What is a Tesseract? Cybercube - the first step into the fourth dimension

Teachings about multidimensional spaces began to appear in the middle of the 19th century. Science fiction borrowed the idea of ​​four-dimensional space from scientists. 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 the egg without damaging the shell, drink a drink without opening the cork of the bottle. The kidnappers retrieved the treasure from the safe through the fourth dimension. Surgeons performed operations on the internal organs without cutting the tissues of the patient's body.

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.

Let's try to imagine how the hypercube will look without leaving the three-dimensional space.
In 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. You will get a square CDBA. Repeating this operation with a plane, we get 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 CDBAGHFEKLJIOPNM hypercube.

Similarly, we can continue the reasoning for hypercubes more dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, the inhabitants of three-dimensional space.

Let's take the 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 direction of the fourth axis. 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 the future 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 six faces of a three-dimensional cube, you can decompose it into a flat figure - a net. It will have a square on each side of the original face, plus one more - the face opposite to it. A three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes that "grow" 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 unfolding of a tesseract, and then, due to an earthquake, "formed" in the fourth dimension and became a "real" tesseract. In the novel Glory Road by Heinlein, a hyperdimensional box is described that was larger on the inside than on the outside.

Henry Kuttner's story "All Borog's Tenals" 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 brought to life the notion of existence parallel worlds. These are realities that exist simultaneously with ours, but independently of it. Parallel world may have various sizes: from a small geographical area to the whole universe. In a parallel world, events take place in their own way, it may differ from our world, both in individual details and in almost everything. At the same time, the physical laws of the parallel world are not necessarily similar to the laws of our Universe.

This topic is fertile ground for science fiction writers.

The painting by Salvador Dali "The Crucifixion" depicts a tesseract. "Crucifixion or Hypercubic Body" is a 1954 painting by the Spanish artist Salvador Dali. Depicts the crucified Jesus Christ on the development of the tesseract. The painting is kept at 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," discovered the existence of parallel worlds for fantasy. 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 Are 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 short story "The Garden of Forking Paths" in his book Fictional Stories. Here the idea of ​​time branching was finally expressed with the utmost clarity.
Despite the appearance of the works listed above, the idea of ​​a multi-world began to develop seriously in science fiction only at the end of the forties of the XX century, approximately at the same time when a similar idea arose in physics.

One of the pioneers of a 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 - having gone from your world to a parallel one, you will not go back, but you will move from one world to the next. However, returning to your own world is also not excluded - for this it is necessary that the system of worlds be closed.

In the novel by Clifford Simak "Ring around the Sun" (1982), numerous planets of the Earth are described, each existing in its own world, but on the same orbit, and these worlds and these planets differ from each other only by a slight (by a microsecond) shift in time . Numerous lands visited by the hero of the novel form single system worlds.

A curious look at the branching of the worlds was expressed by Alfred Bester in the story "The Man Who Killed Mohammed" (1958). “Changing the past,” the hero of the story claimed, “you change it only for yourself.” In other words, after changing the past, a branch of history arises, in which only for the character who made the change, this change exists.

In the story of the Strugatsky brothers "Monday begins on Saturday" (1962), the characters' journeys to different variants described by science fiction writers of the future - in contrast to the travels in science fiction that already existed in various options of the past.

However, even a simple enumeration of all the works that deal with the theme of the parallelism of the 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 in one thing - this is a hypothesis that has the right to exist.
The fourth dimension of the tesseract is still waiting for us to visit.

Victor Savinov

What is a hypercube and four-dimensional space

There are three dimensions in our habitual space. 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 the fourth straight line perpendicular to the three existing ones.

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 the fourth one, such that it will be perpendicular to all three.

A hypercube is just a cube in four dimensions.
Is it possible to imagine a 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, of course, you will not imagine what Jesus saw (he is sitting facing the viewer), especially since you will not smell the garden outside the window and the taste of 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.
Building 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 have only one direction. We shift 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 get a square.

It's a cube in two dimensions.
3 dimensional cube

With the advent of the third dimension, we do the same: we shift the square and get the usual 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 of the previous ones. Let's use it the same way. The 4D 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 will 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 dimensions
1-dimension of space
2-number of vertices
3-number of ribs
4-number of faces

0 (dot) 1 0 0
1 (line) 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

Note that the face of the hypercube is our regular 3D cube. If you look closely at the drawing of the 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. That 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 already a side effect associated with the optics built into our eye.)

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

Thus, with the help of our organs of vision, we are not able to perceive a four-dimensional cube in the same way as an inhabitant of a four-dimensional space would perceive it. Alas. It remains only to rely on the mind's eye and fantasy, which, fortunately, have no physical limitations.

However, when depicting a hypercube on a plane, I simply have to project it onto a two-dimensional space. Keep this in mind when studying drawings.
Edge intersections

Naturally, the edges of the hypercube do not intersect. Intersections appear only in figures. However, this should not come as a surprise, because the edges of an ordinary cube in the figures also intersect.
Rib 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 unfold the hypercube so that all projections have the same length.

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

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 consider a 2D projection of a regular 3D cube (I purposely made it somewhat sketchy).

Is it possible to guess from it that there is some space inside the cube? Yes, but only with imagination. The eye does not see this space. This is 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 that bound the space of the cube overlapped each other. But you can imagine that in the original figure (three-dimensional cube) these squares were located in different planes, and not one on top of the other in the same plane, as it turned out in the figure.

The same is true for the hypercube. The cube-faces of the hypercube do not actually overlap, as it seems to us on the projection, but are located in four-dimensional space.
Reamers

So, a resident of four-dimensional space can see a three-dimensional object simultaneously from all sides. Can we simultaneously see a three-dimensional cube from all sides? 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 sweep.
Unfolding a 3D cube

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

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

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

It looks like the gradual appearance of first the outlines of distorted squares, and then their spreading into place with the simultaneous adoption of the necessary shape.

If you look at an 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 a development is even less clear. Everything looks like crawling out of squares from the initial square (not an unfolded cube).

But the scan is not visual only for the eyes. Just thanks to the imagination, a lot of information can be gleaned from it.
Unfolding a 4D Cube

It is simply impossible to make the animated process of hypercube unfolding 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 spread looks like this.

All eight cubes bounding the hypercube are visible here.

Faces are painted with the same colors, which should be aligned when folding. Faces for which paired ones are not visible are left gray. After folding, the topmost face of the top cube should align with the bottom face of the bottom cube. (Similarly, the development of a three-dimensional cube is collapsed.)

Please note that after folding, all the faces of the eight cubes will come into contact, closing the hypercube. And finally, while representing the process of folding, do not forget that when folding, the cubes are not superimposed, but they wrap around a certain (hypercubic) four-dimensional area.

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

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

Here we talked about Euclidean four-dimensional space, but space-time has very different properties. In particular, at any rotation, 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

A 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 "tetracube".

Let's try to imagine how the hypercube will look without leaving the three-dimensional space.
In 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. Get the square ABCD. Repeating this operation with a 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 the hypercube ABCDEFGHIJKLMNOP.

The one-dimensional segment AB serves as a 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. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. In a four-dimensional hypercube, therefore, there will be 16 vertices: 8 vertices of the original cube and 8 vertices 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 "draw" eight of its vertices that 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 twelve of its edges.

In a similar way, we can continue the reasoning for hypercubes of more dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, the inhabitants of three-dimensional space. Let us use for this 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 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 the future will look like some rather complex figure. Its part, which remained in “our” space, is drawn with solid lines, and the part that went into hyperspace is dashed. 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 six faces of a three-dimensional cube, you can decompose it into a flat figure - a net. It will have a square on each side of the original face, plus one more - the face opposite to it. A three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes that "grow" from it, plus one more - the final "hyperface". The properties of the tesseract are an extension of the properties geometric shapes lower dimension into a four-dimensional space.

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

10-dimensional space
there in English. who does not know, the pictures are quite clear

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

Points (±1, ±1, ±1, ±1). In other words, it can be represented as the following set:

The tesseract is limited 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 how the hypercube will look without leaving the three-dimensional space.

In 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. You will get a square CDBA. Repeating this operation with a plane, we get 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 CDBAGHFEKLJIOPNM hypercube.

Construction of a tesseract on a plane

The one-dimensional segment AB serves as a side of the two-dimensional square CDBA, the square is the side of the cube CDBAGHFE, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. In a four-dimensional hypercube, therefore, there will be 16 vertices: 8 vertices of the original cube and 8 vertices 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 "draw" eight of its vertices that 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 twelve of its edges.

As the sides of a square are 4 one-dimensional segments, and the sides (faces) of a cube are 6 two-dimensional squares, so for the “four-dimensional cube” (tesseract) the sides are 8 three-dimensional cubes. The spaces of opposite pairs of tesseract cubes (that is, the three-dimensional spaces to which these cubes belong) are parallel. In the figure, these are cubes: CDBAGHFE and KLJIOPNM, CDBAKLJI and GHFEOPNM, EFBAMNJI and GHDCOPLK, CKIAGOME and DLJBHPNF.

In a similar way, we can continue the reasoning for hypercubes of more dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, the inhabitants of three-dimensional space. Let us use for this 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 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 direction of the fourth axis. 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 the future will look like some rather complex figure. The four-dimensional hypercube itself consists of an infinite number of cubes, just as a three-dimensional cube can be “cut” into an infinite number of flat squares.

By cutting 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. A three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes that "grow" from it, plus one more - the final "hyperface".

The properties of a tesseract are an extension of the properties of geometric figures of a smaller dimension into a four-dimensional space.

projections

to two-dimensional space

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

The third picture shows the tesseract in isometry, relative to the construction point. This view is of interest when using the tesseract as the basis for a topological network to link multiple processors in parallel computing.

to three-dimensional space

One of the projections of the tesseract onto three-dimensional space is two nested three-dimensional cubes, the corresponding vertices of which are connected by segments. The inner and outer cubes have different sizes in 3D space, but in 4D space they are equal cubes. To understand the equality of all cubes of the tesseract, a rotating model of the tesseract was created.

• Six truncated pyramids along the edges of the tesseract are images of equal six cubes. However, these cubes are to the tesseract like squares (faces) are to the cube. But in fact, a tesseract can be divided into an infinite number of cubes, just as a cube can be divided into an infinite number of squares, or a square can be divided into an infinite number of segments.

Another interesting projection of the tesseract onto three-dimensional space is a rhombic dodecahedron with its four diagonals drawn, connecting pairs of opposite vertices at large angles of rhombuses. In this case, 14 of the 16 vertices of the tesseract are projected into 14 vertices of the rhombic dodecahedron, and the projections of the remaining 2 coincide in its center. In such a projection onto three-dimensional space, the equality and parallelism of all one-dimensional, two-dimensional and three-dimensional sides are preserved.

stereo pair

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

Tesseract unfolding

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

Tesseract in art

• In Edwine A. Abbott's New Plain, the hypercube is the narrator.
• In one episode of The Adventures of Jimmy Neutron, "boy genius" Jimmy invents a four-dimensional hypercube, identical to the foldbox from the novel Glory Road (1963) by Robert Heinlein.
• Robert E. Heinlein has mentioned hypercubes in at least three science fiction stories. In The House of Four Dimensions (The House That Teel Built), he described a house built as an unfolding of a tesseract, and then, due to an earthquake, "formed" in the fourth dimension and became a "real" tesseract.
• In the novel Glory Road by Heinlein, a hyperdimensional box is described that was larger on the inside than on the outside.
• Henry Kuttner's story "All Borog's Tenals" describes an educational toy for children from the distant future, similar in structure to a tesseract.
• In Alex Garland's novel ( ), the term "tesseract" is used for the three-dimensional unfolding of a four-dimensional hypercube, rather than the hypercube itself. This is a metaphor designed to show that the cognizing system should be wider than the cognizable one.
• The plot of The Cube 2: Hypercube centers on eight strangers trapped in a "hypercube", or network of linked cubes.
• The TV series Andromeda uses tesseract generators as a conspiracy device. They are primarily meant to control space and time.
• Painting " Crucifixion"(Corpus Hypercubus) by Salvador Dali ().
• The Nextwave comic book depicts a vehicle that includes 5 tesseract zones.
• In the album Voivod Nothingface, one of the songs is called "In my hypercube".
• In Anthony Pierce's novel Route Cube, one of IDA's orbital moons is called a tesseract that has been compressed into 3 dimensions.
• In the series "School" Black Hole "" in the third season there is an episode "Tesseract". Lucas presses the secret button and the school begins to "take shape like a mathematical tesseract".
• The term "tesseract" and the term "tesse" derived from it is found in Madeleine L'Engle's story "Wrinkle of Time".
• TesseracT is the name of a British djent group.
• In the Marvel Cinematic Universe film series, the Tesseract is a key plot element, a hypercube-shaped cosmic artifact.
• In Robert Sheckley's story "Miss Mouse and the Fourth Dimension," one esoteric writer, an acquaintance of the author, tries to see the tesseract, looking for hours at the device he designed: a ball on a leg with rods stuck in it, on which cubes are stuck, glued with everyone in a row esoteric symbols. The story mentions Hinton's work.
• In the films The First Avenger, The Avengers. Tesseract is the energy of the entire universe

Other names

• Hexadecachoron (English) Hexadecachoron)
• Octochoron (English) Octachoron)
• tetracube
• 4-cube
• Hypercube (if the number of dimensions is not specified)

Notes

Literature

• Charles H Hinton. Fourth Dimension, 1904. ISBN 0-405-07953-2
• Martin Gardner, Mathmatical Carnival, 1977. ISBN 0-394-72349-X
• Ian Stewart, Concepts of Modern Mathematics, 1995. ISBN 0-486-28424-7

Links

• Transformator4D program. Formation of models of three-dimensional projections of four-dimensional objects (including the Hypercube).
• A program that implements the construction of a tesseract and all its affine transformations, with C++ sources.

In English

• Mushware Limited is a tesseract output program ( Tesseract Trainer, licensed under GPLv2) and a 4D first-person shooter ( Adanaxis; graphics, mostly three-dimensional; there is a GPL version in the OS repositories).

Wikimedia Foundation. 2010 .

Teachings about multidimensional spaces began to appear in the middle of the 19th century in the works of G. Grassmann, A. Cayley, B. Riemann, W. Clifford, L. Schläfli and other mathematicians. At the beginning of the 20th century, with the advent of A. Einstein's theory of relativity and the ideas of G. Minkowski, physics began to use a four-dimensional space-time coordinate system.

Then science fiction writers borrowed the idea of ​​four-dimensional space from scientists. 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 the egg without damaging the shell, drink a drink without opening the cork of the bottle. The kidnappers retrieved the treasure from the safe through the fourth dimension. The links of the chain can be easily disconnected, and the knot on the rope can be untied without touching its ends. Surgeons performed operations on the internal organs without cutting the tissues of the patient's body. The mystics placed the souls of the dead in the fourth dimension. For ordinary person the idea of ​​a four-dimensional space has remained incomprehensible and mysterious, and many generally consider four-dimensional space to be the fruit of the imagination of scientists and science fiction writers, which has nothing to do with reality.

Perception problem

It is traditionally believed that a person cannot perceive and represent four-dimensional figures, since he is a three-dimensional being. The subject perceives three-dimensional figures with the help of the retina, which is two-dimensional. To perceive four-dimensional figures, a three-dimensional retina is needed, but a person does not have such an opportunity.

To get a visual representation of four-dimensional figures, we will use analogies from spaces of lower dimension for extrapolation to figures of higher dimension, use the modeling method, apply methods system analysis to search for patterns between elements of four-dimensional figures. The proposed models should adequately describe the properties of four-dimensional figures, do not contradict each other and give a sufficient idea of ​​a four-dimensional figure and, first of all, of its geometric shape. Since there is no systematic and visual description four-dimensional figures, and there are only their names indicating some properties, we propose to start the study of four-dimensional figures with the simplest - a four-dimensional cube, which is called a hypercube.

Hypercube Definition

hypercubea regular polytope is called, the cell of which is a cube.

Polytop is a four-dimensional figure, the boundary of which consists of polyhedra. An analogue of a cell of a polytope is a face of a polyhedron. The hypercube is analogous to a three-dimensional cube.

We will have an idea about the hypercube if we know its properties. The subject perceives some object, representing it in the form of some model. Let's use this method and present the idea of ​​a hypercube in the form of various models.

Analytical Model

We will consider a one-dimensional space (straight line) as an ordered set of pointsM(x), where xis the coordinate of an arbitrary point on the straight line. Then the unit segment is given by specifying two points:A(0) and B(1).

A plane (two-dimensional space) can be viewed as an ordered set of points M(x; y). The unit square will be completely defined by its four vertices: A(0; 0), B(1; 0), C(1; 1), D(0; 1). The coordinates of the vertices of the square are obtained by adding zero to the coordinates of the segment, and then one.

Three-dimensional space - an ordered set of points M(x; y; z). Eight points are required to define a 3D cube:

A(0; 0; 0), B(1; 0; 0), C(1; 1; 0), D(0; 1; 0),

E(0; 0; 1), F(1; 0; 1), G(1; 1; 1), H(0; 1; 1).

The cube coordinates are obtained from the square coordinates by adding zero and then one.

Four-dimensional space is an ordered set of points M(x; y; z; t). To specify a hypercube, you need to determine the coordinates of its sixteen vertices:

A(0; 0; 0; 0), B(1; 0; 0; 0), C(1; 1; 0; 0), D(0; 1; 0; 0),

E(0; 0; 1; 0), F(1; 0; 1; 0), G(1; 1; 1; 0), H(0; 1; 1; 0),

K(0; 0; 0; 1), L(1; 0; 0; 1), M(1; 1; 0; 1), N(0; 1; 0; 1),

O(0; 0; 1; 1), P(1; 0; 1; 1), R(1; 1; 1; 1), S(0; 1; 1; 1).

The hypercube coordinates are obtained from the coordinates of the 3D cube by adding a fourth coordinate equal to zero and then one.

Using the formulas of analytic geometry for the four-dimensional Euclidean space, one can obtain the properties of a hypercube.
As an example, consider the calculation of the length of the main diagonal of a hypercube. Let it be required to find the distance between points A(0, 0, 0, 0) and R(1, 1, 1, 1). To do this, we use the distance formula in four-dimensional Euclidean space.

In two-dimensional space (on a plane), the distance between points A(x 1 , y 1) and B(x 2 , y 2) is calculated by the formula

This formula follows from the Pythagorean theorem.

The corresponding formula for the distance between points A(x 1 , y 1 , z 1) and B(x 2 , y 2 , z 2) in three-dimensional space has the form

And in one-dimensional space (on a straight line) between points A( x 1) and B( x 2) you can write the corresponding distance formula:

Similarly, the distance between points A(x 1 , y 1 , z 1 , t 1) and B(x 2 , y 2 , z 2 , t 2) in four-dimensional space will be calculated by the formula:

For the proposed example, we find

Thus, the hypercube exists analytically, and its properties can be described no worse than the properties of a three-dimensional cube.

Dynamic Model

The analytical model of the hypercube is very abstract, so let's consider another model - the dynamic one.

A point (a zero-dimensional figure), moving in one direction, generates a segment (a one-dimensional figure). The segment, moving in a direction perpendicular to itself, creates a square (two-dimensional figure). The square, moving in a direction perpendicular to the plane of the square, creates a cube (three-dimensional figure).

The cube, moving perpendicular to the three-dimensional space in which it was originally located, generates a hypercube (four-dimensional figure).

The hypercube boundary is three-dimensional, finite and closed. It consists of a three-dimensional cube in the initial position, a three-dimensional cube in the final position, and six cubes formed by moving the squares of the original cube in the direction of the fourth dimension. The entire boundary of the hypercube consists of 8 three-dimensional cubes (cells).

When moving in the initial position, the cube had 8 vertices and in the final position also 8 vertices. Therefore, the hypercube has a total of 16 vertices.

Four mutually perpendicular edges emanate from each vertex. In total, the hypercube has 32 edges. In the initial position, it had 12 edges, in the final position also 12 edges, and 8 edges formed the tops of the cube when moving in the fourth dimension.

Thus, the border of the hypercube consists of 8 cubes, which consist of 24 squares. Namely, 6 squares in the initial position, 6 in the final position, and 12 squares formed by moving 12 edges in the direction of the fourth dimension.

geometric model

The dynamic model of a hypercube may seem insufficiently clear. Therefore, consider the geometric model of the hypercube. How do we get the geometric model of a 3D cube? We unfold it, and from the unfold we “glue” the cube model. The development of a three-dimensional cube consists of a square, to the sides of which is attached a square plus one more square. We turn adjacent squares around the sides of the square, and connect the adjacent sides of the squares to each other. And we close the remaining four sides with the last square (Fig. 1).

Similarly, consider the unfolding of the hypercube. Its development will be a three-dimensional figure, consisting of the original three-dimensional cube, six cubes adjacent to each face of the original cube, and one more cube. There are eight three-dimensional cubes in total (Fig. 2). In order to obtain a four-dimensional cube (hypercube) from this development, each of the adjacent cubes must be rotated by 90 degrees. These adjoining cubes will be located in a different 3D space. Connect adjacent faces (squares) of cubes to each other. Embed the eighth cube with its faces into the remaining unfilled space. We get a four-dimensional figure - a hypercube, the boundary of which consists of eight three-dimensional cubes.

Hypercube image

It was shown above how to “glue” a hypercube model from a three-dimensional sweep. We get images using projection. The central projection of a three-dimensional cube (its image on a plane) looks like this (Fig. 3). Inside the square is another square. The corresponding vertices of the square are connected by segments. Adjacent squares are depicted as trapezoids, although they are squares in 3D space. The inner and outer squares are different sizes, but in real 3D space they are equal squares.

Similarly, the central projection of a four-dimensional cube onto three-dimensional space will look like this: inside one cube is another cube. The corresponding vertices of the cubes are connected by segments. The inner and outer cubes have different sizes in 3D space, but they are equal cubes in 4D space (Figure 4).

Six truncated pyramids are images of equal six cells (cubes) of a four-dimensional cube.

This three-dimensional projection can be drawn on a plane and you can verify the truth of the properties of the hypercube obtained using the dynamic model.

The hypercube has 16 vertices, 32 edges, 24 faces (squares), 8 cells (cubes). Four mutually perpendicular edges emanate from each vertex. The boundary of the hypercube is a three-dimensional closed convex figure, the volume of which (the side volume of the hypercube) is equal to eight unit three-dimensional cubes. Inside itself, this figure contains a unit hypercube, the hypervolume of which is equal to the hypervolume of the unit hypercube.

Conclusion

In this work, the goal was to give an initial acquaintance with four-dimensional space. This was done on the example of the simplest figure - the hypercube.

The world of four-dimensional space is amazing! In it, along with similar figures in three-dimensional space, there are also figures that have no analogues in three-dimensional space.

Many phenomena of the material world, the macrocosm and the megaworld, despite the grandiose successes in physics, chemistry and astronomy, have remained inexplicable.

There is no single theory that explains all the forces of nature. There is no satisfactory model of the Universe that explains its structure and excludes paradoxes.

By knowing the properties of four-dimensional space and borrowing some ideas from four-dimensional geometry, it will be possible not only to build more rigorous theories and models of the material world, but also to create tools and systems that function according to the laws of the four-dimensional world, then human capabilities will be even more impressive.