Rotation of a four-dimensional cube. Cybercube - the first step into the fourth dimension

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.

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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 shifting the cube in the fourth dimension (perpendicular to the first three!) by a distance L, we get a hypercube.

The one-dimensional segment AB serves as the face of the two-dimensional square ABCD, the square serves as 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, 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 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 a continuation of the properties of geometric figures of lower dimension into 4-dimensional space, presented in the table below.

If you're a fan of the Avengers movies, the first thing that might come to your mind when you hear the word "Tesseract" is the transparent cube-shaped vessel of the Infinity Stone containing limitless power.

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

However, this needs to be said: The Tesseract is an actual geometric concept, or more specifically, a shape that exists in 4D. It's not just a blue cube from the Avengers... it's a real concept.

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

What is "measurement"?

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

Dimension is simply a direction you can go. For example, if you are drawing a line on a piece of paper, you can go either left/right (x-axis) or up/down (y-axis). So we say the paper is two-dimensional because you can only go in two directions.

There is a sense of depth in 3D.

Now, in the real world, besides the two directions mentioned above (left/right and up/down), you can also go "to/from". Consequently, a sense of depth is added to the 3D space. That's why 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 of its faces (which are currently squares) 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 analogue of a cube. This 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 coming 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 like 4D, 5D, 6D, etc. doesn't make much sense to us because we can't do them at all introduce. Our brain cannot understand the 4th dimension in space. We just can't think about it.

However, just because we can't visualize the concept of multidimensional spaces doesn't mean it can't exist.

What is a hypercube and four-dimensional space

Our usual space has three dimensions. From a geometric point of view, this means that three mutually perpendicular lines can be indicated in it. That is, for any line you can find a second line perpendicular to the first, and for a pair you can find a third line perpendicular to the first two. It will no longer be possible to find a fourth line perpendicular to the existing three.

Four-dimensional space differs from ours only in that it has one more additional direction. If you already have three mutually perpendicular lines, then you can find a fourth one, such that it will be perpendicular to all three.

A hypercube is simply a cube in four-dimensional space.
Is it possible to imagine four-dimensional space and a hypercube?

This question is related to the question: “is it possible to imagine the Last Supper by looking at the painting of the same name (1495-1498) by Leonardo da Vinci (1452-1519)?”

On the one hand, you, of course, will not imagine what Jesus saw (he is sitting facing the viewer), especially since you will not smell the garden outside the window and taste the food on the table, you will not hear the birds singing... You will not get a complete picture of what was happening that evening, but it cannot be said that you will not learn anything new and that the picture is of no interest.

The situation is similar with the question of the hypercube. It is impossible to fully imagine it, but you can get closer to understanding what it is like.
Construction of a hypercube
0-dimensional cube

Let's start from the beginning - with a 0-dimensional cube. This cube contains 0 mutually perpendicular faces, that is, it is just a point.

1-dimensional cube

In one-dimensional space, we only have one direction. We move the point in this direction and get a segment.

This is a one-dimensional cube.
2 dimensional cube

We have a second dimension, we shift our one-dimensional cube (segment) in the direction of the second dimension and we get a square.

It is a cube in two-dimensional space.
3 dimensional cube

With the advent of the third dimension, we proceed in a similar way: we move the square and get an ordinary three-dimensional cube.

4-dimensional cube (hypercube)

Now we have a fourth dimension. That is, we have at our disposal a direction perpendicular to all three previous ones. Let's use it exactly the same way. A four-dimensional cube will look like this.

Naturally, three-dimensional and four-dimensional cubes cannot be depicted on a two-dimensional screen plane. What I drew are projections. We'll talk about projections a little later, but for now a few bare facts and figures.
Number of vertices, edges, faces
Characteristics of cubes of various sizes
1-dimension of space
2-number of vertices
3-number of edges
4-number of faces

0 (dot) 1 0 0
1 (segment) 2 1 2 (points)
2 (square) 4 4 4 (segments)
3 (cube) 8 12 6 (squares)
4 (hypercube) 16 32 8 (cubes)
N (general formula) 2N N 2N-1 2 N

Please note that the face of a hypercube is our ordinary three-dimensional cube. If you look closely at the drawing of a hypercube, you can actually find eight cubes.
Projections and vision of an inhabitant of four-dimensional space
A few words about vision

We live in a three-dimensional world, but we see it as two-dimensional. This is due to the fact that the retina of our eyes is located in a plane that has only two dimensions. This is why we are able to perceive two-dimensional pictures and find them similar to reality. (Of course, thanks to accommodation, the eye can estimate the distance to an object, but this is a side effect associated with the optics built into our eyes.)

The eyes of an inhabitant of four-dimensional space must have a three-dimensional retina. Such a creature can immediately see the entire three-dimensional figure: all its faces and interiors. (In the same way, we can see a two-dimensional figure, all its faces and interiors.)

Thus, with the help of our organs of vision, we are not able to perceive a four-dimensional cube the way a resident of a four-dimensional space would perceive it. Alas. All that remains is to rely on your mind's eye and imagination, which, fortunately, have no physical limitations.

However, when depicting a hypercube on a plane, I am simply forced to make its projection onto two-dimensional space. Take this fact into account when studying the drawings.
Edge intersections

Naturally, the edges of the hypercube do not intersect. Intersections appear only in drawings. However, this should not come as a surprise, because the edges of a regular cube in the pictures also intersect.
Ribs lengths

It is worth noting that all faces and edges of a four-dimensional cube are equal. In the figure they are not equal only because they are located at different angles to the direction of view. However, it is possible to rotate a hypercube so that all projections have the same length.

By the way, in this figure eight cubes, which are the faces of a hypercube, are clearly visible.
The hypercube is empty inside

It’s hard to believe, but between the cubes that bound the hypercube, there is some space (a fragment of four-dimensional space).

To understand this better, let's look at a two-dimensional projection of an ordinary three-dimensional cube (I deliberately made it somewhat schematic).

Can you guess from it that there is some space inside the cube? Yes, but only by using your imagination. The eye does not see this space. This happens because the edges located in the third dimension (which cannot be depicted in a flat drawing) have now turned into segments lying in the plane of the drawing. They no longer provide volume.

The squares enclosing the space of the cube overlapped each other. But one can imagine that in the original figure (a three-dimensional cube) these squares were located in different planes, and not one on top of the other in the same plane, as happened in the figure.

The situation is exactly the same with a hypercube. The cubes-faces of a hypercube do not actually overlap, as it seems to us on the projection, but are located in four-dimensional space.
Sweeps

So, a resident of four-dimensional space can see a three-dimensional object from all sides simultaneously. Can we see a three-dimensional cube from all sides at the same time? With the eye - no. But people have come up with a way to depict all the faces of a three-dimensional cube at the same time on a flat drawing. Such an image is called a scan.
Development of a three-dimensional cube

Everyone probably knows how the development of a three-dimensional cube is formed. This process is shown in the animation.

For clarity, the edges of the cube faces are made translucent.

It should be noted that we are able to perceive this two-dimensional picture only thanks to our imagination. If we consider the unfolding phases from a purely two-dimensional point of view, the process will seem strange and not at all clear.

It looks like the gradual appearance of first the outlines of distorted squares, and then their creeping into place while simultaneously taking on the required shape.

If you look at the unfolding cube in the direction of one of its faces (from this point of view the cube looks like a square), then the process of formation of the unfold is even less clear. Everything looks like squares creeping out from the initial square (not the unfolded cube).

But the scan is not visual only for the eyes. It is thanks to your imagination that you can glean a lot of information from it.
Development of a four-dimensional cube

It is simply impossible to make the animated process of unfolding a hypercube at least somewhat visual. But this process can be imagined. (To do this, you need to look at it through the eyes of a four-dimensional being.)

The scan looks like this.

All eight cubes bounding the hypercube are visible here.

The edges that should align when folded are painted with the same colors. Faces for which pairs are not visible are left gray. After folding, the topmost face of the top cube should align with the bottom edge of the bottom cube. (The unfolding of a three-dimensional cube is collapsed in a similar way.)

Please note that after convolution, all the faces of the eight cubes will come into contact, closing the hypercube. And finally, when imagining the process of folding, do not forget that when folding, it is not the overlapping of cubes that occurs, but the wrapping of them around a certain (hypercubic) four-dimensional area.

Salvador Dali (1904-1989) depicted the crucifixion many times, and crosses appear in many of his paintings. The painting “The Crucifixion” (1954) uses a hypercube scan.
Space-time and Euclidean four-dimensional space

I hope you were able to imagine the hypercube. But have you managed to come closer to understanding how the four-dimensional space-time in which we live works? Alas, not quite.

Here we talked about Euclidean four-dimensional space, but space-time has completely different properties. In particular, during any rotations, the segments always remain inclined to the time axis, either at an angle less than 45 degrees, or at an angle greater than 45 degrees.

SOURCE 2

The Tesseract is a four-dimensional hypercube, an analogue of a cube in four-dimensional space. According to the Oxford Dictionary, the word "tesseract" was coined and used in 1888 by Charles Howard Hinton (1853-1907) in his book A New Age of Thought. Later, some people called the same figure a "tetracube".

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 ABCD. Repeating this operation with the plane, we obtain 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 the hypercube ABCDEFGHIJKLMNOP.

The one-dimensional segment AB serves as the face of the two-dimensional square ABCD, the square serves as 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, 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 a four-dimensional hypercube will look for us, residents of three-dimensional space. For this we will use the already familiar method of analogies.
Let's take the wire cube ABCDHEFG and look at it with one eye from the side of the edge. We will see and can draw two squares on the plane (its near and far edges), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic “boxes” inserted into each other and connected by eight edges. In this case, the “boxes” themselves - three-dimensional faces - will be projected onto “our” space, and the lines connecting them will stretch in the 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 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”. The properties of a tesseract represent a continuation of the properties of geometric figures of lower dimension into four-dimensional space.

Other names
Hexadecachoron
Octachoron
Tetracube
4-Cube
Hypercube (if the number of dimensions is not specified)

10-dimensional space
It’s in English. For those who don’t know, the pictures make it quite clear

Http://www.skillopedia.ru/material.php?id=1338

Hypercube and Platonic solids

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

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

Steps for constructing a truncated icosahedron in the Vector system

Figures in 4-dimensional space.

--à

--à ?

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

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

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

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

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

Appendix 2

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

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

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

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

B - P + G = 2,

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

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

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

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

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

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

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

B - (P + 1) + (G "+1) = B – P + G " .

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

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

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

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

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

Consider the second case yourself.

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

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

Appendix 3.

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

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

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

Criticism

If in the first part people imprisoned in a labyrinth tried to help each other, in this film it’s every man for himself. There are a lot of unnecessary special effects (aka traps) that do not logically connect this part of the film with the previous one. That is, it turns out that the film Cube 2 is a kind of labyrinth of the future 2020-2030, but not 2000. In the first part, all types of traps can theoretically be created by a person. In the second part, these traps are some kind of computer program, the so-called “Virtual Reality”.

The doctrine of multidimensional spaces began to appear in the middle of the 19th 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 a 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.

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 perspective will look like some rather 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”.

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.

Parallel world

Mathematical abstractions gave rise to the idea of ​​the existence of 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” is a painting by the 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 his story “The Door in the Wall,” 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 in the novel Men 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 the story “The Garden of Forking Paths” in his book Fictional Stories. 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 in the late 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’ journeys to different versions of the future described by science fiction writers - in contrast to the travels to different 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