Multidimensional music of the spheres by Perelman. One broke, the other lost Three-dimensional sphere in four-dimensional space

Back when I was a first-year student, I had a heated argument with one of my classmates. He said that a four-dimensional cube cannot be represented in any form, but I assured that it can be represented quite clearly. Then I even made a projection of a hypercube onto our three-dimensional space from paper clips... But let's talk about everything in order.

What is a hypercube and four-dimensional space

Our usual space has three dimensions. From a geometric point of view, this means that three mutually perpendicular lines can be indicated in it. That is, for any line you can find a second line perpendicular to the first, and for a pair you can find a third line perpendicular to the first two. It will no longer be possible to find a fourth line perpendicular to the existing three.

Four-dimensional space differs from ours only in that it has one more additional direction. If you already have three mutually perpendicular lines, then you can find a fourth one, such that it will be perpendicular to all three.

Hypercube it's just a cube in four-dimensional space.

Is it possible to imagine four-dimensional space and a hypercube?

This question is akin to the question: “is it possible to imagine the Last Supper by looking at the painting of the same name (1495-1498) by Leonardo da Vinci (1452-1519)?”

On the one hand, you, of course, will not imagine what Jesus saw (he is sitting facing the viewer), especially since you will not smell the garden outside the window and taste the food on the table, you will not hear the birds singing... You will not get a complete picture of what happened at that time evening, but it cannot be said that you will not learn anything new and that the picture is of no interest.

The situation is similar with the question of the hypercube. It is impossible to fully imagine it, but you can get closer to understanding what it is like.

Construction of a hypercube

0-dimensional cube

Let's start from the beginning - with a 0-dimensional cube. This cube contains 0 mutually perpendicular faces, that is, it is just a point.

1-dimensional cube

In one-dimensional space, we only have one direction. We move the point in this direction and get a segment.

This is a one-dimensional cube.

2 dimensional cube

We have a second dimension, we shift our one-dimensional cube (segment) in the direction of the second dimension and we get a square.

It is a cube in two-dimensional space.

3 dimensional cube

With the advent of the third dimension, we do the same: we move the square and get a regular three-dimensional cube.

4-dimensional cube (hypercube)

Now we have a fourth dimension. That is, we have at our disposal a direction perpendicular to all three previous ones. Let's use it exactly the same way. A four-dimensional cube will look like this.

Naturally, three-dimensional and four-dimensional cubes cannot be depicted on a two-dimensional screen plane. What I drew are projections. We'll talk about projections a little later, but for now a few bare facts and figures.

Number of vertices, edges, faces

Please note that the face of a hypercube is our ordinary three-dimensional cube. If you look closely at the drawing of a hypercube, you can actually find eight cubes.

Projections and vision of a resident of four-dimensional space

A few words about vision

We live in a three-dimensional world, but we see it as two-dimensional. This is due to the fact that the retina of our eyes is located in a plane that has only two dimensions. This is why we are able to perceive two-dimensional pictures and find them similar to reality.

(Of course, thanks to accommodation, the eye can estimate the distance to an object, but this is a side effect associated with the optics built into our eyes.)

The eyes of an inhabitant of four-dimensional space must have a three-dimensional retina. Such a creature can immediately see the entire three-dimensional figure: all its faces and interiors. (In the same way, we can see a two-dimensional figure, all its faces and interiors.)

Thus, with the help of our organs of vision, we are not able to perceive a four-dimensional cube the way a resident of a four-dimensional space would perceive it. Alas. All that remains is to rely on your mind's eye and imagination, which, fortunately, have no physical limitations.

However, when depicting a hypercube on a plane, I am simply forced to make its projection onto two-dimensional space. Take this fact into account when studying the drawings.

Edge intersections

Naturally, the edges of the hypercube do not intersect. Intersections appear only in drawings. However, this should not come as a surprise, because the edges of a regular cube in the pictures also intersect.

Edge lengths

It is worth noting that all faces and edges four-dimensional cube are equal. In the figure they are not equal only because they are located at different angles to the direction of view. However, it is possible to rotate a hypercube so that all projections have the same length.

By the way, in this figure eight cubes, which are the faces of a hypercube, are clearly visible.

The hypercube is empty inside

It’s hard to believe, but between the cubes that bound the hypercube, there is some space (a fragment of four-dimensional space).

To understand this better, let's look at a two-dimensional projection of an ordinary three-dimensional cube (I deliberately made it somewhat schematic).

Can you guess from it that there is some space inside the cube? Yes, but only by using your imagination. The eye does not see this space.

This happens because the edges located in the third dimension (which cannot be depicted in a flat drawing) have now turned into segments lying in the plane of the drawing. They no longer provide volume.

The squares enclosing the space of the cube overlapped each other. But one can imagine that in the original figure (a three-dimensional cube) these squares were located in different planes, and not one on top of the other in the same plane, as happened in the figure.

The situation is exactly the same with a hypercube. The cubes-faces of a hypercube do not actually overlap, as it seems to us on the projection, but are located in four-dimensional space.

Sweeps

So, a resident of four-dimensional space can see a three-dimensional object from all sides simultaneously. Can we see a three-dimensional cube from all sides at the same time? With the eye - no. But people have come up with a way to depict all the faces of a three-dimensional cube at the same time on a flat drawing. Such an image is called a scan.

Development of a three-dimensional cube

Everyone probably knows how the development of a three-dimensional cube is formed. This process is shown in the animation.

For clarity, the edges of the cube faces are made translucent.

It should be noted that we are able to perceive this two-dimensional picture only thanks to our imagination. If we consider the unfolding phases from a purely two-dimensional point of view, the process will seem strange and not at all clear.

It looks like the gradual appearance of first the outlines of distorted squares, and then their creeping into place while simultaneously taking on the required shape.

If you look at the unfolding cube in the direction of one of its faces (from this point of view the cube looks like a square), then the process of formation of the unfold is even less clear. Everything looks like squares creeping out from the initial square (not the unfolded cube).

But not visual scan only for eye.

How to understand 4-dimensional space?

It is thanks to your imagination that you can glean a lot of information from it.

Development of a four-dimensional cube

It is simply impossible to make the animated process of unfolding a hypercube at least somewhat visual. But this process can be imagined. (To do this, you need to look at it through the eyes of a four-dimensional being.)

The scan looks like this.

All eight cubes bounding the hypercube are visible here.

The edges that should align when folded are painted with the same colors. Faces for which pairs are not visible are left gray. After folding, the topmost face of the top cube should align with the bottom edge of the bottom cube. (The unfolding of a three-dimensional cube is collapsed in a similar way.)

Please note that after convolution, all the faces of the eight cubes will come into contact, closing the hypercube. And finally, when imagining the process of folding, do not forget that when folding, it is not the overlapping of cubes that occurs, but the wrapping of them around a certain (hypercubic) four-dimensional area.

Salvador Dali (1904-1989) depicted the crucifixion many times, and crosses appear in many of his paintings. The painting “The Crucifixion” (1954) uses a hypercube scan.

Space-time and Euclidean four-dimensional space

I hope you were able to imagine the hypercube. But have you managed to come closer to understanding how the four-dimensional space-time in which we live works? Alas, not quite.

Here we talked about Euclidean four-dimensional space, but space-time has completely different properties. In particular, during any rotations, the segments always remain inclined to the time axis, either at an angle less than 45 degrees, or at an angle greater than 45 degrees.

I devoted a series of notes to the properties of space-time.

Three-dimensionality of the image

The world is three-dimensional. Its image is two-dimensional. An important task of painting and, now, photography is to convey the three-dimensionality of space. The Romans already mastered some techniques, then they were forgotten and began to return to classical painting with the Renaissance.

The main technique for creating three-dimensional space in painting is perspective. The railway rails, moving away from the viewer, visually narrow. In painting, the rails can be physically narrowed. In photography, perspective occurs automatically: the camera will photograph the rails as narrowed as the eye sees them. However, do not allow it to almost close: it will no longer look like a perspective, but a strange figure; There must be a noticeable gap between the rails, the sides of the street, and the banks of the river.

It is important to understand that linear perspective is the most primitive, realistic way of conveying the world.

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It is no coincidence that its appearance is associated with theatrical scenery (Florensky, “Reverse Perspective”). The conventionality and simplicity of conveying a theatrical scene of small depth is very suitable for photography, which lacks the variety of techniques available in painting.

There are perspectives that are much more interesting than the linear one. In the works of Chinese masters there is a floating perspective, when objects are depicted simultaneously from below, above and in front. It was not a technical mistake by incompetent artists: the legendary author of this technique, Guo Xi, wrote that such a display allows one to realize the world in its totality. The technique of Russian icon painting is similar, in which the viewer can see the character’s face and back at the same time. An interesting technique of icon painting, also found among Western European artists, was reverse perspective, in which distant objects, on the contrary, are larger than close ones, emphasizing importance. Only in our days has it been established that such a perspective is correct: unlike distant objects, the close-up is actually perceived in reverse perspective (Rauschenbach). Using Photoshop, you can achieve reverse perspective by enlarging background objects. For a viewer accustomed to the laws of photography, such an image will look strange.

Introducing the corner of a building into the frame, from which the walls diverge in both directions, creates a semblance of an isometric perspective. The brain understands that the walls are at right angles and arranges the rest of the image accordingly. This perspective is more dynamic than the frontal one and more natural for the close-up. Simply introduce the end angles of objects and nearby buildings into the frame.

Due to the expansion, the isometric perspective is major, which is rarely suitable for a classical portrait. Linear perspective, due to narrowing, better conveys minor emotions.

At the shooting stage, the photographer has a number of tools available to him to emphasize perspective. Objects of equal width extending into the distance (tracks, streets, columns, furrows) by their narrowing and even simply moving away indicate to the viewer the three-dimensionality of space. The effect is stronger if you shoot from a low angle to increase perspective distortion. This is enough for landscape photography, but with a shallow image depth for interior photography, the effect is hardly noticeable. It can be enhanced a little in post-processing by narrowing the top of the image (Transform Perspective). However, in a landscape, an exaggerated perspective can look interesting.

Depth can be obvious in the meaning of the image: buildings are separated by a street or river. The diagonal emphasizes three-dimensionality; for example, a bridge over a river.

Objects of a size known to the viewer in the background set the scale and, accordingly, form the perspective. In landscape photography, this object could be a car, but in portrait photography, try bending your leg (away from the camera) under the chair so that it appears smaller while remaining visible. You can even make this leg a little smaller in post-processing.

The ornament conveys perspective by visually reducing the elements. An example would be large tiles on the floor, marking lines on the road.

There is a technique called hypertrophied foreground. Disproportionally large, it creates depth in the image. By comparing the scale of the foreground and the model, the eye comes to the conclusion that the model is much further away than it seems. The exaggeration should remain subtle so that the image is not perceived as an error. This technique works not only for post-processing, but also for shooting: distort the proportions by shooting with a 35 or 50mm lens. Shooting with a wide-angle lens stretches the space, enhancing its three-dimensionality by breaking the proportions. The effect is stronger if you shoot the model at close range, but beware of grotesque proportions: only the authors of religious images can depict a person larger than a building.

The intersection works great. If the apple partially covers the pear, then the brain will not be mistaken: the apple is in front of the pear. The model partially covers the furniture, thereby creating depth in the interior.

The alternation of light and dark spots also gives depth to the image. The brain knows from experience that nearby objects are lit approximately equally, so it interprets differently lit objects as being located at different distances. For this effect, the spots alternate in the direction of the perspective axis - deep into the image, and not across it. For example, when shooting a model lying away from the camera in a dark frame, place highlights near the buttocks and near the legs. You can lighten/darken areas in post-processing.

The sequence of increasingly darker objects is perceived to decrease. By gradually shading objects along the active line, you can get a subtle sense of perspective. Likewise, depth is conveyed by weakening the light: cast a strip of light across the furniture or on the floor.

A three-dimensional image can be obtained due to not only light, but also color contrast. This technique was known to Flemish painters, who placed bright colored spots on their still lifes. A red pomegranate and a yellow lemon next to each other will look three-dimensional even in flat frontal lighting. They will stand out especially well against the background of purple grapes: a warm color against a cold background. Bright colored surfaces emerge well from the darkness even with weak light, typical of still life. Color contrast works better with primary colors: red, yellow, blue, rather than shades.

On a black background, yellow comes forward, blue hides back. On a white background it’s the other way around. Color saturation enhances this effect. Why is this happening? The color yellow is never dark, so the brain refuses to believe that a yellow object can be immersed in a dark background, not illuminated. Blue, on the contrary, is dark.

Enhancing perspective in post-processing comes down to simulating atmospheric perception: distant objects appear lighter, blurrier, with reduced contrast in brightness, saturation and tone.

Besides long distances, atmospheric effects look natural in morning haze, fog, or a smoky bar. Consider the weather: on a cloudy day or at dusk, there may not be a significant difference between the foreground and background.

The strongest factor is brightness contrast. In the settings this is the usual contrast. Reduce the contrast of distant objects, raise the contrast of the foreground - and the image will become convex. We are not talking about the contrast between the foreground and background, but about the contrast of the background, which should be lower than the contrast of the foreground. This method is suitable not only for landscapes and genre photography, but also for studio portraits: raise the contrast of the front of the face, reduce the contrast on the hair, cheekbones, and clothes. Portrait filters do something similar, blurring the model's skin and leaving the eyes and lips harsh.

Contrast adjustment is the easiest way to do 3D image post-processing. Unlike other processes, the viewer will hardly notice any changes, which will allow maintaining maximum naturalness.

Blurring is similar to contrast reduction, but they are different processes. The image can be low contrast while remaining sharp. Due to limited depth of field, blurring distant objects remains the most popular way to convey three-dimensionality in photography, and can easily be enhanced by blurring distant subjects in post-production. Therefore, fewer details should be placed in the background - the brain does not expect distinguishable objects in the distance. Meanwhile, reducing the contrast better corresponds to natural perception: distant mountains are visible in low contrast, and not blurred, because when scanning the landscape, the eye is constantly refocused, and the problem of depth of field is alien to it. By blurring the background, you can at the same time sharpen the foreground. Additionally, in the foreground you can enhance the image lines (High Pass Filter or Clarity). It is the high sharpness of the foreground that explains the characteristic bump in the image of high-quality lenses. Beware: for the sake of a slight increase in three-dimensionality, you may make the image too rigid.

Lighter objects appear further away. This is due to the fact that in nature we see distant objects through the thickness of light-scattering air; distant mountains appear light. In landscape photography, therefore, you should be careful about the placement of light objects in the foreground.

Brighten distant objects. The further away they are, the more they blend in with the brightness and tone of the sky. Please note that horizontal objects (ground, sea) are better illuminated than vertical ones (walls, trees), so do not overdo it with lightening the latter. In any case, objects should remain noticeably lighter than the sky.

Well, if you notice that dodging is another way to reduce the contrast in the brightness of the background. Darken the foreground slightly to enhance the bump effect.

It would seem that in the interior everything is the other way around. If on the street the eye is accustomed to the fact that the distance is bright, then in the room the light is often concentrated on the person, and the interior is immersed in darkness; the brain is accustomed to foreground lighting, not background lighting.

In interior images with shallow scene depth, unlike landscape images, the illuminated model protrudes from a dark background. But there is also an opposite factor: for 99% of his evolution, man observed the perspective in open areas, and with the advent of rooms, the brain had not yet had time to restructure. Vermeer preferred a light background for his portraits, and his portraits are really prominent. Lighting a vertical background, recommended in photography, not only separates the model from it, but also, by lightening the background, gives the image a slight three-dimensionality. Here we are faced with the fact that the brain analyzes the location of objects according to several factors, and they can be conflicting.

Studio lighting looks interesting, in which light spots lie on areas of the model remote from the camera. For example, the breast that is farthest from the camera is highlighted.

Reduce color saturation on distant objects: due to the thickness of the air separating us, distant mountains are desaturated almost to the level of monochrome and covered with a blue haze. The foreground saturation can be increased.

Since yellow is light, and blue and red are dark, the color contrast is also a contrast in brightness.

When desaturating the distant background, do not let it disappear from view. Often, on the contrary, you need to increase the saturation of the background in order to reveal it. This is more important than three-dimensionality.

A lot of 3D photography advice focuses on temperature contrast. In fact, this effect is very weak and is easily interrupted by brightness contrast. In addition, the temperature contrast is annoying and noticeable.

Very distant objects appear cooler in color because the air absorbs warm orange light. When photographing a model on the beach with ships on the horizon in the background, lower the color temperature of the distant sea and ships in post-processing. A model in a red swimsuit emerges from the blue sea, and the model is in a yellow light street lamp- from the bluish twilight.

This is the essence of separate toning: we make the model warmer, the background cooler. The brain understands that there are no different color temperatures in the same plane, and perceives such a three-dimensional image, in which the model protrudes from the background. Split toning adds depth to landscapes: make the foreground warmer, the background cooler.

An important exception to separate toning: at sunrise and sunset, the distant background is not cold at all, but warm, with yellow and red-orange tones. The obvious solution - using a white model in a purple swimsuit - doesn't work because the sunset light casts a warm tint on the model's body as well.

Let's summarize: to give a photo three-dimensionality based on atmospheric effects, it is necessary to contrast the foreground and background. The main contrast is based on the usual contrast: the foreground is high-contrast, the background is low-contrast. The second contrast is in terms of sharpness: the foreground is sharp, the background is blurry. The third contrast is in terms of lightness: the foreground is dark, the background is light. The fourth contrast is in terms of saturation: the foreground colors are saturated, the background colors are desaturated. The fifth contrast is in temperature: the foreground is warm, the background is cold.

The listed factors are often multidirectional. Yellow is brighter than blue, and light objects appear further away from dark ones. It would be natural to expect that yellow would recede and blue would approach the viewer. In fact, it's the other way around: a warm color emerges from a cold background. That is, color turns out to be a stronger factor than brightness. Which, on reflection, is not surprising: yellow and red are clearly distinguishable only at close range, and the viewer does not expect to encounter them at a great distance.

Bottom line: keep the background low contrast, washed out, light, desaturated, bluish. And be prepared for the fact that the viewer, accustomed to the hypertrophied 3D of films, will find the three-dimensionality you created to be barely noticeable or absent.

In portrait photography, it is better to rely on the proven chiaroscuro effect - the play of light and shadow on the model's face, which will make the image quite prominent. In genre photography, perspective gives the most noticeable three-dimensional effect. In a still life, the main factor will be the intersection (overlapping) of objects.

Don't get carried away with the prospect; it is just a background for the frontal plane on which your image flutters. In modern painting, which is far from realism, perspective is not held in high esteem.

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GEOMETRIC IMAGE OF A FOUR-DIMENSIONAL BALL.

Egorov Nester Alexandrovich

4th year student, Department of Algebra and Geometry IMI NEFU, Russian Federation, Yakutsk

E- mail: egrvnester@ mail. ru

Popov Oleg Nikolaevich

scientific supervisor, Ph.D. tech. Sciences, Associate Professor IMI NEFU, Russian Federation, Yakutsk

This paper provides a representation of a four-dimensional ball in four-dimensional space using its three-dimensional sections. To explain the difficulties associated with the perception of objects in four-dimensional space, a method is used that is based on considering spaces with lower dimensions. The relevance of this approach lies in the fact that it allows us to understand the structure of geometric images of four-dimensional space, and also contributes to the development of spatial and abstract thinking. This work is of interest to high school students, students of mathematical and natural sciences, as well as mathematics teachers. It is stated visual method, without using formulas, based only school course geometry.

In scientific and popular literature, in the media, multidimensional spaces and objects are often mentioned. There are various theories about the multidimensionality of our Universe. It is human nature to represent geometric objects in a visual form. Therefore, many, having heard the phrase “four-dimensional ball,” immediately try to visualize it in their imagination. We well imagine a two-dimensional ball (this is a circle lying on a plane), a three-dimensional ball is an object that is often encountered in our lives. But in the four-dimensional case, we cannot in any way construct in our imagination a geometric image of a four-dimensional ball. This is due to the emergence of the fourth dimension, inaccessible to us.

Forming an intuitively understandable idea for the reader about the geometric image of a four-dimensional ball is the goal of our work. It does not use strict definitions or mathematical formulas. All concepts and terms used are understood only intuitively. All material is presented in a popular form.

The relevance of the work lies in the fact that it allows us to understand the structure of geometric images of four-dimensional space, and also contributes to the development of spatial and abstract thinking and is of interest to high school students, students of the faculties of mathematical and natural sciences, as well as mathematics teachers.

Figure 1. a) A straight line in four-dimensional space intersects a three-dimensional ball at only one interior point; b) A straight line on a plane intersects a two-dimensional ball along a segment; c) A straight line located in space intersects a two-dimensional ball at only one point

Four-dimensional space is, to some extent, an unusual space. We know that in three-dimensional space a straight line intersects a limited three-dimensional convex volume (for example, a ball) along a segment. The exception is when a straight line touches a given object. In four-dimensional space, everything can happen differently. A straight line can “pierce” a three-dimensional ball right through, hitting only one internal point without disturbing its surroundings (Fig. 1, a)). This makes it possible for a 4D person (if he existed) to take all our things from the bag without opening or cutting it, which seems very unusual and inexplicable. To understand this, consider a two-dimensional space (a two-dimensional space is a plane embedded in a three-dimensional space). A straight line on the plane will intersect a circle located in the plane along a segment, and a straight line in space lying outside the plane will intersect the circle at only one point (Fig. 1, b), c)).

To make the episode of things missing from a bag more understandable, let’s draw a two-dimensional person on the board, draw his kidneys, a kidney stone. Then we take a rag in our hands and carefully, without touching the kidneys of a two-dimensional person, we wipe off the stone (Fig. 2). Now we can congratulate ourselves on the fact that we have just successfully performed an operation to remove a kidney stone without the use of incisions, and that our patient is healthy. What is beyond the control of a two-dimensional surgeon turns out to be a simple matter for an ordinary three-dimensional person.

Figure 2. Stone removal from a two-dimensional kidney by a three-dimensional doctor without reserves

Next, we will use this technique associated with the transition to a lower dimension to explain the difficulties associated with the perception of objects located in four-dimensional space. The difficulties of perception of a two-dimensional person when he tries to understand a three-dimensional world are similar to ours when perceiving four-dimensional space, since they are connected in both cases by the appearance of a new inaccessible dimension.

Two three-dimensional spaces can intersect or be parallel in four-dimensional space. Let's consider the case when they intersect.

Figure 3. Two three-dimensional spaces intersect in four-dimensional space along a plane.

If two planes x and y intersect along a straight line l (Fig. 4), then the three-dimensional spaces P and Q intersect along a plane α (Fig. 3). For a two-dimensional person, straight line l (if it is opaque) will be a wall dividing his world into two parts. And the half-planes y 1 and y 2 do not exist for him, since they are in the third dimension, inaccessible to him. For a three-dimensional person, such a wall, dividing the entire space into two parts, will be the plane α (Fig. 3).

Next, consider two intersecting planes x and y, along one of which a two-dimensional ball rolls (Fig. 4). Note that a two-dimensional person sees only the line l from the y plane, since it is in his x space. The half-planes y 1 and y 2 are invisible to him, so a two-dimensional person located in the x plane will see a point (the flat ball touched the line), which then splits (the ball crossed the line). Further, as the ball moves, the points will diverge until the straight line of intersection of the planes coincides with the diameter of the ball, then everything will happen in the reverse order.

Figure 4. A two-dimensional person sees only the point of contact of the circle with its plane

Now it is not difficult to understand what we will see, being in three-dimensional space P, in the case when the ball launched by the foot of a football player located in Q crosses our space. First on the α plane. a point will appear, which will immediately transform into a gradually increasing circle, which is the intersection of the α plane and the ball. Having reached its maximum, with a radius equal to the radius of a soccer ball, it will gradually begin to decrease until it degenerates back into a point and disappears from view (Fig. 5). What we will see when the football player himself runs after the ball, we will leave it to the reader to imagine. For fun, let’s imagine what will happen if a football player, in some incredible way, while in space Q, accidentally turns into our space P (see Fig. 6).

Figure 5. View of the ball crossing the observer’s space in dynamics

Figure 6. The appearance of a football player in space P from space Q

In a two-dimensional version, it is easy to imagine two parallel planes. Three-dimensional space can be represented as an infinite collection of parallel “stuck together” planes. This idea can be obtained by looking at a deck of cards, where each card is associated with a plane or a book, where the role of the planes is played by the sheets of this book.

Four-dimensional space also represents a collection of “stuck together”, but already three-dimensional parallel spaces. Try to imagine in your imagination two parallel (sticking together), i.e. located very close to each other, three-dimensional spaces. You won't succeed. The spaces that we want to imagine in our imagination either begin to intersect or do not want to get closer, pushing away from each other. Let's figure out the reason for our failure. To do this, let us analyze how a two-dimensional person living in the x plane will try to imagine two parallel planes y and z lying very close to each other. Since for a two-dimensional person there is no third dimension h (Fig. 7a)), he will be forced to place them in his space, although in reality they will be located perpendicularly (or at some angle) intersecting the x plane (Fig. 7b)). Now it immediately becomes obvious what the reason for our failure is. We are trying to place two three-dimensional spaces in one three-dimensional space in which we are (Fig. 7c)), when they should extend along the fourth dimension, inaccessible to us. It is clear that they cannot seem to stick together.

Note that three-dimensional space can be represented as a trace left by a plane as a result of its movement in a given direction (Fig. 8).

Figure 7. a) A two-dimensional person tries to imagine two parallel planes; b) The actual location of parallel planes; c) We are trying to put two three-dimensional spaces into one three-dimensional space

Figure 8. Three-dimensional space obtained by the movement of a plane

Now, as before, consider the spaces P and Q intersecting along the plane α (Fig. 9a)). Each of the spaces can be obtained by moving the plane α according to the directions of the x and t coordinate axes. Next, let us draw the plane β in space P at a very close distance parallel to the plane α. Obviously, β will not be in the space Q. Let's start moving these planes in the direction t so that at any moment t the moving planes are parallel and close to each other. Then the space Q and the space Q β obtained by the movement of the planes α and β, respectively, are parallel, and will be at a very close distance from each other (at a distance equal to the distance between the planes α and β, along the x dimension). Then two three-dimensional bodies, for example, two balls, located in completely different, but parallel spaces Q and Q β close to each other, can turn out to be very close (“stuck together”) (Fig. 9b)).

Figure 9. a) Plane β from gloss P is close and parallel to the α plane and is not in space Q ; b) Sets of planes obtained by the movement of planes α and β in the direction t , form parallel spaces close to each other Q And Q β The depicted balls located in these spaces are close to each other at all points (“sticky” balls)

All four-dimensional space can be considered as a collection of parallel, very closely spaced (“stuck together”) three-dimensional spaces. If we take time as the fourth dimension, then the movement of a person in a time machine will correspond to the transition from one parallel space to another. In this case, unlike intersecting spaces, when we see only a cross-section of an object that moves through the second space, crossing ours, a time machine with a person sitting in it will suddenly appear in front of us, which will dissolve in the past or future depending on the direction of its movement .

Thus: we understood that three-dimensional spaces intersect along a plane; four-dimensional space can be represented as a set of “stuck together” parallel three-dimensional spaces; got an idea of ​​“sticking together” three-dimensional bodies located in parallel spaces.

What is a four-dimensional ball? To answer this question, let’s analyze how our ordinary three-dimensional ball is structured from the point of view of a two-dimensional person. Of course, he cannot see the entire ball; in his field of vision there is only a two-dimensional sphere - a circle that borders a two-dimensional circle, and is the intersection of the world of a two-dimensional person with the ball (what is inside the circle is not visible to him. Fig. 10 a)). When moving into parallel spaces, the circle will narrow until it degenerates into a point (Fig. 10 b)).

Figure 10. a) A two-dimensional person can see only part of the circle, bordered by the intersection of the plane and the ball; b) When a person moves into parallel planes, the circle will gradually degenerate into a point

In the case of a four-dimensional ball, a person’s field of vision is limited by the space in which he is located. By analogy, we can assume that he sees a sphere bordering the ball, which is the intersection of this three-dimensional space with a four-dimensional ball. When moving into parallel spaces, the sphere will also decrease in radius until it degenerates into a point (Fig. 11 a)). Now we will try to understand in more detail what kind of balls we see and how they form a four-dimensional ball.

Let's consider a three-dimensional ball 2 (Fig. 11 b)) and its sections by parallel planes. The totality of these parallel planes form a three-dimensional space with dimensions y, z, t, in which the desired ball 2 is located. Each of these planes, with its movement in the x direction, form “sticky” three-dimensional spaces. It is in these spaces that three-dimensional balls are located (see ball 1), which we observe during the (described above) transitions to parallel spaces (Fig. 11a)). The combination of these balls will form a four-dimensional ball. Thus, a four-dimensional ball is a collection of balls sticking together at all points, decreasing in size, which forms the geometric image of a four-dimensional ball. However, we cannot see the overall integral picture of the ball, since we cannot see outside our space.

Figure 11. a) Visible by man, during transitions into parallel spaces, balls decrease in size; b) A four-dimensional ball is a collection of decreasing “merged” balls, which are sections of a four-dimensional ball by three-dimensional spaces parallel to space P

Let's look at a four-dimensional ball from different sides. An observer located in three-dimensional space P with dimensions y, z, t and looking in the direction t will see a ball (Fig. 12), which consists of sections of balls forming a four-dimensional ball (in Fig. 11 this is ball 2).

An observer located in space Q and looking in the x direction will also see a three-dimensional ball (Fig. 12). Thus, observers located in spaces P and Q see the same picture - a three-dimensional ball. However, the balls they observe are different geometric objects located in different spaces and intersecting in a two-dimensional circle.

Figure 12. Observers located in intersecting spaces P And Q see a three-dimensional ball. However, in reality they are observing various balls intersecting along a path

Unfortunately, as noted above, our field of vision is limited to three-dimensional space, so we cannot see four-dimensional images as a whole. However, the British mathematician Charles Hinton (1853-1907) developed a special method for constructing models geometric shapes in four-dimensional space along their three-dimensional sections. This method is described in detail in two of his monographs. Hinton claimed that as a result of many years of work, which was based on this special method, he learned to mentally represent geometric images in four-dimensional space. He also believed that a person who mastered this method well enough would gain an intuitive understanding of four-dimensional space.

Bibliography:

1.Hinton Charles H. A New Era of Thought, orig. 1888, reprinted 1900, by Swan Sonnenschein & Co. Ltd., London - p. 240.

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In 1904, Henri Poincaré proposed that any three-dimensional object that has certain properties of a three-dimensional sphere can be converted into a 3-sphere. It took 99 years to prove this hypothesis. (Warning: A three-dimensional sphere is not what you think it is.) Russian mathematician Grigory Perelman proved Poincaré's hundred-year-old conjecture and completed a catalog of shapes in three-dimensional spaces.

Poincaré suggested that the 3-sphere is unique and no other compact 3-manifold (Non-compact manifolds are infinite or have edges. Below only compact manifolds are considered) has the properties that make it so simple. More complex 3-manifolds have boundaries that stand up like a brick wall, or multiple connections between certain areas, like a forest path that branches and then joins again. Any three-dimensional object with the properties of a 3-sphere can be transformed into it itself, so to topologists it appears to be simply a copy of it. Perelman's proof also allows us to answer the third question and classify all existing 3-manifolds.
You'll need a fair amount of imagination to imagine a 3-sphere. Fortunately, it has a lot in common with the 2-sphere, a typical example of which is the rubber of a round balloon: it is two-dimensional, since any point on it is defined by only two coordinates - latitude and longitude. If you examine a fairly small area of ​​it under a powerful magnifying glass, it will seem like a piece of a flat sheet. To a tiny insect crawling on a balloon, it will appear to be a flat surface. But if the booger moves in a straight line long enough, it will eventually return to its point of departure. In the same way, we would perceive a 3-sphere the size of our Universe as “ordinary” three-dimensional space. Having flown far enough in any direction, we would eventually have "circumnavigated" it and ended up back at our starting point.
As you may have guessed, an n-dimensional sphere is called an n-sphere. For example, the 1-sphere is familiar to everyone: it is just a circle.

Mathematicians who prove theorems about higher-dimensional spaces do not have to imagine the object of study: they deal with abstract properties, guided by intuitions based on analogies with fewer dimensions (such analogies must be treated with caution and not taken literally). We will also consider the 3-sphere, based on the properties of objects with fewer dimensions.
1. Let's start by looking at a circle and its enclosing circle. For mathematicians, a circle is a two-dimensional ball, and a circle is a one-dimensional sphere. Further, a ball of any dimension is a filled object, resembling a watermelon, and a sphere is its surface, more like balloon. A circle is one-dimensional because the position of a point on it can be specified by a single number.

2. From two circles we can construct a two-dimensional sphere, turning one of them into the Northern Hemisphere and the other into the Southern Hemisphere. All that remains is to glue them together, and the 2-sphere is ready.

3. Imagine an ant crawling from the North Pole along a large circle formed by the prime and 180th meridians (on the left). If we map its path onto the two original circles (on the right), we see that the insect moves in a straight line (1) to the edge of the northern circle (a), then crosses the border, hits the corresponding point on the southern circle and continues to follow the straight line (2 and 3). Then the ant again reaches the edge (b), crosses it and again finds itself on the northern circle, rushing towards the starting point - the North Pole (4). Note that when traveling around the world on a 2-sphere, the direction of movement is reversed when moving from one circle to another.

4. Now consider our 2-sphere and the volume contained in it (a three-dimensional ball) and do with them the same thing as with a circle and a circle: take two copies of the ball and glue their boundaries together. It is impossible and not necessary to clearly show how balls are distorted in four dimensions and turn into an analogue of hemispheres. It is enough to know that the corresponding points on the surfaces, i.e. 2-spheres are connected to each other in the same way as in the case of circles. The result of connecting two balls is a 3-sphere - the surface of a four-dimensional ball. (In four dimensions, where a 3-sphere and a 4-ball exist, the surface of an object is three-dimensional.) Let's call one ball the northern hemisphere and the other the southern hemisphere. By analogy with circles, the poles are now located in the centers of the balls.

5. Imagine that the balls in question are large empty areas of space. Let's say an astronaut sets off from the North Pole on a rocket. Over time, it reaches the equator (1), which is now a sphere surrounding the northern ball. Crossing it, the rocket hits the southern hemisphere and moves in a straight line through its center - South Pole- To opposite side equator (2 and 3). There the transition to the northern hemisphere occurs again, and the traveler returns to the North Pole, i.e. to the starting point (4). This is the scenario for a trip around the world on the surface of a 4-dimensional ball! The three-dimensional sphere considered is the space referred to in the Poincaré conjecture. Perhaps our Universe is precisely a 3-sphere.

The reasoning can be extended to five dimensions and construct a 4-sphere, but this is extremely difficult to imagine. If you glue two n-balls along the (n-1)-spheres surrounding them, you get an n-sphere bounding the (n+1)-ball.

Half a century passed before the matter of the Poincaré conjecture got off the ground. In the 60s XX century Mathematicians have proven similar statements to her for spheres of five or more dimensions. In each case, the n-sphere is indeed the only and simplest n-manifold. Oddly enough, it turned out to be easier to obtain results for multidimensional spheres than for 3- and 4-spheres. The proof for four dimensions appeared in 1982. And only the original Poincaré conjecture about the 3-sphere remained unconfirmed.
The decisive step was taken in November 2002, when Grigory Perelman, a mathematician from the St. Petersburg branch of the Mathematical Institute. Steklov, sent the article to the website www.arxiv.org, where physicists and mathematicians from all over the world discuss the results of their scientific activity. Topologists immediately grasped the connection between the Russian scientist’s work and the Poincaré conjecture, although the author did not directly mention it.

In fact, Perelman’s proof, the correctness of which no one has yet been able to question, solves a much wider range of issues than the Poincaré conjecture itself. The geometrization procedure proposed by William P. Thurston from Cornell University allows full classification 3-manifold, which is based on the 3-sphere, unique in its sublime simplicity. If the Poincaré conjecture were false, i.e. If there were many spaces as simple as a sphere, then the classification of 3-manifolds would turn into something infinitely more complex. Thanks to Perelman and Thurston, we have a complete catalog of all the mathematically possible forms of three-dimensional space that our Universe could take (if we consider only space without time).

To better understand the Poincaré conjecture and Perelman's proof, you should take a closer look at topology. In this branch of mathematics, the shape of an object does not matter, as if it were made of dough that can be stretched, compressed and bent in any way. Why should we think about things or spaces made from imaginary dough? The fact is that the exact shape of an object - the distance between all its points - refers to a structural level called geometry. By examining an object from a dough, topologists identify its fundamental properties that do not depend on the geometric structure. Studying topology is like finding the most common features, inherent in people, by considering the “plasticine man”, which can be turned into any specific individual.
In popular literature, there is often a hackneyed statement that, from a topological point of view, a cup is no different from a donut. The fact is that a cup of dough can be turned into a donut by simply crushing the material, i.e. without blinding anything or making holes. On the other hand, to make a donut from a ball, you definitely need to make a hole in it or roll it into a cylinder and mold the ends, so a ball is not a donut at all.
Topologists are most interested in the sphere and donut surfaces. Therefore, instead of solid bodies, you should imagine balloons. Their topology is still different because a spherical balloon cannot be converted into a ring-shaped one, which is called a torus. First, scientists decided to figure out how many objects with different topologies exist and how they can be characterized. For 2-manifolds, which we are used to calling surfaces, the answer is elegant and simple: everything is determined by the number of “holes” or, what is the same, the number of handles. TO end of the 19th century V. Mathematicians figured out how to classify surfaces and determined that the simplest of them was the sphere. Naturally, topologists began to think about 3-manifolds: is the 3-sphere unique in its simplicity? The century-long history of searching for an answer is full of missteps and flawed evidence.
Henri Poincaré took up this issue closely. He was one of the two most powerful mathematicians of the early 20th century. (the other was David Gilbert). He was called the last universalist - he successfully worked in all areas of both pure and applied mathematics. In addition, Poincaré made enormous contributions to the development of celestial mechanics, the theory of electromagnetism, as well as to the philosophy of science, about which he wrote several popular books.
Poincaré became the founder of algebraic topology and, using its methods, in 1900 he formulated a topological characteristic of an object, called homotopy. To determine the homotopy of a manifold, you need to mentally immerse a closed loop in it. Then you should find out whether it is always possible to contract the loop to a point by moving it inside the manifold. For a torus, the answer will be negative: if you place a loop around the circumference of the torus, you will not be able to tighten it to a point, because the donut “hole” will get in the way. Homotopy is the number of different paths that can prevent a loop from contracting.

On the n-sphere, any loop, even an intricately twisted one, can always be unraveled and pulled together to a point. (The loop is allowed to pass through itself.) Poincaré assumed that the 3-sphere is the only 3-manifold on which any loop can be contracted to a point. Unfortunately, he was never able to prove his conjecture, which later became known as the Poincaré conjecture.

Perelman's analysis of 3-manifolds is closely related to the geometrization procedure. Geometry deals with the actual shape of objects and manifolds, no longer made of dough, but of ceramics. For example, a cup and a donut are geometrically different because their surfaces are curved differently. It is said that a cup and a donut are two examples of a topological torus that is given different geometric shapes.
To understand why Perelman used geometrization, consider the classification of 2-manifolds. Each topological surface is assigned a unique geometry whose curvature is distributed evenly across the manifold. For example, for a sphere, this is a perfectly spherical surface. Another possible geometry for a topological sphere is an egg, but its curvature is not evenly distributed everywhere: the sharp end is more curved than the blunt end.
2-manifolds form three geometric types. The sphere is characterized by positive curvature. A geometrized torus is flat and has zero curvature. All other 2-manifolds with two or more "holes" have negative curvature. They correspond to a surface similar to a saddle, which curves upward in front and behind, and downward on the left and right. Poincaré developed this geometric classification (geometrization) of 2-manifolds together with Paul Koebe and Felix Klein, after whom the Klein bottle is named.

There is a natural desire to apply a similar method to 3-manifolds. Is it possible to find for each of them a unique configuration in which the curvature would be distributed evenly throughout the entire variety?
It turned out that 3-manifolds are much more complex than their two-dimensional counterparts and most of them cannot be assigned a homogeneous geometry. They should be divided into parts that correspond to one of the eight canonical geometries. This procedure is reminiscent of decomposing a number into prime factors.

How can a manifold be geometrized and given uniform curvature everywhere? You need to take some arbitrary geometry with various protrusions and recesses, and then smooth out all the irregularities. In the early 90s. XX century Hamilton began analyzing 3-manifolds using the Ricci flow equation, named after the mathematician Gregorio Ricci-Curbastro. It is somewhat similar to the heat conduction equation, which describes heat flows flowing in an unevenly heated body until its temperature becomes the same everywhere. In the same way, the Ricci flow equation specifies a change in the curvature of the manifold that leads to the alignment of all protrusions and recesses. For example, if you start with an egg, it will gradually become spherical.

Perelman added a new term to Ricci's flow equation. This change did not eliminate the peculiarity problem, but it did allow for much more in-depth analysis. A Russian scientist has shown that a “surgical” operation can be performed on a dumbbell-shaped manifold: cut off a thin tube on either side of the emerging constriction and seal the open tubes protruding from the balls with spherical caps. Then one should continue changing the “operated” manifold in accordance with the Ricci flow equation, and apply the above procedure to all emerging constrictions. Perelman also showed that cigar-shaped features cannot appear. Thus, any 3-manifold can be reduced to a set of parts with homogeneous geometry.
When Ricci flow and "surgery" are applied to all possible 3-manifolds, any one of them, if it is as simple as a 3-sphere (in other words, characterized by the same homotopy), necessarily reduces to the same homogeneous geometry as and 3-sphere. This means, from a topological point of view, the manifold in question is a 3-sphere. Thus, the 3-sphere is unique.

The value of Perelman's articles lies not only in the proof of the Poincaré conjecture, but also in new methods of analysis. Scientists around the world are already using the results obtained by the Russian mathematician in their work and applying the methods he developed in other areas. It turned out that the Ricci flow is associated with the so-called renormalization group, which determines how the strength of interactions changes depending on the particle collision energy. For example, at low energies the strength of the electromagnetic interaction is characterized by the number 0.0073 (approximately 1/137). However, when two electrons collide head-on at nearly the speed of light, the force approaches 0.0078. The mathematics that describes the change in physical forces is very similar to the mathematics that describes the geometrization of manifolds.
Increasing the collision energy is equivalent to studying the force at smaller distances. Therefore, the renormalization group is similar to a microscope with a variable magnification factor, which allows you to study the process at different levels of detail. Likewise, Ricci flow is a microscope for viewing manifolds. Protrusions and depressions visible at one magnification disappear at another. It is likely that on the Planck length scale (about 10 -35 m), the space in which we live looks like foam with a complex topological structure. Moreover, the equations general theory relativity, which describes the characteristics of gravity and the large-scale structure of the Universe, is closely related to the Ricci flow equation. Paradoxically, the term that Perelman added to the expression used by Hamilton appears in string theory, which claims to be quantum theory gravity. It is possible that in the articles of the Russian mathematician, scientists will find a lot more useful information not only about abstract 3-manifolds, but also about the space in which we live.

Some time ago, two papers appeared on the preprint website arXiv.org, devoted to the problem of the densest packing of balls in spaces of dimensions 8 and 24. Until now, similar results were known only for dimensions 1, 2 and 3 (and not everything is so simple here, but more on that below). The breakthrough - and we are talking about a real revolutionary breakthrough - became possible thanks to the work of Marina Vyazovskaya, a mathematician of Ukrainian origin, who now works in Germany. We will tell the story of this achievement in ten short stories.

1.

In the 16th century, the famous court figure and poet Sir Walter Raleigh lived in England. He was famous, first of all, for the fact that he once threw his expensive cloak into a puddle in front of the queen so that Her Majesty would not get her feet dirty. But that’s not why he’s interesting to us.

Sir Walter Raleigh had a passion - he really loved to rob Spanish ships and look for Eldorado. And then one day Raleigh saw a bunch of stacked cannonballs on the ship. And I thought (this happened to British courtiers), they say, it would be nice if it were possible to find out how many cores are in the heap without counting them. The benefit of such knowledge, especially if you like to plunder the Spanish fleet, is obvious.

Walter Raleigh

Raleigh himself was not very good at mathematics, so he assigned this problem to his assistant Thomas Herriot. He, in turn, was strong in mathematics (Harriott, by the way, is the inventor of the signs “>” and “<» для сравнения численных величин), поэтому довольно быстро решил эту задачу. Но Хэрриот был хорошим математиком, поэтому он задался вопросом: а как лучше всего укладывать ядра? Сам он немного подумал, но решить задачу не смог.

For comments, he turned to the famous mathematician of his time, Johannes Kepler - at that time an assistant to Tycho Brahe. Kepler did not give an answer, but he remembered the problem. In 1611, he published a small brochure in which he discussed four questions: why bees have hexagonal honeycombs, why flower petals are most often grouped in fives ( Kepler probably only meantRosaceae - approx. N+1), why garnet grains have the shape of dodecahedrons (albeit irregular ones) and why, finally, snowflakes have the shape of hexagons.

Johannes Kepler

The brochure was intended as a gift, so it was more of a philosophical and entertaining read than a real scientific work. Kepler associated the answer to the first question with two conditions - there should be no gaps between the cells, and the sum of the areas of the cells should be minimal. The author connected the second question with Fibonacci numbers, and the conversation about snowflakes prompted Kepler to talk about atomic symmetries.

The third question gave rise to the hypothesis that hexagonal close packing(it is in the picture below) is the densest (which means this in a mathematical sense is also below). Of course, Kepler did not consider it necessary to refer to Harriot. Therefore, this statement is called the Kepler hypothesis. Stigler's law - also known as Arnold's principle - is in action.

Yes, 7 years after the publication of this brochure, Sir Walter Raleigh's head was cut off. However, this had nothing to do with the problem of dense packing.

2.

By modern standards, the problem that Harriot solved was not difficult. Therefore, let's analyze it in more detail. And at the same time, we will better understand how hexagonal close packing works.

So, the main condition is that the pile of kernels does not roll out during rolling. So, we lay out the kernels in a row on the deck. We place the kernels in the next row so that the balls are placed in the gaps between the spheres of the first row. If there are n balls in the first row, then there are n - 1 in the second row (because there are one fewer gaps between the balls than the balls themselves). The next row will have one less core. And so on until we get a triangle like this (if you look at the layout from above):

Those who remember what an arithmetic progression is can easily calculate that if there were n balls in the first row, then there are n(n + 1)/2 balls in total in such a triangle. If you look from above, there are convenient grooves between the balls. This is where we will put the second layer of balls. The result is a triangle organized like the first one, only with one less balls on the side. This means we added n(n - 1)/2 balls to the heap.

Let's continue adding layers until we get a layer of one ball. We got a triangular pyramid of nuclei. To find out how many cores there are in total, you need to add up the number of cores in each layer. If the first layer had side n, then we get n layers, which in total will give n(n + 1)(n + 2)/6. The inquisitive reader will notice that this is exactly the binomial coefficient C 3 n + 2. This combinatorial coincidence is not without reason, but we will not delve into it.

By the way, in addition to this task, Herriot was able to determine approximately what proportion the kernels occupy in a sufficiently large container, if we take the shape of the latter as a cube. It turned out that the fraction is π/(3√2) ≈ 0.74048.

3.

What does the word mean densest in the problem statement? Raleigh, Harriot, and Kepler himself did not give an exact answer to this. It meant densest in some reasonable sense. However, this formulation is not suitable for mathematics. It needs to be clarified.

Let's first go down one dimension and see how everything works on the plane. For the two-dimensional case, the problem turns into this: let the plane be given an infinite set of circles that do not intersect in the interior (but possibly touching - that is, having a common point on the boundary). Let's draw a square. Let's calculate the sum of the areas of the pieces of circles that fall inside the square. Let's take the ratio of this sum to the area of ​​the square, and we will increase the side of the square, looking at the change in the ratio.

We get the function f(a), Where a- side of a square. If we are lucky, then this function with growth argument will approach asymptotically to a certain number. This number is called the density of a given package. It is important that the function itself at some point can give a value greater than the density. Indeed, if the square is small, then it fits entirely in a circle and a certain ratio is equal to 1. But we are interested in the density on average, that is, informally speaking, “for a square with a sufficiently large side.”

Among all such densities, the maximum can be found. It is this, as well as the packaging that implements it, that will be called the densest.

“The closest packing is not necessarily the only one (in the asymptotic sense). There are an infinite number of dense packings in 3-dimensional space, and Kepler knew this,” says Oleg Musin from the University of Texas at Brownsville.

Once we have defined the concept of tightest packing, it is easy to understand that such a definition can be easily extended to a space of arbitrary dimension n. Indeed, let us replace the circles with balls of the corresponding dimension, that is, a set of points, the distance from which to a fixed point (called the center) does not exceed a certain value called the radius of the ball. Let us again arrange them so that any two, at best, touch, and at worst, have no common points at all. Let us define the same function as in the previous case, taking the volume of an n-dimensional cube and the sum of the volumes of the corresponding n-dimensional balls.

4.

So, we understand that Kepler's hypothesis is a problem about the densest packing of three-dimensional balls in three-dimensional space. What about the plane (since we started with it)? Or even from a straight line? With a straight line, everything is simple: a ball on a straight line is a segment. A straight line can be completely covered with identical segments intersecting at the ends. With such coverage, the function f(a) is constant and equal to 1.

On the plane everything turned out to be somewhat more complicated. So, let’s start with a set of points on the plane. We say that this set of points forms a lattice if we can find a pair of vectors v and w such that all points are obtained as N*v + M*w, where N and M are integers. In a similar way, a lattice can be defined in a space of arbitrarily large dimensions - it just requires more vectors.

Lattices are important for many reasons (for example, lattice sites are where atoms prefer to be located when it comes to solid materials), but for mathematicians they are good because they are very convenient to work with. Therefore, from all the packages, a class is separately distinguished in which the centers of the balls are located at the lattice nodes. If we limit ourselves to this case, then there are only five types of lattices on the plane. The densest packing of them is produced by one in which the points are arranged at the vertices of regular hexagons - like honeycombs in bees or atoms in graphene. This fact was proven by Lagrange in 1773. More precisely: Lagrange was not interested in dense packings, but was interested in quadratic forms. Already in XX it became clear that from his results on forms a result on the packing density for two-dimensional lattices follows.

“In 1831, Ludwig Sieber wrote a book on ternary quadratic forms. This book put forward a conjecture that is equivalent to Kepler's conjecture for lattice packings. Sieber himself was able to prove only a weak form of his hypothesis and test it for a large number of examples. This book was reviewed by the great Carl Friedrich Gauss. In this review, Gauss provides a truly amazing proof, which fits into 40 lines. This, as we now say, “Olympiad” proof is understandable to a high school student. Many mathematicians have tried to find the hidden meaning in Gauss’s proof, but so far no one has succeeded,” says Oleg Musin.

What happens, however, if we abandon the network condition? Here everything turns out to be somewhat more complicated. The first full-fledged attempt to deal with this case was made by the Norwegian mathematician Axel Thue. If you look at the page dedicated to Thue on Wikipedia, you won’t find anything about tight packaging there. This is understandable - Thue published two works that were more reminiscent of essays than normal mathematical works, in which, as it seemed to him, he completely solved the problem of dense packing. The only problem was that no one except Thue himself was convinced by his reasoning.

Laszlo Fejes Toth

Danzer, Ludwig / Wikimedia Commons

The problem was finally solved by the Hungarian mathematician Laszlo Fejes Toth in 1940. It turned out, by the way, that the arrangement of circles on a plane that realizes the most dense packing is the only one.

5.

Closely related to the close packing problem is the contact number problem. Let's look again at a circle on a plane. How many circles of the same radius can be placed around it so that they all touch the central one? The answer is six. Indeed, let's look at two neighboring circles touching our central one. Let's look at the distance from the center of the central circle to the centers of these two. It is equal 2R, Where R- radius of the circle. The distance between the centers of adjacent circles does not exceed 2R. Calculating the angle at the center of the central circle using the cosine theorem, we find that it is no less than 60 degrees. The sum of all central angles should give 360 ​​degrees, which means there can be no more than 6 such angles. And we know the location of circles with six angles.

The resulting number is called the plane contact number. A similar question can be asked for spaces of any dimension. Let the simplicity of the solution on a plane not mislead the reader - the problem of contact numbers, if simpler than the problem of close packing, is not much simpler. But more results have actually been obtained in this direction.

For three-dimensional space, the contact number became the subject of a public dispute between Isaac Newton himself and James Gregory in 1694. The first believed that the contact number should be 12, and the second - that 13. The thing is that it is not difficult to place 12 balls around the central one - the centers of such balls lie at the vertices of a regular icosahedron (he has exactly 12 of them). But these balls don't touch! At first glance, it seems that they can be moved so that one more, the 13th ball, can fit through. This is almost true: if the balls are moved apart a little, making the distance between their centers and the center of the central one 2R, but in total 2.06R, then 13 balls will already fit. But for touching balls Gregory was wrong - this fact was proved by van der Waarden and Schutte in 1953.

For dimension 4, this problem was solved by Oleg Musin in 2003. There the contact number turned out to be 24.

6.

In addition to these dimensions 1, 2, 3 and 4, contact numbers are also known in dimensions 8 and 24. Why these particular dimensions? The fact is that there are very interesting lattices for them, called E8 and the Leach lattice.

So, we have already found out what a lattice is. An important characteristic of a lattice for mathematics is its symmetry. By symmetry we mean, of course, not subjective sensations (and who would imagine this lattice in dimensions, for example, four?), but the number of different types of movements of space that translate this lattice into themselves. Let's explain with an example.

Let's take the same hexagonal lattice that realizes the closest packing on a plane. It is easy to understand that the lattice turns into itself if you shift it by the vectors v and w that were in the definition. But, in addition, the lattice can be rotated around the center of the hexagon. And there are 6 such rotations: 0, 60, 120, 180, 240, 300 degrees. In addition, the lattice can be displayed symmetrically about any axis of symmetry of the composite hexagon. A little exercise shows that, not counting the shifts, we get 12 transformations. Other lattices have fewer such transformations, so we say they are less symmetrical.

So, E8 and the Leach lattice are incredibly symmetrical lattices. E8 is located in 8-dimensional space. This lattice was invented in 1877 by Russian mathematicians Korkin and Zolotarev. It consists of vectors, all of whose coordinates are integers, and their sum is even. Such a lattice, minus shifts, has 696,729,600 transformations. The Lich Grid exists in twenty-four dimensional space. It consists of vectors with integer coordinates and the condition - the sum of the coordinates minus any coordinate multiplied by 4 is divided by 8. It has a simply colossal number of symmetries - 8,315,553,613,086,720,000 pieces.

So, in 8-dimensional and 24-dimensional space, the balls located at the vertices of these same lattices touch 240 and 19650 balls, respectively. Surprisingly, these are precisely the contact numbers (see point 5) for spaces of the corresponding dimension.

7.

Now let's return to the three-dimensional case and Kepler's hypothesis (the one we talked about at the very beginning). This task turned out to be many times more difficult than its predecessors.

Let's start with the fact that there are infinitely many packings with the same density as the hexagonal dense one. We began to lay it out, starting with the balls laid out at the nodes of the hexagonal lattice. But you can do it differently: for example, at the first level, fold the balls into a square, that is, so that the vertices of the balls are located at the nodes of an already square lattice. In this case, each ball touches four neighbors. The second layer, as in the case of the hexagonal one, will be placed on top in the gaps between the balls of the first layer. This packaging is called face-centered cubic packing. This, by the way, is the only densest lattice packaging in space.

At first glance, it seems that this packing should be worse, because the gaps between the four balls in the first layer are much larger (feels like) than the gaps in the hexagonal dense packing. But when we place the second row, the balls - precisely because the gaps are larger - sink deeper. As a result, as it turns out, the density is the same as before. In fact, of course, the trick is that such a package is obtained if you look at the hexagonal one from a different angle.

It turns out that in three-dimensional space there are no such beautiful unique lattices as, for example, hexagonal on a plane or E8 in 8-dimensional space. At first glance, it is completely unclear how to search for the closest packing in three-dimensional space.

8.

The solution to Kepler's hypothesis was born in several stages.

First, Fejes Tóth, the same Hungarian who solved the problem of close packing in a non-plane, put forward the following hypothesis: in order to understand whether the packing is close or not, it is enough to consider finite clusters of balls. As we found out, unlike a plane, if the central ball touches 12 neighbors, then there are gaps between them. Therefore, Fejes Toth proposed studying clusters consisting of a central ball, its neighbors, and neighbors of neighbors.

The thing is that this assumption was made in the 60s of the last century. And the problem of minimizing the volume of such a cluster is essentially a nonlinear optimization problem for a function of approximately 150 variables (each ball has a center, it is specified by three coordinates). Roughly speaking, such a function needs to find a minimum under some additional conditions. On the one hand, the task has become finite, but on the other hand, it is completely insurmountable from a computational point of view for humans. But Fejes Toth was not upset and said that very soon computers would have the necessary computing power. They will help.

Mathematicians really liked Fejes Thoth's hypothesis and they began to actively work in this direction. By the beginning of the 90s, estimates of the maximum packing density of spheres in three-dimensional space were gradually decreasing. The idea was that at some point the estimate would be equal to the density of a face-centered cubic packing and, therefore, Kepler's hypothesis would be proven. During this time, mathematician Thomas Hales published his first papers on packaging. For his work, he chose an object called Delaunay stars (after the Soviet mathematician Boris Delaunay). This was a bold step - at that moment, the effectiveness of such objects for studying the packaging problem was questionable.

After just 8 years of hard work, in 1998, Hales completed the proof of the Kepler hypothesis. He reduced the proof to a finite combinatorial search of different structures such as Delaunay stars. For each such combinatorial structure, it was necessary to maximize the density. Since the computer works normally only with integer numbers (simply because in mathematics numbers are most often infinite fractions), then for each case Delaunay automatically built an approximation from above using symbolic rational calculations (rational numbers, after all, if you do not convert them to decimal fractions, just a couple of integers). With this approximation, he received an estimate for the maximum density from above. As a result, all estimates turned out to be less than those given by the face-centered cubic packing.

Many mathematicians, however, were confused by the situation in which a computer was built to construct the approximation. To prove that he had no errors in the computer part of the proof, Hales began formalization and verification, albeit also with the help of a computer. This work, which was carried out by a fairly large international team, was completed in August 2014. No errors were found in the proof.

9.

The proofs for dimensions 8 and 24 do not require a computer and are somewhat simpler. Some time ago, very good estimates were obtained to estimate the maximum packing density in these dimensions. This was done by mathematicians Kohn and Elkies in 2003. By the way, this estimate (also called the Kohn-Elkies boundary) was found by the Russian mathematician Dmitry Gorbachev from Tula a couple of years before Kohn and Elkies themselves. However, he published this work in Russian and in the Tula magazine. Kon and Elkies did not know about this work, and when they were told, they, by the way, referred to it.

“The Kohn-Elkies boundary appeared on the basis of the work of Jean-Frederic Delsarte and our wonderful mathematicians Grigory Kabatyansky and Vladimir Levenshtein. The asymptotic (in terms of the space dimension) estimate of the packing density of balls in n-dimensional space, obtained by Kabatyansky and Levenshtein, has been “standing” since 1978. By the way, it was Levenshtein and independently the Americans Odlyzhko and Sloan who solved the problem of contact numbers in dimensions 8 and 24 in 1979. They directly used the Delsarte-Kabatyansky-Levenshtein method,” says Oleg Musin.

Kohn and Elkies' estimates are actually correct for all packings, but in dimensions 8 and 24 they give a very good approximation. For example, the mathematicians' estimate is only about 0.0001 percent greater than the density of E8 in eight-dimensional space. Therefore, the task arose to improve this assessment - after all, the solution, it would seem, is already nearby. Moreover, in 2012, the same Dmitry Gorbachev applied for (and won) a grant from the Dynasty Foundation. In the application, he explicitly stated that he planned to prove the packing density of E8 in eight-dimensional space.

They say that Gorbachev was prompted to make such a bold statement by another mathematician, Andrei Bondarenko, essentially a mentor, one of the scientific supervisors of Marina Vyazovskaya, the one who solved the problem for 8-dimensional space (and co-authored, for 24-dimensional space). It is Bondarenko who she thanks at the end of her breakthrough work. So, Bondarenko and Gorbachev didn’t succeed, but Vyazovskaya did. Why?

Marina Vyazovskaya

Humboldt University of Berlin

The Kohn-Elkies estimate relates packing density to a property of some function from a suitable set. Roughly speaking, an estimate is constructed for each such function. That is, the main task is to find a suitable function so that the resulting estimate turns out to be the one we need. So, the key ingredient in the construction of Vyazovskaya is modular forms. We have already mentioned them in relation to the proof of Fermat's Last Theorem, for which. This is a rather symmetrical object that constantly appears in various branches of mathematics. It was this toolkit that allowed us to find the desired function.

In 24-dimensional space, the estimate was obtained in the same way. This work has more authors, but it is based on the same achievement of Vyazovskaya (albeit, of course, slightly adapted). By the way, another remarkable fact was proven in the work: the Leach lattice realizes the only periodic closest packing. That is, all other periodic packages have a density less than this. According to Oleg Musin, a similar result for periodic packings can be true in dimensions 4 and 8.

10.

From an application point of view, the problem of dense packing in high-dimensional spaces is primarily a problem of optimal error-correcting coding.

Let's imagine that Alice and Bob are trying to communicate using radio signals. Alice says that she will send Bob a signal consisting of 24 different frequencies. Bob will measure the amplitude of each frequency. As a result, he will have a set of 24 amplitudes. They, of course, define a point in 24-dimensional space - after all, there are 24 of them. Bob and Alice take, say, the Dahl dictionary and assign each word its own set of 24 amplitudes. It turns out that we encoded words from Dahl's dictionary with points of 24-dimensional space.

In an ideal world, nothing else is needed. But real data channels add noise, which means that during decoding Bob may receive a set of amplitudes that does not correspond to a single word. But then he can look at the word closest to the deciphered version. If there is one, then that means most likely this is it. In order to always be able to do this, it is necessary that the points of space be located as far as possible from each other. That is, for example, if the noise level is such that a distortion is introduced that shifts the result by a vector of length no more than one, then the two code points must be exactly at a distance of at least two. Then, even with distortions, Bob’s result will always be close to one single word - the one that is needed.

At the same time, I also don’t really want to inflate a lot of words - we have a rather limited range in which we can transmit information. For example, it will be strange (and not very effective) if Alice and Bob begin to communicate in the X-ray range. Therefore, ideally, the distance between adjacent code words should be exactly two. And this means that the words are located at the vertices of balls of radius 1, tightly packed in a 24-dimensional space.