseract, a rotating tesseract model was created.

The six truncated pyramids at the edges of the tesseract are images of equal six cubes.
Stereo pair

A stereopair of a tesseract is depicted as two projections onto three-dimensional space. This tesseract image was designed to represent depth as a fourth dimension. A stereopair is viewed so that each eye sees only one of these images, a stereoscopic picture appears that reproduces the depth of the tesseract.

Unfolding the tesseract

The surface of a tesseract can be expanded into eight cubes (similar to how the surface of a cube can be expanded into six squares). There are 261 different tesseract unfolding. The unfolding of the tesseract can be calculated by drawing connected corners on the graph.

Tesseract in art

In Edwine A.'s New Abbott Plains, the hypercube is the storyteller.
In one episode of The Adventures of Jimmy Neutron: Genius Boy Jimmy invents a four-dimensional hypercube identical to the foldbox from Heinlein's 1963 novel Road of Glory.
Robert E. Heinlein has mentioned hypercubes in at least three science fiction stories. In The House of Four Dimensions (The House That Teale Built) (1940), he described a house built as an unfolding of a tesseract.
Heinlein's novel Road of Glory describes an oversized dish that was larger on the inside than on the outside.
Henry Kuttner's story "Mimsy Were the Borogoves" describes an educational toy for children from the distant future, similar in structure to a tesseract.
In the novel by Alex Garland (1999), the term "tesseract" is used for a three-dimensional unfolding of a four-dimensional hypercube, not the hypercube itself. This is a metaphor designed to show that the cognizing system should be broader than the cognizable one.
Cube 2: Hypercube focuses on eight strangers trapped in a hypercube, or network of interconnected cubes.
The TV series Andromeda uses tesseract generators as a conspiracy device. They are primarily designed to manipulate space and time.
Painting "Crucifixion" (Corpus Hypercubus) by Salvador Dali (1954)
The Nextwave comic book depicts a vehicle that includes 5 tesseract zones.
On the Voivod Nothingface album, one of the songs is called “In my hypercube”.
In the novel by Anthony Pierce "Route Cuba" one of the orbiting moons of the International Development Association is called a tesseract, which was compressed into 3 dimensions.
In the series "School" Black Hole "" in the third season there is a series "Tesseract". Lucas pushes a secret button and the school begins to take shape like a mathematical tesseract.
The term "tesseract" and the term "tesserate" derived from it is found in Madeleine L'Engle's story "The Fold of Time"

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, 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 imagine, 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, you can get images 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 joining 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 the 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 "The 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, possibly 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.

Let's start by explaining what 4-space is.

This is a one-dimensional space, that is, just the OX axis. Any point on it is characterized by one coordinate.

Now let's draw the OY axis perpendicular to the OX axis. So we got a two-dimensional space, that is, the XOY plane. Any point on it is characterized by two coordinates - abscissa and ordinate.

Let's draw the OZ axis perpendicular to the OX and OY axes. The result is a three-dimensional space, in which any point has an abscissa, ordinate and applicate.

It is logical that the fourth axis, OQ, should be perpendicular to the OX, OY and OZ axes at the same time. But we cannot precisely build such an axis, and therefore it remains only to try to imagine it. Each point in four-dimensional space has four coordinates: x, y, z, and q.

Now let's see how the four-dimensional cube came about.

The picture shows a figure of one-dimensional space - a line.

If you make a parallel translation of this line along the OY axis, and then connect the corresponding ends of the two resulting lines, you get a square.

Similarly, if you make a parallel translation of the square along the OZ axis and connect the corresponding vertices, you get a cube.

And if we make a parallel translation of the cube along the OQ axis and connect the vertices of these two cubes, we get a four-dimensional cube. By the way, it's called tesseract.

To draw a cube on a plane, you need it project... It looks like this:

Imagine that in the air above the surface hangs wireframe model cube, that is, as if "made of wire", and above it - a light bulb. If you turn on the light bulb, trace the shadow from the cube with a pencil, and then turn off the light bulb, then the projection of the cube will be displayed on the surface.

Let's move on to a little more complex. Look again at the drawing with the light bulb: as you can see, all the rays converge at one point. It is called vanishing point and is used to build perspective projection(and sometimes it is parallel, when all the rays are parallel to each other. The result is that the sensation of volume is not created, but it is lighter, and if the vanishing point is far enough away from the projected object, then the difference between these two projections is hardly noticeable). To project a given point onto a given plane using the vanishing point, you need to draw a straight line through the vanishing point and this point, and then find the intersection point of the resulting line and the plane. And in order to project a more complex figure, say, a cube, you need to project each of its vertices, and then connect the corresponding points. It should be noted that subspace projection algorithm can be generalized to 4D-> 3D case, not just 3D-> 2D.

As I said, we cannot imagine exactly what the OQ axis looks like, nor does the tesseract. But we can get a limited idea of ​​it if we project it onto the volume, and then draw it on the computer screen!

Now let's talk about the tesseract projection.

On the left is the projection of the cube onto the plane, and on the right is the tesseract on the volume. They are quite similar: the projection of a cube looks like two squares, small and large, one inside the other, and whose corresponding vertices are connected by lines. And the tesseract projection looks like two cubes, small and large, one inside the other, and in which the corresponding vertices are connected. But we have all seen a cube, and we can say with confidence that both the small square and the large one, and the four trapezoids above, below, to the right and to the left of the small square, are in fact squares, and at what equal. And the tesseract is the same. And the big cube, and the small cube, and the six truncated pyramids on the sides of the small cube - these are all cubes, and they are equal.

My program can not only draw a projection of a tesseract onto a volume, but also rotate it. Let's see how this is done.

To begin with, I will tell you what it is rotation parallel to the plane.

Imagine that the cube rotates around the OZ axis. Then each of its vertices describes a circle around the OZ axis.

A circle is a flat figure. And the planes of each of these circles are parallel to each other, and in this case are parallel to the XOY plane. That is, we can talk not only about rotation around the OZ axis, but also about rotation parallel to the XOY plane. As you can see, at points that rotate parallel to the XOY axis, only the abscissa and ordinate change, the applicate remains unchanged And, in fact, we we can talk about rotation around a straight line only when we are dealing with three-dimensional space. In two-dimensional space, everything revolves around a point, in four-dimensional space - around a plane, in five-dimensional space we are talking about rotation around a volume. And if we can imagine rotation around a point, then rotation around a plane and volume is something unthinkable. And if we talk about rotation parallel to the plane, then in any n-dimensional space a point can rotate parallel to the plane.

Many of you have probably heard of the rotation matrix. Multiplying a point by it, we get a point rotated parallel to the plane at an angle phi. For two-dimensional space, it looks like this:

How to multiply: x of a point rotated by an angle phi = cosine of the angle phi * x of the original point minus the sine of the angle phi * y of the original point;
angle phi = sine of the angle phi * x of the original point plus the cosine of the angle phi * y of the original point.
Xa` = cosph * Xa - sinph * Ya
Ya` = sinph * Xa + cosph * Ya
, where Xa and Ya are the abscissa and ordinate of the point to be rotated, Xa` and Ya` are the abscissa and ordinate of the already rotated point

For three-dimensional space, this matrix is ​​generalized as follows:

Rotation parallel to the XOY plane. As you can see, the Z coordinate does not change, but only X and Y change.
Xa` = cosph * Xa - sinph * Ya + Za * 0
Ya` = sinph * Xa + cosph * Ya + Za * 0
Za` = Xa * 0 + Ya * 0 + Za * 1 (in fact, Za` = Za)

Rotation parallel to the XOZ plane. Nothing new,
Xa` = cosph * Xa + Ya * 0 - sinph * Za
Ya` = Xa * 0 + Ya * 1 + Za * 0 (in fact, Ya` = Ya)
Za` = sinph * Xa + Ya * 0 + cosph * Za

And the third matrix.
Xa` = Xa * 1 + Ya * 0 + Za * 0 (in fact, Xa` = Xa)
Ya` = Xa * 0 + cosph * Ya - sinph * Za
Za` = Xa * 0 + sinph * Ya + cosph * Za

And for the fourth dimension, they look like this:


I think you have already figured out what to multiply, so I will not describe it once again. But note that it does the same thing as a matrix for rotating parallel to a plane in three-dimensional space! Both this and this change only the ordinate and applicate, and the rest of the coordinates do not touch, therefore it can be used in the three-dimensional case, simply not paying attention to the fourth coordinate.

But the projection formula is not so simple. No matter how many forums I read, none of the projection methods came up to me. Parallel did not suit me, since the projection will not look three-dimensional. In some projection formulas, to find a point, you need to solve a system of equations (and I don’t know how to teach a computer to solve them), others I simply didn’t understand ... In general, I decided to come up with my own method. Consider a 2D-> 1D projection for this.

pov means "Point of view", ptp means "Point to project", and ptp` is the desired point on the OX axis.

The angles povptpB and ptpptp`A are equal as corresponding (the dashed line is parallel to the OX axis, the straight line povptp is the secant).
The x of the point ptp` is equal to the x of the point ptp minus the length of the segment ptp`A. This segment can be found from the triangle ptpptp`A: ptp`A = ptpA / tangent of the angle ptpptp`A. We can find this tangent from the triangle povptpB: the tangent of the angle ptpptp`A = (Ypov-Yptp) (Xpov-Xptp).
Answer: Xptp` = Xptp-Yptp / ptpptp`A.

I did not begin to describe this algorithm in detail here, since there are a lot of special cases when the formula changes somewhat. Who cares about it - look in the source code of the program, everything is described there in the comments.

In order to project a point in three-dimensional space onto a plane, just consider two planes - XOZ and YOZ, and for each of them we will solve this problem. In the case of four-dimensional space, there are already three planes to consider: XOQ, YOQ and ZOQ.

And finally, about the program. It works like this: initialize sixteen vertices of the tesseract -> depending on the commands entered by the user, rotate it -> project to the volume -> depending on the commands entered by the user, rotate its projection -> project to the plane -> draw.

I wrote the projections and turns myself. They work according to the formulas I just described. The OpenGL library draws lines and also does color mixing. And the coordinates of the vertices of the tesseract are calculated as follows:

The coordinates of the vertices of the line with the center at the origin and length 2 - (1) and (-1);
- "-" - a square - "-" - and an edge length 2:
(1; 1), (-1; 1), (1; -1) and (-1; -1);
- "-" - cube - "-" -:
(1; 1; 1), (-1; 1; 1), (1; -1; 1), (-1; -1; 1), (1; 1; -1), (-1; 1; -1), (1; -1; -1), (-1; -1; -1);
As you can see, a square is one line above the OY axis and one line below the OY axis; a cube is one square in front of the XOY plane, and one behind it; The tesseract is one cube on the other side of the XOYZ volume, and one on this one. But it is much easier to perceive this alternation of ones and minus ones if you write them down in a column

1; 1; 1
-1; 1; 1
1; -1; 1
-1; -1; 1
1; 1; -1
-1; 1; -1
1; -1; -1
-1; -1; -1

In the first column, one and minus one alternate. In the second column, first there are two pluses, then two minuses. In the third, four plus ones, and then four minus ones. These were the tops of a cube. The tesseract has twice as many of them, and therefore it was necessary to write a loop to declare them, otherwise it is very easy to get confused.

My program can also draw anaglyph. Happy owners of 3D glasses can see a stereoscopic picture. There is nothing tricky in drawing a picture, it just draws two projections onto a plane, for the right and left eyes. But the program becomes much more visual and interesting, and most importantly - it gives a better idea of ​​the four-dimensional world.

Less significant functions - highlighting one of the edges in red, so that you can better see the turns, as well as minor amenities - regulation of the coordinates of the "eye" points, increasing and decreasing the speed of rotation.

Archive with the program, source code and instructions for use.

As soon as I was able to lecture after the operation, the very 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 4D 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 position 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.
In the same way as we did with the 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.
This means that 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 "now cube" and up to the front face of the "future cube" - it is red. Rear, respectively, purple.

They are the hardest to spot because other cubes get tangled under your feet, which constrain the hypercube in a different projected coordinate. But note that the cubes are still different! Here's another picture, where the "cube now" and "cube of the future" 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 respective 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 there is a sweep of a 3-dimensional cube onto a plane (this is how you need to cut a sheet of paper in order to get a cube when folding), there is also a sweep 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)?

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 separates us 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 the sake of 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's 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 axis of measurement 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 have common faces 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.

Inhabitants of two-dimensional space built on a plane a frame of the projection of a cube on 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. 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 such an 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.