Cubes in space. Cybercube is the first step into the fourth dimension. To three-dimensional space
In geometry hypercube- This n-dimensional analogy of a square ( n= 2) and cube ( n= 3). It is a closed convex figure consisting of groups of parallel lines located on opposite edges of the figure, and connected to each other at right angles.
This figure is also known as tesseract(tesseract). The tesseract is to the cube as the cube is to the square. More formally, a tesseract can be described as a regular convex four-dimensional polytope (polyhedron) whose boundary consists of eight cubic cells.
According to the Oxford English Dictionary, the word "tesseract" was coined in 1888 by Charles Howard Hinton and used in his book "A New Era of Thought." The word was derived from the Greek "τεσσερες ακτινες" ("four rays"), in the form of four coordinate axes. In addition, in some sources, the same figure was called tetracube(tetracube).
n-dimensional hypercube is also called n-cube.
A point is a hypercube of dimension 0. If you shift the point by a unit of length, you get a segment of unit length - a hypercube of dimension 1. Further, if you shift the segment by a unit of length in a direction perpendicular to the direction of the segment, you get a cube - a hypercube of dimension 2. Shifting the square by a unit of length in the direction perpendicular to the plane of the square, a cube is obtained - a hypercube of dimension 3. This process can be generalized to any number of dimensions. For example, if you move a cube by one unit of length in the fourth dimension, you get a tesseract.
The hypercube family is one of the few regular polyhedra that can be represented in any dimension.
Elements of a hypercube
Dimension hypercube n has 2 n“sides” (a one-dimensional line has 2 points; a two-dimensional square has 4 sides; a three-dimensional cube has 6 faces; a four-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is 2 n(for example, for a cube - 2 3 vertices).
Quantity m-dimensional hypercubes on the boundary n-cube equals
For example, on the boundary of a hypercube there are 8 cubes, 24 squares, 32 edges and 16 vertices.
| n-cube | Name | Vertex (0-face) |
Edge (1-face) |
Edge (2-face) |
Cell (3-face) |
(4-face) | (5-face) | (6-sided) | (7-face) | (8-face) |
|---|---|---|---|---|---|---|---|---|---|---|
| 0-cube | Dot | 1 | ||||||||
| 1-cube | Line segment | 2 | 1 | |||||||
| 2-cube | Square | 4 | 4 | 1 | ||||||
| 3-cube | Cube | 8 | 12 | 6 | 1 | |||||
| 4-cube | Tesseract | 16 | 32 | 24 | 8 | 1 | ||||
| 5-cube | Penteract | 32 | 80 | 80 | 40 | 10 | 1 | |||
| 6-cube | Hexeract | 64 | 192 | 240 | 160 | 60 | 12 | 1 | ||
| 7-cube | Hepteract | 128 | 448 | 672 | 560 | 280 | 84 | 14 | 1 | |
| 8-cube | Octeract | 256 | 1024 | 1792 | 1792 | 1120 | 448 | 112 | 16 | 1 |
| 9-cube | Eneneract | 512 | 2304 | 4608 | 5376 | 4032 | 2016 | 672 | 144 | 18 |
Projection onto a plane
The formation of a hypercube can be represented in the following way:
- Two points A and B can be connected to form a line segment AB.
- Two parallel segments AB and CD can be connected to form a square ABCD.
- Two parallel squares ABCD and EFGH can be connected to form a cube ABCDEFGH.
- Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to form the hypercube ABCDEFGHIJKLMNOP.
The latter structure is not easy to visualize, but it is possible to depict its projection into two-dimensional or three-dimensional space. Moreover, projections onto a two-dimensional plane can be more useful by allowing the positions of the projected vertices to be rearranged. In this case, it is possible to obtain images that no longer reflect the spatial relationships of the elements within the tesseract, but illustrate the structure of the vertex connections, as in the examples below.
The first illustration shows how, in principle, a tesseract is formed by joining two cubes. This scheme is similar to the scheme for creating a cube from two squares. The second diagram shows that all the edges of the tesseract are the same length. This scheme also forces you to look for cubes connected to each other. In the third diagram, the vertices of the tesseract are located in accordance with the distances along the faces relative to the bottom point. This scheme is interesting because it is used as a basic scheme for the network topology of connecting processors when organizing parallel computing: the distance between any two nodes does not exceed 4 edge lengths, and there are many different paths for balancing the load.
Hypercube in art
The hypercube has appeared in science fiction literature since 1940, when Robert Heinlein, in the story “And He Built a Crooked House,” described a house built in the shape of a tesseract scan. In the story, this Next, this house collapses, turning into a four-dimensional tesseract. After this, the hypercube appears in many books and short stories.
The movie Cube 2: Hypercube is about eight people trapped in a network of hypercubes.
Salvador Dali's painting "Crucifixion (Corpus Hypercubus)", 1954, depicts Jesus crucified on a tesseract scan. This painting can be seen in the Metropolitan Museum of Art in New York.
Conclusion
A hypercube is one of the simplest four-dimensional objects, from which one can see the complexity and unusualness of the fourth dimension. And what looks impossible in three dimensions is possible in four, for example, impossible figures. So, for example, the bars of an impossible triangle in four dimensions will be connected at right angles. And this figure will look like this from all viewing points, and will not be distorted, unlike the implementations of an impossible triangle in three-dimensional space (see.
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 clear, let's say we want to solve specific task, 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 the three-dimensional face of the hypercube where the cube came at the end of the construction of the hypercube.
Why are we so confident that we have received the correct description of the construction of 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.
Residents two-dimensional space built on a plane a frame for the projection of a cube onto a plane and invited us, three-dimensional inhabitants, 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 the entire internal frame, and they are left with only 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.
Bakalyar Maria
Methods of introducing the concept are studied four-dimensional cube(tesseract), its structure and some properties. The question is being solved of what three-dimensional objects are obtained when a four-dimensional cube is intersected by hyperplanes parallel to its three-dimensional faces, as well as hyperplanes perpendicular to its main diagonal. The apparatus of multidimensional analytical geometry used for research is considered.
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Introduction……………………………………………………………………………….2
Main part……………………………………………………………..4
Conclusions………….. ………………………………………………………..12
References………………………………………………………..13
Introduction
Four-dimensional space has long attracted the attention of both professional mathematicians and people far from studying this science. Interest in the fourth dimension may be due to the assumption that our three-dimensional world is “immersed” in four-dimensional space, just as a plane is “immersed” in three-dimensional space, the straight line is “immersed” in the plane, and the point is in the straight line. In addition, four-dimensional space plays an important role in the modern theory of relativity (the so-called space-time or Minkowski space), and can also be considered as a special casedimensional Euclidean space (with).
A four-dimensional cube (tesseract) is an object in four-dimensional space that has the maximum possible dimension (just as an ordinary cube is an object in three-dimensional space). Note that it is also of direct interest, namely, it may appear in optimization problems linear programming (as an area in which the minimum or maximum is sought linear function four variables), and is also used in digital microelectronics (when programming the operation of an electronic watch display). In addition, the very process of studying a four-dimensional cube contributes to the development of spatial thinking and imagination.
Consequently, the study of the structure and specific properties of a four-dimensional cube is quite relevant. It is worth noting that in terms of structure, the four-dimensional cube has been studied quite well. Much more interesting is the nature of its sections by various hyperplanes. Thus, the main goal of this work is to study the structure of the tesseract, as well as to clarify the question of what three-dimensional objects will be obtained if a four-dimensional cube is dissected by hyperplanes parallel to one of its three-dimensional faces, or by hyperplanes perpendicular to its main diagonal. A hyperplane in four-dimensional space will be called a three-dimensional subspace. We can say that a straight line on a plane is a one-dimensional hyperplane, a plane in three-dimensional space is a two-dimensional hyperplane.
The goal determined the objectives of the study:
1) Study the basic facts of multidimensional analytical geometry;
2) Study the features of constructing cubes of dimensions from 0 to 3;
3) Study the structure of a four-dimensional cube;
4) Analytically and geometrically describe a four-dimensional cube;
5) Make models of developments and central projections of three-dimensional and four-dimensional cubes.
6) Using the apparatus of multidimensional analytical geometry, describe three-dimensional objects resulting from the intersection of a four-dimensional cube with hyperplanes parallel to one of its three-dimensional faces, or hyperplanes perpendicular to its main diagonal.
The information obtained in this way will allow us to better understand the structure of the tesseract, as well as to identify deep analogies in the structure and properties of cubes of different dimensions.
Main part
First, we describe the mathematical apparatus that we will use during this study.
1) Vector coordinates: if, That
2) Equation of a hyperplane with a normal vector looks like Here
3) Planes and are parallel if and only if
4) The distance between two points is determined as follows: if, That
5) Condition for orthogonality of vectors:
First of all, let's find out how to describe a four-dimensional cube. This can be done in two ways - geometric and analytical.
If we talk about the geometric method of specifying, then it is advisable to trace the process of constructing cubes, starting from zero dimension. A cube of zero dimension is a point (note, by the way, that a point can also play the role of a ball of zero dimension). Next, we introduce the first dimension (the x-axis) and on the corresponding axis we mark two points (two zero-dimensional cubes) located at a distance of 1 from each other. The result is a segment - a one-dimensional cube. Let us immediately note a characteristic feature: The boundary (ends) of a one-dimensional cube (segment) are two zero-dimensional cubes (two points). Next, we introduce the second dimension (ordinate axis) and on the planeLet's construct two one-dimensional cubes (two segments), the ends of which are at a distance of 1 from each other (in fact, one of the segments is an orthogonal projection of the other). By connecting the corresponding ends of the segments, we obtain a square - a two-dimensional cube. Again, note that the boundary of a two-dimensional cube (square) is four one-dimensional cubes (four segments). Finally, we introduce the third dimension (applicate axis) and construct in spacetwo squares in such a way that one of them is an orthogonal projection of the other (the corresponding vertices of the squares are at a distance of 1 from each other). Let's connect the corresponding vertices with segments - we get a three-dimensional cube. We see that the boundary of a three-dimensional cube is six two-dimensional cubes (six squares). The described constructions allow us to identify the following pattern: at each stepthe dimensional cube “moves, leaving a trace” ine measurement at a distance of 1, while the direction of movement is perpendicular to the cube. It is the formal continuation of this process that allows us to arrive at the concept of a four-dimensional cube. Namely, we will force the three-dimensional cube to move in the direction of the fourth dimension (perpendicular to the cube) at a distance of 1. Acting similarly to the previous one, that is, by connecting the corresponding vertices of the cubes, we will obtain a four-dimensional cube. It should be noted that geometrically such a construction in our space is impossible (since it is three-dimensional), but here we do not encounter any contradictions from a logical point of view. Now let's move on to the analytical description of a four-dimensional cube. It is also obtained formally, using analogy. So, the analytical specification of a zero-dimensional unit cube has the form:
The analytical task of a one-dimensional unit cube has the form:
The analytical task of a two-dimensional unit cube has the form:
The analytical task of a three-dimensional unit cube has the form:
Now it is very easy to give an analytical representation of a four-dimensional cube, namely:
As we can see, both the geometric and analytical methods of defining a four-dimensional cube used the method of analogies.
Now, using the apparatus of analytical geometry, we will find out what the structure of a four-dimensional cube is. First, let's find out what elements it includes. Here again we can use an analogy (to put forward a hypothesis). The boundaries of a one-dimensional cube are points (zero-dimensional cubes), of a two-dimensional cube - segments (one-dimensional cubes), of a three-dimensional cube - squares (two-dimensional faces). It can be assumed that the boundaries of the tesseract are three-dimensional cubes. In order to prove this, let us clarify what is meant by vertices, edges and faces. The vertices of a cube are its corner points. That is, the coordinates of the vertices can be zeros or ones. Thus, a connection is revealed between the dimension of the cube and the number of its vertices. Let us apply the combinatorial product rule - since the vertexmeasured cube has exactlycoordinates, each of which is equal to zero or one (independent of all others), then in total there ispeaks Thus, for any vertex all coordinates are fixed and can be equal to or . If we fix all the coordinates (putting each of them equal or , regardless of the others), except for one, we obtain straight lines containing the edges of the cube. Similar to the previous one, you can count that there are exactlythings. And if we now fix all the coordinates (putting each of them equal or , regardless of the others), except for some two, we obtain planes containing two-dimensional faces of the cube. Using the rule of combinatorics, we find that there are exactlythings. Next, similarly - fixing all the coordinates (putting each of them equal or , independently of the others), except for some three, we obtain hyperplanes containing three-dimensional faces of the cube. Using the same rule, we calculate their number - exactlyetc. This will be sufficient for our research. Let us apply the results obtained to the structure of a four-dimensional cube, namely, in all the derived formulas we put. Therefore, a four-dimensional cube has: 16 vertices, 32 edges, 24 two-dimensional faces, and 8 three-dimensional faces. For clarity, let us define analytically all its elements.
Vertices of a four-dimensional cube:
Edges of a four-dimensional cube ():
Two-dimensional faces of a four-dimensional cube (similar restrictions):
Three-dimensional faces of a four-dimensional cube (similar restrictions):
Now that the structure of a four-dimensional cube and the methods for defining it have been described in sufficient detail, let us proceed to the implementation of the main goal - to clarify the nature of the various sections of the cube. Let's start with the elementary case when the sections of a cube are parallel to one of its three-dimensional faces. For example, consider its sections with hyperplanes parallel to the faceFrom analytical geometry it is known that any such section will be given by the equationLet us define the corresponding sections analytically:
As we can see, we have obtained an analytical specification for a three-dimensional unit cube lying in a hyperplane
To establish an analogy, let us write the section of a three-dimensional cube by a plane We get:
This is a square lying in a plane. The analogy is obvious.
Sections of a four-dimensional cube by hyperplanesgive completely similar results. These will also be single three-dimensional cubes lying in hyperplanes respectively.
Now let's consider sections of a four-dimensional cube with hyperplanes perpendicular to its main diagonal. First, let's solve this problem for a three-dimensional cube. Using the above-described method of defining a unit three-dimensional cube, he concludes that as the main diagonal one can take, for example, a segment with ends And . This means that the vector of the main diagonal will have coordinates. Therefore, the equation of any plane perpendicular to the main diagonal will be:
Let's determine the limits of parameter change. Because , then, adding these inequalities term by term, we obtain:
Or .
If , then (due to restrictions). Likewise - if, That . So, when and when cutting plane and cube have exactly one common point ( And respectively). Now let's note the following. If(again due to variable limitations). The corresponding planes intersect three faces at once, because, otherwise, the cutting plane would be parallel to one of them, which does not take place according to the condition. If, then the plane intersects all faces of the cube. If, then the plane intersects the faces. Let us present the corresponding calculations.
Let Then the planecrosses the line in a straight line, and . The edge, moreover. Edge the plane intersects in a straight line, and
Let Then the planecrosses the line:
edge in a straight line, and .
edge in a straight line, and .
edge in a straight line, and .
edge in a straight line, and .
edge in a straight line, and .
edge in a straight line, and .
This time we get six segments that have sequentially common ends:
Let Then the planecrosses the line in a straight line, and . Edge the plane intersects in a straight line, and . Edge the plane intersects in a straight line, and . That is, we get three segments that have pairwise common ends:Thus, for the specified parameter valuesthe plane will intersect the cube along regular triangle with peaks
So, here is a comprehensive description of the plane figures obtained when a cube is intersected by a plane perpendicular to its main diagonal. The main idea was as follows. It is necessary to understand which faces the plane intersects, along which sets it intersects them, and how these sets are related to each other. For example, if it turned out that the plane intersects exactly three faces along segments that have pairwise common ends, then the section is an equilateral triangle (which is proven by directly calculating the lengths of the segments), the vertices of which are these ends of the segments.
Using the same apparatus and the same idea of studying sections, the following facts can be deduced in a completely analogous way:
1) The vector of one of the main diagonals of a four-dimensional unit cube has the coordinates
2) Any hyperplane perpendicular to the main diagonal of a four-dimensional cube can be written in the form.
3) In the equation of a secant hyperplane, the parametercan vary from 0 to 4;
4) When and a secant hyperplane and a four-dimensional cube have one common point ( And respectively);
5) When the cross section will produce a regular tetrahedron;
6) When in cross-section the result will be an octahedron;
7) When the cross section will produce a regular tetrahedron.
Accordingly, here the hyperplane intersects the tesseract along a plane on which, due to the restrictions of the variables, a triangular region is distinguished (an analogy - the plane intersected the cube along a straight line, on which, due to the restrictions of the variables, a segment was distinguished). In case 5) the hyperplane intersects exactly four three-dimensional faces of the tesseract, that is, four triangles are obtained that have pairwise common sides, in other words, forming a tetrahedron (how this can be calculated is correct). In case 6), the hyperplane intersects exactly eight three-dimensional faces of the tesseract, that is, eight triangles are obtained that have sequentially common sides, in other words, forming an octahedron. Case 7) is completely similar to case 5).
Let us illustrate what has been said concrete example. Namely, we study the section of a four-dimensional cube by a hyperplaneDue to variable restrictions, this hyperplane intersects the following three-dimensional faces: Edge intersects along a planeDue to the limitations of the variables, we have:We get a triangular area with verticesFurther,we get a triangleWhen a hyperplane intersects a facewe get a triangleWhen a hyperplane intersects a facewe get a triangleThus, the vertices of the tetrahedron have the following coordinates. As is easy to calculate, this tetrahedron is indeed regular.
conclusions
So, in the process of this research, the basic facts of multidimensional analytical geometry were studied, the features of constructing cubes of dimensions from 0 to 3 were studied, the structure of a four-dimensional cube was studied, a four-dimensional cube was analytically and geometrically described, models of developments and central projections of three-dimensional and four-dimensional cubes were made, three-dimensional cubes were analytically described objects resulting from the intersection of a four-dimensional cube with hyperplanes parallel to one of its three-dimensional faces, or with hyperplanes perpendicular to its main diagonal.
The conducted research made it possible to identify deep analogies in the structure and properties of cubes of different dimensions. The analogy technique used can be applied in research, for example,dimensional sphere ordimensional simplex. Namely,a dimensional sphere can be defined as a set of pointsdimensional space equidistant from a given point, which is called the center of the sphere. Further,a dimensional simplex can be defined as a partdimensional space limited by the minimum numberdimensional hyperplanes. For example, a one-dimensional simplex is a segment (a part of one-dimensional space, limited by two points), a two-dimensional simplex is a triangle (a part of two-dimensional space, limited by three lines), a three-dimensional simplex is a tetrahedron (a part of three-dimensional space, limited by four planes). Finally,we define the dimensional simplex as the partdimensional space, limitedhyperplane of dimension.
Note that, despite the numerous applications of the tesseract in some areas of science, this research is still largely a mathematical study.
Bibliography
1) Bugrov Ya.S., Nikolsky S.M. Higher mathematics, vol. 1 – M.: Bustard, 2005 – 284 p.
2) Quantum. Four-dimensional cube / Duzhin S., Rubtsov V., No. 6, 1986.
3) Quantum. How to draw dimensional cube / Demidovich N.B., No. 8, 1974.
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 New era thoughts". 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), a cube has 6 of them (two faces from the moved square and four more that describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from its twelve edges.
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 for us, residents of three-dimensional space. For this we will use the already familiar method of analogies.
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 space three dimensions will look like two cubic “boxes” inserted into each other and connected by eight edges. In this case, the “boxes” themselves - three-dimensional faces - will be projected onto “our” space, and the lines connecting them will stretch in the fourth dimension. You can also try to imagine the cube not in projection, but in a spatial image.
Just as a three-dimensional cube is formed by a square shifted by the length of its face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in perspective will look like some rather complex figure. The part that remained in “our” space is drawn with solid lines, and the part that went into hyperspace is drawn with dotted lines. The four-dimensional hypercube itself consists of an infinite number of cubes, just as a three-dimensional cube can be “cut” into an infinite number of flat squares.
By cutting the six faces of a three-dimensional cube, you can decompose it into a flat figure - a development. It will have a square on each side of the original face, plus one more - the face opposite to it. And the three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes “growing” from it, plus one more - the final “hyperface”.
The properties of 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 a 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 “Black Hole School” 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.
Let's start by explaining what four-dimensional space is.
This is a one-dimensional space, that is, simply the OX axis. Any point on it is characterized by one coordinate.
Now let's draw the OY axis perpendicular to the OX axis. So we get a two-dimensional space, that is, the XOY plane. Any point on it is characterized by two coordinates - abscissa and ordinate.
Let's draw the OZ axis perpendicular to the OX and OY axes. The result is a three-dimensional space in which any point has an abscissa, ordinate and applicate.
It is logical that the fourth axis, OQ, should be perpendicular to the OX, OY and OZ axes at the same time. But we cannot accurately construct such an axis, and therefore we can only try to imagine it. Every point in four-dimensional space has four coordinates: x, y, z and q.
Now let's see how the four-dimensional cube appeared.
The picture shows a figure in one-dimensional space - a line.
If you make a parallel translation of this line along the OY axis, and then connect the corresponding ends of the two resulting lines, you will get a square.
Similarly, if you make a parallel translation of the square along the OZ axis and connect the corresponding vertices, you will get a cube.
And if we make a parallel translation of the cube along the OQ axis and connect the vertices of these two cubes, then we will get a four-dimensional cube. By the way, it's called tesseract.
To draw a cube on a plane, you need it project. Visually it looks like this: 
Let's imagine that it is hanging in the air above the surface wireframe model cube, that is, as if “made of wire,” and above it is a light bulb. If you turn on the light bulb, trace the shadow of the cube with a pencil, and then turn off the light bulb, a projection of the cube will be depicted on the surface.
Let's move on to something a little more complex. Look again at the drawing with the light bulb: as you can see, all the rays converge at one point. It is called vanishing point and is used to build perspective projection(and it can also be parallel, when all the rays are parallel to each other. The result is that the sensation of volume is not created, but it is lighter, and moreover, if the vanishing point is quite far removed from the projected object, then the difference between these two projections is little noticeable). To project this point on a given plane, using the vanishing point, you need to draw a straight line through the vanishing point and the given point, and then find the intersection point of the resulting straight line and the plane. And in order to project a more complex figure, say, a cube, you need to project each of its vertices, and then connect the corresponding points. It should be noted that algorithm for projecting space onto subspace can be generalized to the case of 4D->3D, not just 3D->2D.
As I said, we can't imagine exactly what the OQ axis looks like, just like the tesseract. But we can get a limited idea of it if we project it onto a volume and then draw it on a computer screen!
Now let's talk about the tesseract projection.
On the left is the projection of the cube onto the plane, and on the right is the tesseract onto the volume. They are quite similar: the projection of a cube looks like two squares, small and large, one inside the other, and whose corresponding vertices are connected by lines. And the projection of the tesseract looks like two cubes, small and large, one inside the other, and whose corresponding vertices are connected. But we have all seen a cube, and we can say with confidence that both a small square and a large one, and four trapezoids above, below, to the right and left of small square, are actually squares, and they are equal. And the tesseract has the same thing. And a large cube, and a small cube, and six truncated pyramids on the sides of a small cube - these are all cubes, and they are equal.
My program can not only draw the projection of a tesseract onto a volume, but also rotate it. Let's look at how this is done.
First, I'll tell you what it is rotation parallel to the plane.
Imagine that the cube rotates around the OZ axis. Then each of its vertices describes a circle around the OZ axis. 
A circle is a flat figure. And the planes of each of these circles are parallel to each other, and in in this case parallel to the XOY plane. That is, we can talk not only about rotation around the OZ axis, but also about rotation parallel to the XOY plane. As we see, for points that rotate parallel to the XOY axis, only the abscissa and ordinate change, while the applicate remains unchanged. And, in fact, we we can talk about rotation around a straight line only when we are dealing with three-dimensional space. In two-dimensional space everything rotates around a point, in four-dimensional space everything rotates around a plane, in five-dimensional space we talk about rotation around a volume. And if we can imagine rotation around a point, then rotation around a plane and volume is something unthinkable. And if we talk about rotation parallel to the plane, then in any n-dimensional space a point can rotate parallel to the plane.
Many of you have probably heard of the rotation matrix. Multiplying the point by it, we get a point rotated parallel to the plane by an angle phi. For two-dimensional space it looks like this: 
How to multiply: x of a point rotated by an angle phi = cosine of the angle phi*ix of the original point minus sine of the angle phi*ig of the original point;
ig of a point rotated by an angle phi = sine of the angle phi * ix of the original point plus cosine of the angle phi * ig of the original point.
Xa`=cosф*Xa - sinф*Ya
Ya`=sinф*Xa + cosф*Ya
, where Xa and Ya are the abscissa and ordinate of the point to be rotated, Xa` and Ya` are the abscissa and ordinate of the already rotated point
For three-dimensional space, this matrix is generalized as follows: 
Rotation parallel to the XOY plane. As you can see, the Z coordinate does not change, but only X and Y change
Xa`=cosф*Xa - sinф*Ya + Za*0
Ya`=sinф*Xa +cosф*Ya + Za*0
Za`=Xa*0 + Ya*0 + Za*1 (essentially, Za`=Za)
Parallel rotation XOZ plane. Nothing new,
Xa`=cosф*Xa + Ya*0 - sinф*Za
Ya`=Xa*0 + Ya*1 + Za*0 (essentially, Ya`=Ya)
Za`=sinф*Xa + Ya*0 + cosф*Za
And the third matrix.
Xa`=Xa*1 + Ya*0 + Za*0 (essentially, Xa`=Xa)
Ya`=Xa*0 + cosф*Ya - sinф*Za
Za`=Xa*0 + sinф*Ya + cosф*Za
And for the fourth dimension they look like this: 
I think you already understand what to multiply by, so I won’t go into detail again. But I note that it does the same thing as a matrix for rotation parallel to a plane in three-dimensional space! Both of them change only the ordinate and the applicate, and do not touch the other coordinates, so it can be used in the three-dimensional case, simply not paying attention to the fourth coordinate.
But with the projection formula, not everything is so simple. No matter how many forums I read, none of the projection methods worked for me. The parallel one was not suitable for me, since the projection would not look three-dimensional. In some projection formulas, to find a point you need to solve a system of equations (and I don’t know how to teach a computer to solve them), others I simply didn’t understand... In general, I decided to come up with my own way. For this purpose, consider the 2D->1D projection.
pov means "Point of view", ptp means "Point to project" (the point to be projected), and ptp` is the desired point on the OX axis.
Angles povptpB and ptpptp`A are equal as corresponding (the dotted line is parallel to the OX axis, the straight line povptp is a secant).
The x of the point ptp` is equal to the x of the point ptp minus the length of the segment ptp`A. This segment can be found from the triangle ptpptp`A: ptp`A = ptpA/tangent of angle ptpptp`A. We can find this tangent from the triangle povptpB: tangent ptpptp`A = (Ypov-Yptp)(Xpov-Xptp).
Answer: Xptp`=Xptp-Yptp/tangent of angle ptpptp`A.
I did not describe this algorithm in detail here, since there are a lot of special cases when the formula changes somewhat. If anyone is interested, look at the source code of the program, everything is described there in the comments.
In order to project a point in three-dimensional space onto a plane, we simply consider two planes - XOZ and YOZ, and solve this problem for each of them. In the case of four-dimensional space, it is necessary to consider three planes: XOQ, YOQ and ZOQ.
And finally, about the program. It works like this: initialize sixteen vertices of the tesseract -> depending on the commands entered by the user, rotate it -> project it onto the volume -> depending on the commands entered by the user, rotate its projection -> project onto the plane -> draw.
I wrote the projections and rotations myself. They work according to the formulas I just described. The OpenGL library draws lines and also handles color mixing. And the coordinates of the tesseract vertices are calculated in this way:
Coordinates of the vertices of a line centered at the origin and length 2 - (1) and (-1);
- " - " - square - " - " - and an edge of length 2:
(1; 1), (-1; 1), (1; -1) and (-1; -1);
- " - " - cube - " - " -:
(1; 1; 1), (-1; 1; 1), (1; -1; 1), (-1; -1; 1), (1; 1; -1), (-1; 1; -1), (1; -1; -1), (-1; -1; -1);
As you can see, a square is one line above the OY axis and one line below the OY axis; a cube is one square in front of the XOY plane, and one behind it; The tesseract is one cube on the other side of the XOYZ volume, and one on this side. But it is much easier to perceive this alternation of ones and minus ones if they are written in a column
1; 1; 1
-1; 1; 1
1; -1; 1
-1; -1; 1
1; 1; -1
-1; 1; -1
1; -1; -1
-1; -1; -1
In the first column, one and minus one alternate. In the second column, first there are two pluses, then two minuses. In the third - four plus ones, and then four minus ones. These were the vertices of the cube. The tesseract has twice as many of them, and therefore it was necessary to write a loop to declare them, otherwise it is very easy to get confused.
My program can also draw anaglyph. Happy owners of 3D glasses can observe a stereoscopic image. There is nothing tricky about drawing a picture; you simply draw two projections onto the plane, for the right and left eyes. But the program becomes much more visual and interesting, and most importantly, it gives a better idea of the four-dimensional world.
Less significant functions are the illumination of one of the edges in red so that turns can be better seen, as well as minor conveniences - regulation of the coordinates of the “eye” points, increasing and decreasing the turning speed.
Archive with the program, source code and instructions for use.
