What is a cube in 4 dimensions called? Cybercube is the first step into the fourth dimension. Hypercube and Platonic solids

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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 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 flat figure- scan. 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 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, as 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,” opened up the existence of parallel worlds to 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' travels to different variants described by science fiction writers of the future - in contrast to the travel to the world that already existed in science fiction various options of the past.

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

Hypercube and Platonic solids

Model a truncated icosahedron (“ soccer ball»)
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, proven 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:

Polyhedron name
Triangular pyramid
Quadrangular pyramid
Triangular prism
Quadrangular prism
n-coal pyramid	n+1	2n	n+1
n-carbon prism	2n	3n	n+2
n-coal truncated pyramid	2n	3n	n+2

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 death 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 exact time destruction 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 in no way 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”.

If you're a fan of the Avengers movies, the first thing that might come to 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 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. Therefore we say that real life 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.

A 3D projection of a tesseract performing a double rotation around two orthogonal planes.
Image: Jason Hise

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.

Mathematically, the tesseract is a perfectly precise shape. Likewise, all forms are more high dimensions, that is, 5D and 6D, are also mathematically plausible.

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

Surprising and incomprehensible, isn't it?

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

The evolution of the human brain took place in three-dimensional space. Therefore, it is difficult for us to imagine spaces with dimensions greater than three. In fact, the human brain cannot imagine geometric objects with dimensions greater than three. And at the same time, we can easily imagine geometric objects with dimensions not only three, but also with dimensions two and one.

The difference and analogy between one-dimensional and two-dimensional spaces, as well as the difference and analogy between two-dimensional and three-dimensional spaces allow us to slightly open the screen of mystery that fences us off from spaces of higher dimensions. To understand how this analogy is used, consider a very simple four-dimensional object - a hypercube, that is, a four-dimensional cube. To be specific, let’s say we want to solve a specific problem, namely, count the number of square faces of a four-dimensional cube. All further consideration will be very lax, without any evidence, purely by analogy.

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

To build a cube from a subcube, you need to extend the subcube in a direction perpendicular to the plane of the subcube in the direction of the third dimension. In this case, from each side of the initial subcube a subcube will grow, which is the side two-dimensional face of the cube, which will limit the three-dimensional volume of the cube on four sides, two perpendicular to each direction in the plane of the subcube. And along the new third axis there are also two subcubes that limit the three-dimensional volume of the cube. This is the two-dimensional face where our subcube was originally located and that two-dimensional face of the cube where the subcube came at the end of the construction of the cube.

What you have just read is presented in excessive detail and with a lot of clarifications. And for good reason. Now we will do such a trick, we will formally replace some words in the previous text in this way:
cube -> hypercube
subcube -> cube
plane -> volume
third -> fourth
two-dimensional -> three-dimensional
four -> six
three-dimensional -> four-dimensional
two -> three
plane -> space

As a result, we get the following meaningful text, which no longer seems overly detailed.

To build a hypercube from a cube, you need to stretch the cube in a direction perpendicular to the volume of the cube in the direction of the fourth dimension. In this case, a cube will grow from each side of the original cube, which is the lateral three-dimensional face of the hypercube, which will limit the four-dimensional volume of the hypercube on six sides, three perpendicular to each direction in the space of the cube. And along the new fourth axis there are also two cubes that limit the four-dimensional volume of the hypercube. This is the three-dimensional face where our cube was originally located and that three-dimensional face of the hypercube where the cube came at the end of the construction of the hypercube.

Why do we have such confidence that we have received correct description building a hypercube? Yes, because by exactly the same formal substitution of words we get a description of the construction of a cube from a description of the construction of a square. (Check it out for yourself.)

Now it is clear that if another three-dimensional cube should grow from each side of the cube, then a face should grow from each edge of the initial cube. In total, the cube has 12 edges, which means that an additional 12 new faces (subcubes) will appear on those 6 cubes that limit the four-dimensional volume along the three axes of three-dimensional space. And there are two more cubes left that limit this four-dimensional volume from below and above along the fourth axis. Each of these cubes has 6 faces.

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

The following picture shows the logical structure of a hypercube. This is like a projection of a hypercube onto three-dimensional space. This produces a three-dimensional frame of ribs. In the figure, naturally, you see the projection of this frame onto a plane.

On this frame, the inner cube is like the initial cube from which the construction began and which limits the four-dimensional volume of the hypercube along the fourth axis from the bottom. We stretch this initial cube upward along the fourth axis of measurement and it goes into the outer cube. So the outer and inner cubes from this figure limit the hypercube along the fourth axis of measurement.

And between these two cubes you can see 6 more new cubes, which touch common faces with the first two. These six cubes bound our hypercube along the three axes of three-dimensional space. As you can see, they are not only in contact with the first two cubes, which are the inner and outer cubes on this three-dimensional frame, but they are also in contact with each other.

You can count directly in the figure and make sure that the hypercube really has 24 faces. But this question arises. This hypercube frame in three-dimensional space is filled with eight three-dimensional cubes without any gaps. To make a real hypercube from this three-dimensional projection of a hypercube, you need to turn this frame inside out so that all 8 cubes bound a 4-dimensional volume.

It's done like this. We invite a resident of four-dimensional space to visit us and ask him to help us. He grabs the inner cube of this frame and moves it in the direction of the fourth dimension, which is perpendicular to our three-dimensional space. In our three-dimensional space, we perceive it as if the entire internal frame had disappeared and only the frame of the outer cube remained.

Further, our four-dimensional assistant offers his assistance in maternity hospitals for painless childbirth, but our pregnant women are frightened by the prospect that the baby will simply disappear from the stomach and end up in parallel three-dimensional space. Therefore, the four-dimensional person is politely refused.

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

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

The inhabitants of two-dimensional space built a frame on a plane for the projection of a cube onto a plane and invited us, three-dimensional residents, to turn this frame inside out. We take the four vertices of the inner square and move them perpendicular to the plane. Two-dimensional residents see the complete disappearance of everything internal frame, and they only have the frame of the outer square. With such an operation, all the squares that were in contact with their edges continue to touch with the same edges.

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

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

Tesseract (from ancient Greek τέσσερες ἀκτῖνες - four rays) is a four-dimensional hypercube - an analogue of a cube in four-dimensional space.

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

According to the Oxford Dictionary, the word "tesseract" was coined and used in 1888 by Charles Howard Hinton (1853–1907) in his book A New Age of Thought. Later, some people called the same figure a "tetracube".

Geometry

An ordinary tesseract in Euclidean four-dimensional space is defined as a convex hull of 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, the tesseract has 8 3D faces, 24 2D faces, 32 edges and 16 vertices.

Popular 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 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.
http://upload.wikimedia.org/wikipedia/ru/1/13/Construction_tesseract.PNG

The one-dimensional segment AB serves as the side of the two-dimensional square ABCD, the square - 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, and 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), the 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.

Tesseract unwrapping

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 remains 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 the tesseract are an extension of the properties geometric shapes smaller dimension into four-dimensional space.

Projections

To two-dimensional space

This structure is difficult to imagine, but it is possible to project a tesseract into two-dimensional or three-dimensional spaces. In addition, projecting 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:

To three-dimensional space

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

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

A stereo pair of a tesseract is depicted as two projections onto three-dimensional space. This image 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 appears that reproduces the depth of the tesseract.

Tesseract unwrapping

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 tesseract designs. The unfolding of a tesseract can be calculated by plotting the connected angles on a graph.

Tesseract in art

In Edwina A.'s "New Abbott Plain", the hypercube acts as a narrator.
In one episode of The Adventures of Jimmy Neutron: "Boy Genius", Jimmy invents a four-dimensional hypercube identical to the foldbox from Heinlein's 1963 novel Glory Road.
Robert E. Heinlein has mentioned hypercubes in at least three science fiction stories. In The House of Four Dimensions (The House That Teal Built) (1940), he described a house built like an unwrapped tesseract.
Heinlein's novel Glory Road describes hyper-sized dishes that were larger on the inside than on the outside.
Henry Kuttner's story "Mimsy Were the Borogoves" describes an educational toy for children from the distant future, similar in structure to a tesseract.
In the novel by Alex Garland (1999), the term "tesseract" is used for the three-dimensional unfolding of a four-dimensional hypercube, rather than the hypercube itself. This is a metaphor designed to show that the cognitive system must be broader than the knowable.
The plot of Cube 2: Hypercube centers on eight strangers trapped in a "hypercube", or network of connected cubes.
The television series Andromeda uses tesseract generators as a plot device. They are primarily designed to manipulate space and time.
Painting “The Crucifixion” (Corpus Hypercubus) by Salvador Dali (1954)
The Nextwave comic book depicts a vehicle that includes 5 tesseract zones.
In the album Voivod Nothingface one of the compositions is called “In my hypercube”.
In Anthony Pearce's novel Route Cube, one of the International Development Association's orbiting 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 a secret button and the school begins to take shape like a mathematical tesseract.
The term “tesseract” and its derivative term “tesserate” are found in the story “A Wrinkle in Time” by Madeleine L’Engle.