Why do we live in three-dimensional space. Three-dimensional space: vectors, coordinates Where three-dimensional space is used

Launches the Question to the Scientist project, in which experts will answer interesting, naive or practical questions. In this issue, Candidate of Physical and Mathematical Sciences Ilya Shchurov talks about 4D and whether it is possible to enter the fourth dimension.

What is four-dimensional space ("4D")?

Ilya Shchurov

Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Higher Mathematics, National Research University Higher School of Economics

Let's start with the simplest geometric object - a point. The point is zero-dimensional. It has no length, no width, no height.

Now let's move the point along a straight line by some distance. Let's say that our point is the tip of a pencil; when we moved it, it drew a line. A segment has a length, and no more dimensions - it is one-dimensional. The segment "lives" on a straight line; the line is a one-dimensional space.

Now let's take a segment and try to move it, as before a point. (You can imagine that our segment is the base of a wide and very thin brush.) If we go beyond the line and move in a perpendicular direction, we get a rectangle. A rectangle has two dimensions - width and height. The rectangle lies in some plane. The plane is a two-dimensional space (2D), on it you can enter a two-dimensional coordinate system - each point will correspond to a pair of numbers. (For example, a Cartesian coordinate system on a blackboard, or latitude and longitude on a geographic map.)

If you move a rectangle in a direction perpendicular to the plane in which it lies, you get a "brick" (rectangular parallelepiped) - a three-dimensional object that has a length, width and height; it is located in three-dimensional space - in the one in which we live. Therefore, we have a good idea of what three-dimensional objects look like. But if we lived in a two-dimensional space - on a plane - we would have to pretty much stretch our imagination to imagine how we can move the rectangle so that it goes out of the plane in which we live.

It is also quite difficult for us to imagine a four-dimensional space, although it is very easy to describe mathematically. Three-dimensional space is a space in which the position of a point is given by three numbers (for example, the position of an aircraft is given by longitude, latitude, and altitude). In four-dimensional space, a point corresponds to four numbers-coordinates. A "four-dimensional brick" is obtained by shifting an ordinary brick along some direction that does not lie in our three-dimensional space; it has four dimensions.

In fact, we encounter four-dimensional space every day: for example, when making a date, we indicate not only the meeting place (it can be set with a triple of numbers), but also the time (it can be set with a single number - for example, the number of seconds that have passed since a certain date). If you look at a real brick, it has not only length, width and height, but also a length in time - from the moment of creation to the moment of destruction.

The physicist will say that we live not just in space, but in space-time; the mathematician will add that it is four-dimensional. So the fourth dimension is closer than it seems.

Tasks:

Give some other example of the implementation of four-dimensional space in real life.

Define what five-dimensional space (5D) is. What should a 5D movie look like?

Please send answers by e-mail: [email protected]

Even from the school course of algebra and geometry, we know about the concept of three-dimensional space. If you look, the term "three-dimensional space" itself is defined as a coordinate system with three dimensions (everyone knows this). In fact, any volumetric object can be described using length, width and height in the classical sense. However, let's, as they say, dig a little deeper.

What is 3D space

As it has already become clear, the understanding of three-dimensional space and the objects that can exist inside it is determined by three basic concepts. True, in the case of a point, these are exactly three values, and in the case of straight lines, curves, broken lines or volumetric objects, there may be more corresponding coordinates.

In this case, everything depends on the type of object and the coordinate system used. Today, the most common (classical) system is considered to be the Cartesian system, which is sometimes also called rectangular. She and some other varieties will be discussed a little later.

Among other things, here it is necessary to distinguish between abstract concepts (if I may say so, shapeless) like points, lines or planes and figures that have finite dimensions or even volume. Each of these definitions has its own equations describing their possible position in three-dimensional space. But now is not about that.

The concept of a point in three-dimensional space

First, let's define what a point in three-dimensional space is. In general, it can be called a certain basic unit that defines any flat or three-dimensional figure, straight line, segment, vector, plane, etc.

The point itself is characterized by three main coordinates. For them, in a rectangular system, special guides are used, called the X, Y and Z axes, with the first two axes used to express the horizontal position of the object, and the third refers to the vertical setting of coordinates. Naturally, for the convenience of expressing the position of an object relative to zero coordinates, positive and negative values are taken in the system. However, other systems can be found today.

Varieties of coordinate systems

As already mentioned, the rectangular coordinate system created by Descartes is the main one today. Nevertheless, in some methods for specifying the location of an object in three-dimensional space, some other varieties are also used.

The most famous are cylindrical and spherical systems. The difference from the classical one is that when setting the same three values that determine the location of a point in three-dimensional space, one of the values is an angular one. In other words, such systems use a circle corresponding to an angle of 360 degrees. Hence the specific assignment of coordinates, including such elements as radius, angle and generatrix. Coordinates in a three-dimensional space (system) of this type obey somewhat different laws. Their task in this case is controlled by the right hand rule: if you align your thumb and forefinger with the X and Y axes, respectively, the remaining fingers in a bent position will point in the direction of the Z axis.

The concept of a straight line in three-dimensional space

Now a few words about what a straight line is in three-dimensional space. Based on the basic concept of a straight line, this is a kind of infinite line drawn through a point or two, not counting the set of points located in a sequence that does not change the direct passage of the line through them.

If you look at a straight line drawn through two points in three-dimensional space, you will have to take into account three coordinates of both points. The same applies to segments and vectors. The latter determine the basis of the three-dimensional space and its dimension.

Definition of vectors and basis of three-dimensional space

Note that these can only be three vectors, but you can define as many triplets of vectors as you like. The space dimension is determined by the number of linearly independent vectors (three in our case). And a space in which there is a finite number of such vectors is called finite-dimensional.

Dependent and independent vectors

With regard to the definition of dependent and independent vectors, it is customary to consider vectors that are projections linearly independent (for example, vectors of the X axis projected onto the Y axis).

As it is already clear, any fourth vector is dependent (the theory of linear spaces). But three independent vectors in three-dimensional space must not necessarily lie in the same plane. In addition, if independent vectors are defined in three-dimensional space, they cannot be, so to speak, one continuation of the other. As is already clear, in the case we are considering with three dimensions, according to the general theory, it is possible to construct exclusively only triples of linearly independent vectors in a certain coordinate system (no matter what type).

Plane in 3D space

If we consider the concept of a plane, without going into mathematical definitions, for a simpler understanding of this term, such an object can be considered exclusively as two-dimensional. In other words, it is an infinite collection of points for which one of the coordinates is constant (constant).

For example, a plane can be called any number of points with different X and Y coordinates, but the same Z coordinates. In any case, one of the three-dimensional coordinates remains unchanged. However, this is, so to speak, the general case. In some situations, three-dimensional space can be intersected by a plane along all axes.

Are there more than three dimensions

The question of how many dimensions can exist is quite interesting. It is believed that we do not live in a three-dimensional space from a classical point of view, but in a four-dimensional one. In addition to the length, width and height known to everyone, such a space also includes the lifetime of the object, and time and space are interconnected quite strongly. This was proved by Einstein in his theory of relativity, although this applies more to physics than to algebra and geometry.

It is also interesting that today scientists have already proved the existence of at least twelve dimensions. Of course, not everyone will be able to understand what they are, since this refers rather to a certain abstract area that is outside the human perception of the world. Nevertheless, the fact remains. And it is not for nothing that many anthropologists and historians argue that our ancestors could have some specific developed sense organs like a third eye, which helped to perceive multidimensional reality, and not exclusively three-dimensional space.

By the way, today there are quite a lot of opinions about the fact that extrasensory perception is also one of the manifestations of the perception of a multidimensional world, and quite a lot of evidence can be found for this.

Note that it is also not always possible to describe multidimensional spaces that differ from our four-dimensional world with modern basic equations and theorems. Yes, and science in this area refers more to the field of theories and assumptions than to what can be clearly felt or, so to speak, touched or seen with one's own eyes. Nevertheless, indirect evidence of the existence of multidimensional worlds, in which four or more dimensions can exist, is beyond doubt today.

Conclusion

In general, we have very briefly reviewed the basic concepts related to three-dimensional space and basic definitions. Naturally, there are many special cases associated with different coordinate systems. In addition, we tried not to go too far into the mathematical jungle to explain the basic terms only so that the question related to them is understandable to any student (so to speak, the explanation is “on the fingers”).

Nevertheless, it seems that even from such simple interpretations one can draw a conclusion about the mathematical aspect of all the components included in the basic school course of algebra and geometry.

In which we ask our scientists to answer rather simple, at first glance, but controversial questions from readers. For you, we have selected the most interesting answers from PostNauka experts.

Everyone is familiar with the abbreviation 3D, meaning "three-dimensional" (the letter D - from the word dimension - measurement). For example, when choosing a movie marked 3D in a cinema, we know for sure that you will have to wear special glasses to watch it, but the picture will not be flat, but three-dimensional. What is 4D? Does “four-dimensional space” exist in reality? Is it possible to enter the "fourth dimension"?

To answer these questions, let's start with the simplest geometric object - a point. The point is null. It has no length, no width, no height.

// 8-cell simple

Now let's move the point along a straight line by some distance. Let's say that our point is the tip of a pencil; when we moved it, it drew a line. A segment has a length, and no more dimensions: it is one-dimensional. The segment "lives" on a straight line; the line is a one-dimensional space.

Now let's take a segment and try to move it as before a point. You can imagine that our segment is the base of a wide and very thin brush. If we go beyond the line and move in a perpendicular direction, we get a rectangle. A rectangle has two dimensions - width and height. The rectangle lies in some plane. The plane is a two-dimensional space (2D), on it you can enter a two-dimensional coordinate system - each point will correspond to a pair of numbers. (For example, a Cartesian coordinate system on a blackboard, or latitude and longitude on a geographic map.)

If you move a rectangle in a direction perpendicular to the plane in which it lies, you get a "brick" (rectangular parallelepiped) - a three-dimensional object that has a length, width and height; it is located in a three-dimensional space, in the same one in which we live. Therefore, we have a good idea of what three-dimensional objects look like. But if we lived in a two-dimensional space - on a plane - we would have to pretty much stretch our imagination to imagine how we can move the rectangle so that it goes out of the plane in which we live.

In fact, we encounter four-dimensional space on a daily basis: for example, when setting up a date, we indicate not only the meeting place (it can be set with a triple of numbers), but also the time (it can be set with a single number, for example, the number of seconds that have passed since a certain date). If you look at a real brick, it has not only length, width and height, but also a length in time - from the moment of creation to the moment of destruction.

The physicist will say that we live not just in space, but in space-time; the mathematician will add that it is four-dimensional. So the fourth dimension is closer than it seems.

Three-dimensional space - has three homogeneous dimensions: height, width and length. This is a geometric model of our material world.

To understand the nature of physical space, one must first answer the question of the origin of its dimension. Therefore, the value of dimension, as can be seen, is the most significant characteristic of physical space.

Dimension of space

Dimension is the most general quantifiable property of space-time. At present, a physical theory that claims to be a spatio-temporal description of reality takes the value of dimension as an initial postulate. The concept of the number of dimensions, or the dimension of space, is one of the most fundamental concepts of mathematics and physics.

Modern physics has come close to answering the metaphysical question that was posed in the works of the Austrian physicist and philosopher Ernst Mach: “Why is space three-dimensional?”. It is believed that the fact of the three-dimensionality of space is associated with the fundamental properties of the material world.

The development of a process from a point generates space, i.e. the place where the implementation of the development program should take place. "The generated space" is for us the form of the Universe, or the form of matter in the Universe.

So it was believed in ancient times ...

Even Ptolemy wrote on the topic of the dimension of space, where he argued that in nature there cannot be more than three spatial dimensions. In his book On the Sky, another Greek thinker, Aristotle, wrote that only the presence of three dimensions ensures the perfection and completeness of the world. One dimension, Aristotle reasoned, forms a line. If we add another dimension to the line, we get a surface. The addition of a surface with one more dimension forms a three-dimensional body.

It turns out that “it is no longer possible to go beyond the limits of a volumetric body to something else, since any change occurs due to some kind of deficiency, and there is none here. The given way of Aristotle's thought suffers from one significant weakness: it remains unclear for what reason exactly a three-dimensional three-dimensional body possesses completeness and perfection. At one time, Galileo rightly ridiculed the opinion that "the number "3" is a perfect number and that it is endowed with the ability to communicate perfection to everything that has a trinity."

What determines the dimension of space

Space is infinite in all directions. However, at the same time, it can be measured only in three directions independent of each other: in length, width and height; we call these directions dimensions of space and say that our space has three dimensions, that it is three-dimensional. In this case, “in this case, we call an independent direction a line lying at a right angle to another. Such lines, i.e. lying simultaneously at right angles to one another and not parallel to each other, our geometry knows only three. That is, the dimension of our space is determined by the number of possible lines in it, lying at right angles to one another. There cannot be another line on a line - this is a one-dimensional space. Two perpendiculars are possible on the surface - this is a two-dimensional space. In "space" three perpendiculars are three-dimensional space.

Why is space three-dimensional?

Rare in earthly conditions, the experience of materialization of people often has a physical impact on eyewitnesses ...

But, in the ideas about space and time, there is still a lot of obscurity, giving rise to ongoing discussions of scientists. Why does our space have three dimensions? Can multidimensional worlds exist? Is it possible for material objects to exist outside of space and time?

The assertion that physical space has three dimensions is just as objective as the assertion, for example, that there are three physical states of matter: solid, liquid, and gaseous; it describes a fundamental fact of the objective world. I. Kant stressed that the reason for the three-dimensionality of our space is still unknown. P. Ehrenfest and J. Whitrow showed that if the number of space dimensions were more than three, then the existence of planetary systems would be impossible - only in the three-dimensional world can there be stable orbits of planets in planetary systems. That is, the three-dimensional order of matter is the only stable order.

But the three-dimensionality of space cannot be asserted as some kind of absolute necessity. It is a physical fact like any other and, as a consequence, is subject to the same kind of explanation.

The question of why our space is three-dimensional can be solved either from the standpoint of teleology, based on the unscientific assertion that “the three-dimensional world is the most perfect of all possible worlds”, or from scientific materialistic positions, based on fundamental physical laws.

Opinion of contemporaries

Modern physics says that the characteristic of three-dimensionality is that it, and only it, makes it possible to formulate continuous causal laws for physical reality. But, “modern concepts do not reflect the true state of the physical picture of the world. In our time, scientists consider space as a kind of structure, consisting of many levels, which are also indefinite. And therefore, it is no coincidence that modern science cannot answer the question why our space, in which we live and which we survey, is three-dimensional.

Theory of connected spaces

In parallel worlds, events happen in their own way, they can ...

“Attempts to search for an answer to this question, remaining only within the limits of mathematics, are doomed to failure. The answer may lie in a new, underdeveloped area of physics.” Let's try to find an answer to this question based on the provisions of the considered physics of bound spaces.

According to the theory of connected spaces, the development of an object proceeds in three stages, with each stage developing along its chosen direction, i.e. along its axis of development.

At the first stage, the development of the object goes along the initial selected direction, i.e. has one axis of development. At the second stage, the system formed at the first stage is rotated by 90°, i.e. there is a change in the direction of the spatial axis, and the development of the system begins to go along the second selected direction, perpendicular to the original one. At the third stage, the development of the system again turns by 90°, and it begins to develop along the third selected direction, perpendicular to the first two. As a result, three nested spheres of space are formed, each of which corresponds to one of the axes of development. Moreover, all three of these spaces are connected into a single stable formation by a physical process.

And because this process is implemented at all scale levels of our world, then all systems, including the coordinates themselves, are built according to the triadic (three-coordinate) principle. It follows that as a result of passing through the three stages of the development of the process, a three-dimensional space is naturally formed, formed as a result of the physical process of development by three coordinate axes of three mutually perpendicular directions of development!

These intelligent entities arose at the very dawn of the existence of the Universe ...

No wonder Pythagoras, who, apparently, could have this knowledge, owns the expression: "All things consist of three." The same is said by N.K. Roerich: “The symbol of the Trinity is of great antiquity and is found all over the World, therefore it cannot be limited to any sect, organization, religion or tradition, as well as personal or group interests, because it represents the evolution of consciousness in all its phases ... The sign of the trinity turned out to be scattered all over the world ... If you put together all the imprints of the same sign, then perhaps it will turn out to be the most common and oldest among human symbols. No one can claim that this sign belongs to only one belief or is based on one folklore.

It is not for nothing that even in ancient times our world was presented as a triune deity (three merged into one): something one, whole and indivisible, in its sacred significance far exceeding the original values.

We traced the spatial specialization (distribution along the coordinate directions of space) within a single system, but we can see exactly the same distribution in any society from an atom to galaxies. These three varieties of space are nothing but the three coordinate states of the geometric space.

How many dimensions does the space of the world in which we live have?

What's question! Of course, an ordinary person will say three and he will be right. But there is still a special breed of people who have an acquired property to doubt obvious things. These people are called "scientists" because they are specifically taught to do so. For them, our question is not so simple: the measurement of space is an elusive thing, they cannot simply be counted by pointing a finger: one, two, three. It is impossible to measure their number with any instrument like a ruler or an ammeter: space has 2.97 plus or minus 0.04 measurements. We have to think this question deeper and look for indirect ways. Such searches have proved fruitful: modern physics believes that the number of dimensions of the real world is closely related to the deepest properties of matter. But the path to these ideas began with a revision of our everyday experience.

It is usually said that the world, like any body, has three dimensions, which correspond to three different directions, say, "height", "width" and "depth". It seems clear that the "depth" depicted on the plane of the drawing is reduced to "height" and "width", is in a sense a combination of them. It is also clear that in a real three-dimensional space all conceivable directions are reduced to some three pre-selected ones. But what does “reduced”, “are a combination” mean? Where will these "width" and "depth" be if we find ourselves not in a rectangular room, but in weightlessness somewhere between Venus and Mars? Finally, who can guarantee that "height", say, in Moscow and New York is the same "measurement"?

The trouble is that we already know the answer to the problem we are trying to solve, and this is far from always useful. Now, if you could find yourself in a world whose number of dimensions is not known in advance, and look for them one at a time Or, at least, so renounce the available knowledge of reality in order to look at its initial properties in a completely new way.

Cobblestone mathematician's tool

In 1915, the French mathematician Henri Lebesgue figured out how to determine the number of dimensions of space without using the concepts of height, width and depth. To understand his idea, it is enough to look closely at the cobbled pavement. On it you can easily find places where the stones converge in threes and fours. You can pave the street with square tiles, which will adjoin each other in two or four; if you take the same triangular tiles, they will adjoin two or six. But not a single master can pave the street so that the cobblestones everywhere adjoin each other only two by two. This is so obvious that it's ridiculous to suggest otherwise.

Mathematicians differ from normal people precisely in that they notice the possibility of such absurd assumptions and are able to draw conclusions from them. In our case, Lebesgue reasoned as follows: the surface of the pavement is, of course, two-dimensional. At the same time, it inevitably has points where at least three boulders converge. Let's try to generalize this observation: let's say that the dimension of some area is equal to N, if during its tiling it is not possible to avoid the contact of N + 1 or more "cobblestones". Now any bricklayer will confirm the three-dimensionality of space: after all, when laying out a thick wall in several layers, there will definitely be points where at least four bricks will touch!

However, at first glance it seems that one can find, as mathematicians put it, a "counterexample" to Lebesgue's definition of dimension. This is a plank floor in which the floorboards touch exactly two by two. Why not tiling? Therefore, Lebesgue also demanded that the "pebbles" used in the definition of dimension be small. This is an important idea, and we will return to it again at the end from an unexpected perspective. And now it is clear that the condition of a small size of "cobblestones" saves Lebesgue's definition: say, short parquet floors, in contrast to long floorboards, at some points will necessarily come into contact in threes. This means that the three dimensions of space are not just the ability to arbitrarily choose some three “different” directions in it. Three dimensions is a real limitation of our possibilities, which is easy to feel after playing a little with cubes or bricks.

The dimension of space through the eyes of Stirlitz

Another limitation associated with the three-dimensionality of space is well felt by a prisoner locked in a prison cell (for example, Stirlitz in Muller's basement). What does this camera look like from his point of view? Rough concrete walls, a tightly closed steel door - in a word, one two-dimensional surface without cracks and holes, enclosing the enclosed space where he is located on all sides. There is really nowhere to go from such a shell. Is it possible to lock a person inside a one-dimensional circuit? Imagine how Muller draws a circle around Stirlitz with chalk on the floor and goes home: this does not even look like a joke.

From these considerations, one more way is extracted to determine the number of dimensions of our space. Let's formulate it this way: it is possible to enclose an area of N-dimensional space from all sides only with an (N-1)-dimensional "surface". In two-dimensional space the “surface” will be a one-dimensional contour, in one-dimensional space it will be two zero-dimensional points. This definition was invented in 1913 by the Dutch mathematician Brouwer, but it became known only eight years later, when it was independently rediscovered by our Pavel Uryson and the Austrian Karl Menger.

Here our paths with Lebesgue, Brouwer and their colleagues diverge. They needed a new definition of dimension in order to construct an abstract mathematical theory of spaces of any dimension up to infinite. This is a purely mathematical construction, a game of the human mind, which is strong enough even to cognize such strange objects as infinite-dimensional space. Mathematicians do not try to find out whether there really are things with such a structure: this is not their profession. On the contrary, our interest in the number of dimensions of the world in which we live is physical: we want to know how many there really are and how to feel their number “on our own skin”. We need phenomena, not pure ideas.

It is characteristic that all the examples given were borrowed more or less from the architecture. It is this area of human activity that is most closely connected with space, as it appears to us in ordinary life. In order to advance further in the search for dimensions of the physical world, an exit to other levels of reality will be required. They are available to man thanks to modern technology, and therefore physics.

What is the speed of light here?

Let's briefly return to Stirlitz, who was left in the cell. To get out of the shell that reliably separated him from the rest of the three-dimensional world, he took advantage of the fourth dimension, which is not afraid of two-dimensional barriers. Namely, he thought for a while and found himself a suitable alibi. In other words, the new mysterious dimension that Stirlitz used is time.

It is difficult to say who first noticed the analogy between time and the dimensions of space. They already knew about it two centuries ago. Joseph Lagrange, one of the creators of classical mechanics, the science of the motions of bodies, compared it with the geometry of the four-dimensional world: his comparison sounds like a quote from a modern book on General Relativity.

Lagrange's train of thought, however, is easy to understand. In his time, graphs of the dependence of variables on time were already known, like the current cardiograms or graphs of the monthly course of temperature. Such graphs are drawn on a two-dimensional plane: along the ordinate axis, the path traveled by the variable is plotted, and along the abscissa axis, the elapsed time. At the same time, time really becomes just “another” geometric dimension. In the same way, you can add it to the three-dimensional space of our world.

But is time really like spatial dimensions? On the plane with the drawn graph, there are two selected "meaningful" directions. And directions that do not coincide with any of the axes do not make sense, they do not depict anything. On the usual geometric two-dimensional plane, all directions are equal, there are no distinguished axes.

Time can truly be considered the fourth coordinate only if it is not distinguished from other directions in the four-dimensional "space-time". It is necessary to find a way to "rotate" space-time so that time and space dimensions "mix" and can, in a certain sense, pass into each other.

This method was found by Albert Einstein, who created the theory of relativity, and Hermann Minkowski, who gave it a rigorous mathematical form. They took advantage of the fact that in nature there is a universal speed the speed of light.

Let's take two points in space, each at its own moment in time, or two "events" in the jargon of the theory of relativity. If we multiply the time interval between them, measured in seconds, by the speed of light, we get a certain distance in meters. We will assume that this imaginary segment is “perpendicular” to the spatial distance between the events, and together they form the “legs” of some right-angled triangle, the “hypotenuse” of which is a segment in space-time connecting the selected events. Minkowski suggested: to find the square of the length of the "hypotenuse" of this triangle, we will not add the square of the length of the "spatial" leg to the square of the length of the "temporal", but subtract it. Of course, this can lead to a negative result: then they consider that the "hypotenuse" has an imaginary length! But what's the point?

When a plane is rotated, the length of any segment drawn on it is preserved. Minkowski realized that it is necessary to consider such "rotations" of space-time, which preserve the "length" of segments between events proposed by him. This is how you can achieve that the speed of light in the constructed theory is universal. If two events are connected by a light signal, then the “Minkowski distance” between them is zero: the spatial distance coincides with the time interval multiplied by the speed of light. The "rotation" proposed by Minkowski keeps this "distance" zero, no matter how space and time are mixed during the "rotation".

This is not the only reason why Minkowski's "distance" has a real physical meaning, despite the definition, which is extremely strange for an unprepared person. Minkowski's "distance" provides a way to construct the "geometry" of space-time in such a way that both spatial and temporal intervals between events can be made equal. Perhaps this is the main idea of the theory of relativity.

So, the time and space of our world are so closely connected with each other that it is difficult to understand where one ends and the other begins. Together they form something like a stage on which the play “The History of the Universe” is being played. Actors particles of matter, atoms and molecules, from which galaxies, nebulae, stars, planets, and on some planets even living intelligent organisms are assembled (the reader should be aware of at least one such planet).

Based on the discoveries of his predecessors, Einstein created a new physical picture of the world, in which space and time turned out to be inseparable from each other, and reality became truly four-dimensional. And in this four-dimensional reality, one of the two “fundamental interactions” known to science at that time “dissolved”: the law of universal gravitation was reduced to the geometric structure of the four-dimensional world. But Einstein could not do anything with another fundamental interaction electromagnetic.

Space-time takes on new dimensions

The general theory of relativity is so beautiful and convincing that immediately after it became known, other scientists tried to follow the same path further. Einstein reduced gravity to geometry? So, it remains for his followers to geometrize the electromagnetic forces!

Since Einstein exhausted the possibilities of the four-dimensional space metric, his followers began to try to somehow expand the set of geometric objects from which such a theory could be constructed. It is quite natural that they wanted to increase the number of dimensions.

But while theorists were engaged in the geometrization of electromagnetic forces, two more fundamental interactions were discovered - the so-called strong and weak. Now it was necessary to combine already four interactions. At the same time, a lot of unexpected difficulties arose, to overcome which new ideas were invented, leading scientists further and further away from the visual physics of the last century. They began to consider models of worlds that have tens and even hundreds of dimensions, and infinite-dimensional space came in handy. To tell about these searches, it would be necessary to write a whole book. Another question is important for us: where are all these new dimensions located? Can we feel them in the same way that we feel time and three-dimensional space?

Imagine a long and very thin tube for example, a fire hose empty inside, reduced a thousand times. It is a two-dimensional surface, but its two dimensions are unequal. One of them, length, is easy to notice this is a "macroscopic" measurement. The perimeter, however, the “transverse” dimension, can only be seen under a microscope. Modern multidimensional models of the world are similar to this tube, although they have not one, but four macroscopic dimensions - three spatial and one temporal. The remaining measurements in these models cannot be seen even under an electron microscope. To detect their manifestations, physicists use accelerators, very expensive but crude "microscopes" for the subatomic world.

While some scientists perfected this impressive picture, brilliantly overcoming one obstacle after another, others had a tricky question:

Can the dimension be fractional?

Why not? To do this, it is necessary to “simply” find a new dimension property that could connect it with non-integer numbers, and geometric objects that have this property and have a fractional dimension. If we want to find, for example, a geometric figure that has one and a half dimensions, then we have two ways. You can either try to subtract half a dimension from a 2D surface or add a half dimension to a 1D line. To do this, let's first practice adding or subtracting an entire dimension.

There is such a famous children's trick. The magician takes a triangular sheet of paper, makes an incision on it with scissors, folds the sheet along the incision line in half, makes another incision, folds again, cuts one last time, and ap! in his hands is a garland of eight triangles, each of which is completely similar to the original one, but eight times smaller than it in area (and the square root of eight times in size). Perhaps this trick was shown in 1890 to the Italian mathematician Giuseppe Peano (or maybe he himself liked to show it), in any case, it was then that he noticed this. Let's take perfect paper, perfect scissors and repeat the sequence of cutting and folding an infinite number of times. Then the dimensions of the individual triangles obtained at each step of this process will tend to zero, and the triangles themselves will shrink into points. Therefore, we will get a one-dimensional line from a two-dimensional triangle, without losing a single piece of paper! If you do not stretch this line into a garland, but leave it as “crumpled”, as we did when cutting, then it will fill the entire triangle. Moreover, no matter under what strong microscope we consider this triangle, magnifying its fragments any number of times, the resulting picture will look exactly the same as unenlarged: scientifically speaking, the Peano curve has the same structure at all magnification scales, or is “scaled invariant".

So, having bent countless times, the one-dimensional curve could, as it were, acquire a dimension of two. This means that there is hope that a less "crumpled" curve will have a "dimension", say, one and a half. But how do you find a way to measure fractional dimensions?

In the "cobblestone" definition of dimension, as the reader remembers, it was necessary to use sufficiently small "cobblestones", otherwise the result could turn out wrong. But a lot of small "cobblestones" will be required: the more, the smaller their size. It turns out that in order to determine the dimension, it is not necessary to study how the “cobblestones” adjoin each other, but it is enough just to find out how their number increases with decreasing size.

Take a straight line segment 1 decimeter long and two Peano curves, together filling a square measuring decimeter by decimeter. We will cover them with small square "cobblestones" with a side length of 1 centimeter, 1 millimeter, 0.1 millimeter, and so on down to a micron. If we express the size of the “cobblestone” in decimeters, then the number of “cobblestones” equal to their size to the power of minus one is required for the segment, and for the Peano curves to the size to the power of minus two. Moreover, the segment definitely has one dimension, and the Peano curve, as we have seen, has two. This is not just a coincidence. The exponent in the ratio linking the number of "cobblestones" with their size is indeed equal (with a minus sign) to the dimension of the figure that is covered by them. It is especially important that the exponent can be a fractional number. For example, for a curve intermediate in its “crumpledness” between an ordinary line and sometimes densely filling the square of Peano curves, the value of the exponent will be greater than 1 and less than 2. This opens the way we need to determine fractional dimensions.

It was in this way that, for example, the dimension of the coastline of Norway was determined a country that has a very indented (or “crumpled” as you like) coastline. Of course, the paving of the coast of Norway with cobblestones did not take place on the ground, but on a map from a geographical atlas. The result (not absolutely accurate due to the impossibility in practice to reach infinitely small “cobblestones”) was 1.52 plus or minus one hundredth. It is clear that the dimension could not be less than one, since we are still talking about a "one-dimensional" line, and more than two, since the coastline of Norway is "drawn" on the two-dimensional surface of the globe.

Man as the measure of all things

Fractional dimensions are fine, the reader might say here, but what do they have to do with the question of the number of dimensions of the world we live in? Can it happen that the dimension of the world is fractional and not exactly equal to three?

The examples of the Peano curve and the coast of Norway show that a fractional dimension is obtained if the curved line is strongly “crumpled”, embedded in infinitely small folds. The process of determining the fractional dimension also includes the use of infinitely decreasing "cobblestones" with which we cover the curve under study. Therefore, the fractional dimension, scientifically speaking, can only manifest itself “on sufficiently small scales”, that is, the exponent in the ratio connecting the number of “cobblestones” with their size can only reach its fractional value in the limit. On the contrary, one huge boulder can cover a fractal an object of fractional dimension of finite dimensions is indistinguishable from a point.

For us, the world in which we live is, first of all, the scale on which it is available to us in everyday reality. Despite the amazing achievements of technology, its characteristic dimensions are still determined by the sharpness of our vision and the range of our walks, the characteristic periods of time by the speed of our reaction and the depth of our memory, the characteristic quantities of energy by the strength of those interactions that our body enters into with surrounding things. We have not surpassed the ancients by much, and is it worth striving for this? Natural and technological disasters somewhat expand the scale of "our" reality, but do not make them cosmic. The microworld is all the more inaccessible in our everyday life. The world open before us is three-dimensional, “smooth” and “flat”, it is perfectly described by the geometry of the ancient Greeks; the achievements of science, in the final analysis, should serve not so much to expand as to protect its borders.

So what is the answer to people who are waiting for the discovery of the hidden dimensions of our world? Alas, the only dimension available to us, which the world has beyond the three spatial ones, is time. Is it a little or a lot, old or new, wonderful or ordinary? Time is simply the fourth degree of freedom, and it can be used in many different ways. Let us recall once again the same Stirlitz, by the way, a physicist by education: every moment has its own reason

Andrey Sobolevsky