Four-dimensional cube. Cybercube - the first step into the fourth dimension

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 more than three dimensions. 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, and 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. Let, for definiteness, let's say we want to solve a specific problem, namely, to count the number of square faces of a four-dimensional cube. All consideration below will be very loose, without any evidence, purely by analogy.

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

To construct a cube from a subcube, it is necessary to extend the subcube in a direction perpendicular to the plane of the subcube in the direction of the third dimension. At the same time, a subcube will grow from each side of the original subcube, which is a two-dimensional side face of the cube, which will limit the three-dimensional volume of the cube from four sides, two perpendicular to each direction in the plane of the subcube. And along the new third axis, there are also two subcubes that limit the three-dimensional volume of the cube. This is the two-dimensional face where our subcube was originally located and the two-dimensional face of the cube where the subcube came at the end of the cube construction.

What you have just read is set out in excessive detail and with a lot of clarifications. And not casually. Now we will do such a trick, we will replace some words in the previous text formally in this way:
cube -> hypercube
subcube -> cube
plane -> volume
third -> fourth
2D -> 3D
four -> six
three-dimensional -> four-dimensional
two -> three
plane -> space

As a result, we get the following meaningful text, which no longer seems too 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. At the same time, 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 from six sides, three perpendicular to each direction in the space of the cube. And along the new fourth axis, there are also two cubes that limit the four-dimensional volume of the hypercube. This is the three-dimensional face where our cube was originally located and the three-dimensional face of the hypercube, where the cube came at the end of the hypercube construction.

Why do we have such confidence that we have received the correct description of the construction of the hypercube? Yes, because by exactly the same formal replacement 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 must grow from each edge of the initial cube. In total, the cube has 12 edges, which means that there will be an additional 12 new faces (subcubes) for those 6 cubes that limit the four-dimensional volume along the three axes of three-dimensional space. And there are two more cubes that limit this four-dimensional volume from below and from above along the fourth axis. Each of these cubes has 6 faces.

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

The following picture shows the logical structure of a hypercube. It is like a projection of a hypercube onto three-dimensional space. In this case, a three-dimensional frame of ribs is obtained. In the figure, of course, you see the projection of this frame also onto a plane.

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

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

You can calculate directly in the figure and make sure that the hypercube really has 24 faces. But here comes the question. This 3D hypercube frame is filled with eight 3D cubes without any gaps. In order to make a real hypercube from this 3D projection of a hypercube, it is necessary to turn this frame inside out so that all 8 cubes limit the 4D volume.

It is done like this. We invite a resident of the four-dimensional space to visit and ask him to help us. It grabs the inner cube of this framework and shifts it towards the fourth dimension, which is perpendicular to our 3D space. We in our three-dimensional space perceive it as if the entire inner frame had disappeared and only the frame of the outer cube remained.

Next, our 4D assistant offers to help in maternity hospitals for a painless birth, but our pregnant women are terrified at the prospect of the baby simply disappearing from the abdomen and ending up in a parallel 3D space. Therefore, the fourfold is politely refused.

And we're wondering if some of our cubes got unstuck when the hypercube frame was turned inside out. After all, if some three-dimensional cubes surrounding the hypercube touch their neighbors on the frame, will they also touch the same faces if the four-dimensional one turns the frame inside out.

Let us again turn to the analogy with spaces of lower dimension. Compare the image of the hypercube wireframe with the projection of the 3D cube onto the plane shown in the following picture.

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

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

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

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

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

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

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

What is a "measurement"?

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

A dimension is simply a direction you can go. For example, if you are drawing a line on a piece of paper, you can either go left/right (x-axis) or up/down (y-axis). So we say the paper is two-dimensional since you can only walk 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 "in/out". Consequently, a sense of depth is added in 3D space. Therefore we say that real life 3-dimensional.

A point can represent 0 dimensions (because it doesn't 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 face (which is currently a square) 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 this is the 4D equivalent 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; so each of its corners has 2 lines extending from it at 90 degrees to each other. The cube is 3D, so each of its corners has 3 lines coming off of 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?

Because we as humans have evolved to visualize objects in three dimensions, anything that goes into extra dimensions like 4D, 5D, 6D, etc. has no meaning for us. great sense because we can't imagine them at all. 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.

September 19th, 2009
Tesseract (from other Greek τέσσερες ἀκτῖνες - four rays) - a four-dimensional hypercube - an analogue of a cube in four-dimensional space.

The image is a projection (perspective) four-dimensional cube to 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 " new era thoughts". Later, some people called the same figure "tetracube".

Geometry

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

The tesseract is 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. 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.
http://upload.wikimedia.org/wikipedia/ru/1/13/Build_tesseract.PNG

The one-dimensional segment AB serves as a side of the two-dimensional square ABCD, the square is the side of the cube ABCDHEFG, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 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 a larger number of dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, inhabitants of three-dimensional space. Let us use for this the already familiar method of analogies.

Tesseract unfolding

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 pretty complex figure. Its part, remaining in "our" space, is drawn solid lines, and what went into hyperspace is dotted. 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, one can decompose it into flat figure- a sweep. It will have a square on each side of the original face, plus one more - the face opposite to it. A three-dimensional 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.

projections

to two-dimensional space

This structure is hard 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, images can be obtained that no longer reflect the spatial relationships within the tesseract, but which illustrate the vertex link structure, as in the following examples:


to three-dimensional space

The projection 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.
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 Heinlein's 1963 Glory Road.
Robert E. Heinlein has mentioned hypercubes in at least three science fiction stories. In The House of Four Dimensions (The House That Teel Built) (1940), he described a house built as an unfolding of a tesseract.
In Heinlein's novel Glory Road, hyper-sized dishes are described that were larger on the inside than on the outside.
Henry Kuttner's short story "Mimsy Were the Borogoves" describes an educational toy for children from the distant future, similar in structure to the 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 cognizing system should be wider than the cognizable one.
The plot of Cube 2: Hypercube centers on eight strangers trapped in a "hypercube", or network of connected cubes.
The TV series Andromeda uses tesseract generators as a conspiracy device. They are primarily intended to control space and time.
The painting "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 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"

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

This figure is also known as tesseract(tesseract). The tesseract is to the cube as the cube is to the square. More formally, a tesseract can be described as a regular convex four-dimensional polytope (polytope) whose boundary consists of eight cubic cells.

According to the Oxford English Dictionary, the word "tesseract" was coined in 1888 by Charles Howard Hinton and used in his book A New Era of Thought. The word was formed from the Greek "τεσσερες ακτινες" ("four rays"), is in the form of four coordinate axes. In addition, in some sources, the same figure was called tetracube(tetracube).

n-dimensional hypercube is also called n-cube.

A point is a hypercube of dimension 0. If you shift a point by a unit of length, you get a segment of unit length - a hypercube of dimension 1. Further, if you shift a segment by a unit of length in a direction perpendicular to the direction of the segment, you get a cube - a hypercube of dimension 2. Shifting the square by a unit of length in the direction perpendicular to the plane of the square, a cube is obtained - a hypercube of dimension 3. This process can be generalized to any number of dimensions. For example, if you shift a cube by a unit of length in the fourth dimension, you get a tesseract.

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

Hypercube elements

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

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

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

Elements of hypercubes

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

Plane projection

The formation of a hypercube can be represented in the following way:

• Two points A and B can be connected to form line segment AB.
• Two parallel segments AB and CD can be connected to form a square ABCD.
• Two parallel squares ABCD and EFGH can be joined to form the cube ABCDEFGH.
• Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to form a hypercube ABCDEFGHIJKLMNOP.

The latter structure is not easy to imagine, but it is possible to depict its projection onto two or three dimensions. Moreover, projections onto a 2D plane can be more useful by rearranging the positions of the projected vertices. In this case, images can be obtained that no longer reflect the spatial relationships of the elements within the tesseract, but illustrate the structure of the vertex connections, as in the examples below.

The first illustration shows how a tesseract is formed in principle by joining two cubes. This scheme is similar to the scheme for creating a cube from two squares. The second diagram shows that all the edges of the tesseract have the same length. This scheme is also forced to look for cubes connected to each other. In the third diagram, the vertices of the tesseract are located in accordance with the distances along the faces relative to the bottom point. This scheme is interesting because it is used as the basic scheme for the network topology of connecting processors in organizing parallel computing: the distance between any two nodes does not exceed 4 edge lengths, and there are many different ways to balance the load.

Hypercube in art

The hypercube has appeared in science fiction since 1940, when Robert Heinlein, in the story "The House That Teal Built" ("And He Built a Crooked House"), described a house built in the shape of a tesseract unfold. In the story, this Further, this house is folded up, turning into a four-dimensional tesseract. After that, the hypercube appears in many books and novels.

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

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

Conclusion

The hypercube is one of the simplest four-dimensional objects, on the example of which you can see all the complexity and unusualness of the fourth dimension. And what looks impossible in three dimensions is possible in four, for example, impossible figures. So, for example, the bars of an impossible triangle in four dimensions will be connected at right angles. And this figure will look like this from all viewpoints, and will not be distorted, unlike the implementations of the impossible triangle in three-dimensional space (see Fig.