Blog of the ideal wife. Program for drawing a four-dimensional cube

Let's start by explaining what a 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 got a two-dimensional space, that is, the XOY plane. Any point on it is characterized by two coordinates - the abscissa and the ordinate.

Let's draw the OZ axis perpendicular to the axes OX and OY. You will get a three-dimensional space in which any point has an abscissa, an ordinate and an applicate.

It is logical that the fourth axis, OQ, should be perpendicular to the axes OX, OY and OZ at the same time. But we cannot accurately construct such an axis, and therefore it remains only to 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 of 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 get a square.

Similarly, if we make a parallel translation of the square along the OZ axis and connect the corresponding vertices, we 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:

Imagine that in the air above the surface hangs wireframe model cube, that is, as if "made of wire", and above it - 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, then a projection of the cube will be shown on the surface.

Let's move on to something a little more complicated. Look again at the drawing with the light bulb: as you can see, all the rays converged at one point. It is called vanishing point and is used to build perspective projection(and sometimes parallel, when all the rays are parallel to each other. The result is that there is no sense of volume, but it is lighter, and if the vanishing point is far enough away from the projected object, then the difference between these two projections is hardly noticeable). To project given 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 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 space-to-subspace projection algorithm can be generalized to 4D->3D, not just 3D->2D.

As I said, we can't imagine exactly what the OQ axis looks like, and neither can 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 projection of the tesseract.

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, with corresponding vertices 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 the cube, and we can say with confidence that both the small square and the large one, and the four trapezoids above, below, to the right and left of the small square, are in fact squares, moreover, they are equal. The same goes for the Tesseract. And a big 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 the tesseract onto the volume, but also rotate it. Let's see how this is done.

First, I'll tell you what 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 this case they are 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 you can 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 2D everything revolves around a point, in 4D everything revolves around a plane, in 5D space we are talking about rotation around a volume. And if we can imagine the rotation around a point, then the rotation around the 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 a point by it, we get a point rotated parallel to the plane by an angle phi. For a two-dimensional space, it looks like this:

How to multiply: x of a point rotated by an angle phi = cosine of the angle phi*x of the original point minus the sine of the angle phi*y of the original point;
y of the point rotated by the angle phi=sine of the angle phi*x of the original point plus cosine of the angle phi*y 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 a 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)

Rotation parallel to the XOZ plane. Nothing new,
Xa`=cosФ*Xa + Ya*0 - sinФ*Za
Ya`=Xa*0 + Ya*1 + Za*0 (in fact, 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 understood what to multiply by, so I won’t paint it again. But I note that it does the same as the matrix for rotating parallel to the plane in three-dimensional space! Both that and this one change only the ordinate and the applicate, and the rest of the coordinates are not touched, therefore it can be used in the three-dimensional case, simply ignoring the fourth coordinate.

But with the projection formula, not everything is so simple. No matter how much I read the forums, none of the projection methods suited me. Parallel did not suit me, since the projection will 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), I simply didn’t understand others ... In general, I decided to come up with my own way. Consider for this the projection 2D->1D.

pov means "Point of view" (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 (dashed line is parallel to axis OX, line povptp is secant).
The x of ptp` is equal to the x of ptp minus the length of 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 triangle povptpB: tangent of angle 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 where the formula changes somewhat. Who cares - look in the source code of the program, everything is written 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 a four-dimensional space, it is necessary to consider already 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 onto the volume -> depending on the commands entered by the user, rotate its projection -> project onto a plane -> draw.

Projections and rotations I wrote myself. They work according to the formulas that I just described. The OpenGL library draws lines and also mixes colors. And the coordinates of the vertices of the tesseract are calculated in this way:

Line vertex coordinates centered at the origin and length 2 - (1) and (-1);
- "-" - a 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, the 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; a 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 units and minus units 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 one, and then four minus one. These were the tops of the cube. The tesseract has twice as many of them, and therefore it was necessary to write a cycle for declaring them, otherwise it is very easy to get confused.

My program also knows how to draw anaglyph. Happy owners of 3D glasses can watch a stereoscopic picture. There is nothing tricky in drawing a picture, it just draws two projections on a plane, for the right and left eyes. But the program becomes much more visual and interesting, and most importantly - gives a better idea of ​​the four-dimensional world.

Less significant functions - highlighting one of the faces in red, so that you can better see the turns, as well as minor conveniences - adjusting the coordinates of the "eye" points, increasing and decreasing the speed of rotation.

Archive with the program, source code and instructions for use.

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

The tesseract is limited by eight hyperplanes, the intersection of which with the tesseract itself defines its three-dimensional faces (which are ordinary cubes). Each pair of non-parallel 3D faces intersect to form 2D faces (squares), and so on. Finally, a tesseract has 8 3D faces, 24 2D, 32 edges and 16 vertices.

Popular Description

Let's try to imagine how the hypercube will look without leaving the three-dimensional space.

In one-dimensional "space" - on a line - we select a segment AB of length L. On a two-dimensional plane at a distance L from AB, we draw a segment DC parallel to it and connect their ends. You will get a square CDBA. Repeating this operation with a plane, we get a three-dimensional cube CDBAGHFE. And by shifting the cube in the fourth dimension (perpendicular to the first three) by a distance L, we get the CDBAGHFEKLJIOPNM hypercube.

Construction of a tesseract on a plane

The one-dimensional segment AB serves as a side of the two-dimensional square CDBA, the square is the side of the cube CDBAGHFE, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 vertices shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and 8 more edges "draw" eight of its vertices that have moved into the fourth dimension. The same reasoning can be done for the faces of the hypercube. In two-dimensional space, it is one (the square itself), the cube has 6 of them (two faces from the moved square and four more will describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from twelve of its edges.

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

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

Let's take the wire cube ABCDHEFG and look at it with one eye from the side of the face. We will see and can draw two squares on the plane (its near and far faces), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic "boxes" inserted into each other and connected by eight edges. In this case, the "boxes" themselves - three-dimensional faces - will be projected onto "our" space, and the lines connecting them will stretch in the direction of the fourth axis. You can also try to imagine a cube not in projection, but in a spatial image.

Just as a three-dimensional cube is formed by a square shifted by the length of a face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in the future will look like some rather complex figure. The four-dimensional hypercube itself consists of an infinite number of cubes, just as a three-dimensional cube can be “cut” into an infinite number of flat squares.

By cutting six faces of a three-dimensional cube, you can decompose it into a flat figure - a development. It will have a square on each side of the original face, plus one more - the face opposite to it. A three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes that "grow" from it, plus one more - the final "hyperface".

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

projections

to two-dimensional space

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

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

to three-dimensional space

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

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

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

stereo pair

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

Tesseract unfolding

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

Tesseract in art

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

Other names

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

Notes

Literature

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

Links

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

In English

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

Wikimedia Foundation. 2010 .

Hypercube and Platonic Solids

Simulate a truncated icosahedron (“soccer ball”) in the “Vector” system
where each pentagon is bounded by hexagons

Truncated icosahedron can be obtained by cutting 12 vertices to form faces in the form of regular pentagons. In this case, the number of vertices of the new polyhedron increases 5 times (12×5=60), 20 triangular faces turn into regular hexagons (total faces becomes 20+12=32), A the number of edges increases to 30+12×5=90.

Steps for constructing a truncated icosahedron in the Vector system

Figures in 4-dimensional space.

--à

--à ?

For example, given a cube and a hypercube. There are 24 faces in a hypercube. This means that a 4-dimensional octahedron will have 24 vertices. Although no, the hypercube has 8 faces of cubes - in each center is a vertex. This means that a 4-dimensional octahedron will have 8 vertices of that one easier.

4-dimensional octahedron. It consists of eight equilateral and equal tetrahedra,
connected four at each vertex.

Rice. An attempt to simulate
hyperball-hypersphere in the "Vector" system

Front - back faces - balls without distortion. Another six balls - can be specified through ellipsoids or quadratic surfaces (through 4 contour lines as generators) or through faces (first defined through generators).

More tricks to "build" a hypersphere
- the same "soccer ball" in 4-dimensional space

Annex 2

For convex polyhedra, there is a property relating the number of its vertices, edges, and faces, proved in 1752 by Leonhard Euler, and called Euler's theorem.

Before formulating it, consider the polyhedra known to us and fill in the following table, in which B is the number of vertices, P - edges and G - faces of a given polyhedron:

It is directly seen from this table that for all the chosen polyhedra the equality B - P + T = 2 holds. It turns out that this equality is true not only for these polyhedra, but also for an arbitrary convex polyhedron.

Euler's theorem. For any convex polyhedron, the equality

V - R + G \u003d 2,

where B is the number of vertices, P is the number of edges, and G is the number of faces of the given polyhedron.

Proof. To prove this equality, imagine the surface of a given polyhedron made of an elastic material. Let's delete (cut out) one of its faces and stretch the remaining surface on a plane. We get a polygon (formed by the edges of the removed face of the polyhedron), divided into smaller polygons (formed by the remaining faces of the polyhedron).

Note that polygons can be deformed, enlarged, reduced, or even bent their sides, as long as the sides do not break. The number of vertices, edges and faces will not change.

Let us prove that the resulting partition of a polygon into smaller polygons satisfies the equality

(*) V - R + G "= 1,

where B is the total number of vertices, P is the total number of edges, and Г "is the number of polygons included in the partition. It is clear that Г" \u003d Г - 1, where Г is the number of faces of this polyhedron.

Let us prove that the equality (*) does not change if we draw a diagonal in some polygon of the given partition (Fig. 5, a). Indeed, after drawing such a diagonal, the new partition will have B vertices, P + 1 edges, and the number of polygons will increase by one. Therefore, we have

V - (R + 1) + (G "+1) \u003d V - R + G" .

Using this property, we draw diagonals dividing the incoming polygons into triangles, and for the resulting partition we show that the equality (*) is satisfied (Fig. 5, b). To do this, we will consistently remove the outer edges, reducing the number of triangles. In this case, two cases are possible:

a) to remove the triangle ABC it is required to remove two ribs, in our case AB And BC;

b) to remove the triangleMKNit is required to remove one edge, in our caseMN.

In both cases, the equality (*) will not change. For example, in the first case, after removing the triangle, the graph will consist of B - 1 vertices, R - 2 edges and G "- 1 polygon:

(B - 1) - (P + 2) + (G "- 1) \u003d B - R + G".

Consider the second case for yourself.

Thus, removing one triangle does not change the equality (*). Continuing this process of removing triangles, we will eventually arrive at a partition consisting of a single triangle. For such a partition, B \u003d 3, P \u003d 3, Г "= 1 and, therefore, B - Р + Г" = 1. Hence, equality (*) also holds for the original partition, from which we finally obtain that for a given polygon partition equality (*) holds true. Thus, for the original convex polyhedron, the equality B - P + G = 2 is true.

An example of a polyhedron for which the Euler relation does not hold is shown in Figure 6. This polyhedron has 16 vertices, 32 edges and 16 faces. Thus, for this polyhedron, the equality B - P + G = 0 is satisfied.

Appendix 3

Movie Cube 2: Hypercube "(eng. Cube 2: Hypercube) - a fantasy film, a continuation of the movie" Cube ".

Eight strangers wake up in cube-shaped rooms. The rooms are inside a four-dimensional hypercube. The rooms are constantly moving by "quantum teleportation", and if you climb into the next room, then it is unlikely to return to the previous one. Parallel worlds intersect in the hypercube, time flows differently in some rooms, and some rooms are death traps.

The plot of the picture largely repeats the story of the first part, which is also reflected in the images of some characters. In the rooms of the hypercube dies Nobel laureate Rosenzweig, who calculated the exact time of the destruction of the hypercube.

Criticism

If in the first part people imprisoned in a labyrinth tried to help each other, in this film it's every man for himself. There are a lot of extra special effects (they are also traps) that do not logically connect this part of the film with the previous one. That is, it turns out the film Cube 2 is a kind of labyrinth of the future 2020-2030, but not 2000. In the first part, all types of traps can theoretically be created by a person. In the second part, these traps are a program of some kind of computer, the so-called "Virtual Reality".

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

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

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

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

What is a "measurement"?

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

A dimension is simply a direction you can go. For example, if you are drawing a line on a piece of paper, you can either go left/right (x-axis) or up/down (y-axis). So we say the paper is two-dimensional since you can only walk in two directions.

There is a sense of depth in 3D.

Now, in real world, besides the two directions mentioned above (left/right and up/down), you can also go "in/out". Consequently, a sense of depth is added in 3D space. Therefore we say that real life 3-dimensional.

A point can represent 0 dimensions (because it doesn't move in any direction), a line represents 1 dimension (length), a square represents 2 dimensions (length and width), and a cube represents 3 dimensions (length, width and height).

Take a 3D cube and replace each face (which is currently a square) with a cube. And so! The shape you get is the tesseract.

What is a tesseract?

Simply put, a tesseract is a cube in 4-dimensional space. You can also say that this is the 4D equivalent of a cube. This is a 4D shape where each face is a cube.

Here's a simple way to conceptualize dimensions: a square is two-dimensional; so each of its corners has 2 lines extending from it at 90 degrees to each other. The cube is 3D, so each of its corners has 3 lines coming off of it. Likewise, the tesseract is a 4D shape, so each corner has 4 lines extending from it.

Why is it difficult to imagine a tesseract?

Since we as humans have evolved to render 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 visualize 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.

Tesseract - a four-dimensional hypercube - a cube in four-dimensional space.
According to the Oxford Dictionary, the word tesseract was coined and used in 1888 by Charles Howard Hinton (1853-1907) in his book " new era thoughts". Later, some people called the same figure a tetracube (Greek τετρα - four) - a four-dimensional cube.
An ordinary tesseract in Euclidean four-dimensional space is defined as the convex hull of points (±1, ±1, ±1, ±1). In other words, it can be represented as the following set:
[-1, 1]^4 = ((x_1,x_2,x_3,x_4) : -1 = A tesseract is bounded by eight hyperplanes x_i= +- 1, i=1,2,3,4 , whose intersection with the tesseract itself defines it 3D faces (which are regular cubes) Each pair of non-parallel 3D faces intersect to form 2D faces (squares), etc. Finally, the tesseract has 8 3D faces, 24 2D, 32 edges, and 16 vertices.
Popular Description
Let's try to imagine how the hypercube will look without leaving the three-dimensional space.
In one-dimensional "space" - on a line - we select a segment AB of length L. On a two-dimensional plane at a distance L from AB, we draw a segment DC parallel to it and connect their ends. You will get a square CDBA. Repeating this operation with a plane, we get a three-dimensional cube CDBAGHFE. And by shifting the cube in the fourth dimension (perpendicular to the first three) by a distance L, we get the CDBAGHFEKLJIOPNM hypercube.
The one-dimensional segment AB serves as a side of the two-dimensional square CDBA, the square is the side of the cube CDBAGHFE, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 vertices shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and 8 more edges "draw" eight of its vertices that have moved into the fourth dimension. The same reasoning can be done for the faces of the hypercube. In two-dimensional space, it is one (the square itself), the cube has 6 of them (two faces from the moved square and four more will describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from twelve of its edges.
As the sides of a square are 4 one-dimensional segments, and the sides (faces) of a cube are 6 two-dimensional squares, so for the “four-dimensional cube” (tesseract) the sides are 8 three-dimensional cubes. The spaces of opposite pairs of tesseract cubes (that is, the three-dimensional spaces to which these cubes belong) are parallel. In the figure, these are cubes: CDBAGHFE and KLJIOPNM, CDBAKLJI and GHFEOPNM, EFBAMNJI and GHDCOPLK, CKIAGOME and DLJBHPNF.
In a similar way, we can continue the reasoning for hypercubes of a larger number of dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, inhabitants of three-dimensional space. Let us use for this the already familiar method of analogies.
Let's take the wire cube ABCDHEFG and look at it with one eye from the side of the face. We will see and can draw two squares on the plane (its near and far faces), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic "boxes" inserted into each other and connected by eight edges. In this case, the "boxes" themselves - three-dimensional faces - will be projected onto "our" space, and the lines connecting them will stretch in the direction of the fourth axis. You can also try to imagine a cube not in projection, but in a spatial image.
Just as a three-dimensional cube is formed by a square shifted by the length of a face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in the future will look like some rather complex figure. The four-dimensional hypercube itself consists of an infinite number of cubes, just as a three-dimensional cube can be “cut” into an infinite number of flat squares.
By cutting six faces of a three-dimensional cube, you can decompose it into a flat figure - a net. It will have a square on each side of the original face, plus one more - the face opposite to it. A three-dimensional development of a four-dimensional hypercube will consist of the original cube, six cubes that "grow" from it, plus one more - the final "hyperface".
The properties of a tesseract are an extension of the properties of geometric figures of a smaller dimension into a four-dimensional space.