Teseract. Four-dimensional space. Hypercube

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.

Elements of hypercubes

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 circuit is interesting because it is used as basic diagram 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 ways to balance 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.

Bakalyar Maria

Methods for introducing the concept of a four-dimensional cube (tesseract), its structure and some properties are studied. The question 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 is addressed. 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, a straight line is “immersed” in a plane, and a point is in a straight line. In addition, four-dimensional space plays important role V modern theory 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 can appear in optimization problems linear programming(as an area in which the minimum or maximum of a linear function of four variables is found), and is also used in digital microelectronics (when programming the operation of the display electronic watch). 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 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 , independently 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 methods for defining it have been described in sufficient detail, let’s proceed to implementation main goal– clarifying 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 get:

Or .

If , then (due to restrictions). Likewise - if, That . So, when and when the cutting plane and the 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 limitations of the variables, a triangular region is allocated (an analogy - the plane intersected the cube along a straight line, on which, due to the constraints of the variables, a segment was allocated). 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 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.

September 19th, 2009
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 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 the future will look like some kind of pretty 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 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”.

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 consists of 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 Voivod Nothingface album 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.

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

Then 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. The links of the chain can be easily disconnected, and the knot on the rope can be undone without touching its ends. Surgeons performed operations on internal organs without cutting the patient's body tissue. Mystics placed the souls of the departed in the fourth dimension. For ordinary person the idea of ​​four-dimensional space has remained incomprehensible and mysterious, and many generally consider four-dimensional space to be a figment of the imagination of scientists and science fiction writers, which has nothing to do with reality.

Problem of perception

It is traditionally believed that a person cannot perceive and imagine four-dimensional figures, since he is a three-dimensional being. The subject perceives three-dimensional figures using the retina, which is two-dimensional. To perceive four-dimensional figures, a three-dimensional retina is needed, but humans do not have this ability.

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

Definition of a hypercube

Hypercubeis a regular polytope whose cell is a cube.

Polytope is a four-dimensional figure whose boundary consists of polyhedra. An analogue of a polytope cell is the face of a polyhedron. A hypercube is an analogue of a three-dimensional cube.

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

Analytical model

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

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

Three-dimensional space - an ordered set of points M(x; y; z). To define a three-dimensional cube, eight points are required:

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

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

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

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

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

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

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

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

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

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

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

This formula follows from the Pythagorean theorem.

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

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

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

For the proposed example we find

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

Dynamic model

The analytical model of a hypercube is very abstract, so let’s consider another model – a dynamic one.

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

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

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

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

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

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

Geometric model

The dynamic model of a hypercube may not seem clear enough. Therefore, let's consider the geometric model of a hypercube. How do we obtain a geometric model of a 3D cube? We make a development of it, and from the development we “glue together” a model of the cube. The development of a three-dimensional cube consists of a square, to the sides of which is attached a square plus another square. We rotate the adjacent squares around the sides of the square, and connect the adjacent sides of the squares to each other. And we close the remaining four sides with the last square (Fig. 1).

Let us similarly consider the development of a hypercube. Its development will be a three-dimensional figure consisting of the original three-dimensional cube, six cubes adjacent to each face of the original cube and one more cube. There are eight three-dimensional cubes in total (Fig. 2). To obtain a four-dimensional cube (hypercube) from this development, you need to rotate each of the adjacent cubes by 90 degrees. These adjacent cubes will be located in a different 3D space. Connect adjacent faces (squares) of cubes to each other. Place the eighth cube with its faces in the remaining empty space. We get a four-dimensional figure - a hypercube, the boundary of which consists of eight three-dimensional cubes.

Image of a hypercube

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

Similarly, the central projection of a four-dimensional cube onto three-dimensional space will look like this: inside one cube there is another cube. The corresponding vertices of the cubes 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 (Fig. 4).

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

This three-dimensional projection can be drawn on a plane and verified that the properties of the hypercube obtained using the dynamic model are true.

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

Conclusion

The goal of this work was to provide an initial introduction to four-dimensional space. This was done using the example of the simplest figure - a hypercube.

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

Many phenomena of the material world, the macroworld and the megaworld, despite the tremendous successes in physics, chemistry and astronomy, remained unexplained.

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

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

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 version 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.