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

What is three-dimensional space

As it has already become clear, the understanding of three-dimensional space and 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, the corresponding coordinates may be greater.

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

Among other things, here it is necessary to distinguish between abstract concepts (if I may say so, formless) like points, lines or planes and figures that have finite dimensions or even volume. For each of these definitions, there are also equations that describe 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 is in three-dimensional space. In general, it can be called a certain basic unit that defines any plane or volumetric 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 serving 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 the object relative to the zero coordinates in the system, positive and negative values ​​are taken. 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. However, some techniques for specifying the location of an object in three-dimensional space use some other varieties.

The most famous are the cylindrical and spherical systems. The difference from the classical one is that when specifying the same three values ​​that determine the location of a point in three-dimensional space, one of the values ​​is angular. In other words, such systems use a circle corresponding to an angle of 360 degrees. Hence the specific assignment of coordinates, including elements such as radius, angle and generatrix. Coordinates in three-dimensional space (system) of this type obey somewhat different laws. Their task in this case is controlled by the rule of the right hand: if you align your thumb and forefinger with the X and Y axes, respectively, the remaining fingers in a curved position will indicate 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, it is a kind of endless line drawn through a point or two, not counting the many 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 goes for lines and vectors. The latter determine the basis of three-dimensional space and its dimension.

Determination of vectors and basis of three-dimensional space

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

Dependent and independent vectors

Regarding the definition of dependent and independent vectors, vectors that are projections (for example, vectors of the X axis projected onto the Y axis) are considered linearly independent.

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 you define independent vectors in three-dimensional space, they cannot be, so to speak, one continuation of the other. As it is already clear, in the case we are considering with three dimensions, according to the general theory, it is possible to construct only triplets 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 where one of the coordinates is constant (constant).

For example, a plane can be called any number of points with different coordinates along the X and Y axes, but the same coordinates along the Z axis. In any case, one of the three-dimensional coordinates remains unchanged. However, this is, so to speak, a 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 the classical point of view, but in a four-dimensional one. In addition to the well-known length, width and height, such space also includes the lifetime of an object, and time and space are interconnected quite strongly. This was proved by Einstein in his theory of relativity, although this is more related to physics than to algebra and geometry.

An interesting fact is that today scientists have already proven the existence of at least twelve dimensions. Of course, not everyone can 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 the third eye, which helped to perceive multidimensional reality, and not exclusively three-dimensional space.

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

Note that it is not always possible to describe multidimensional spaces that differ from our four-dimensional world with modern basic equations and theorems. And science in this area belongs rather to the field of theories and assumptions, rather than something that can be clearly felt or, so to speak, touched or seen with one's own eyes. Nevertheless, no one doubts the indirect evidence of the existence of multidimensional worlds in which four or more dimensions can exist.

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 into the mathematical jungle to explain the basic terms only so that the question associated with them would be clear to any student (so to speak, an explanation "on the fingers").

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

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

What's question! Of course, three - an ordinary person will say and he will be right. But there is also a special breed of people who have the acquired quality of doubting the obvious. These people are called "scientists" because they are specifically taught to do this. For them, our question is not so simple: the measurement of space is an elusive thing, they cannot be simply counted by pointing with your finger: one, two, three. You cannot measure their number with any device like a ruler or ammeter: space has 2.97 plus or minus 0.04 measurements. We have to think about this issue deeper and look for indirect ways. Such searches turned out to be 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 real three-dimensional space, all conceivable directions are reduced to some three pre-selected ones. But what does "reduce", "are a combination" mean? Where will these "width" and "depth" be if we find ourselves not in a rectangular room, but in zero gravity somewhere between Venus and Mars? Finally, who can guarantee that the "height", say, in Moscow and New York is one and the same "dimension"?

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

Cobblestone - a tool of the mathematician

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 three and four. 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 in two or six. But no master will be able to pave the street so that the cobblestones everywhere adjoin each other only 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 argued as follows: the pavement surface is undoubtedly two-dimensional. At the same time, there are inevitably points on it where at least three cobblestones converge. Let's try to generalize this observation: let's say that the dimension of some region is equal to N, if, when tiling it, it is impossible to avoid collisions of N + 1 or more "cobblestones". Now the three-dimensionality of space will be confirmed by any bricklayer: after all, when laying out a thick wall, in several layers, there will certainly be points where at least four bricks will touch!

However, at first glance it seems that one can find, as mathematicians say, a "counterexample" to Lebesgue's definition of dimension. This is a plank floor in which the floorboards touch exactly two at a time. Isn't it tiling? Therefore, Lebesgue also demanded that the "cobblestones" used in the definition of dimensions be small. This is an important idea, and at the end we will come back to it again from an unexpected perspective. And now it is clear that the condition of a small size of "cobblestones" saves Lebesgue's definition: for example, short parquet floors, unlike long floorboards, will necessarily touch three at some points. This means that 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 capabilities, which is easy to feel after playing a little with cubes or bricks.

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 Mueller's basement). What does this camera look like from his point of view? Rough concrete walls, a tightly locked steel door - in a word, one two-dimensional surface without cracks and holes, enclosing from all sides the enclosed space where it is located. There really is nowhere to go from such a shell. Is it possible to lock a person inside a one-dimensional contour? Imagine how Mueller draws a circle on the floor around Stirlitz with chalk and goes home: this does not even make a joke.

From these considerations, another way is derived to determine the number of dimensions of our space. Let us formulate it as follows: it is possible to enclose a region of N-dimensional space on 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 - 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 diverge with Lebesgue, Brouwer and their colleagues. 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 if there really are things with this 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 architecture. It is this area of ​​human activity that is most closely related to space, as it appears to us in ordinary life. To move further in the search for the dimensions of the physical world, an exit to other levels of reality is required. They are accessible to humans thanks to modern technology, which means physics.

What does the speed of light have to do with it?

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

It is difficult to say who was the first to notice the analogy between time and the dimensions of space. They already knew about it two centuries ago. Joseph Lagrange, one of the founders of classical mechanics, the science of body movements, 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 line 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, the path traveled by the variable is plotted, and the elapsed time along the abscissa. In this case, time really becomes just “one more” geometric dimension. Likewise, 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 highlighted "meaningful" directions. And directions that do not coincide with any of the axes do not make sense, they do not represent anything. On the usual geometric two-dimensional plane, all directions are equal, there are no selected axes.

At present, time can be considered the fourth coordinate only if it is not distinguished from the rest of the directions in the four-dimensional "space-time". It is necessary to find a way to "rotate" space-time so that time and spatial 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 nature has a universal speed - the speed of light.

Take two points in space, each at its own moment in time, or two "events" in the jargon of the theory of relativity. If you multiply the time interval between them, measured in seconds, by the speed of light, you get a certain distance in meters. We will assume that this imaginary segment is “perpendicular” to the spatial distance between events, and together they form the “legs” of a 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 "temporary" leg, but subtract it. Of course, this may result in a negative result: then it is believed that the "hypotenuse" has an imaginary length! But what's the point?

When you rotate the plane, the length of any line drawn on it is preserved. Minkowski understood that it is necessary to consider such "rotations" of space-time, which preserve the "length" of the intervals between events proposed by him. This is how it is possible to achieve that the speed of light is universal in the constructed theory. If two events are connected by a light signal, then the "Minkowski distance" between them is equal to zero: the spatial distance coincides with the time interval multiplied by the speed of light. "Rotation", proposed by Minkowski, keeps this "distance" zero, no matter how space and time mix during the "rotation".

This is not the only reason why Minkowski's "distance" has 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 precisely the main idea of ​​the theory of relativity.

So, the time and space of our world are so closely related to each other that it is difficult to understand where one ends and another begins. Together they form something like a stage on which the play "History of the Universe" is played out. The characters are particles of matter, atoms and molecules from which galaxies, nebulae, stars, planets are assembled, and on some planets even living intelligent organisms (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 were inseparable from each other, and reality became truly four-dimensional. And in this four-dimensional reality one of the two “fundamental interactions” known to the science of that time “dissolved”: the law of universal gravitation was reduced to the geometric structure of the four-dimensional world. But Einstein could do nothing with another fundamental interaction - electromagnetic.

Space-time takes on new dimensions

General 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? This means that it remains for his followers to geometrize electromagnetic forces!

Since Einstein exhausted the possibilities of the metric of four-dimensional space, his followers began to try to somehow expand the set of geometric objects from which such a theory could be constructed. Quite naturally, 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, which led scientists further and further away from the visual physics of the last century. They began to consider models of worlds with tens and even hundreds of dimensions, and infinite-dimensional space came in handy. A whole book would have to be written to tell about this quest. Another question is important for us: where are all these new dimensions located? Can we feel them in the same way as we experience time and three-dimensional space?

Imagine a long and very thin tube - for example, an empty fire hose 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" dimension. The perimeter, 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 rest of the 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 were perfecting 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-integers, and geometric objects possessing this property and having 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 try to either subtract half a dimension from a two-dimensional surface, or add half a dimension to a one-dimensional line. To do this, let's first practice adding or subtracting an entire dimension.

There is such a well-known children's trick. The magician takes a triangular piece of paper, makes an incision on it with scissors, folds the sheet along the incision line in half, makes another incision, bends it again, cuts it 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 in area (and eight times the square root 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. Take perfect paper, perfect scissors, and repeat the cutting and folding sequence an infinite number of times. Then the sizes of individual triangles obtained at each step of this process will tend to zero, and the triangles themselves will contract into points. Therefore, we will get a one-dimensional line from a two-dimensional triangle without losing a 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 how strong a microscope we look at this triangle, enlarging its fragments any number of times, the resulting picture will look exactly the same as an unmagnified one: 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 was able, as it were, to acquire dimension two. This means that there is hope that the less "crumpled" curve will have a "dimension" of, say, one and a half. But how do you find a way to measure fractional dimensions?

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

Take a straight line segment 1 decimeter long and two Peano curves that together fill a decimeter by decimeter square. We will cover them with small square "boulders" 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 decimetres, then the number of "cobblestones" equal to their size to the power of minus one is required per segment, and the size to the power of minus two for the Peano curves. In this case, 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 connecting the number of "cobblestones" with their size is really 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 that is intermediate in its “crumpledness” between a regular 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 with a very rugged (or "crumpled" - as you like) coast. Of course, the cobblestones of the coast of Norway did not take place on the ground, but on a map from a geographic atlas. The result (not absolutely accurate due to the impossibility in practice to reach infinitely small "boulders") was 1.52 plus or minus one hundredth. It is clear that the dimension could not have turned out 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 a two-dimensional surface of the globe.

Man as a measure of all things

Fractional dimensions are great, the reader might say here, but how do they relate to the question of the number of dimensions of the world in which we live? Could it happen that the dimension of the world is fractional and not exactly equal to three?

Examples of the Peano curve and the coast of Norway show that the fractional dimension is obtained if the curved line is strongly "crumpled", laid in infinitesimal folds. The process of determining fractional dimensions also involves the use of infinitely decreasing "cobblestones" with which we cover the studied curve. Therefore, the fractional dimension, scientifically speaking, can manifest itself only "on a sufficiently small scale", that is, the exponent in the ratio connecting the number of "cobblestones" with their size can only go to its fractional value in the limit. On the contrary, one huge cobblestone can cover a fractal - an object of fractional dimension - of finite size 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 advances in technology, its characteristic dimensions are still determined by our visual acuity and the range of our walks, characteristic time intervals - by the speed of our reaction and the depth of our memory, characteristic values ​​of energy - by the strength of those interactions that our body enters into with the surrounding things. We have not surpassed the ancients by much here, 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 microcosm is all the more inaccessible in our daily 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 should ultimately 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 three spatial dimensions, is time. Is it little or much, old or new, wonderful or ordinary? Time is just the fourth degree of freedom, and you can use it in very 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

Three-dimensional space - has three uniform 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 dimension value is, as you can see, the most significant characteristic of physical space.

Dimension of space

Dimension is the most general quantitatively expressed property of space-time. At present, the physical theory, which 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 back in the works of the Austrian physicist and philosopher Ernst Mach: "Why is space three-dimensional?" It is believed that the fact of 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 antiquity ...

Even Ptolemy wrote about the dimension of space, where he argued that in nature there can be no more than three spatial dimensions. In his book On Heaven, another Greek thinker, Aristotle, wrote that only the presence of three dimensions ensures the perfection and completeness of the world. One dimension, argued Aristotle, forms a line. If we add another dimension to the line, we get a surface. Complementing the surface with another dimension forms a solid.

It turns out that “it is no longer possible to go beyond the bounds of a volumetric body to something else, since any change occurs due to some kind of deficiency, and there is no such thing here. The given train of thought of Aristotle suffers from one essential weakness: it remains unclear why it is the three-dimensional volumetric body that has completeness and perfection. At one time Galileo justly ridiculed the opinion that "the number '3' is the number perfect and that it is endowed with the ability to communicate perfection to everything that has a Trinity."

What determines the dimensionality of space

Space is infinite in all directions. However, it can be measured only in three directions independent of each other: length, width and height; we call these directions the dimensions of space and say that our space has three dimensions, that it is three-dimensional. In this case, "an independent direction, we in this case call a line lying at right angles to the other. 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 dimensionality of our space is determined by the number of possible lines in it, lying at right angles to one another. There can be no other line on the line - this is one-dimensional space. On the surface, 2 perpendiculars are possible - this is a two-dimensional space. In "space", three perpendiculars are three-dimensional space. "

Why is space three-dimensional?

The experience of materialization of people, rare in earthly conditions, often has a physical effect on eyewitnesses ...

But, in the concepts of space and time, there is still a lot that is unclear, giving rise to incessant 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 statement that physical space has three dimensions is just as objective as the statement, 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 emphasized that the reason for the three-dimensionality of our space is still unknown. P. Ehrenfest and J. Whitrow showed that if the number of dimensions of space were more than three, then the existence of planetary systems would be impossible - only in a 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 affirmed as an absolute necessity. It is a physical fact like any other and, as a consequence, it is subject to the same kind of explanation.

The question of why our space is three-dimensional can be solved either from the position of teleology, proceeding from the unscientific statement that “the three-dimensional world is the most perfect of possible worlds”, or from a scientific materialistic point of view, based on fundamental physical laws.

Contemporaries' opinion

Modern physics says that the characteristic of three-dimensionality consists in the fact 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. Nowadays, 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 give an answer to the question of why our space, in which we live and which we observe, is three-dimensional. "

Theory of connected spaces

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

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

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

At the first stage, the development of the object proceeds along the initial selected direction, i.e. has one development axis. At the second stage, the system formed at the first stage rotates 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. At the third stage, the system's development rotates by 90 ° again, 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 indicated spaces are connected into a single stable formation by a physical process.

And because this process is implemented at all large-scale levels of our world, then all systems, including the coordinates themselves, are built according to the triad (three-coordinate) principle. It follows from this that as a result of passing through 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 appeared at the very dawn of the Universe ...

It is not for nothing that Pythagoras, who, as you can see, could have this knowledge, belongs to the expression: "All things consist of three." N.K. Roerich: “The Trinity symbol is of great antiquity and is found throughout the World, therefore it cannot be limited by any sect, organization, religion or tradition, as well as personal or group interests, because it represents the evolution of consciousness in all its phases ... turned out to be scattered all over the world ... If you collect together all the prints of the same sign, then, perhaps, it will turn out to be the most widespread 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 represented as a triune deity (three merged into one): something one, whole and indivisible, in its sacred significance far exceeding the original values.

We have 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 the atom to the galaxies. These three types of space are nothing more than three coordinate states of geometric space.

Launches the project "Ask a Scientist", in which specialists will answer interesting, naive or practical questions. In this issue, Ilya Shchurov, PhD in Physics and Mathematics, talks about 4D and whether it is possible to enter the fourth dimension.

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

Ilya Shchurov

PhD in Physics and Mathematics, 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 neither length, nor width, nor height.

Now let's move the point along a straight line some distance. Let's say our point is the tip of a pencil; when we moved it, it drew a line. A segment has a length, and no more measurements - it is one-dimensional. The segment "lives" on a straight line; the straight 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 straight line and move in the perpendicular direction, we get a rectangle. A rectangle has two dimensions - width and height. The rectangle lies in a certain plane. A plane is a two-dimensional space (2D), on it you can enter a two-dimensional coordinate system - a pair of numbers will correspond to each point. (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 same space 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 strain our imagination in order to imagine how a rectangle can be shifted so that it leaves the plane in which we live.

Imagining a four-dimensional space is also quite difficult for us, although it is very easy to describe mathematically. Three-dimensional space is a space in which the position of a point is specified by three numbers (for example, the position of an airplane is specified by longitude, latitude and altitude above sea level). In four-dimensional space, a point corresponds to four numbers-coordinates. "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 are faced with a four-dimensional space every day: for example, when making a date, we indicate not only the meeting place (it can be set with three numbers), but also the time (it can be set with one 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 a four-dimensional space in real life.

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

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

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

Everyone is familiar with the abbreviation 3D, which means three-dimensional (the letter D stands for dimension). For example, when choosing a movie marked 3D in a cinema, we know for sure: to watch it, you will have to wear special glasses, but the picture will not be flat, but three-dimensional. What is 4D? Does "four-dimensional space" exist in reality? And 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 zero-dimensional. It has neither length, nor width, nor height.

// 8-cell-simple

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

Now let's take a segment and try to move it as the point did before. You can imagine that our segment is the base of a wide and very thin brush. If we go beyond the straight line and move in the perpendicular direction, we get a rectangle. A rectangle has two dimensions - width and height. The rectangle lies in a certain plane. A plane is a two-dimensional space (2D), on it you can enter a two-dimensional coordinate system - a pair of numbers will correspond to each point. (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 such a way that we live with you. 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 strain our imagination in order to imagine how a rectangle can be shifted so that it leaves the plane in which we live.

Imagining a four-dimensional space is also quite difficult for us, although it is very easy to describe mathematically. Three-dimensional space is a space in which the position of a point is specified by three numbers (for example, the position of an airplane is specified by longitude, latitude and altitude above sea level). In four-dimensional space, a point corresponds to four numbers-coordinates. "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 are faced with a four-dimensional space every day: for example, when making a date, we indicate not only the meeting place (it can be set with three numbers), but also the time (it can be set with one number, for example, the number of seconds elapsed since a certain date). If you look at a real brick, it has not only length, width and height, but also an extension 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.