Four-dimensional cube. The Ideal Wife's Blog

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

This figure is also known as tesseract(tesseract). The tesseract is to the cube as the cube is to the square. More formally, a tesseract can be described as a regular convex four-dimensional polytope (polyhedron) 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 derived from the Greek "τεσσερες ακτινες" ("four rays"), in the form of four coordinate axes. 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 shift the point by a unit of length, you get a segment of unit length - a hypercube of dimension 1. Further, if you shift the segment by a unit of length in a direction perpendicular to the direction of the segment, you get a cube - a hypercube of dimension 2. Shifting the 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 by one unit of length in the fourth dimension, you get a tesseract.

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

Elements of a hypercube

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

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

For example, on the boundary of a hypercube there are 8 cubes, 24 squares, 32 edges and 16 vertices.

Elements of hypercubes

n-cube	Name	Vertex (0-face)	Edge (1-face)	Edge (2-face)	Cell (3-face)	(4-face)	(5-face)	(6-sided)	(7-face)	(8-face)
0-cube	Dot	1
1-cube	Line segment	2	1
2-cube	Square	4	4	1
3-cube	Cube	8	12	6	1
4-cube	Tesseract	16	32	24	8	1
5-cube	Penteract	32	80	80	40	10	1
6-cube	Hexeract	64	192	240	160	60	12	1
7-cube	Hepteract	128	448	672	560	280	84	14	1
8-cube	Octeract	256	1024	1792	1792	1120	448	112	16	1
9-cube	Eneneract	512	2304	4608	5376	4032	2016	672	144	18

Projection onto a plane

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 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 into two-dimensional or three-dimensional space. Moreover, projections onto a two-dimensional plane can be more useful by allowing the positions of the projected vertices to be rearranged. In this case, it is possible to obtain images that no longer reflect the spatial relationships of the elements within the tesseract, but illustrate the structure of the vertex connections, as in the examples below.

The first illustration shows how, in principle, a tesseract is formed by joining two cubes. This scheme is similar to the scheme for creating a cube from two squares. The second diagram shows that all the edges of the tesseract are the same length. This scheme also forces you to look for cubes connected to each other. In the third diagram, the vertices of the tesseract are located in accordance with the distances along the faces relative to the bottom point. This scheme is interesting because 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 paths for balancing 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 scan. In the story, this Next, this house collapses, turning into a four-dimensional tesseract. After this, the hypercube appears in many books and short stories.

The movie Cube 2: Hypercube is about eight people trapped in a network of hypercubes.

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

Conclusion

A hypercube is one of the simplest four-dimensional objects, from which one can see the complexity and unusualness of the fourth dimension. And what looks impossible in three dimensions is possible 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 viewing points, and will not be distorted, unlike the implementations of an impossible triangle in three-dimensional space (see.

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 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'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 is from the plane drawing is coming"to 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 the same 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 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 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 doctrine of multidimensional spaces began to appear in mid-19th century century. The idea of ​​four-dimensional space was borrowed from scientists by science fiction writers. In their works they told the world about the amazing wonders of the fourth dimension.

The heroes of their works, using the properties of four-dimensional space, could eat the contents of an egg without damaging the shell, and drink a drink without opening the bottle cap. The thieves removed the treasure from the safe through the fourth dimension. Surgeons performed operations on internal organs without cutting the patient's body tissue.

Tesseract

In geometry, a hypercube is an n-dimensional analogy of a square (n = 2) and a cube (n = 3). The four-dimensional analogue of our usual 3-dimensional cube is known as the tesseract. The tesseract is to the cube as the cube is to the square. More formally, a tesseract can be described as a regular convex four-dimensional polyhedron whose boundary consists of eight cubic cells.

Each pair of non-parallel 3D faces intersect to form 2D faces (squares), and so on. Finally, the tesseract has 8 3D faces, 24 2D faces, 32 edges and 16 vertices.
By the way, according to the Oxford Dictionary, the word tesseract was coined and began to be used in 1888 by Charles Howard Hinton (1853-1907) in his book “ New era thoughts". Later, some people called the same figure a tetracube (Greek tetra - four) - a four-dimensional cube.

Construction and 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.

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.

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 the future will look like some kind of pretty complex figure. The four-dimensional hypercube itself can be divided into 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 a flat figure - a development. 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”.

Hypercube in art

The Tesseract is such an interesting figure that it has repeatedly attracted the attention of writers and filmmakers.
Robert E. Heinlein mentioned hypercubes several times. In The House That Teal Built (1940), he described a house built as an unwrapped tesseract and then, due to an earthquake, "folded" in the fourth dimension to become a "real" tesseract. Heinlein's novel Glory Road describes a hyper-sized box that was larger on the inside than on the outside.

Henry Kuttner's story "All Tenali Borogov" describes an educational toy for children from the distant future, similar in structure to a tesseract.

The plot of Cube 2: Hypercube centers on eight strangers trapped in a "hypercube", or network of connected cubes.

A parallel world

Mathematical abstractions gave rise to the idea of ​​existence parallel worlds. These are understood as realities that exist simultaneously with ours, but independently of it. A parallel world can have different sizes: from a small geographical area to an entire universe. In a parallel world, events occur in their own way; it may differ from our world, both in individual details and in almost everything. Moreover, the physical laws of a parallel world are not necessarily similar to the laws of our Universe.

This topic is fertile ground for science fiction writers.

Salvador Dali's painting "The Crucifixion" depicts a tesseract. “Crucifixion or Hypercubic Body” - painting Spanish artist Salvador Dali, painted in 1954. Depicts the crucified Jesus Christ on a tesseract scan. The painting is kept in the Metropolitan Museum of Art in New York

It all started in 1895, when H.G. Wells With the story “The Door in the Wall,” he discovered the existence of parallel worlds for science fiction. In 1923 Wells returned to the idea of ​​parallel worlds and placed in one of them a utopian country, where the characters of the novel “People Like Gods” go.

The novel did not go unnoticed. In 1926, G. Dent’s story “The Emperor of the Country “If”” appeared. In Dent’s story, for the first time, the idea arose that there could be countries (worlds) whose history could go differently from the history of real countries in our world. And worlds these are no less real than ours.

In 1944, Jorge Luis Borges published in his book " Fictional stories"story "The Garden of Forking Paths." Here the idea of ​​branching time was finally expressed with utmost clarity.
Despite the appearance of the works listed above, the idea of ​​many worlds began to seriously develop in science fiction only at the end of the forties of the 20th century, approximately at the same time when a similar idea arose in physics.

One of the pioneers of the new direction in science fiction was John Bixby, who suggested in the story “One Way Street” (1954) that between worlds you can only move in one direction - once you go from your world to a parallel one, you will not return back, but you will move from one world to the next. However, returning to one’s own world is also not excluded - for this it is necessary that the system of worlds be closed.

Clifford Simak's novel A Ring Around the Sun (1982) describes numerous planets Earth, each existing in its own world, but in the same orbit, and these worlds and these planets differ from each other only by a slight (microsecond) shift in time . The numerous Earths that the hero of the novel visits form a single system of worlds.

Alfred Bester expressed an interesting view of the branching of worlds in his story “The Man Who Killed Mohammed” (1958). “By changing the past,” the hero of the story argued, “you change it only for yourself.” In other words, after a change in the past, a branch of history arises in which only for the character who made the change does this change exist.

The Strugatsky brothers' story "Monday Begins on Saturday" (1962) describes the characters' travels to different variants the future described by science fiction writers - in contrast to the travels to various versions of the past that already existed in science fiction.

However, even a simple listing of all the works that touch on the theme of parallel worlds would take too much time. And although science fiction writers, as a rule, do not scientifically substantiate the postulate of multidimensionality, they are right about one thing - this is a hypothesis that has a right to exist.
The fourth dimension of the tesseract is still waiting for us to visit.

Victor Savinov

A universe of four dimensions, or four coordinates, is as unsatisfactory as a universe of three. We can say that we do not have all the data necessary to construct the universe, since neither the three coordinates of the old physics nor the four coordinates of the new are sufficient to describe Total variety of phenomena in the universe.

Let us consider in order “cubes” of various dimensions.

A one-dimensional cube on a line is a segment. Two-dimensional - a square. The border of the square consists of four points - peaks And four segments - ribs Thus, a square has two types of elements on its boundary: points and segments. The border of a three-dimensional cube contains elements of three types: vertices - there are 8 of them, edges (segments) - there are 12 of them and faces (squares) - there are 6 of them. The one-dimensional segment AB serves as the face 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.

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.

Cube dimension	Border dimension
2 square
4 tesseract

Coordinates infour-dimensional space.

A point on a line is defined as a number, a point on a plane as a pair of numbers, a point in three-dimensional space as a triple of numbers. Therefore, it is completely natural to construct the geometry of four-dimensional space by defining a point in this imaginary space as a quadruple of numbers.

A two-dimensional face of a four-dimensional cube is a set of points for which any two coordinates can take all possible values from 0 to 1, and the other two are constant (equal to either 0 or 1).

Three-dimensional face A four-dimensional cube is a set of points in which three coordinates take all possible values ​​from 0 to 1, and one is constant (equal to either 0 or 1).

Developments of cubes of various dimensions.

We take a segment, place one segment on all sides, and attach another one to any one, in this case to the right segment.

We got a square scan.

We take a square, place a square on all sides, attach another one to any one, in this case to the bottom square.

This is a development of a three-dimensional cube.

Four dimensional cube

We take a cube, place a cube on all sides, attach another one to any one in this lower cube.

Development of a four-dimensional cube

Let's imagine that a four-dimensional cube is made of wire and an ant sits at the vertex (1;1;1;1), then the ant will have to crawl from one vertex to another along the edges.

Question: how many edges will he have to crawl along to get to the vertex (0;0;0;0)?

Along 4 edges, that is, the vertex (0;0;0;0) is a 4th order vertex, by passing along 1 edge he can get to a vertex that has one of the coordinates 0, this is a 1st order vertex, by passing along 2 edges he can get to vertices where there are 2 zeros are vertices of the 2nd order, there are 6 such vertices, passing along 3 edges, he will get to the vertices that have 3 coordinates zero, these are vertices of the third order.

There are other cubes in multidimensional space. In addition to the tesseract, you can build cubes with a large number of dimensions. The model of a five-dimensional cube is a penteract. A penteract has 32 vertices, 80 edges, 80 faces, 40 cubes and 10 tesseracts.

Artists, directors, sculptors, scientists represent the multidimensional cube in different ways. Here are some examples:

Many science fiction writers describe the tesseract in their works. For example, Robert Anson Heinlein (1907–1988) mentioned hypercubes in at least three of his non-fiction stories. In "The House of Four Dimensions" he described a house built like the unfolding of a tesseract.

The plot of Cube 2 centers on eight strangers trapped in a hypercube.

« Crucifixion" by Salvador Dali 1954 (1951). Dali's surrealism sought points of contact between our reality and the otherworldly, in particular, the 4-dimensional world. Therefore, on the one hand, it is amazing, and, on the other hand, nothing surprising in the fact that a geometric figure made of cubes, forming christian cross, is an image of a 3-dimensional development of a 4-dimensional cube or tesseract.

On October 21, the Department of Mathematics of the Pennsylvania State University held an opening unusual sculpture called "Octacube". It is an image of a four-dimensional geometric object in three-dimensional space. According to the author of the sculpture, Professor Adrian Ocneanu, such beautiful figure this kind did not exist in the world, either virtually or physically, although three-dimensional projections of four-dimensional figures had been made before.

In general, mathematicians easily operate with four-, five-, and even more multidimensional objects, but it is impossible to depict them in three-dimensional space. The "Octacube", like all similar figures, is not truly four-dimensional. It can be compared to a map - a projection of a three-dimensional surface globe on a flat sheet of paper.

A three-dimensional projection of a four-dimensional figure was obtained by Okneanu using radial stereography using a computer. At the same time, the symmetry of the original four-dimensional figure was preserved. The sculpture has 24 vertices and 96 faces. In four-dimensional space, the edges of a figure are straight, but in projection they are curved. The angles between the faces of the three-dimensional projection and the original figure are the same.

The Octacube was made from stainless steel in the engineering workshops of Pennsylvania State University. The sculpture was installed in the renovated McAllister building of the Faculty of Mathematics.

Multidimensional space was of interest to many scientists, such as Rene Descartes and Hermann Minkowski. In our days go by increasing knowledge on this topic. It helps mathematicians, researchers and inventors of our time to achieve their goals and advance science. A step into multidimensional space is a step into a new, more developed era of humanity.

τέσσαρες ἀκτίνες - four rays) - 4-dimensional Hypercube- analogue in 4-dimensional space.

The image is a projection () of a four-dimensional cube onto three-dimensional space.

A generalization of the cube to cases with more than 3 dimensions is called hypercube or (en:measure polytopes). Formally, a hypercube is defined as four equal segments.

This article mainly describes the 4-dimensional hypercube, called tesseract.

Popular description

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

In one-dimensional “space” - on a line - we select AB with length L. In two-dimensional space, at a distance L from AB, we draw a segment DC parallel to it and connect their ends. The result is a square ABCD. Repeating this operation with the plane, we obtain a three-dimensional cube ABCDHEFG. And by moving the cube in the fourth dimension (perpendicular to the first three!) by a distance L, we get a hypercube.

A one-dimensional segment AB serves as a face of a two-dimensional square ABCD, the square serves as a side of a cube ABCDHEFG, which, in turn, will be a side of a 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.

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 it will look for us, residents of three-dimensional space. four-dimensional hypercube. 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 fourth dimension. 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 part that remained in “our” space is drawn with solid lines, and the part that went into hyperspace is drawn with dotted lines. 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 eight faces of a three-dimensional cube, you can decompose it into a flat figure - a development. 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 dimensions into 4-dimensional space, presented in the table below.