Three-dimensional world. A three-dimensional world in which we do not live. And yet, why three

First of all, I want to say thank you for the comment in the previous post about the tetracube. Its length should be taken equal to 299792458 m (this is how much light passes in a second). This will be true for a cube of any volume in order for it to become correct.

Now let's get down to business. And don’t criticize the pictures too much; I drew most of them myself in paint).

I would like to start with a brief repetition of the first post, as this is necessary for understanding the subsequent arguments.

To begin with, let’s separate the concepts of “measurement” and “n-dimensional world”. We call a measurement a straight line (coordinate axis) such that all points can be assigned points on this straight line. For example, dimensions can be called axes in the coordinate plane. An N-dimensional world means a world in which each of its points can be assigned only n coordinates, that is, having n dimensions.

Let's consider a zero-dimensional world - a point that has no coordinates. If you put an infinite number of zero-dimensional ones in a row, you get a line - a one-dimensional world with length. In it, each zero-dimensional world will have a corresponding coordinate in the first dimension. Let us repeat, the one-dimensional world consists of an INFINITE number of zero-dimensional worlds.

Now consider a plane - a two-dimensional world. It can be represented as an INFINITE number of lines (one-dimensional worlds), and they can be either parallel or intersect at different angles.

As is already obvious, space consists of planes, which can also be parallel, perpendicular, or intersect at different angles. All these worlds are simple and understandable to us.

But what can we say about the four-dimensional world? As we have already found out, it must consist of many three-dimensional worlds, of many spaces. Now let's think about the fact that at every moment of time space changes. Every moment the space is new, although similar to the previous one. Thus, we can say that we live in a FOUR-DIMENSIONAL world, since during our lives we pass through an INFINITE number of spaces that replace each other over time. Time will be the fourth dimension of this world, because we can assign a time coordinate to each space. It doesn't seem to be difficult either. Let's move on.

Based on this logic, the fifth-dimensional world must consist of many four-dimensional worlds. This is where we can no longer imagine this right away; we need a more complex analysis.

The now popular standard model and superstring theory divides dimensions into spatial and temporal. Modern M-theory speaks of the existence of 10 spatial dimensions. It’s not even a matter of the ideologists of string theory not understanding the ESSENCE of measurements and multidimensional worlds, not that for some reason there are 10 of them, but that it says that the “twisted” dimensions have a SPATIAL EXTENSION that is different from zero. It turns out that the coordinate in these dimensions depends on the coordinate in three dimensions. This generally contradicts the idea that the dimensions should be unrelated to each other, and that one can move in one of them while remaining motionless in the others. This is a good attempt to explain the structure of the world, but it is too complex, the equations of string theory are sometimes not able to be solved even by supercomputers, and the essence of the research boils down to the fact that, based on already existing laws, find out the FORM of curled up dimensions in order to use this form to explain these same laws .

What is the four-dimensional world? This is the whole world in which we live, which existed long before the existence of humanity and will exist for an unknown amount of time, which lasts from the so-called “Big Bang” and goes to infinity (there are opinions that the universe is spatially limited, for now we will not touch on this issue ) and eternity. If the fifth-dimensional world consists of many four-dimensional worlds, this means that it consists of parallel or intersecting worlds with ours.

Nature has laws that, as we know, are unchangeable. That is, if we take space from our four-dimensional world at some early point in time (let it be moment T), and moving forward along the time dimension, using these laws, we will ALWAYS get the space in which we are now. Even all our thoughts and actions are caused by chemical and electrostatic reactions in our brain, which work according to the same laws by which all interactions in the universe work.

But what will happen if at time T we apply other laws, for example, just changing the fact that gravitational interaction causes masses not to APPROACH, but to REPEND, and leave the other laws unchanged. Then, moving forward in time, we will get a completely different world in which there may be no humanity at all, and if intelligent life forms arise, they will try to explain why all bodies repel in the same way as we try to explain why they attract. Thanks to this thought experiment, we imagined another four-dimensional world that INTERSECTS OURS at a point with coordinate T in the fourth dimension.

Let's consider intersecting planes. For them to intersect, they must be in SPACE (in a three-dimensional world), they themselves are two-dimensional, and they will intersect along a LINE, in a one-dimensional world.

Likewise, four-dimensional worlds, being in the fifth, can (purely theoretically) intersect in three-dimensional worlds. That is, using different laws from different three-dimensional positions, we can, over time, come to the SAME three-dimensional world. For example, if you drop a cannonball from a tower and fire a cannonball from a loophole in that tower, at some point in time (not necessarily the same for the two cases) both cannonballs will be in the same position, despite the fact that they were initially were in different positions and different forces were applied to them.

In example 1 there are 2 parallel four-dimensional worlds. In the second example, 2 worlds intersect at a point (in our case, four-dimensional worlds have one identical spatial structure in certain moment time). In the third example, there is a world that endlessly turns into itself thanks to its laws, and even intersects with another at two points.

Now think about an example of a four-dimensional world, which will always be absolutely identical to ours, but in which completely different laws of physics will operate. At first glance, this is impossible, because if we take the same three-dimensional world and apply different laws to it, we will get completely different four-dimensional dimensions. But such an example exists. It is quite simple in OUR four-dimensional world to move not forward in time, but backward. Obviously, each position of space in such a world will correspond to space in our familiar world.

Now let’s look at some postulates that will be observed in this new world, which is identically equal to ours. Take, for example, four provisions from modern world, which are recognized by most physicists:

1) The universe is expanding

2) Gravity causes bodies to attract

3) A body that is not affected by forces maintains inertial motion

4) To break a connection, you need to expend energy, and when a connection is created, energy is released

Now let’s compare each position from our world to the position of a world identical to ours, where time flows in reverse

1) The universe will shrink as we go down in time

2) With gravity, everything is not so obvious. For example, if you throw a ball from an airplane, it will fly faster and faster towards the ground, and then come to rest on the ground. Let's consider this process in reverse in time. The ball, under the influence of gravity, first takes off sharply from a resting position, and as it gains height, it flies upward more and more slowly. From this we can conclude that in a world opposite to ours, gravity causes bodies to repel. But then let's look at a person going to the store. If you look at this process in the opposite direction, then the person will also walk out of the store, ATTRACTED to the ground, and not flying away from it under the influence of gravity. That is, in this case, gravity forces bodies to ATTRACT. From the two situations we received completely different conclusions, from which it follows that the laws in this world will not be opposite to ours, they will simply be DIFFERENT, not polar opposite.

3) Consider the flight of a pebble in space. If we imagine this process in reverse, then the pebble will also fly in space. This law is CONSERVED in the world opposite to ours.

4) Imagine that a karateka breaks a brick. It expends energy to destroy the internal connection of the brick. Let's consider this process in reverse: the brick is assembled into a single whole and the karateka receives energy. We obtained the same law, in which energy is released when a bond is formed. This law also remains in force.

Thus, in the new world, some laws of our world are preserved (3,4), some are changed to the opposite (1), and some are transformed into those that we cannot immediately describe (2). If a civilization existed in such a world, it would try to find and most likely would find an explanation for all these processes, but for us this is not important. We will call such four-dimensional worlds, in which we obtain the same space structures from different positions with different laws, COINCIDENT.

Thus, the fifth dimension is the LAWS OF PHYSICS. Indeed, in the standard coordinate system, all axes must be perpendicular. Then, for each point in the five-dimensional world, we can actually assign 3 spatial coordinates, one time coordinate, and also a fifth coordinate, which will denote the LAWS that are applied to this point. Let us note that some paradoxes, which those who are especially attentive can find in such a coordinate system, arise due to the DIRECTIONALITY of our four-dimensional world (time moves forward for us, and most of us do not have the opportunity to slow it down through enormous spatial speed).

Even more attentive may say that the laws themselves can be divided into an infinitely large number of dimensions. For example, each interaction force from an infinitely possible number of these forces can be associated with an infinitely large direction of influence. Each set of force and direction can then be associated with an infinite number of possible elementary particles. And vice versa, each set of elements corresponds to an infinity of directions, and to each direction an infinity of forces, etc. Thus, instead of the fifth dimension, we can distinguish an infinite number of dimensions, independent of each other.

What can this theory explain? It can explain that the laws of physics did not come from anywhere. They are simply the way they are, there are an infinite number of combinations of other laws in other worlds that, perhaps with the help of technology, or perhaps with the help of evolution, humanity will someday gain access to.

Ask your questions! Any criticism is welcome except “you’re wrong, and I’m right, and in general, everything is wrong.” The next article will be about the theory of infinite nesting of matter, it will be quite interesting)

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

What a question! Of course, an ordinary person will say three and be right. But there is also a special breed of people who have the acquired ability to doubt obvious things. These people are called "scholars" because they are specifically taught this. For them, our question is not so simple: the measurement of space is an elusive thing, they cannot simply be counted by pointing with a finger: one, two, three. It is impossible to measure their number with any device like a ruler or an ammeter: space has 2.97 plus or minus 0.04 dimensions. We have to think through this issue more deeply and look for indirect methods. Such searches turned out to be a fruitful endeavor: 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 drawing plane is reduced to “height” and “width”, and is in some 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 this “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 “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 only we could find ourselves in a world whose number of dimensions is not known in advance, and look for them one by one Or, at least, so renounce existing knowledge about reality in order to look at its original properties in a completely new way.

Cobblestone mathematics tool

In 1915, 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, just look closely at the cobblestone pavement. You can easily find places where the stones come together in threes and fours. You can pave the street with square tiles, which will be adjacent to each other in twos or fours; if you take identical triangular tiles, they will be adjacent in groups of two or six. But not a single master can pave the street so that the cobblestones everywhere adjoin each other only in twos. This is so obvious that it is 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, 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 area is equal to N if, when tiling it, it is not possible to avoid contacts of N + 1 or more “cobblestones”. Now any mason will confirm the three-dimensionality of space: after all, when laying out a thick wall with 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 call it, a “counterexample” to Lebesgue’s definition of dimension. This is a plank floor in which the floorboards touch exactly two at a time. Why not paving? Therefore, Lebesgue also demanded that the “cobblestones” used in determining the 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 the small size of the “cobblestones” saves Lebesgue’s definition: let’s say, short parquet floors, unlike long floorboards, at some points will necessarily touch in threes. This means that three dimensions of space are not just the ability to arbitrarily choose some three “different” directions in it. Three dimensions are a real limitation of our capabilities, which can be easily felt by 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 Müller’s basement). What does this camera look like from his point of view? Rough concrete walls, tightly locked steel door in a word, one two-dimensional surface without cracks and holes, enclosing on all sides the closed space where it is located. There is really nowhere to escape from such a shell. Is it possible to lock a person inside a one-dimensional circuit? Imagine how Müller draws a circle on the floor with chalk around Stirlitz and goes home: this doesn’t even amount to a joke.

From these considerations, another way is derived to determine the number of dimensions of our space. Let us formulate it this way: 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 there will be two zero-dimensional points. This definition was invented in 1913 by the Dutch mathematician Brouwer, but it became famous only eight years later, when it was independently rediscovered by our Pavel Uryson and the Austrian Carl Menger.

Here we part ways with Lebesgue, Brouwer and their colleagues. They needed a new definition of dimension in order to build an abstract mathematical theory of spaces of any dimension up to infinity. This is a purely mathematical construction, a game of the human mind, which is strong enough even to comprehend such strange objects as infinite-dimensional space. Mathematicians do not try to find out whether things with such structure actually exist: that 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 find out how many there really are and how to feel their number “in 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 connected with space, as it appears to us in ordinary life. To move further in the search for dimensions of the physical world, access to other levels of reality will be required. They are available to humans thanks to modern technology, and therefore physics.

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

Let's return briefly 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 used 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 took advantage of was time.

It is difficult to say who was the first to notice the analogy between time and the dimensions of space. Two centuries ago they already knew about this. Joseph Lagrange, one of the creators 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 today’s cardiograms or graphs of monthly temperature variations. Such graphs are drawn on a two-dimensional plane: the path traveled by the variable is plotted along the ordinate axis, and the elapsed time is plotted along the abscissa axis. In this case, time really becomes just “another” geometric dimension. In the same way you can add it to three-dimensional space 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 have no meaning, they do not represent anything. On an ordinary geometric two-dimensional plane, all directions are equal, there are no designated axes.

Time can truly be considered a fourth coordinate only if it is not distinguished from other directions in four-dimensional “space-time”. We need to find a way to “rotate” space-time so that time and spatial dimensions “mix” and can, in a certain sense, transform into each other.

This method was found by Albert Einstein, who created the theory of relativity, and Hermann Minkowski, who gave it a strict 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 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 some right triangle, the “hypotenuse” of which is a segment in space-time connecting the selected events. Minkowski proposed: in order 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” leg, but subtract it. Of course, this may result in a negative result: then the “hypotenuse” is considered to have an imaginary length! But what's the point?

When the plane is rotated, the length of any segment drawn on it is preserved. Minkowski realized that it was necessary to consider such “rotations” of space-time that preserve the “length” of segments between events that he proposed. This is how it is possible to ensure 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 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 its extremely strange definition for an untrained person. Minkowski's “distance” provides a way to construct the “geometry” of space-time so 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, 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 performed. 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 know 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 science at that time “dissolved”: the law of universal gravitation was reduced to geometric structure four-dimensional world. But Einstein could not do anything with the other 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. Did Einstein reduce gravity to geometry? This means that it remains for his followers to geometrize electromagnetic forces!

Since Einstein had exhausted the possibilities of the metrics 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. 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 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 also came in handy. To talk about these searches, one would have to write a whole book. Another question is important to us: where are all these new dimensions located? Is it possible to feel them the same way we feel time and three-dimensional space?

Imagine a long and very thin tube - for example, an empty fire hose, reduced in size a thousand times. It is a two-dimensional surface, but its two dimensions are unequal. One of them, length, is easy to notice - it is a “macroscopic” measurement. 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 remaining dimensions 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, you just need to “simply” find a new property of dimension 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 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 famous children's trick. The magician takes a triangular piece of paper, makes a cut on it with scissors, bends the sheet in half along the cut line, makes another cut, bends it again, cuts last time, and up! 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 the square root of eight times in size). Perhaps this trick was shown to the Italian mathematician Giuseppe Peano in 1890 (or maybe he himself loved 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 sizes of individual triangles obtained at each step of this process will tend to zero, and the triangles themselves will shrink to 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 it, then it will fill the triangle entirely. Moreover, under whatever powerful microscope we examine this triangle, magnifying its fragments any number of times, the resulting picture will look exactly the same as the 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 could, as it were, 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 can we find a way to measure fractional dimensions?

In the “cobblestone” determination of dimension, as the reader remembers, it was necessary to use fairly small “cobblestones”, otherwise the result could be incorrect. But you will need a lot of small “cobblestones”: the smaller their size, the more. 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 as the size decreases.

Let's 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 a “cobblestone” in decimeters, then a segment will require a number of “cobblestones” equal to their size to the power of minus one, and for Peano curves equal to their 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 relation connecting the number of “cobblestones” with their size is indeed equal (with a minus sign) to the dimension of the figure that is covered with them. It is especially important that the exponent can be fractional number. For example, for a curve that is intermediate in its “crumpiness” between an ordinary line and sometimes densely filling a square of Peano curves, the value of the indicator will be more 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 size of the coastline of Norway was determined, a country that has a very rugged (or “crumpled”, as you prefer) 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 of reaching infinitesimal “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 great, the reader may say here, but what do they have to do with 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 Norwegian coast show that a fractional dimension is obtained if the curved line is strongly “crumpled”, embedded in infinitesimal 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 cobblestone can cover a fractal an object of fractional dimension of finite dimensions indistinguishable from a point.

For us, the world in which we live is, first of all, the scale on which it is accessible to us in everyday reality. Despite the amazing achievements of technology, its characteristic dimensions are still determined by the acuity of our vision and the distance of our walks, the characteristic periods of time by the speed of our reaction and the depth of our memory, the characteristic amounts of energy by the strength of the interactions that our body enters into with surrounding things. We have not much surpassed the ancients 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 microworld is even more inaccessible in our everyday life. The world open to 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 that the world has beyond three spatial ones is time. Is it little or much, 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 training: every moment has its own reason…

Andrey Sobolevsky