Fourth dimension. How easy and understandable to explain what a four-dimensional space is

The parallel universe of higher dimensions boasts the longest history of scientific discussions of all types of parallel universes. Common sense and sense organs tell us that we live in three dimensions - length, width and height. No matter how we move an object in space, its position can always be described by these three coordinates. In general, with these three numbers, a person can determine the exact position of any object in the Universe, from the tip of his nose to the most distant galaxies.

At first glance, the fourth spatial dimension is contrary to common sense. For example, when smoke fills an entire room, we don't see it disappear into another dimension. Nowhere in our universe do we see objects that would suddenly disappear or float away to another universe. This means that higher dimensions, if they exist, must be smaller than an atom.

Three spatial dimensions form the foundation, the basis of Greek geometry. For example, Aristotle wrote in his treatise On Heaven:

"A quantity divisible in one dimension is a line, in two it is a plane, in three it is a body, and apart from these there is no other quantity, since three measurements the essence of everything measurements".

In 150 AD e. Ptolemy of Alexandria offered the first "proof" that higher dimensions are "impossible". In the treatise "On Distance" he argues as follows. Let's draw three mutually perpendicular straight lines (like the lines that form the corner of the room). Obviously, it is impossible to draw a fourth line perpendicular to the first three, therefore, the fourth dimension is impossible.

In fact, he managed to prove in this way only one thing: our brain is not able to visualize the fourth dimension. On the other hand, computers are constantly doing calculations in hyperspace.

For two millennia, any mathematician who dared to talk about the fourth dimension risked ridicule. In 1685, the mathematician John Wallis, in a polemic about the fourth dimension, called it "a monster in nature, no more possible than a chimera or a centaur." In the 19th century, the "king of mathematicians" Carl Gauss developed the mathematics of the fourth dimension to a large extent, but was afraid to publish the results for fear of a backlash. He himself, however, was experimenting and trying to determine whether purely three-dimensional Greek geometry really correctly describes the universe. In one experiment, he placed three assistants on the tops of three adjacent hills. Each assistant had a lantern; the light of all three lanterns formed a gigantic triangle in space. Gauss himself carefully measured all the angles of this triangle and, to his own disappointment, found that the sum of the interior angles of the triangle is indeed 180°. From this, the scientist concluded that if deviations from standard Greek geometry exist, then they are so small that they cannot be detected by similar methods.

Painting: Rob Gonsalves, Canada, magical realism style

As a result, the honor of describing and publishing the foundations of higher-dimensional mathematics fell to Georg Bernhard Riemann, a student of Gauss. (After a few decades, this mathematics was wholly incorporated into Einstein's general theory of relativity.) In his famous lecture in 1854, Riemann overturned in one fell swoop 2,000 years of Greek geometry and established the foundations of the mathematics of higher, curvilinear dimensions; We still use this math today.

AT late XIX in. the remarkable discovery of Riemann thundered throughout Europe and aroused the widest interest of the public; the fourth dimension has created a real sensation among artists, musicians, writers, philosophers and artists. For example, art historian Linda Dalrymple Henderson believes that Picasso's Cubism was partly inspired by the fourth dimension. (Picasso's portraits of women, with the eyes facing forward and the nose on the side, are an attempt to present a four-dimensional perspective, because when viewed from the fourth dimension, you can see the face, nose and back of the head of a woman at the same time.) Henderson writes: “Like a black hole, the fourth the dimension had mysterious properties that could not be fully understood even by the scientists themselves. And yet the fourth dimension was much more understandable and conceivable than black holes or any other scientific hypothesis after 1919, with the exception of the theory of relativity.

But historically, physicists have viewed the fourth dimension as just a fun curiosity. There was no evidence of the existence of higher dimensions. The situation began to change in 1919, when the physicist Theodor Kaluza wrote a very controversial paper in which he hinted at the existence of higher dimensions. Starting with Einstein's general theory of relativity, he placed it in a five-dimensional space (four spatial dimensions and the fifth one is time; since time has already established itself as the fourth dimension of space-time, physicists now refer to the fourth spatial dimension as the fifth). If you make the universe smaller and smaller along the fifth dimension, the equations magically fall into two parts. One part describes Einstein's standard theory of relativity, but the other part turns into Maxwell's theory of light!

This was an amazing revelation. Perhaps the secret of light is hidden in the fifth dimension! This decision shocked even Einstein; it seemed to provide an elegant unification of light and gravity. (Einstein was so shocked by Kaluza's suggestion that he hesitated for two years before agreeing to publish his paper.) Einstein wrote to Kaluza: I liked your idea extremely... The formal unity of your theory is amazing.”

For many years, physicists have wondered: if light is a wave, then what, in fact, oscillates? Light can travel billions of light-years of empty space, but empty space is a vacuum, there is no substance in it. So what oscillates in a vacuum? Kaluza's theory made it possible to put forward a specific assumption about this: light is real waves in the fifth dimension. Maxwell's equations, which accurately describe all the properties of light, are obtained in it simply as equations for waves that move in the fifth dimension.

Imagine fish swimming in a shallow pond. Perhaps they are not even aware of the existence of the third dimension, because their eyes look to the sides, and they can only swim forward or backward, right or left. Perhaps the third dimension even seems impossible to them. But now imagine rain on the surface of a pond. Fish cannot see the third dimension, but they can see shadows and ripples on the surface of the pond. Similarly, Kaluza's theory explains light as ripples that move through the fifth dimension.

Kaluza also gave an answer to the question of where the fifth dimension is. Since we do not see any signs of its existence around, it must be "rolled up" to such a small extent that it is impossible to notice it. (Take a two-dimensional sheet of paper and roll it tightly into a cylinder. From afar, the cylinder will appear as a one-dimensional line. It turns out that you have rolled up a two-dimensional object and made it one-dimensional.)

For several decades, Einstein began to work on this theory from time to time. But after his death in 1955, the theory was quickly forgotten and turned into a funny footnote in the history of physics.

An excerpt from Peter D. Uspensky's book "A New Model of the Universe":

The idea of the existence of hidden knowledge, superior to the knowledge that a person can achieve by one's own efforts, grows and strengthens in the minds of people when they understand the insolubility of many issues and problems facing them.

A person can deceive himself, he can think that his knowledge is growing and increasing, that he knows and understands more than he knew and understood before; however, sometimes he becomes sincere with himself and sees that in relation to the basic problems of existence he is as helpless as a savage or a child, although he has invented many clever machines and tools that have complicated his life, but not made it clearer.
Speaking even more frankly with himself, a person may recognize that all his scientific and philosophical systems and theories are similar to these machines and tools, because they only complicate problems without explaining anything.

Among the unsolvable problems surrounding a person, two occupy special position- the problem of the invisible world and the problem of death.

Without exception, all religious systems, from such theologically developed to the smallest detail as Christianity, Buddhism, Judaism, to the completely degenerate religions of "savages" that seem "primitive" to modern knowledge - all of them invariably divide the world into visible and invisible. In Christianity: God, angels, devils, demons, souls of the living and the dead, heaven and hell. In paganism: deities personifying the forces of nature - thunder, sun, fire, spirits of mountains, forests, lakes, spirits of water, spirits of houses - all this belongs to the invisible world.
Philosophy recognizes the world of phenomena and the world of causes, the world of things and the world of ideas, the world of phenomena and the world of noumenons. In Indian philosophy (especially in some of its schools) the visible or phenomenal world, maya, is an illusion which means false concept oh no visible world, is generally considered non-existent.

In science, the invisible world is the world of very small quantities, and also, oddly enough, of very large quantities. The visibility of the world is determined by its scale. The invisible world is, on the one hand, the world of microorganisms, cells, the microscopic and ultramicroscopic world; it is followed by the world of molecules, atoms, electrons, "vibrations"; on the other hand, it is a world of invisible stars, distant solar systems, unknown universes.

The microscope expands the boundaries of our vision in one direction, the telescope in another, but both are very small compared to what remains invisible.

Physics and chemistry give us the opportunity to investigate phenomena in such small particles and in such distant worlds that will never be available to our vision. But this only reinforces the idea that there is a huge invisible world around a small visible one.
Mathematics goes even further. As has already been pointed out, it calculates such ratios between quantities and such ratios between these ratios that have no analogies in the visible world around us. And we are forced to admit that the invisible world differs from the visible world not only in size, but also in some other qualities that we are unable to determine or understand and which show us that the laws found in the physical world cannot apply to world invisible.
Thus, the invisible worlds of religious, philosophical and scientific systems are, after all, more closely connected with each other than it seems at first glance. And such invisible worlds of different categories have the same properties common to all. These properties are. First, they are incomprehensible to us; incomprehensible from the ordinary point of view or for ordinary means of knowledge; secondly, they contain the causes of the phenomena of the visible world.

The idea of causes is always connected with the invisible world. In the invisible world of religious systems, invisible forces control people and visible phenomena. In the invisible world of science, the causes of visible phenomena stem from the invisible world of small quantities and "fluctuations".
In philosophical systems, the phenomenon is only our concept of the noumenon, i.e. an illusion, the true cause of which remains hidden and inaccessible to us.

Thus, at all levels of his development, man understood that the causes of visible and observable phenomena are beyond the scope of his observations. He found that among the phenomena available to observation, some facts can be considered as the causes of other facts; but these conclusions were not sufficient for understanding everything that happens to him and around him. To explain the causes, an invisible world is needed, consisting of "spirits", "ideas" or "vibrations".

Arguing by analogy with existing dimensions, it should be assumed that if the fourth dimension existed, it would mean that right here, next to us, there is some other space that we do not know, do not see and cannot go into. Into this "region of the fourth dimension" from any point of our space it would be possible to draw a line in a direction unknown to us, which we cannot determine or comprehend. If we could imagine the direction of this line coming from our space, then we would see the "area of the fourth dimension."

Geometrically, this means the following. One can imagine three mutually perpendicular lines to each other. With these three lines we measure our space, which is therefore called three-dimensional. If there is an "area of the fourth dimension" lying outside our space, then, in addition to the three perpendiculars known to us, which determine the length, width and height of objects, there must be a fourth perpendicular, which determines some kind of incomprehensible to us, new extension. The space measured by these four perpendiculars will be four-dimensional.

It is impossible to define geometrically or imagine this fourth perpendicular, and the fourth dimension remains extremely mysterious to us. There is an opinion that mathematicians know about the fourth dimension something inaccessible to mere mortals. It is sometimes said, and this can be found even in the press, that Lobachevsky "discovered" the fourth dimension. In the past twenty years, the discovery of the "fourth" dimension has often been attributed to Einstein or Minkowski.

In fact, mathematics has very little to say about the fourth dimension. There is nothing in the fourth dimension hypothesis that makes it mathematically unacceptable. It does not contradict any of the accepted axioms and therefore does not meet with special opposition from mathematics. Mathematics fully admits the possibility of establishing the relations that must exist between four-dimensional and three-dimensional space, i.e. some properties of the fourth dimension. But she does all this in the most general and indefinite form. There is no exact definition of the fourth dimension in mathematics.

The fourth dimension can be considered geometrically proven only in the case when the direction of the unknown line going from any point of our space to the area of the fourth dimension is determined, i.e. found a way to construct the fourth perpendicular.

It is difficult even approximately to outline what significance the discovery of the fourth perpendicular in the universe would have for our entire life. The conquest of the air, the ability to see and hear at a distance, the establishment of relations with other planets and star systems - all this would be nothing compared to the discovery of a new dimension. But so far it hasn't. We must admit that we are powerless before the mystery of the fourth dimension - and try to consider the issue within the limits that are available to us.

Upon a closer and more precise study of the problem, we come to the conclusion that for existing conditions it is impossible to solve it. Purely geometric at first glance, the problem of the fourth dimension is not solved geometrically. Our geometry of three dimensions is not enough to investigate the question of the fourth dimension, just as planimetry alone is not enough to investigate questions of stereometry. We must discover the fourth dimension, if it exists, purely by experience - and also find a way to represent it in perspective in three-dimensional space. Only then can we create a geometry of four dimensions.

The most superficial acquaintance with the problem of the fourth dimension shows that it must be studied from the side of psychology and physics.

The fourth dimension is incomprehensible. If it exists, and if, nevertheless, we are not able to cognize it, then, obviously, something is missing in our psyche, in our perceiving apparatus, in other words, the phenomena of the fourth dimension are not reflected in our sense organs. We must figure out why this is so, what defects cause our immunity, and find the conditions (at least theoretically) under which the fourth dimension becomes understandable and accessible. All these questions belong to psychology, or perhaps to the theory of knowledge.

We know that the area of the fourth dimension (again, if it exists) is not only unknowable for our mental apparatus, but is inaccessible purely physically. It no longer depends on our defects, but on the special properties and conditions of the area of the fourth dimension. We need to figure out what conditions make the area of the fourth dimension inaccessible to us, find the relationship of the physical conditions of the area of the fourth dimension of our world and, having established this, see if there is anything similar to these conditions in the world around us, if there are any relations similar to relations between 3D and 4D regions.

Generally speaking, before constructing the geometry of four dimensions, it is necessary to create the physics of four dimensions, i.e. find and determine the physical laws and conditions that exist in the space of four dimensions.

"We can't solve problems using the same mindsets that we used to create problems." (Albert Einstein)

via quantum-tech. ru and blogs.mail.ru/chudatrella.

The current stage of the evolution of mankind is characterized by the absence of the overwhelming majority of people of the ability to perceive the four-dimensional world - the "second sight", as well as the underdevelopment of an aspect of consciousness that is more perfect than the intellect - intuition.

The disclosure and subsequent development of a new (sixth) sense organ is the future of a person of a new (sixth) race. In the meantime, humanity is going through a transitional period on the way to new opportunities, which is confirmed by the emergence of so-called psychics.

In this regard, only a small part of the planet's population has experience of interaction with the world of higher dimensions. The majority modern people living in really multidimensional world, still perceives and realizes only its most primitive part - the three-dimensional physical world.

This circumstance favors the invention of various fantastic images attributed to worlds of higher dimensions. This, in turn, is reflected not only in the works of science fiction writers, but also in science.

Examples of such scientific fantasies are the 4D continuum, dark matter, wormholes, tesseracts, simplices, superstrings, branes... complete unsuitability three-dimensional mathematical apparatus for understanding and describing multidimensional spaces.

COMMENT. What is called "multidimensional" spaces in mathematics has nothing to do with reality, since they do not take into account such properties of truly multidimensional spaces as materiality and permeability; space is endowed with non-spatial properties, and the property of extension, contrary to common sense, extends beyond the limits of three dimensions.

3D illusions about multidimensionality

The main trouble with mathematics is that it tends more towards orthodox beliefs than towards science, since it is built not on updated knowledge about the world, but on Inviolable Sacred Dogmas, which neither absurdity, nor paradoxes, nor scientific discoveries, nor a series of crises, nor a millennium of struggle against dogmatism.

Below we list only a part of the most odious Dogmas (and their consequences), which makes the knowledge of the multidimensional structure of the world around us with the help of SUCH mathematics fundamentally impossible.

In mathematics, there supposedly really exist spaces with dimensions less than three; while 0D-"space" is a point, 1D-"space" is a line, 2D-"space" is a surface;
Math point size zero, but it allegedly exists;
Allegedly, there really exists an empty space - the "space" of a dimensionless point;
The sizes of bodies are inexplicably determined by the sum of the sizes of dimensionless points;
From the zero size of a point, its non-materiality also follows;
From the non-materiality of a point (0D-"space"), the non-materiality of any space follows;
It follows from the non-materiality of space that space is not recognized as an attribute (an integral property) of matter;
From the misunderstanding of the inseparable connection between space and matter, the most ridiculous delusion follows, allowing the “transfer” of 3D entities to higher-dimensional spaces:
firstly, because 3D objects already contain the matter of all higher dimensions, that is, they are already available to all higher spatial entities;
secondly, complete belonging to a higher-dimensional space requires the complete elimination of the lower 3D material shell, which is tantamount to death in a 3D world.
The consequence of the previous delusions is the absence in mathematics of the concept of "spatial environment";
From the misunderstanding of the incomparability of the properties of matter of different dimensions, the absurdity of the requirement of orthogonality of spatial "axes", the operation of adding vectors and finding scalar sums for a set of different-sized spaces follows.
The last delusion manifests itself, in particular, in an attempt to sum the velocity vector of 4D light with the velocity vector of its 3D source moving in another space;
A striking evidence of the complete misunderstanding of the essence of multidimensionality by mathematicians is the widespread identification of multicomponent 3D vectors (x 1 , x 2 , x 3 , ... x n) with supposedly multidimensional mathematical constructions.
Let's show it on the example of a vector of properties of a 3D-piece of sugar with the following vector components: length x 1 ; width x 2; height x 3; weight x 4; color x 5; flavor x 6; production time x 7 . In terms of mathematics, we get a 7-dimensional (!) vector. However, there will be only three spatial dimensions in this 7-component construction.
This example also makes it easy to understand that the usual three-dimensional space, given out in relativism as Minkowski's 4D space-time, has nothing to do with the fourth spatial dimension.

For the above and other reasons, practically all currently known attempts to model 4D space by means of three-dimensional mathematics are nothing more than 3D fantasies on the topic of multidimensionality that is inaccessible to dogmatic thinking.

Where to look for the fourth dimension

So, if all the above attempts at scientific understanding of multidimensional spaces are nothing more than science fiction, then several reasonable questions arise:

Where, then, is hidden at least the closest real 4D space to us?
And does it exist at all?
And if it exists, why don't we see it?

First of all, it should be said that the four-dimensional space is the same reality as the three-dimensional space we observe.

To the question "Then why don't we see him?" the easiest way to answer is with another question: “Why doesn’t anyone bother that we don’t see the contents of computer disks, electricity, radio waves, radiation, our own aura, other people’s thoughts”? Even ghosts can only be seen in photographs.

It will be more difficult to understand the answer to the question: "Where is the four-dimensional space"?

However, the correct answer is: “We are all inside 4D space; it not only surrounds us, it surrounds and fills us and the entire 3D Universe, including outer space and the space inside atoms; in this case, nucleons are formed by particles of 4D matter.”

The matter of four-dimensional space is called physical ether, in modern physics, most often - the physical vacuum.

According to one of the hypotheses, an ether particle (amer) is an electron-positron pair. Thus, in the unexcited state, an amer, like an atom, is electrically neutral, but unlike an atom, it does not contain a nucleus.

Nuclear-free 4D ethereal matter plays the role of an intermediary (layer) between the atomic 3D physical and 5D astral worlds:

an ether particle is approximately 8 orders of magnitude thinner than a physical atom;
the astral atom is approximately 8 orders of magnitude thinner than the ethereal particle;
relative to the physical atom, the astral atom is thinner by 16 orders of magnitude.

At the atomic level of matter structuring, a difference of 8 orders means a transition to a new dimension:

3D physical atom ≈ 10 -8 cm;
4D particle of ether ≈ 10 -16 cm;
5D-astral atom ≈ 10 -24 cm.

In the real world, a quantitative change in the size of matter within one dimension (for atoms of the same dimension) is periodically accompanied by dialectical abrupt transitions to new ones. quality levels, for example:

physical atom → physical body → physical celestial body...;
astral atom → astral body → astral planet and so on.

Mathematics, ignoring the law of the transition of quantitative changes into qualitative and other fundamental laws of the Universe, produces only illusory-mystical conjectures about multidimensionality, based solely on the quantitative, a continuous and linear increase in the size of matter from a non-existent zero to an imaginary infinity.

This mathematical lawlessness contains another reason for scientific fantasies about multidimensional worlds and spaces.

The hypothesis of the multidimensional organization of the Universe mentioned above is in good agreement with observations and everyday experience, psychic data and experimental results, as well as with information from Eastern spiritual practices, occult, theosophical and esoteric sources.

Properties of the fourth dimension

Trying to represent the properties of a hypothetical 4D space, one cannot replace common sense with three-dimensional mathematical dogmas. Otherwise, unpleasant surprises await us.

Is a 4th orthogonal axis possible?

For most of us, three-dimensional space is associated with the three axes of the Cartesian coordinate system. Therefore, many readily (without bothering with doubts and reflections) agree with the unsubstantiated dogma of the orthogonality of N coordinate axes for a space of N dimensions.

At the same time, for some reason, the simplest thought is completely forgotten: “After all, if we cannot even imagine “something”, that is, mentally create an appropriate image, then this “something” does not exist in principle!

Mathematicians explain the fact that we do not understand the flight of their multidimensional fantasies by the limitations of our mental abilities, since, they say, the world around us is three-dimensional. However, in fact, all the talk about the limitations of our imagination is a deliberate lie, since a person can easily construct at least 6-dimensional images from the 7-dimensional matter of thought.

This means only one thing: mathematicians could well explain their “multidimensional visions” to us, of course, if there was at least a drop of reality in them. In the meantime, we are all doomed to worship the dogma of the "fourth orthogonal axis", without even the slightest explanation about its construction.

Thus, another false dogma of "four perpendiculars" to one point turns into another stumbling block on the way to understanding the real multidimensional world.

What do measurements measure?

Why exactly three spatial dimensions, no more and no less? Obviously, because the atom, and with it all the rest of matter, has strictly three spatial characteristics: length, width and height.

What characterize these three characteristics of space? Of course, length material objects in three possible directions: forward↔back, left↔right, up↔down.

Is it possible to specify some other additional characteristics of the length? Not! Common sense categorically rejects such fantasies. Only three extension characteristics can be represented for matter of any dimension.

Does matter have other properties besides extension? Of course, there is: color, viscosity, temperature... But three-dimensional matter has only one spatial property - extension.

Perhaps 4D matter has an additional spatial property? Exactly! The 4D amer, due to its “subtlety”, has an additional spatial property in relation to the 3D atom – permeability. In the work, the fourth dimension of space is called " depth».

According to the author, both terms cannot be considered successful. The term "permeability" can be erroneously attributed to 3D matter, since it is permeable to matter of all higher dimensions. The term "depth" coincides with the terminology of Euclid to characterize a completely different property (length) of the body.

In this regard, the term " nesting”, more precisely conveying the essence of the immersion of the higher spaces of the real world into the lower ones. Let's demonstrate a combination of spatial characteristics of extent and nesting using the example of a 5D space:

three length characteristics (forward↔back, left↔right, up↔down);
two nesting characteristics (in↔out of 3D space, in↔out of 4D space).

It is clear that the 7D space will have the same three length characteristics, and there will be two more nesting characteristics, that is, four, and in general - 3 + 4 - seven.

It is easy to see that the above interpretation of the multidimensionality of the real world excludes the orthogonality of the directions of extension with the directions of nesting, and the latter also among themselves. This allows us to stop conjectures on the topic of multiple orthogonality for high-dimensional spaces.

What is invested in what?

A huge number of publications tell us that the speculative two-dimensional "space" is embedded in the three-dimensional one. The most common example of a 2D "space" is a sheet of a book. Well, then a “brilliant” conclusion is made about the nesting of the already real 3D space in the space of four dimensions and then in a similar way. As a result, fantastic pseudo-multidimensional constructions appear in the form of tesseracts, simplices, and other pseudo-hyper-polyhedra.

It is completely useless to appeal to common sense here, because the entire queen of sciences is built on an unshakable faith in the reality of “spaces” with dimensions less than three. Therefore, in order to expose such manipulations with false spaces, let's take note of two fundamentally important points that took place:

The lower space in the example with the book was mentally "invested" in the higher, that is, in a space with a larger number of dimensions;
All the spaces appearing in the example are filled one type of matter, that is, the three-dimensional substance of paper.

If we now move from the religious dogmas of mathematics to examples from real life, then we will see that a 4D electron is embedded in a 3D atom, a 4D radio wave is embedded in a 3D radio receiver. In this case, everything happens exactly the opposite, previously taken note of the points:

In real life, the higher space is embedded in the lower;
The matter of real spaces of different dimensions is different.

If we acted in accordance with the rules of mathematics from the first example, then it would turn out that an atom can be embedded in an electron, and a radio receiver in a radio wave, which, of course, is absurd, as well as mathematical "spaces" with dimensions less than three.

conclusions

Understanding multidimensional spaces within the framework of modern (three-dimensional) mathematics is fundamentally impossible.
For the study of multidimensional spaces, it is necessary to develop a new section of "Multidimensional Mathematics".
The exit of mathematics from the crisis is impossible without the rejection of thousands of years of dogmatism in favor of a revised scientific paradigm.

Literature

Mikisha A. M., Orlov V. B. Explanatory Mathematical Dictionary: Basic Terms. – M.: Rus. yaz., 1989. - 244 p.
Minkowski space: From Wikipedia. – http://ru.wikipedia.org/wiki/Minkowski_Space
Alexander Kotlin. How to understand four-dimensional space? -
Alexander Kotlin. Cosmic octaves are the key to a new understanding of the World. -
Alexander Kotlin. Fundamentals of mathematics - lawlessness cubed. – 02/27/2014. -
Blavatsky H. P. Secret Doctrine In: The Synthesis of Science, Religion and Philosophy. Volume 1: Cosmogenesis. - L .: Ecopolis and culture, 1991. - 361 p.
Nikolay Uranov. Bring joy. Fragments of letters. 1965-1981. - Riga: World of Fire, 1998. - 477 p.
The beginning of Euclid. Books XI-XV. Translation from Greek and comments by D. D. Mordukhai-Boltovsky with the participation of M. Ya. Vygodsky and I. N. Veselovsky. - Mrs. Publishing house of technical-theoretical. literature, M.-L.: 1950. - 335 p.
Alexander Kotlin. How to understand 10-dimensional space? -

FOURTH DIMENSION

The idea of hidden knowledge. – The problem of the invisible world and the problem of death. – The invisible world in religion, philosophy, science. - The problem of death and its various explanations. – The idea of the fourth dimension. - Various approaches To her. - Our position in relation to the "field of the fourth dimension." – Methods of studying the fourth dimension. - Hinton's ideas. – Geometry and the fourth dimension. - Morozov's article. - An imaginary world of two dimensions. - A world of eternal wonder. - Phenomena of life. – Science and phenomena of the immeasurable. - Life and thought. - Perception of flat beings. - Different stages of understanding the world of a flat being. – Hypothesis of the third dimension. – Our attitude towards the “invisible”. – The world of the immeasurable is around us. – Unreality of three-dimensional bodies. “Our own fourth dimension. - Imperfection of our perception. – Properties of perception in the fourth dimension. - Unexplained phenomena our world. - The mental world and attempts to explain it. – Thought and the fourth dimension. – Expansion and contraction of bodies. - Growth. - Phenomena of symmetry. - Drawings of the fourth dimension in nature. – Movement from the center along the radii. - Laws of symmetry. - States of matter. - Relationship between time and space in matter. – Theory of dynamic agents. - The dynamic nature of the universe. “The fourth dimension is within us. - "Astral sphere" - Hypothesis about the subtle states of matter. - Transformation of metals. - Alchemy. - Magic. – Materialization and dematerialization. - The predominance of theories and the absence of facts in astral hypotheses. - The need for a new understanding of "space" and "time".

Speaking even more frankly with himself, a person may recognize that all his scientific and philosophical systems and theories are similar to these machines and tools, because they only complicate problems without explaining anything.

Among the insoluble problems surrounding man, two occupy a special position - the problem of the invisible world and the problem of death.

Throughout the history of human thought, in every form without exception that thought has ever taken, men have divided the world into visible and invisible; they have always understood that the visible world, accessible to direct observation and study, is something very small, perhaps even non-existent in comparison with the vast invisible world.

Such a statement, i.e. the division of the world into the visible and the invisible existed always and everywhere; at first it may seem strange; but in reality everything general schemes worlds, from the most primitive to the most subtle and elaborate, divide the world into the visible and the invisible - and cannot free themselves from it. The division of the world into visible and invisible is the basis of human thinking about the world, no matter what names and definitions he gives to such a division.

This fact becomes clear if we try to enumerate different systems thinking about the world.

First of all, let's divide these systems into three categories: religious, philosophical, scientific.

Philosophy recognizes the world of phenomena and the world of causes, the world of things and the world of ideas, the world of phenomena and the world of noumenons. In Indian philosophy (especially in some of its schools), the visible or phenomenal world, Maya, an illusion, which means a false concept of the invisible world, is generally considered non-existent.

In science, the invisible world is the world of very small magnitudes, and also, oddly enough, the world of very large magnitudes. The visibility of the world is determined by its scale. The invisible world is, on the one hand, the world of microorganisms, cells, the microscopic and ultramicroscopic world; it is followed by the world of molecules, atoms, electrons, "vibrations"; on the other hand, it is the world of invisible stars, distant solar systems, unknown universes. The microscope expands the boundaries of our vision in one direction, the telescope in another, but both are very small compared to what remains invisible. Physics and chemistry give us the opportunity to investigate phenomena in such small particles and in such distant worlds that will never be available to our vision. But this only reinforces the idea that there is a huge invisible world around a small visible one.

Mathematics goes even further. As has already been pointed out, it calculates such ratios between quantities and such ratios between these ratios that have no analogies in the visible world around us. And we have to admit that invisible the world differs from the visible not only in size, but also in some other qualities that we are unable to determine or understand, and which show us that the laws found in the physical world cannot apply to the invisible world.

Thus, the invisible worlds of religious, philosophical and scientific systems are, after all, more closely connected with each other than it seems at first glance. And such invisible worlds of different categories have the same properties common to all. These properties are. First, they are incomprehensible to us; incomprehensible from the ordinary point of view or for ordinary means of knowledge; secondly, they contain the causes of the phenomena of the visible world.

The idea of causes is always connected with the invisible world. In the invisible world of religious systems, invisible forces control people and visible phenomena. In the invisible world of science, the causes of visible phenomena stem from the invisible world of small quantities and "fluctuations". In philosophical systems, the phenomenon is only our concept of the noumenon, i.e. an illusion, the true cause of which remains hidden and inaccessible to us.

Thus, at all levels of his development, man understood that the causes of visible and observable phenomena are beyond the scope of his observations. He found that among the phenomena available to observation, some facts can be considered as the causes of other facts; but these findings were insufficient to understand Total what happens to him and around him. To explain the causes, an invisible world is needed, consisting of "spirits", "ideas" or "vibrations".

Another problem that attracted people's attention by its insolubility, a problem that by the very form of its approximate solution predetermined the direction and development of human thought, was the problem of death, i.e. explanations of death, the idea of a future life, an immortal soul - or the absence of a soul, etc.

Man has never been able to convince himself of the idea of death as disappearance - too much contradicted it. There were too many traces of the dead in him: their faces, words, gestures, opinions, promises, threats, the feelings they aroused, fear, envy, desires. All this continued to live in him, and the fact of their death was more and more forgotten. A person saw in a dream a dead friend or enemy; and they seemed to him exactly the same as they had been before. Obviously they somewhere lived and could come from somewhere at night.

So it was very difficult to believe in death, and man always needed theories to explain the afterlife.

On the other hand, the echo of esoteric teachings about life and death sometimes reached a person. He could hear that the visible, earthly, observable life of a person is only a small part of his life. And of course, a person understood the fragments of the esoteric teaching that reached him in his own way, changed them according to his own taste, adapted them to his level and understanding, built from them theories of a future existence similar to the earthly one.

Most religious teachings about the future life associate it with a reward or punishment, sometimes in an overt and sometimes in a veiled form. Heaven and hell, transmigration of souls, reincarnations, the wheel of lives - all these theories contain the idea of reward or retribution.

But religious theories often do not satisfy a person, and then, in addition to the recognized, orthodox ideas about life after death, there are other, as if not legalized ideas about the afterlife, about the world of spirits, which provide much more freedom to the imagination.

No religious doctrine, no religious system alone can satisfy people. There is always some other, more ancient system of folk beliefs, which is hidden behind it or hidden in its depths. Behind external Christianity, behind external Buddhism, there are ancient pagan beliefs. In Christianity, these are remnants of pagan ideas and customs, in Buddhism - the "cult of the devil." Sometimes they leave a deep imprint on the outward forms of religion. For example, in modern Protestant countries, where traces of ancient paganism have completely died out, systems of almost primitive ideas about the afterlife, such as spiritualism and related teachings, have arisen under the outward mask of rational Christianity.

All theories of the afterlife are connected with the theories of the invisible world; the former are necessarily based on the latter.

All this refers to religion and pseudo-religion, there are no philosophical theories of the afterlife. And all theories about life after death can be called religious or, more correctly, pseudo-religious.

In addition, it is difficult to consider philosophy as something integral - individual philosophical systems are so different and contradictory. It is possible to some extent to accept as the standard of philosophical thinking the point of view that affirms the unreality of the phenomenal world and human existence in the world of things and events, the unreality of the separate existence of a person and the incomprehensibility for us of the forms of true existence, although this point of view is based on various grounds, both materialistic and idealistic. In both cases, the question of life and death acquires a new character, it cannot be reduced to the naive categories of ordinary thinking. For this point of view, there is no special difference between life and death, because, strictly speaking, it does not take as proven a separate existence, separate lives.

No and cannot be scientific theories of existence after death, because there are no facts confirming the reality of such an existence, while science - successfully or unsuccessfully - wants to deal exclusively with facts. In the fact of death, the most important point for science is the change in the state of the organism, the cessation of vital functions and the decomposition of the body that follow death. Science recognizes no human mental life, independent of vital functions, and from a scientific point of view, all theories of life after death are pure fiction.

Modern attempts at "scientific" research into spiritualistic and similar phenomena lead to nothing and cannot lead, because here there is an error in the very formulation of the problem.

Despite the differences between the various theories of the future life, they all have one thing in common. They either depict the afterlife like the earthly one, or completely deny it. They do not try to understand life after death in new forms or new categories. This is what makes conventional theories of life after death unsatisfactory. Philosophical and strictly scientific thought require a revision of this problem from a completely new point of view. Some hints that have come down to us from esoteric teachings point to the same thing.

It becomes obvious that the problem of death and life after death must be approached from a completely new angle. Similarly, the question of the invisible world requires a new approach. Everything we know, everything we have thought so far, demonstrates to us the reality and vital importance of these problems. Until questions about the invisible world and about life after death are somehow answered, a person cannot think of something else without creating a whole series of contradictions. Man must construct for himself some kind of explanation, right or wrong. He must base his solution to the problem of death either on science, or on religion, or on philosophy.

But for thinking person both the “scientific” denial of the possibility of life after death and its pseudo-religious assumption (for we know nothing but pseudo-religions), as well as all kinds of spiritualistic, theosophical and similar theories, seem equally naive.

Cannot satisfy a person and abstract philosophical views. These views are too far from life, from direct, genuine sensations. It is impossible for them to live. In relation to the phenomena of life and their possible causes, which are unknown to us, philosophy is like astronomy in relation to distant stars. Astronomy calculates the movements of stars located at great distances from us. But for her everything celestial bodies are the same - they are nothing more than moving points.

So, philosophy is too far from concrete problems, such as the problem of the future life; science does not know the afterlife; pseudo-religion creates it in the image of the earthly world.

The helplessness of man in the face of the problems of the invisible world and death becomes especially evident when we begin to understand that the world is much larger and more complex than we have hitherto thought; and what we thought we knew occupies the least place among what we do not know.

The foundations of our concept of the world must be expanded. We already feel and realize that we can no longer trust the eyes with which we see and the hands with which we feel something. The real world eludes us during such attempts to ascertain its existence. More subtle methods, more effective means are needed.

The idea of a "fourth dimension", the idea of a "multidimensional space" indicates the way in which we can come to the expansion of our concept of the world.

The expression "fourth dimension" is often found in conversations and literature, but very rarely anyone understands and can determine what is meant by this expression. Usually the "fourth dimension" is used as a synonym for the mysterious, wonderful, "supernatural", incomprehensible, incomprehensible, as general definition phenomena of the "superphysical" or "supersensible" world.

"Spiritists" and "occultists" of various directions often use this expression in their literature, referring all the phenomena of the "higher planes", the "astral sphere", the "other world" to the area of the fourth dimension. What this means, they do not explain; and from what they say, only one property of the "fourth dimension" becomes clear - its incomprehensibility.

The connection of the idea of the fourth dimension with the existing theories of the invisible or otherworldly world is, of course, completely fantastic, for, as already mentioned, all religious, spiritualistic, theosophical and other theories of the invisible world first of all endow it with an exact resemblance to the visible, i.e. "three-dimensional" world.

That is why mathematics quite rightly rejects the common view of the fourth dimension as something inherent in the “other world”.

The very idea of the fourth dimension arose, probably, in close connection with mathematics, or, more precisely, in close connection with the measurement of the world. It undoubtedly was born from the assumption that in addition to the three dimensions of space known to us: length, width and height, there may be a fourth dimension that is inaccessible to our perception.

Logically, the assumption of the existence of a fourth dimension can come from the observation in the world around us of such things and phenomena for which the measurements of length, width and height are insufficient, or which generally elude measurements, because there are things and phenomena whose existence is beyond doubt, but which cannot be expressed in terms of any dimensions. Such, for example, are the various manifestations of vital and mental processes; such are all ideas, all images and memories; such are dreams. Considering them as really, objectively existing, we can assume that they have some other dimension, in addition to those that are available to us, some extension that is immeasurable to us.

There are attempts at a purely mathematical definition of the fourth dimension. They say, for example, like this: “In many questions of pure and applied mathematics, there are formulas and mathematical expressions that include four or more variables, each of which, independently of the others, can take positive and negative values between +? and -?. And since every mathematical formula, every equation has a spatial expression, from here they derive the idea of space in four or more dimensions.

The weak point of this definition lies in the provision accepted without proof that every mathematical formula, every equation can have a spatial expression. In fact, such a position is completely groundless, and this makes the definition meaningless.

Arguing by analogy with existing dimensions, it should be assumed that if the fourth dimension existed, it would mean that right here, next to us, there is some other space that we do not know, do not see and cannot go into. From any point in our space, it would be possible to draw a line into this “region of the fourth dimension” in a direction unknown to us, which we cannot determine or comprehend. If we could imagine the direction of this line coming from our space, then we would see the "area of the fourth dimension."

Geometric means the following. One can imagine three mutually perpendicular lines to each other. With these three lines we measure our space, which is therefore called three-dimensional. If there is an “area of the fourth dimension” lying outside our space, then, in addition to the three perpendiculars known to us, which determine the length, width and height of objects, there must be a fourth perpendicular, which determines some kind of incomprehensible to us, new extension. The space measured by these four perpendiculars will be four-dimensional.

It is impossible to define geometrically or imagine this fourth perpendicular, and the fourth dimension remains extremely mysterious to us. There is an opinion that one hundred mathematicians know something about the fourth dimension that is inaccessible to mere mortals. It is sometimes said, and this can be found even in the press, that Lobachevsky "discovered" the fourth dimension. In the past twenty years, the discovery of the "fourth" dimension has often been attributed to Einstein or Minkowski.

In fact, Lobachevsky considered the geometry of Euclid, i.e. geometry of three-dimensional space, as a special case of geometry in general, which is applicable to the space of any number of dimensions. But this is not mathematics in the strict sense of the word, but only metaphysics on mathematical topics; and it is impossible to formulate mathematical conclusions from it - or it can only be done in specially selected conditional expressions.

Other mathematicians found that the axioms accepted in Euclid's geometry were artificial and unnecessary - and tried to refute them, mainly on the basis of some conclusions from Lobachevsky's spherical geometry, for example, to prove that parallel lines intersect, etc. They argued that the generally accepted axioms are true only for three-dimensional space and, based on reasoning that refuted these axioms, built new geometry many dimensions.

But all this is not the geometry of four dimensions.

With a closer and more accurate study of the problem, we come to the conclusion that under the existing conditions it is impossible to solve it. Purely geometric at first glance, the problem of the fourth dimension is not solved geometrically. Our geometry of three dimensions is not enough to investigate the question of the fourth dimension, just as planimetry alone is not enough to investigate questions of stereometry. We must discover the fourth dimension, if it exists, purely by experience - and also find a way to represent it in perspective in three-dimensional space. Only then can we create a geometry of four dimensions.

The most superficial acquaintance with the problem of the fourth dimension shows that it must be studied from the side of psychology and physics.

We know that the region of the fourth dimension (again, if it exists) is not only unknowable to our psychic apparatus, but unavailable purely physically. It no longer depends on our defects, but on the special properties and conditions of the area of the fourth dimension. We need to figure out what conditions make the area of the fourth dimension inaccessible to us, find the relationship of the physical conditions of the area of the fourth dimension of our world and, having established this, see if there is anything similar to these conditions in the world around us, if there are any relations similar to relations between 3D and 4D regions.

Many people have worked on the problem of the fourth dimension.

Fechner wrote a lot about the fourth dimension. From his reasoning about the worlds of one, two, three and four dimensions follows a very interesting method of studying the fourth dimension by constructing analogies between the worlds of different dimensions, i.e. between the imaginary world on the plane and our world, and between our world and the world of four dimensions. This method is used by almost everyone involved in the question of higher dimensions. We have yet to get to know him.

Professor Zolner derived the theory of the fourth dimension from observations of "mediumistic" phenomena, mainly of the phenomena of the so-called "materialization". But his observations are now considered doubtful due to the insufficiently rigorous setting of experiments (Podmore and Hislop).

A very interesting summary of almost everything that has been written about the fourth dimension (by the way, and attempts to determine it mathematically), we find in the books of K.Kh. Hinton. They also contain many of Hinton's own ideas, but unfortunately, along with valuable thoughts, they contain a lot of unnecessary "dialectics", such as usually happens in connection with the question of the fourth dimension.

Hinton makes several attempts to define the fourth dimension both in terms of physics and psychology. A fair place in his books is occupied by a description of the method he proposed for accustoming consciousness to the comprehension of the fourth dimension. This is a long series of exercises in the apparatus of perceptions and representations with a series of multi-colored cubes, which must be remembered first in one position, then in another, in a third, and then imagined in various combinations.

Hinton's main idea, which guided him in developing his method, is that in order to awaken the "higher consciousness" it is necessary to "destroy oneself" in the representation and cognition of the world, i.e. to learn to cognize and imagine the world not from a personal point of view (as is usually the case), but as it is. At the same time, first of all, one must learn to imagine things not as they seem, but as they are, even if only in simple terms. geometric sense; after which the ability to cognize them will appear, i.e. to see them as they are, and also from points of view other than geometric.

the first exercise given by Hinton: the study of a cube, consisting of 27 smaller cubes, which are colored in different colors and have specific names. Having firmly studied a cube made up of cubes, you need to turn it over and study (i.e. try to remember) in reverse order. Then turn the cubes over again and remember in this order, etc. As a result, as Hinton says, it is possible to completely destroy the concepts of the cube under study: top and bottom, right and left, etc., and to know it regardless of the relative position of its constituent cubes, i.e., probably, to represent it simultaneously in various combinations. This is the first step in destroying the subjective element in the idea of a cube. Further, a whole system of exercises is described with a series of multi-colored and variously named cubes, from which all kinds of figures are composed, all with the same goal of destroying the subjective element in the representation and thus developing a higher consciousness. The destruction of the subjective element, according to Hinton, is the first step towards the development of higher consciousness and comprehension of the fourth dimension.

Hinton argues that if there is the ability to see in the fourth dimension, if it is possible to see the objects of our world from the fourth dimension, then we will see them in a completely different way, not as usual.

Usually we see objects above or below us, or on the same level with us, to the right, to the left, behind us, or in front of us, always on the same side facing us and in perspective. Our eye is an extremely imperfect apparatus: it gives us a highly incorrect picture of the world. What we call perspective is, in essence, the distortion of visible objects, produced by a poorly constructed optical apparatus - the eye. We see objects distorted and we imagine them in the same way. But all this is solely due to the habit of seeing them distorted, i.e. due to habit caused by our defective vision, which has weakened our ability to imagine.

But, according to Hinton, we have no need to imagine the objects of the external world necessarily distorted. The faculty of representation is by no means limited to the faculty of sight. We see things distorted, but we know them for what they are. We can get rid of the habit of representing things as they appear to us, and learn to imagine them as we know they are. Hinton's idea is that before thinking about developing the ability to see in the fourth dimension, you need to learn to imagine objects as they would be seen from the fourth dimension, i.e. not in perspective, but from all sides at once, as our "consciousness" knows them. It is this ability that Hinton's exercises develop. The development of the ability to imagine objects from all sides at once destroys the subjective element in representations. According to Hinton, "the destruction of the subjective element in representations leads to the destruction of the subjective element in perception." Thus, the development of the ability to imagine objects from all sides is the first step to the development of the ability to see objects as they are in a geometric sense, i.e. to the development of what Hinton calls "higher consciousness".

In all this there is much that is true, but there is also much far-fetched, artificial. First, Hinton does not take into account the differences between different mental types of people. A method that is satisfactory for himself may not produce any results or even cause negative consequences for other people. Secondly, the very psychological basis of Hinton's system is too unreliable. Usually, he does not know where to stop, his analogies lead too far, thereby depriving many of his conclusions of any value.

From the point of view of geometry, the question of the fourth dimension can be considered according to Hinton in the following way.

We know geometric figures three genera:

one dimension - a line, two dimensions - a plane, three dimensions - a body.

At the same time, we consider a line as a trace from the movement of a point in space, a plane as a trace from the movement of a line in space, a body as a trace from the movement of a plane in space.

Imagine a line segment bounded by two points, and denote it by the letter a. Suppose this segment moves in space in a direction perpendicular to itself and leaves a trail behind it. When it has traveled a distance equal to its length, its trail will look like a square, the sides of which are equal to the segment a, i.e. a2.

Let this square move in space in a direction perpendicular to two adjacent sides of the square and leave a trail behind it. When he has traveled a distance equal to the length of the side of the square, his trail will look like a cube, a3.

Now, if we assume the movement of the cube in space, then what form will its trace have, i.e. figure a4?

Considering the relations of figures of one, two and three dimensions, i.e. lines, planes and bodies, we can deduce the rule that each figure of the next dimension is a trace of the movement of the figure of the previous dimension. Based on this rule, we can consider the figure a4 as a trace from the movement of the cube in space.

But what is this movement of the cube in space, the trace of which turns out to be a figure of four dimensions? If we consider how the movement of a lower dimensional figure creates a higher dimensional figure, we will find several common properties, general patterns.

Namely, when we consider a square as a trace from the movement of a line, we know, we know that all the points of the line moved in space; when we consider the cube as a trace of the movement of the square, then we know that all the points of the square moved. In this case, the line moves in a direction perpendicular to itself; a square is in a direction perpendicular to its two dimensions.

Therefore, if we consider the figure a4 as a trace from the movement of the cube in space, then we must remember that all the points of the cube moved in space. At the same time, by analogy with the previous one, we can conclude that the cube moved in space in a direction not contained in itself, i.e. in a direction perpendicular to its three dimensions. This direction is the fourth perpendicular, which does not exist in our space and in our geometry of three dimensions.

The line can then be viewed as an infinite number of points; square - as an infinite number of lines; a cube is like an infinite number of squares. Likewise, figure a4 can be thought of as an infinite number of cubes. Further, looking at the square, we see only lines; looking at the cube - its surfaces or even one of these surfaces.

It must be assumed that the figure a4 will be presented to us in the form of a cube. In other words, the cube is what we see when we look at the figure. a4. Further, a point can be defined as a section of a line; line - as a section of the plane; plane - as a section of the volume; in the same way, a three-dimensional body can be defined as a section of a four-dimensional body. Generally speaking, when looking at a four-dimensional body, we will see its three-dimensional projection, or section. A cube, a ball, a cone, a pyramid, a cylinder - may turn out to be projections, or sections, of some four-dimensional bodies unknown to us.

In 1908, I came across a curious article about the fourth dimension in Russian, published in the journal Modern World.

It was a letter written in 1891 by N.A. Morozov* to fellow prisoners in the Shlisselburg Fortress. It is interesting mainly because it very figuratively sets out the main provisions of the method of reasoning about the fourth dimension by analogy, which was mentioned earlier.

* ON THE. Morozov, a scientist by education, belonged to the revolutionaries of the 70s and 80s. He was arrested in connection with the assassination of Emperor Alexander II and spent 23 years in prison, mainly in the Shlisselburg Fortress. Released in 1905, he wrote several books: one about the Revelation of the Apostle John, another about alchemy, magic, etc., which found very numerous readers in the pre-war period. It is curious that the public in Morozov's books liked not what he wrote, but what about what he wrote. His real intentions were very limited and strictly corresponded to the scientific ideas of the 70s of the XIX century. He tried to present "mystical objects" rationally; for example, he announced that in the Revelation of John only a description of a hurricane was given. But, being a good writer, Morozov expounded the subject very vividly, and sometimes added little-known material to this. Therefore, his books produced completely unexpected results; after reading them, many became interested in mysticism and mystical literature. After the revolution, Morozov joined the Bolsheviks and remained in Russia. As far as is known, he did not take a personal part in their destructive activities and did not write anything else, but on solemn occasions he unfailingly expressed his admiration for the Bolshevik regime.

The beginning of Morozov's article is very interesting, but in his conclusions about what could be in the area of the fourth dimension, he departs from the method of analogies and refers to the fourth dimension only the "spirits" that are called up at spiritualistic sessions. And then, rejecting spirits, he also denies the objective meaning of the fourth dimension.

In the fourth dimension, the existence of prisons and fortresses is impossible, and, probably, therefore, the fourth dimension was one of the favorite topics of conversations that were conducted in the Shlisselburg fortress by tapping. Letter to N.A. Morozov is the answer to the questions posed to him in one of these conversations. He's writing:

My dear friends, our short Shlisselburg summer is ending, and mysterious dark autumn nights are coming. In these nights, descending like a black veil over the roof of our dungeon and enveloping our little island with its ancient towers and bastions in impenetrable darkness, it involuntarily seems that the shadows of the comrades who died here and our predecessors fly invisibly around these cells, look into our windows and join us. , still alive, in mysterious intercourse. And are we ourselves not shadows of what we once were? Haven't we already turned into some kind of knocking spirits that appear at seances and invisibly talk to each other through the stone walls separating us?

All this day I have been thinking about your dispute today about the fourth, fifth and other dimensions of the space of the universe that are inaccessible to us. I tried with all my might to imagine in my imagination at least a fourth dimension of the world, the very one along which, according to metaphysicians, all our closed objects can suddenly be open, and along which creatures capable of moving without movement can penetrate them. only according to our three, but also according to this fourth dimension, which is unusual for us.

You demand from me a scientific treatment of the question. For the time being, we will talk about the world of only two dimensions, and then we will see if it will not give us the opportunity to draw any conclusions about the other worlds.

Suppose that some plane, at least the one that separates the surface of Lake Ladoga on this quiet autumn evening from the atmosphere above it, is a special world, a world of two dimensions, inhabited by its own creatures that can only move along this plane, like those the shadows of swallows and seagulls that run in all directions on the smooth surface of the water surrounding us, but never visible to us behind these bastions.

Suppose that, having escaped behind our Shlisselburg bastions, you went swimming in the lake.

As beings of three dimensions, you also have those two that lie on the surface of the water. You will take a certain place in this world of shadowy creatures. All parts of your body above and below the water level will be imperceptible to them, and only that contour of yours, which is surrounded by the surface of the lake, will be completely accessible to them. Your contour should seem to them the object of their own world, but only extremely amazing and wonderful. The first miracle, from their point of view, will be your unexpected appearance among them. It can be said with full confidence that the effect that you have produced by this is in no way inferior to the unexpected appearance between us of some spirit from an unknown world. The second miracle is the extraordinary variability of your species. When you sink to the waist, your shape will be almost elliptical to them, since only that circle will be noticeable to them, which on the surface of the water covers your waist and is impenetrable to them. When you begin to swim, you will take on the shape of a human outline in their eyes. When you come to a shallow place, so that the surface they inhabit is bordered only by your feet, you will seem to them turned into two round-shaped beings. If, wanting to keep you in a certain place, they surrounded you on all sides, you could step over them and find yourself free in a way incomprehensible to them. You would be omnipotent beings for them, inhabitants of a higher world, like those supernatural beings about which theologians and metaphysicians narrate.

Now, if we assume that in addition to these two worlds, flat and ours, there is also a world of four dimensions, higher than ours, then it is clear that its inhabitants in relation to us will be the same as we were now for the inhabitants of the plane. They should just as unexpectedly appear before us and arbitrarily disappear from our world, leaving for the fourth or some other, higher dimensions.

In a word, a complete analogy so far, but only so far. Further in the same analogy, we will find a complete refutation of all our assumptions.

Indeed, if the beings of the four dimensions were not our invention, their appearance among us would be ordinary, everyday occurrences.

Further, Morozov analyzes the question of whether we have any reason to think that such "supernatural beings" really exist, and comes to the conclusion that we have no reason for this if we are not ready to believe the stories.

The only worthy indications of such beings can be found, according to Morozov, in the teachings of spiritualists. But his experiences with "spiritualism" convinced him that despite the presence mysterious phenomena which undoubtedly take place in séances, the "spirits" take no part in it. The so-called "automatic writing", usually cited as evidence of participation in the sessions of the intelligent forces of the other world, according to his observations, is the result of mind reading. The "medium" consciously or unconsciously "reads" the thoughts of those present and thus receives answers to their questions. ON THE. Morozov was present at many sessions and did not meet the case that in the received answers something unknown to everyone was reported, or that the answers were in a language unfamiliar to everyone. Therefore, without doubting the sincerity of most spiritualists, N.A. Morozov concludes that the spirits have nothing to do with it.

According to him, his practice with spiritualism finally convinced him many years ago that the phenomena he attributed to the fourth dimension did not really exist. He says that in such seances, the answers are given unconsciously by those present themselves, and therefore all assumptions about the existence of the fourth dimension are pure fantasy.

These conclusions of Morozov are completely unexpected, and it is difficult to understand how he arrived at them. Nothing can be objected to his opinion about spiritualism. The psychic side of spiritual phenomena is, of course, quite "subjective". But it is completely incomprehensible why N.A. Morozov sees the "fourth dimension" exclusively in spiritualistic phenomena and why, denying spirits, he denies the fourth dimension. This looks like a ready-made solution offered by that official “positivism” to which N.A. Morozov and from which he could not move away. His foregoing reasoning leads quite differently. In addition to "spirits", there are many phenomena that are quite real for us, i.e. habitual and daily, but not explainable without the help of hypotheses that bring these phenomena closer to the world of four dimensions. We are only too accustomed to these phenomena and do not notice their “wonderfulness”, we do not understand that we live in a world of eternal miracle, in a world of the mysterious, inexplicable, and most importantly, immeasurable.

ON THE. Morozov describes how wonderful our three-dimensional bodies will be for flat creatures, how they will appear from nowhere and disappear from nowhere, like spirits emerging from an unknown world.

But aren't we ourselves the same fantastic creatures that change their appearance for any immovable object, for a stone, for a tree? Don't we have the properties of "higher beings" for animals? And do not phenomena exist for ourselves, such as, for example, all manifestations of life, about which we do not know where they came from and where they go: the appearance of a plant from a seed, the birth of living beings, and the like; or natural phenomena: thunderstorm, rain, spring, autumn, which we are not able to explain or interpret? Isn't each of them, taken separately, something of which we grope only a little, only a part, like the blind in an old oriental tale, each of them defining the elephant in his own way: one by the legs, the other by the ears, the third by the tail?

Continuing the reasoning of N.A. Morozov about the relation of the world of three dimensions to the world of four dimensions, we have no reason to look for the latter only in the field of "spiritualism".

Let's take a living cell. It can be absolutely equal - in length, width and height - to another, dead cell. And yet there is something in a living cell that is not in a dead cell, something that we cannot measure.

We call this something "life force" and try to explain it as a kind of movement. But, in essence, we do not explain anything, but only give a name to a phenomenon that remains inexplicable.

According to some scientific theories, the vital force must be decomposed into physical and chemical elements, into the simplest forces. But none of these theories can explain how one passes into the other, in what relation one stands to the other. We are not able to express the simplest manifestation of living energy in the simplest physical and chemical form. And while we are not able to do this, we strictly logically have no right to consider life processes identical with physical and chemical ones.

We can recognize philosophical "monism", but we have no reason to accept the physico-chemical monism that is constantly being imposed on us, which identifies vital and mental processes with physical and chemical ones. Our mind can come to an abstract conclusion about the unity of physical-chemical, vital and mental processes, but for science, for exact knowledge, these three kinds of phenomena stand completely apart.

For science, three kinds of phenomena—mechanical force, vital force, and psychic force—only partly pass one into the other, apparently without any proportionality, without yielding to any calculation. Therefore, scientists will only then have the right to explain life and mental processes as a kind of movement when they come up with a way to translate movement into vital and psychic energy and vice versa and take this transition into account. In other words, to know how many calories contained in a certain amount of coal are needed for the emergence of life in one cell, or how much pressure is needed to form one thought, one logical conclusion. While it is not known, the physical, biological and mental phenomena studied by science occur on different planes. One can, of course, guess about their unity, but it is impossible to assert this.

The fourth bad excuse is "No one wants to go with me, but I can't go alone." Are you reading this book or are you just flipping through it?

From book New model Universe author Uspensky Petr Demyanovich

THE FOURTH DIMENSION The idea of hidden knowledge. – The problem of the invisible world and the problem of death. – The invisible world in religion, philosophy, science. - The problem of death and its various explanations. – The idea of the fourth dimension. – Different approaches to it. - Our position in relation to

From the book Strategic Family Therapy author Madanes Claudio

Fourth Interview At this meeting, which took place exactly one week later, a man came, an unspoken member of this family. His visit was prepared by the insistence of the therapist. The mother mentioned the existence of this man in the first minutes of the show, and in the next -

From the book She. Deep aspects of female psychology author Johnson Robert

Interview 4 Belson: So, how well did your wife cope with the role of the stalker? What did she achieve? Husband: Ah, she did pretty well with her, very well indeed. Belson: What did she do? Husband: We made love twice in the last few days. She led

From the book Homo Gamer. Psychology computer games author Burlakov Igor

The fourth task The fourth task turned out to be the most important and most difficult for Psyche. Few women reach this stage in their development, so what will be discussed next may seem strange and have nothing to do with you. If this task is not for you,

From the book Almighty Mind or Simple and Effective Self-Healing Techniques author Vasyutin Alexander Mikhailovich

The Fourth Dimension of Doom Games The world of Doom Games is full of wonders. Some are fantastic physical properties: scary monsters, powerful weapon and colossal machinery. Another type of miracle is the properties of space: an aggressive labyrinth has more than three dimensions

From the book The Way to the Fool. Book one. Philosophy of Laughter. author Kurlov Grigory

Exercise Four If you have ever tried to suck air out of a bottle, you probably know that after a while the rarefaction of the air inside the bottle will not allow you to continue this activity. The same can happen when doing exercises

From the book The Self-Releasing Game author Demchog Vadim Viktorovich

Fourth movement. "Swing" Stand up straight, feet shoulder-width apart. In the first phase of the movement, while inhaling, passionately push the pelvis forward, hold your breath for 5 seconds, while contracting the muscles of the pelvic floor and trying to raise the testicle as high as possible. Then slowly, as you exhale, relax

From the book Conflict Management author Sheinov Viktor Pavlovich

32. Love is the fourth "PA" or GRANDBATMAN! In order to scan this beast, it must be introduced into rigid schematic boundaries from the very beginning. Following the imagery of the GAME, there are four types of love: 1) ROLE LOVE, or DEMONIC, DISCRETE LOVE. 2) LOVE ACTOR, or

Lesson 4 Girls, my dears, good evening! Write to me how you, I hope that everyone today with red roses has not been forgotten. Because we're going to have amazing practice with them. And tell me, how was your week? What did you do? What didn't you do? pamper yourself or

Flatland: A Novel of the Fourth Dimension

I am [Square]. But taking me with him to the Land of Three Dimensions. Your
Lordship showed me the entrails of my countrymen
in the Land of Two Dimensions. What could be easier than taking
your humble servant on a second journey, to a blessed
area of the Fourth Dimension from where I could look
to the Land of Three Dimensions... Sphere. But where
is this Country of Four Dimensions located?
I. I do not know, but to my highly esteemed
The instructor should be aware of this.
Edwin E. Abbott "Flatland"»

Flatland: A Novel of the Fourth Dimension is without a doubt the book that has made the greatest contribution to spreading and popularizing the idea of the fourth dimension among mathematicians, scientists and students, as well as thinkers, artists and the general public. It was published in 1884 and is still popular today. The book continues to arouse sincere interest, new editions continue to be printed, despite the fact that the text is freely available on the Internet.
This is not so much a popular science book as a work of fiction, which, with the help of analogies, introduces the reader to fascinating world fourth and other dimensions. The author invites us, in the form of a two-dimensional being, to explore the flat world in which such beings live, in order to then lead us to the idea that there are worlds of greater and lesser dimensions - three-dimensional and one-dimensional. This allows the reader to experience the complexity of presenting reality with more dimensions than those perceived by our senses. At the same time, it also proves that such imperceptible dimensions may well exist. The author suggests thought experiment, which will help us imagine a fourth dimension that exists outside of our three-dimensional world[…].

The second part of the book, entitled "Other Worlds", touches on the problems of multidimensional analogies and theological aspects, although social satire is present throughout the book. First, the Square in a strange dream finds itself in Lineland, whose world is an infinite straight line and therefore is one-dimensional. It is inhabited by line segments (men) and points (women). While out of Lineland, the Square addresses the king of this world, who at first cannot understand who or what he is talking to. The square tries to explain to the king that he himself lives in a two-dimensional world and perceives everything in two dimensions, but the king does not understand him, and the square does not know how to explain it all. He begins to describe the situation when a point, moving along a one-dimensional Lineland, forms a segment - which is obvious to the king - but if the segment moves "up", then a square is obtained. However, the king is unable to understand either the meaning of the expression "up" or the concept of "square". The two-dimensional mathematician then decides to cross Lineland to show the king that he is a two-dimensional being. But the king does not believe that the segments he sees are different sections of the square, and not some kind of Linelander with an incomprehensible ability to appear and disappear.
The next day after waking up, the Square meets the Sphere, who lives in Spaceland - a world with three dimensions, which contains Flatland. As with the King of Lineland, Square is at first unable to figure out where the voice is coming from. This time, the Sphere tries to describe the nature of three-dimensional space to a Flatlander by giving the analogy that if a square figure grows in the "up" direction, then a cube will be obtained, which has three dimensions. When the student is unable to understand these arguments, the Sphere decides to cross Flatland so that its flat sections, which are circles, are visible. But Square thinks that this is a priest who appeared in some magical way, then quickly grew, as if time had accelerated, and then mysteriously shrank and disappeared.
Continuing a series of analogies regarding different dimensions and social structure, the 3D visitor makes an argument based on the number of vertices (corners) and faces. The number of vertices of a point, a segment, and a square form a geometric progression of 1, 2, 4, which continues with the number 8, which, as the Sphere explains to the Square, is the number of vertices of a cube. Also, points have no faces, a line segment has two (its two ends), and a square has four faces (its four sides). It turns out arithmetic progression 0. 2, 4, which continues with the number 6, equal to the number of faces of the cube.

The sphere, convinced of the futility of its explanations, takes drastic measures and takes our hero out of Flatland, which is possible due to the fact that Flatland and all its inhabitants have a constant thickness in three-dimensional space. Seeing your world from the outside. The square understands the meaning of the third intention of space, which his teacher spoke about. All the arguments presented immediately became clear, but that's not all. As a good mathematician, he understands that these arguments allow him to go further. After thinking for a while, he explains to the Sphere that if you use the same analogy with dimensions, then perhaps there is a four-dimensional space containing the world of the Sphere, now the Sphere itself becomes confused, refusing to recognize this argument and the Fact of the existence of a four-dimensional space: “Such no country. The very idea that it exists is devoid of any meaning.
As we have said, Abbott did not believe in miracles and believed that Christians should not base their faith on them. This idea is also reflected in "Flatland", where what seems like a miracle to two-dimensional beings is actually easily explained when moving into the third dimension[…]
Abbott's best friend, math teacher Howard Candler, who has an extensive correspondence with him, taught at Uppingham School. By the way, the English mathematician Charles Hinton, one of the main specialists in the fourth dimension, also taught at this school. It is possible that Abbott met Hinton at Uppingham or learned about these ideas through his friend Candler. In any case, he understood the concept of the fourth dimension clearly enough to use it as a metaphor for the social and theological structure of the class-divided society of Victorian England[…].

Charles Hinton and the Philosophy of the Fourth Dimension

The young Charles Hinton was heavily influenced by a group of intellectuals with progressive social and political views. Among them were sexologist Havelock Ellis. , founder of mathematical logic George Boole and his wife, mathematician Maria Everest Boole. However, the most radical of them was Charles's father, James Hinton, who worked as a surgeon before becoming a famous writer and philosopher. Several books were published from his pen, both in medicine (James Hinoton was considered the best otolaryngologist of his time) and in social philosophy.
The mathematician Charles Hinton was one of those who did much to popularize the fourth dimension. He was interested in various fields: mathematics and physics, philosophy and religion, as well as the visualization of four-dimensional space, in particular the hypercube. He also published works on other interesting topics.
Charles Hinton was born in London in 1853. He studied mathematics at Oxford, from which he graduated in 1877, and received his master's degree there in 1886. He then began working as a science teacher at Uppingham School. From an early age, Hinton was interested in the problem of visualization. At Oxford, he received a decent mathematical knowledge, but it was not enough for him. At that time, he began working with a cubic yard (91.5 cm), consisting of 36 x 36 x 36 = 46,656 cubes, each of which had the corresponding name on Latin such as Collis Nebula. When Hinton wanted to visualize a four-dimensional object, he mentally unrolled it and placed it inside a cube. After that, he could study the structure of the object by analyzing the cubes that made up its three-dimensional unfolding. Hinton also developed a system to reduce the amount of detail that had to be remembered. This seemingly absurd idea materialized into a kind of converter - a converter of four-dimensional objects into three-dimensional ones - and became another step towards understanding the fourth dimension. Hinton's cube was a kind of four-dimensional eye, which inspired him to invent the famous colored cubes.

Hinton's interest in the fourth dimension continued to grow, and in 1880 he published "What is the Fourth Dimension" in the Journal of the University of Dublin, which was reprinted in 1883 in the journal of Cheltenham College. The following year saw the pamphlet What Are Ghosts published by Swan Sonnenschein & Co., which produced nine pamphlets, essays, and science fiction stories about the fourth dimension. They were later collected together under the title "Scientific Romances". Among these was the short story "The Flat World" (1884) with an idea similar to Abbott's "Flatland", although Hinton was more interested in physical aspects a two-dimensional world that is the surface of a sphere, not a plane.
Heaton's life was prosperous, to some extent he even achieved social success. But in 1885 everything collapsed: he was arrested for bigamy. Hinton lost his job, his career was ruined, and after the sentence, after spending three days in prison, he moved with his family to Japan, where he worked as a teacher. high school in Yokohama. From there he sent his friends the manuscript A New Age of Thought, which was published in 1888. The first part of the work was devoted to the question of the awareness of the fourth dimension, as well as the philosophical and religious aspects associated with the fourth dimension. The second part related to the visualization of the hypercube, and it contained a description of the colored cubes and instructions for their use.
In 1893 Hinton came to North America. There he worked at the universities of Princeton, Minnesota and then in Washington, DC, as well as at the US Naval Observatory and the Patent Office. He also spread the ideas of the fourth dimension in the United States and was considered in intellectual circles as a recognized and respected person. Hinton wrote numerous articles and lectured on a wide range of subjects, including poetry. In 1904, he published The Fourth Dimension, which included all his reflections on the subject, as well as a new story about the two-dimensional universe, The Incident in Flatland. Hinton died in 1907.

Gods and ghosts

From the fact that we do not hear high or low frequencies and do not
we distinguish colors outside the visible spectrum, it does not at all follow that they
does not exist. Isn't it possible, isn't it the same
it is probable that there is a fourth dimension which is not
open to our eyes, in which our souls can live so
called dead people and through which
will we ever be able to communicate with them?
And this one new world around is also ours - this world
an endless variety of colors and sounds.
Charles Paterson. new skies and new earth, or the Path to Eternal Life(1909)

The fourth dimension had all the necessary qualities so that at the end of the 19th and beginning of the 20th centuries. attract the attention of people of various beliefs: both adherents of traditional religions and adherents of new religious movements, sectarians, lovers of paranormal phenomena, occultism and spiritualism, philosophers, theologians, mystics and so on. This topic was discussed very seriously in the religious world, we see it in the books and articles published at that time. However, if you search the Internet and in books, you will find that in our time, the fourth intention still fascinates a huge number of people.

Spiritualism and ghosts from the fourth dimension

Spiritualism, or the belief that the souls of the dead are with us and can be contacted, originated in Europe in the 19th century. as a religious and philosophical movement. It soon became very popular in the US, leading to an avalanche of paranormal reports. At the same time, a huge number of mediums began to organize sessions with spirits, staging performances and playing on the feelings, religious and mystical beliefs of those who came to them to talk with their loved ones. The activity of mediums was more connected with psychology than with contacts with spirits, and most often came down to tricks and theatrical performances. Mediums were often accused of fraud, and information about them was colorful anecdotes and a complete lack of scientific information.
Only a few scientists were interested in the world of spirits. Among them were those, as we shall see later, who tried to prove the existence of spirits. One of the most prominent proponents of scientific spiritualism was the English chemist William Crookes (1832-1919), inventor of the cathode ray tube. , on the basis of which the first televisions and computer monitors were made.
There were two opinions about the nature of the spirits themselves. The first, more common among spiritualists, was that spirits are immaterial, three-dimensional beings composed of energy, ectoplasm, or some other type of supernatural substance. But if they were intangible, how could they move objects during the sessions? Another opinion, which became popular towards the end of the 19th century, was that spirits are material, but we cannot see them because they exist outside our space and visit us when they want. They are, for example, beings living in the fourth dimension. Then the materialization of spirits is nothing more than their passage through our three-dimensional space. Some spiritualists have criticized this materialistic version, arguing that if spirits were material, they could not pass through doors or walls. However, for beings from hyperspace, this is possible through the fourth dimension, as described in the previous chapter.
The idea that spirits are beings from the fourth dimension was popularized mainly by the American medium Henry Slade and the German physicist Johann Zöllner. As we have already mentioned, the fourth dimension became widely known after Slade was accused of fraud. But his studies in the field of spiritualism interested the Russian prince Constantine, and Slade was invited by Colonel Olcott and Madame Blavatsky, founders of the Theosophical Society in New York. The séances organized by Slade became extremely popular among spiritualists and high society in London. However, Slade was soon accused of fraud. During one session, it was discovered that the board on which the spirits used to leave their messages already contained notes before the start of the session. The court sentenced Slade to three months hard labor. But the sentence was finally overturned, and Slade left England.
Slade's criminal case hit the papers and became a hot topic. It caused a great scandal in English high society, and although there were other processes associated with spiritualism, it was Slade's case that became the most famous, because many eminent scientists around the world came to his defense. Among them were Johann Zöllner, William Crookes, the German physicist Wilhelm Weber (1804 - 1891) - a colleague of Gauss and Riemann's mentor, English physicist Joseph Thomson (1856-1940), who soon became laureate Nobel Prize for the discovery of the electron, and the English physicist Lord Rayleigh (1842-1919), also a future Nobel Prize winner for his studies of the density of various gases and the discovery of argon. These luminaries of science have confirmed that spirits exist and that the paranormal phenomena for which Slade was accused are quite possible in four-dimensional space. Ghosts, they said, were beings that lived in the fourth dimension.
A year after escaping from London, Henry Slade appeared in Leipzig at the invitation of Zöllner, who, together with a number of colleagues, including Weber and Fechner (the author of the story “Space has four dimensions”), decided to conduct a series of experiments. These experiments were supposed to prove once and for all that spirits are four-dimensional beings and thus a fourth dimension exists. Zöllner, while doing physical research, was familiar with the theory of multidimensional spaces, and also studied the work of Gays, Riemann and Helmholtz and understood that these theories could be used to explain paranormal phenomena.
The Leipzig group held séances for several months, and then Zöllner published two papers in London: an article "On Four-Dimensional Space" in 1878 and a translation of the third book of the series Wissenschaftlicbc Abhancllungcn ("Transcendental Physics") in 1880. This book, summarizing the results experiments, was very popular, becoming a desktop for all those interested in spirits: theosophists and some artists, including the Russian expressionist painter Wassily Kandinsky.
The first experiment of an American medium was with a rope tied in a noose. After Slade put his hand on the rope, four knots appeared on it. Since the rope is a closed loop, it was impossible to tie these knots in 3D without cutting the ropes. However, this is quite accessible to a being from the fourth dimension, although in order to tie a knot, the creature had to move the rope to ana or kata. For Zöllner, the result of this experiment proved the existence of spirits from the fourth dimension.
The book Transcendental Physics contains details of many of the paranormal experiments Slade performed at the Leipzig group meetings, in addition to the series of experiments personally designed by Zöllner to prove the four-dimensional nature of spirits. For example:

1. In one of the experiments, spirits connected two wooden rings through the fourth dimension without breaking them.
2. In nature, a property of a certain orientation is often found, for example, a snail shell. When passing through the fourth dimension, this orientation could change.
3. On a rope connected in the form of a loop, the spirits tied a knot.

But were Zollner and Slade's experiments really successful? Zöllner thought so, but from the point of view of the scientific approach, the experiments themselves were erroneous. The spirits did not do what Zöllner expected from them in accordance with the planned plan of his experiments. Instead, the rings were put on the leg of the stand, the snail moved from the table to the floor, and two additional loops were formed on the rope.
Not everyone was satisfied with Zöllner's explanations, and the experiments sparked a fierce debate among intellectuals. Especially strong criticism came from scientists such as Helmholtz. A physicist who had departed from spiritualism believed that a scientist was not the best specialist for evaluating the actions of a magician, since, observing his right hand, he does not see what tricks the left one is doing. In the end, everyone came to the conclusion that Zöllner allowed himself to be misled and may have gone insane.

The result of Zöllnsr's work was that the fourth dimension turned into a joke, far from any scientific facts. However, at the end of the XIX century. English Protestant priest Edwin Abyott once again returned to the idea that spirits are beings from the fourth dimension, Abbott had nothing to do with mediums and used this concept for theological discussions. In addition, specialists such as Hinton continued to work on the more serious aspects of the fourth dimension.

Theology and the fourth dimension

In theological matters, there have been two approaches to the fourth dimension. On the one hand, we have already mentioned Abbott's position: We cannot reach God through the fourth dimension, through science.". However, many other believers, such as some Christians, have enthusiastically accepted the idea that heaven, hell, souls, angels, and God himself can be "situated" in the fourth dimension. These ideas can be found in the book of the English doctor and writer Alfred Taylor Schofield (1846-1929) "The Other World, or the Fourth Dimension":
«... Therefore, we can conclude that another world not only can exist, but is even quite probable. Secondly, such a world can be considered as a space of four dimensions, and thirdly. the spiritual world is governed mainly by its own mysterious laws, has a strange language for us, is full of miraculous manifestations of the high level omniscience and omnipresence and so on, which by analogy are the laws, language and properties of the fourth dimension... ...Although our beautiful material Universe goes far beyond our knowledge, despite the use of the most powerful telescopes, this does not interfere with the other world and its beings, as well as heaven and hell, to be very close to us».
Two brief remarks on Schofield's ideas. Contrary to popular belief, if angels or souls could pass through our world as four-dimensional beings, this does not mean at all that they would look like a person, as we said in the fourth chapter.
Besides, why did God in his perfection choose the fourth dimension for himself? Why not fifth or sixth or higher? A two-dimensional plane is in three-dimensional space, which in turn is in four-dimensional space, and so on, up to an infinite number of dimensions. For such a perfect, all-powerful and all-seeing being as God, a space of infinite dimension would be more suitable. Philosophers of the fourth dimension made similar conclusions in the 19th century.
The British theologian and Protestant pastor Arthur Willink (1850-1913) shared this view. In his work “The Invisible World”, he wrote that God lives in a space of infinite dimension:
« But now we can go further and consider a generalization of the idea in dimensions, which is by no means exhausted by the concept of a space of four dimensions ... If we recognize the existence of a space of four dimensions, it is no longer so difficult to come to the idea of the existence of a space of five dimensions, and so on up to infinite-dimensional spaces ... And although it is impossible to even imagine how a cap looks like a material object of our space for an observer from a world of higher dimension, it is still obvious that he sees a more beautiful view in its entirety than an observer from a space of lower dimension. From a higher world, more perfect images are visible, including the hidden and secret sides of phenomena and objects.
This especially emphasizes the aspect of God's omniscience. For He, living in the highest world, not only perfectly sees all the components of our being, but He is also infinitely close to every point and particle of our soul and body. So even in the strictest physical sense we all live, move and have our being in Him».
At the same time, the German mathematicians Richard Dedekind (1631 - 1916) and above all Georg Cantor (1845-1918) studied the concept of infinity with the strictest mathematical precision. Subsequently, at the beginning of the XX century. the German mathematician David Hil6ert (1862-1943) introduced the concept of infinite-dimensional spaces in which it was possible to measure the distance, so smeared Hilbert spaces.
Philosopher and mathematician William Granville (1864-1943), author of the article "The Fourth Dimension and the Bible", also shared the belief that God dwells in infinite space. However, he believed that the fourth dimension and other higher intentions are heaven, while the two-dimensional and one-dimensional worlds are hell. Thus, when a person dies, their soul goes to a higher or lower dimensional world.

Mysticism, Theosophy and the Astral Universe

Russian philosopher and writer Pyotr Demyanovich Uspensky (1878-1947) remarks in his essay " The Fourth Dimension” that, contrary to our beliefs, we are not three-dimensional beings at all. In his opinion, the existence of a fourth intention inevitably means one of two things: either we are four-dimensional beings, or we have only three dimensions. However, in the latter case, we would not physically exist.
For if there is a fourth dimension, and we are three-dimensional beings, this means that we do not really exist: we would be conditional, non-material beings, like points that have no length on a straight line, or straight lines that have no width on a plane , or planes, which have no volume in 3D space. Thus, we would exist only in the mind of a higher being, whether we call him God or otherwise, and all our actions, thoughts and feelings would be just a product of the imagination of this being.
If we do not believe that we are in an imaginary world that depends on a higher being and his whims, then we will have to recognize our four-dimensional reality. That is, that not only spirits or ghosts, but we ourselves are four-dimensional beings. However, only one part of us lives in the three-dimensional universe we observe, and we are aware of only that part of our being, as in Plato's cave myth.
For Hinton and Ouspensky, the fourth dimension was not only a conceptual space, but also a special knowledge of a higher reality. Their mathematical study of the fourth dimension was based on a mystical approach, which can be formulated as follows: the world is one and unknowable.
Through the mystical oneness, we can achieve universal unity. This is a superspace that unites everything (near and far, past and future, real and imaginary) in one (the One, as mystics call it; mathematicians call it hyperspace, and others call it God, the Absolute, or otherwise) cannot be represented as human-readable symbols. This explains the second part of the approach: "The One is the unknowable." But what does this approach mean? From the point of view of mystics, we can understand and realize the One in the sense of how we can feel the space around us or how we can open our hearts to feel life, beauty, love. However rationally the One is unknowable.
Rudy Rooker in The Fourth Dimension (1984) uses the following analogy to explain this. Consider an infinite set, for example the set natural numbers N - (1, 2, 3, 4, ...). Having a definition of a number, we can understand what N is, but complete knowledge, that is, a list of all natural numbers, is not available to us. Therefore, the set N is unknowable.
Theosophists also tended to be very interested in the fourth dimension, although Madame Blavatsky, the founder of the Theosophical Society herself, showed no interest in it (theosophists, like fourth-dimensional supporters such as Hinton and Ouspensky, shared a mystical belief in the One, as well as in the occult. Thus, there was a certain connection between Theosophy and Spiritualism.In addition, many Theosophists, such as the Anglican priest Charles Leadbeater (1854-1934), believed that the fourth dimension was the astral world, parallel to our visible universe, and that the idea of this world is good. explained with the help of the fourth dimension: "... the theory of the fourth dimension gives a more accurate and more complete explanation of the astral world."

SIR WILLIAM CROOKES, SPIRITUAL SCIENTIST

The English chemist, who also worked in the field of physics, was one of the most important scientists in Europe at that time. Among his works are the invention of the cathode ray tube, the study of electrical conductivity, the discovery of thallium, the development of an amalgamation process to separate gold and silver from other minerals, the invention of chemical dyes for the textile industry, and such research on the production of industrial diamonds. In addition to this, Crookes was one of the pioneers of research into psychic phenomena and also served as president of the Society for Psychical Research. In 1870 he wrote one of his most famous articles"Spiritualism in the Light of Modern Science". Crookes studied the materialization of spirits and the work of a number of well-known mediums such as Daniel Home, Cathy Fox, and Florence Cooke. The last of them is a young lady from London who knew how to summon and materialize spirits. Her most famous materialization session was to summon the spirit of Katie King, daughter of the pirate Henry Morgan. Crookes managed to take 44 photographs of Cathy, as well as feel her pulse and cut off a strand of her hair. It is said that the scientist fell in love with a ghost. All this, published in his book "Studies in the Phenomena of Spiritualism", caused a great scandal, which was further aggravated by the arrest of a woman who looked like the spirit of Katie King.

Raul Ibanez. Fourth dimension. Is our world a shadow of another Universe? (Volume 6; The World of Mathematics in 40 volumes) - M.: De Agostini, 2014

Translation

You probably know that the planets move around the sun in elliptical orbits. But why? In fact, they move in circles in four-dimensional space. And if you project these circles onto three-dimensional space, they turn into ellipses.

In the figure, the plane represents 2 of the 3 dimensions of our space. The vertical direction is the fourth dimension. The planet moves in a circle in four-dimensional space, and its "shadow" in three-dimensional space moves in an ellipse.

What is this 4th dimension? It looks like time, but it's not exactly time. This is such a special time that flows at a speed inversely proportional to the distance between the planet and the sun. And relative to this time, the planet moves at a constant speed in a circle in 4 dimensions. And in normal time, its shadow in three dimensions moves faster when it is closer to the sun.

Sounds strange - but it's just an unusual way of representing ordinary Newtonian physics. This method has been known since at least 1980 thanks to the work of the mathematical physicist Jurgen Moser. And I found out about this when I received by email a paper by Jesper Goranson called "Symmetries in the Kepler problem" (March 8, 2015).

The most interesting thing about this work is that this approach explains one interesting fact. If we take any elliptical orbit, and rotate it in 4-dimensional space, then we get another valid orbit.

Of course, it is possible to rotate an elliptical orbit around the sun and in ordinary space, obtaining a valid orbit. The interesting thing is that this can be done in 4-dimensional space, for example, by narrowing or expanding the ellipse.

In general, any elliptical orbit can be turned into any other. All orbits with the same energy are circular orbits on the same sphere in 4-dimensional space.

Kepler's problem

Let's say we have a particle that moves according to the inverse square law. Its equation of motion will be

Where r- position as a function of time, r is the distance from the center, m is the mass, and k determines the force. From this we can derive the law of conservation of energy

For some constant E that depends on the orbit but does not change with time. If this force is an attraction, then k > 0, and on an elliptical orbit E< 0. Будем звать частицу планетой. Планета двигается вокруг солнца, которое настолько тяжело, что его колебаниями можно пренебречь.

We will study orbits with one energy E. Therefore, the units of mass, length and time can be taken as any. Let's put

M=1, k=1, E=-1/2

This will save us from unnecessary letters. Now the equation of motion looks like

And the conservation law says

Now, following Moser's idea, let's move on from ordinary time to new. Let's call it s and require that

Such time passes more slowly as you move away from the sun. Therefore, the speed of the planet with distance from the sun increases. This compensates for the planets' tendency to move more slowly as they move away from the sun in normal time.

Now let's rewrite the conservation law using the new time. Since I used a dot for derivatives with respect to ordinary time, let's use a prime for derivatives with respect to s. Then for example:

Using such a derivative, Goranson shows that the conservation of energy can be written as

And this is nothing but the equation of a four-dimensional sphere. The proof will come later. Now let's talk about what this means to us. To do this, we need to combine the usual time coordinate t and spatial coordinates (x, y, z). Dot

Moves in 4D space as the s parameter changes. That is, the speed of this point, namely

Moves in a 4D sphere. It is a sphere of radius 1 centered at a point

Additional calculations show other interesting facts:

T""" = -(t" - 1)

These are the usual harmonic oscillator equations, but with an additional derivative. The proof will be later, but for now let's think about what this means. In words, this can be described as follows: 4-dimensional speed v makes simple harmonic vibrations around the point (1,0,0,0).

But since v at the same time remains on the sphere centered at this point, then we can conclude that v moves with constant speed in a circle on this sphere. And this implies that the average value of the spatial components of the 4-dimensional velocity is 0, and the average t is 1.

The first part is clear: our planet, on average, does not fly away from the Sun, so its average speed is zero. The second part is more complicated: the usual time t moves forward with an average speed of 1 relative to the new time s, but the rate of its change fluctuates sinusoidally.

By integrating both parts

We'll get

a. The equation says that position r oscillates harmonically around a point a. Because the a does not change with time, it is a conserved quantity. This is called the Laplace-Runge-Lenz vector.

Often people start with the inverse square law, show that angular momentum and the Laplace-Runge-Lenz vector are conserved, and use these conserved quantities and Noether's theorem to show the existence of a 6-dimensional symmetry group. For negative energy solutions, this turns into a group of rotations in 4 dimensions, SO(4). With a little more work, you can see how the Kepler problem is paired with a harmonic oscillator in 4 dimensions. This is done through time reparametrization.

I liked Gorasnon's approach better because it starts with time reparametrization. This makes it possible to effectively show that the elliptical orbit of a planet is a projection of a circular orbit in four-dimensional space onto three-dimensional space. Thus, 4-dimensional rotational symmetry becomes apparent.

Goranson extends this approach to the inverse square law in n-dimensional space. It turns out that elliptical orbits in n dimensions are projections of circular orbits from n + 1 dimensions.

He also applies this approach to positive-energy orbits, which are hyperbolas, and to zero-energy orbits (parabolas). Hyperbolas get the symmetry of the Lorentz groups, and parabolas get the symmetry of the Euclidean groups. This is a known fact, but it is remarkable how easy it is to derive with the new approach.

Mathematical details

Because of the abundance of equations, I will put boxes around the important equations. The basic equations are conservation of energy, force, and change of variables, which give:

Let's start with conservation of energy:

Then we use

To obtain

A little algebra - and we get

This shows that the 4-dimensional speed

Remains on a sphere of unit radius centered at (1,0,0,0).

The next step is to take the equation of motion

And rewrite it using strokes (derivatives of s), not dots (derivatives of t). Starting with

And we differentiate to get

Now we use another equation for

And we get

Now it would be nice to get a formula for r"". Let's count first

And then we differentiate

Connecting the formula for r", something will be reduced, and we get

Recall that the conservation law says

And we know that t" = r. Therefore,

We get

Since t" = r, it turns out

As we need.

Now we get a similar formula for r""". Let's start with

And differentiate

Connect the formulas for r"" and r""". Something shrinks and remains

We integrate both parts and get

For some constant vector a. It means that r oscillates harmonically about a. Interestingly, the vector r and its norm r oscillate harmonically.

The quantum version of a planetary orbit is a hydrogen atom. Everything that we have calculated can be used in the quantum version. See Greg Egan for details.