multidimensional space
a space that has more than three dimensions (dimension). Real space is three-dimensional. Through each of its points it is possible to draw three mutually perpendicular lines, but it is no longer possible to draw four. If we take these three straight lines as coordinate axes, then the position of each point in space is determined by specifying three real numbers - its rectangular coordinates. Generalizing this position, we call n-dimensional Euclidean space a collection of all possible systems of n numbers of “points” of this space.
Multidimensional space
a space that has more than three dimensions (dimension). Ordinary Euclidean space, studied in elementary geometry, is three-dimensional; planes are ≈ two-dimensional, straight lines are ≈ one-dimensional. The emergence of the concept of geometry is associated with the process of generalization of the subject of geometry itself. At the heart of this process is the discovery of relationships and forms, similar to spatial ones, for numerous classes of mathematical objects (often not having a geometric nature). During this process, the idea of abstract mathematical space gradually crystallized as a system of elements of any nature, between which relationships were established that were similar to certain important relationships between points in ordinary space. This idea found its most general expression in concepts such as topological space and, in particular, metric space.
The simplest space spaces are n-dimensional Euclidean spaces, where n can be any natural number. Just as the position of a point in ordinary Euclidean space is determined by specifying its three rectangular coordinates, a “point” in n-dimensional Euclidean space is specified by n “coordinates” x1, x2, ..., xn (which can take any real value); the distance r between two points M(x1, x2, ..., xn) and M"(y1, y2, ..., yn) is determined by the formula
similar to the formula for the distance between two points in ordinary Euclidean space. While maintaining the same analogy, other geometric concepts are generalized to the case of n-dimensional space. Thus, in the magnetic field, not only two-dimensional planes are considered, but also k-dimensional planes (k< n), которые, как и в обычном евклидовом пространстве, определяются линейными уравнениями (или системами таких уравнений).
The concept of n-dimensional Euclidean space has important applications in the theory of functions of many variables, allowing one to treat a function of n variables as a function of a point in this space and thereby apply geometric concepts and methods to the study of functions of any number of variables (not just one, two or three). This was the main incentive to formalize the concept of n-dimensional Euclidean space.
Other spatial concepts also play an important role. Thus, when expounding the physical principle of relativity, four-dimensional space is used, the elements of which are the so-called. "world points". At the same time, the concept of a “world point” (as opposed to a point in ordinary space) combines a certain position in space with a certain position in time (that’s why “world points” are specified by four coordinates instead of three). The square of the “distance” between the “world points” М▓(х▓, y▓, z▓, t▓) and М▓▓(х▓▓, y▓▓, z▓▓, t▓▓) (where the first three “ coordinates" ≈ spatial, and the fourth ≈ temporal) it is natural to consider here the expression
(M▓ M▓▓)2 = (x▓ - x▓▓)2 + (y▓ ≈ y▓▓)2 + (z▓ ≈ z▓▓)2 ≈ c2(t▓ ≈ t▓▓)2,
where c ≈ speed of light. The negativity of the last term makes this space “pseudo-Euclidean”.
In general, an n-dimensional space is a topological space that at each point has dimension n. In the most important cases, this means that every point has a neighborhood homeomorphic to an open ball of n-dimensional Euclidean space.
Read more about the development of the concept of mechanical structure, the geometry of mechanical structure, as well as lit. see Art. Geometry.
UDC 115
© 2006 ., A.V. Korotkov, V.S. Churakov
Multidimensional space concepts
and time (space-time)
Speaking about seven-dimensional space, we should clarify why we are talking about seven-dimensional space and not about n -dimensional space, multidimensional space. The fact is that the three-dimensional Hamilton-Grassmann vector calculus gives only three conservation laws, but in the physics of elementary particles new conservation laws have been discovered for the baryon number, lepton number, parity, and a whole series of conservation laws. It became clear (at least in the field of elementary particle physics) that physics must be significantly refined, expanded to a multidimensional version. The question arises: what dimension should we use – 4, 5, 6, 8, 129 or 1000001? This is not an idle question. In addition, even if the dimension of physical space is clarified, which is practically impossible to obtain from experiment, the question will arise: what kind of mathematics should be used to describe phenomena in this space of this dimension, which is not equal to three?
Therefore, one should proceed, first of all, from number theory. Pythagoras also noted that everything that exists is a number, i.e. physics, theoretical physics is essentially a theory of numbers, a theory of three-dimensional vector numbers. Field theory is completely and entirely built on three-dimensional vector calculus. Quantum mechanics including. All branches of theoretical physics use the apparatus of three-dimensional vector algebra of three-dimensional vector calculus. Attempts to expand space lead to an analysis, therefore, of the very concept of number as such.
A one-dimensional vector number is a space on a ruler, a space of numbers on a ruler. A three-dimensional vector number, a three-dimensional vector space, is now well understood by all of us since the time of Hamilton, but not before that. A multidimensional vector space defined by linear vector algebra, as required by three-dimensional vector calculus, can be obtained by extending three-dimensional vector spaces, three-dimensional vector algebra. Thus, we must introduce the vector and scalar products of two vectors in a linear vector space. This, in fact, is the main task of the theory of multidimensional numbers - to introduce and define the scalar, first and second vector product of two vectors. There are few approaches to this definition. In general terms, the definition of these concepts gives nothing but confusion.
We should proceed from the principles that Hamilton used when constructing three-dimensional vector calculus. He first constructed a quaternion algebra by expanding complex numbers, and then from it he obtained the scalar vector product of two vectors in a three-dimensional vector space, i.e. in the space of vector quaternions. If you follow this path, you should expand, double the quaternion system to the octanion system, which Cayley did in 1844, but use further transformations the same as Hamilton used to obtain a three-dimensional vector number and a four-dimensional quaternion number. If we follow this path, then the only possible algebra that can be obtained from quaternion algebra is a seven-dimensional vector algebra with a scalar, Euclidean character and the vector product of two vectors.
That is, the answer to two questions is immediately given: what dimension should space be? And this is exactly seven, not four, not five, not six. And secondly, the scalar and vector product of two vectors is strictly given. This allows you to expand the algebra, i.e. obtain the properties of algebra arising from these two fundamental concepts, which was once put into practice. Thus, we obtain a seven-dimensional Euclidean vector algebra with seven vectors of an orthogonal coordinate system, possibly orthogonal, in which a seven-dimensional vector is constructed. A whole series of new concepts immediately arise, completely new to algebra, such as: the vector product of not only two vectors, but also three, four, five, six vectors. These are invariant quantities, which in turn give certain conservation laws. Among scalar quantities, invariant quantities also appear, as functions not only of two vectors of the scalar product of two vectors, but also as functions of a larger number of vectors. These are mixed products of three vectors, four vectors, seven vectors. At least these functions have been found, their properties have been clarified, and these functions provide invariant concepts such as conservation laws - laws of conservation of these quantities. That is, it becomes possible to obtain completely new laws of conservation of quantities, physical quantities, when using seven-dimensional vector algebra instead of three-dimensional algebra. The three-dimensional laws of conservation of energy, momentum and angular momentum follow from this algebra simply as a special case. They take place, are preserved, do not disappear anywhere, they are fundamental, just like the new conservation laws that appear when considering seven-dimensional spaces.
Speaking about multidimensionality in general, one should clarify: is it not possible to construct algebras of higher dimension—vector algebra of higher dimension? The answer is - you can! But the properties of these algebras are completely different, although they include three-dimensional seven-dimensional algebras as a special case, as subalgebras. Their properties change. For example, the well-known law for the double vector product will be formulated completely differently. This will no longer be Maltsev’s algebra, it will be fifteen dimensions - a completely different algebra, and for thirty-one dimensions the question has not been studied at all. What can we say about 15 or 31-dimensional space, when the concept of seven-dimensional space has not yet gained a strong fundamental position in the minds of scientists. First of all, you need to be based on the analysis of the seven-dimensional option as the next option after the three-dimensional vector calculus. It should be noted that vector algebra inherently does not use the concept of division, i.e. even three-dimensional algebra is algebra without division - it is impossible to associate a vector with an inverse vector, or to find its opposite, i.e. find the inverse vector. And in vector algebra there is no concept of a unit as such, a scalar unit that could be divided by its reciprocal number, obtaining a vector. Therefore, this removes the restrictions in terms of the fact that we have only four division algebras - four-dimensional, two-dimensional, one-dimensional, eight-dimensional. Further expansion would simply be impossible. But since vector algebras are algebras without division, one can try to go further along this path, constructing multidimensional algebras.
The second aspect is that since we are working with algebras without division, we can use algebras that can be obtained by expanding real numbers without using the division procedure. In the two-dimensional version these are double and dual numbers, in the four-dimensional version - pseudoquaternions and dual quaternions, in the eight-dimensional version - pseudooctanions and dual octanions. From them, using the same Hamilton procedure, one can obtain three-dimensional pseudo-Euclidean index 2 and seven-dimensional pseudo-Euclidean index 4 vector algebras. Again the question is about the three-dimensional and seven-dimensional version. It should be noted that a dual extension is also possible, but a dual extension, in turn, is characterized by the fact that it does not have an isomorphic transformation group. Pseudo-Euclidean algebras three-dimensional and seven-dimensional, as it turns out, have groups that can be described by group properties of transformations of these vector quantities. At the same time, dual quantities are transformed into each other using matrices, singular square matrices, i.e. These matrices have a determinant that is not equal to zero. And this sharply limits the possibilities of such algebras for application. However, they can be built. But transformation groups are degenerate. This concept leads, therefore, to the expansion of the concept of a real number of one-dimensional vector quantities, three-dimensional vector quantities, dual Euclidean, pseudo-Euclidean and proper Euclidean and seven-dimensional vector quantities - proper Euclidean, dual Euclidean, pseudo-Euclidean.
The mathematics of such spaces is already defined, and there are no problems with using transformations and expressions in these spatial relationships. The only slightly more complex option is seven-dimensionality rather than three-dimensionality. But computer technology makes it possible to carry out these transformations without problems. Thus, we fix the concepts of one-dimensional, three-dimensional and seven-dimensional space, Euclidean proper, as the main of these spaces, pseudo-Euclidean, as the existing possibility of non-degenerate spatial transformations with the corresponding group of pseudo-Euclidean transformations and dual Euclidean ones. The result is a set of nine vector algebras that can be considered for physical applications. At least six quantities proper Euclidean and pseudo-Euclidean, probably a little inaccurately, not nine, but seven - and as a result, not six, but four quantities, five quantities, five algebras will take place for possible physical applications. So, it bears repeating: the basis for now, the main spatial transformation of spatial vector algebra is seven-dimensional Euclidean algebra. This is the basis. If you study, master, and apply this foundation, it will be a lot. And it will allow you to quickly and easily master the basic vector transformations of vector algebra.
Seven-dimensional space is characterized by the fact that all spatial directions are exactly the same, i.e. space is isotropic in its properties. At the same time, we have not only the concepts of vectors, but also the concepts of changes in vectors, the position of at least vectors in space. Consequently, it is necessary to evaluate the nature of the change in these vector positions in space - and this necessarily leads to the use of the concept of time as a scalar quantity by which vector quantities can be differentiated. Therefore, a more correct concept would probably be to consider not just seven-dimensional space, but eight-dimensional space - time. Seven completely identical spatial coordinates plus a time coordinate as a scalar component. That is, consider an eight-dimensional radius vector Ctr, where r is a seven-component quantity, and t – time is a one-component scalar quantity. This was done in exactly the same way in the four-dimensional Minkowski space-time and therefore does not cause any complaints or negative considerations and emotions. Eight-dimensional space-time, just like the special theory of relativity, connects time with spatial relationships. There is a relativity between the concepts of spatial quantities and temporal quantities. The same Lorentz transformations take place if we do not use YZ , equal to zero, and all six other components, except the first, equal to zero. That is, the particular theory of relativity of four-dimensional Minkowski space-time is simply a special case of the transformation of eight-dimensional space-time. That, in fact, is probably all that should be noted. The only thing worth adding or repeating is that in seven-dimensional space completely new laws of conservation of quantities take place, and in eight-dimensional space-time these quantities appear in the same way as conserved fundamental quantities and variants during the transition from one system of eight-dimensional space-time to another - another reference system.
Anything else worth noting? When using the actual Euclidean seven-dimensional space, an eight-dimensional space-time of index 1 is obtained, in fact, or some authors, on the contrary, take three negative components of the radius vector, so we can talk about index 3, because the square of the speed, or the square of the radius vector is determined the sum of the squares of the components in the Euclidean space proper. In seven-dimensional space, this tendency is practically preserved entirely, if we use the actual Euclidean vector algebra. However, a seven-dimensional space can also be constructed using a seven-dimensional pseudo-Euclidean vector algebra of index 4, and this suggests that the square of the radius-vector interval, the square of the radius-vector, or better yet, the square of the modulus of the radius-vector can be not only positive, but also zero and even a negative value, the square of the modulus of the radius vector of seven-dimensional pseudo-Euclidean space. In exactly the same way, we can talk about the square of any vector, in particular the velocity vector. Therefore, the concept of speed of a pseudo-Euclidean seven-dimensional vector algebra is completely different than in the seven-dimensional Euclidean space proper. And this leads to serious changes in the physical plane, if you build a physical theory on the basis of such algebras. In mathematical terms, there are no complaints, and algebra can be the foundation for building multidimensional physics and, without problems, multidimensional physics is being built. The perception of these quantities is more difficult. That is, speed is a quantity, in this case the speed of light, as a fundamental quantity, can only occur as a concept of the speed of propagation of electromagnetic waves. Based on eight-dimensional pseudo-Euclidean algebra using seven-dimensional pseudo-Euclidean algebra, the speed can be not only a positive value, but also negative and zero.
This, in turn, requires additional consideration of such physical spaces, awareness of their presence in the real world and an attempt to explain the theory of fields not only electromagnetic, but others, in particular gravitational, weak, strong. The currently available vector multidimensional algebras allow us to make a deeper analysis than the presence of only three-dimensional vector algebra and, moreover, only the actual Euclidean Hamilton–Grassmann vector algebra.
Bibliography
1. Gott, V.S. Space and time of the microworld / V.S. Gott. – M.: Publishing house “Knowledge”, 1964. – 40 p.
2. Korotkov, A.V. Elements of seven-dimensional vector calculus. Algebra. Geometry. Field theory / A.V. Korotkov. – Novocherkassk: Nabla, 1996. – 244 p.
3. Rumer, Y.B. Principles of conservation and properties of space and time / Yu.B. Rumer // Space, time, movement. – M.: Publishing house “Nauka”, 1971. – P. 107-125.
Multidimensional spaces - myth or reality? It is impossible for most of us, or perhaps all of us, to imagine a world consisting of more than three spatial dimensions. Is it correct to say that such a world cannot exist? Or is it simply that the human mind is incapable of imagining extra dimensions—dimensions that might be as real as other things we can't see?
We quite often hear something like “three-dimensional space”, or “multidimensional space”, or “four-dimensional space”. You may know that we live in four-dimensional spacetime. What does this mean and why is it interesting, why do mathematicians and not only mathematicians study such spaces?
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Ilya Shchurov- Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Higher Mathematics at the National Research University Higher School of Economics.
Jason Hise- Physics programmer at Ready at Dawn Studios, 4D geometry enthusiast. Author of the animated models presented in this article.
Ashgrowen- Pikabushnik, who illustrated the construction of a tesseract and a hypercube in this article. |
Let's start simple - let's start with one-dimensional space. Let's imagine that we have a city that is located along a road, and in this city there is only one street. Then we can encode each house on this street with one number - the house has a number, and this number uniquely determines which house we mean. People who live in such a city can be considered to live in such a one-dimensional space. Living in one-dimensional space is quite boring, and people usually do not live in one-dimensional space.
For example, if we are talking about cities, then we can move from one-dimensional space to two-dimensional space. An example of a two-dimensional space is a plane, and if we continue our analogy with cities, then this is a city in which streets can be laid out, say, perpendicular to each other, as is done in New York, in the center of New York. There is a “street” and an avenue, each of which has its own number, and you can specify a location on the plane, specify two numbers. Again, we all know the Cartesian coordinate system, familiar from school - each point is specified by two numbers. That's an example two-dimensional space.
But if we are talking about a city like the center of New York, then in fact it is a three-dimensional space, because it is not enough for you to specify, for example, a specific house, even if you define it by the intersection of some “street” and some avenue, - you will also need to specify the floor on which the apartment you need is located. This will give you a third dimension - height. You can do it three dimensional space, in which each point is specified by three numbers.
Question: what is it four-dimensional space? It is not so easy to imagine it, but you can think of it as a space in which each point is given by four numbers. In fact, you and I really live in four-dimensional space-time, because the events of our lives are encoded by just four numbers - in addition to position in space, there is also time. For example, if you are making a date, then you can do it like this: you can specify three numbers that will correspond to a point in space, and be sure to indicate the time, which is usually given in hours, minutes, seconds, but could be encoded in one number . For example, the number of seconds that have passed since a certain date is also one number. This results in four-dimensional space-time.
It is not very easy to imagine the geometry of this four-dimensional space-time. For example, you and I are accustomed to the fact that in our ordinary three-dimensional space two planes can intersect in a straight line or be parallel. But it does not happen that two planes intersect at one point. Two lines can intersect at one point, but on a plane they cannot in three-dimensional space. And in four-dimensional space, two planes can and most often intersect at one point. You can imagine, although this is quite difficult, a space of greater dimension. In fact, mathematicians, when working with high-dimensional spaces, most often say simply: let’s say a five-dimensional space is a space in which a point is specified by five numbers, five coordinates. Of course, mathematicians have developed various methods that allow us to understand something about the geometry of such a space.
Why is it important? Why were such spaces needed? Firstly, four-dimensional space is important to us because it is used in physics, because we live in it. Why do we need spaces of higher dimensions? Let's imagine that we are studying some objects that have a large number of parameters. For example, we study countries, and each country has a territory, population, gross domestic product, number of cities, some coefficients, indices, something like that. We can imagine each country as a single point in some space of a fairly high dimension. And it turns out that, from a mathematical point of view, this is the correct way to think about it.
In particular, the transition to the geometry of multidimensional space makes it possible to analyze various complex objects with a large number of parameters.
In order to study such objects, methods developed in a science called linear algebra are used. Although it is algebra, it is actually the science of the geometry of multidimensional spaces. Of course, since it is quite difficult to imagine them, mathematicians use formulas in order to study such spaces.
It is quite difficult to imagine a four-, five- or six-dimensional space, but mathematicians are not afraid of difficulties, and even one-hundred-dimensional spaces are not enough for them. Mathematicians came up with infinite-dimensional space - a space containing an infinite number of dimensions. An example of such a space is the space of all possible functions defined on a segment or line.
It turns out that the methods that were developed for finite-dimensional spaces carry over in many ways to cases that are extremely complex in terms of simply trying to represent them all.
Linear algebra has numerous applications not only in mathematics, but also in a variety of sciences, from physics to, for example, economics or political science. In particular, linear algebra is the basis for multivariate statistics, which is precisely used to isolate relationships between various parameters in some data sets. In particular, the now popular term Big Data is often associated with solving data processing problems that are represented by a large number of points in a space of some finite dimension. Most often, such problems can be reformulated and reasonably perceived in geometric terms.
From school years, mathematics is divided into algebra and geometry. But in fact, if we think about how modern mathematics works, we will understand that those problems that are now being solved, in particular, using linear algebra methods, are in fact a very distant continuation of those problems that were thought about many thousands of years ago, for example Pythagoras or Euclid, developing the same school geometry that is now in any school textbook. It is surprising that the task of analyzing big data turns out to be in some sense a descendant of the seemingly completely meaningless - at least from a practical point of view - exercises of the ancient Greeks in drawing lines or circles on a plane or mentally drawing lines or planes in three-dimensional space.
What is four-dimensional space (“4D”)?
Tesseract - four-dimensional cube
Everyone knows the abbreviation 3D, meaning “three-dimensional” ( letter D - from the word dimension - dimension ). For example, when choosing a film marked 3D in a cinema, we know for sure: to watch it we will have to wear special glasses, but the picture will not be flat, but three-dimensional. What is 4D? Does “four-dimensional space” exist in reality? And is it possible to go out "Fourth dimension"?
To answer these questions, let's start with the simplest geometric object - a point. The point is zero-dimensional. It has no length, no width, no height.
Now let's move the point along a straight line some distance. Let's say that our point is the tip of a pencil; when we moved it, it drew a line. A segment has a length and no other dimensions: it is one-dimensional. The segment “lives” on a straight line; a straight line is a one-dimensional space.
Tesseract - four-dimensional cube
Now let’s take a segment and try to move it the way we moved a point before. You can imagine that our segment is the base of a wide and very thin brush. If we go beyond the line and move in a perpendicular direction, we will get a rectangle. A rectangle has two dimensions - width and height. A rectangle lies in a certain plane. A plane is a two-dimensional space (2D), on it you can introduce a two-dimensional coordinate system - each point will correspond to a pair of numbers. (For example, the Cartesian coordinate system on a blackboard or latitude and longitude on a geographic map.)
If you move a rectangle in a direction perpendicular to the plane in which it lies, you get a “brick” (a rectangular parallelepiped) - a three-dimensional object that has length, width and height; it is located in three-dimensional space, the same in which you and I live. Therefore, we have a good idea of what three-dimensional objects look like. But if we lived in two-dimensional space - on a plane - we would have to strain our imagination quite a bit to imagine how we could move the rectangle so that it would come out of the plane in which we live.
Tesseract - four-dimensional cube
It is also quite difficult for us to imagine four-dimensional space, although it is very easy to describe mathematically. Three-dimensional space is a space in which the position of a point is given by three numbers (for example, the position of an airplane is given by longitude, latitude and altitude above sea level). In four-dimensional space, a point corresponds to four coordinate numbers. A “four-dimensional brick” is obtained by shifting an ordinary brick along some direction that does not lie in our three-dimensional space; it has four dimensions.
In fact, we encounter four-dimensional space every day: for example, when making a date, we indicate not only the meeting place (it can be specified by three numbers), but also the time (it can be specified by a single number, for example, the number of seconds that have passed since a certain date). If you look at a real brick, it has not only length, width and height, but also an extension in time - from the moment of creation to the moment of destruction.
A physicist will say that we live not just in space, but in space-time; the mathematician will add that it is four-dimensional. So the fourth dimension is closer than it seems.
Representation of other dimensions
From 2D to 3D
An early attempt to explain the concept of extra dimensions appeared in 1884 with the publication of the Flat Earth novel Edwin A. Abbott "Flatland: A Romance of Many Dimensions"". The action in the novel takes place in a flat world called “Flatland”, and the story is told from the perspective of an inhabitant of this world - a square. One day in a dream, a square finds itself in a one-dimensional world - Lineland, whose inhabitants (triangles and other two-dimensional objects are represented as lines) and tries to explain to the ruler of this world the existence of the 2nd dimension, however, comes to the conclusion that it is impossible to force him to go beyond framework of thinking and imagining only straight lines.
The square describes his world as a plane populated by lines, circles, squares, triangles and pentagons.
One day a ball appears in front of the square, but he cannot comprehend its essence, since the square in its world can only see a slice of the sphere, only the shape of a two-dimensional circle.
The sphere is trying to explain to the square the structure of the three-dimensional world, but the square only understands the concepts of “up/down” and “left/right”; it is not able to comprehend the concepts of “forward/backward”.
It is only after the sphere pulls the square out of his two-dimensional world and into its three-dimensional world that he finally understands the concept of three dimensions. From this new perspective, the square becomes able to see the forms of its compatriots.
Square, armed with his new knowledge, begins to realize the possibility of a fourth dimension. He also comes to the conclusion that the number of spatial dimensions cannot be limited. In an effort to convince the sphere of this possibility, the square uses the same logic that the sphere uses to argue for the existence of three dimensions. But now, of the two, the sphere becomes the “myopic” one, which cannot understand this and does not accept the arguments and arguments of the square - just as most of us “spheres” today do not accept the idea of extra dimensions.
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Review of the book Flatland Taking into account the exclusivity of both the genre, which with some imagination and the existence of other representatives of it, could be called a mathematical novel, and the book itself, I don’t want to criticize it too much. However, the only thing that deserves praise here is the unusual presentation, which is close in spirit to the works of Lewis Carroll, but, unlike him, has much fewer points of contact with real life. This book, as correctly noted in the preface to the publication, is not similar to any popularization; however, it is not entirely clear to the reader why it is compared with popularizations, because, although mathematical truths are certainly touched upon in it, some It is impossible to definitely consider the book as a popularization. Here's why: Here is a unique example of combining artistic imagination with mathematical ideas. And to a fan of mathematics who loves to read, the idea initially seems wonderful: like mathematical postulates, introduce a number of abstract objects into consideration, endow them with certain properties, set the rules of the game in the described space, and then, again imitating the thoughts of a researcher observing the interactions of these speculative objects, monitor their transformation. But, since the book is still artistic, there is no place for scientific willpower here, therefore, for the self-sufficiency of the world presented for everyone to see, the objects here are endowed with consciousness and motivation for any interactions with each other, after which they are transferred to the previously abstract world, divorced from everyday life pure ideas bring social interactions with a whole bunch of problems that always accompany any relationship. All sorts of tensions that arise in the book on social grounds, in the opinion of the viewer, are completely unnecessary in the book: they are practically not disclosed and cannot be taken seriously, and at the same time distract the reader from truly those things for which the book was written. Even taking into account the assurances of both authors about the leisurely narrative, supposedly more comfortable for the reader when acquiring any knowledge (this is where the comparison with popularizations is made), the pace of the narrative seemed to the viewer to be extremely drawn out and slow, and the repetition of the same explanation several times the same words made me doubt that the narrator adequately assessed his mental abilities. And ultimately it is unclear who this book is for. For people unaccustomed to mathematics, a description of a generally interesting phenomenon in such a free form is unlikely to bring pleasure, but for those familiar with mathematics, it will be much more pleasant to pick up a high-quality popularization, where the greatness and beauty of mathematics is not diluted with flat fairy tales. |
From 3D to 4D
It's difficult for us to accept this idea because when we try to imagine even one additional spatial dimension, we hit a brick wall of understanding. It seems that our mind cannot go beyond these boundaries.
Imagine, for example, that you are in the center of an empty sphere. The distance between you and each point on the surface of the sphere is equal. Now try moving in a direction that allows you to move away from all points on the surface of the sphere while maintaining equidistance. You won't be able to do this..
A Flatlander would face the same problem if he were in the center of the circle. In his two-dimensional world, he cannot be in the center of a circle and move in a direction that allows him to remain equidistant to every point on the circle's circumference, unless he moves into the third dimension. Alas, we do not have a guide to four-dimensional space like in Abbott's novel to show us the way to 4D.
What is a hypercube? Construction of a tesseract
Types of hypercubes and their names1. Point - zero dimension 2. A segment is a one-dimensional space
3. Square - two-dimensional space (2D)
4. Cube - three-dimensional space (3D)
5. Tesseract - four-dimensional space (4D)
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Hypercube is a general name for a cube in a derived number of dimensions. There are ten dimensions in total, plus a point (zero dimension).
Accordingly, there are eleven types of hypercube. Let's consider the construction of a tesseract - a fourth-dimensional hypercube:
First, let's build point A (Fig. 1):
Afterwards, we connect it to point B. We obtain vector AB (Fig. 2):
Let's construct a vector parallel to the vector AB and call it CD. By connecting the beginnings and ends of the vectors, we obtain a square ABDC (Fig. 3):
Now let's construct another square A1B1D1C1, which lies in a parallel plane. By connecting the points in a similar way, we get a cube (Fig. 4):
We have a cube. Imagine that the position of a cube in three-dimensional space has changed over time. Let's fix its new location (Fig. 5):
And now, we draw vectors that connect the locations of points in the past and in the present. We get a tesseract (Fig. 6):
Rice. 6 Tesseract (construction)
The rest of the hypercubes are constructed in a similar way; of course, the meaning of the space in which the hypercube is located is taken into account.
How about 10D?
In 1919, a Polish mathematician Theodore Kaluza suggested that the existence of a fourth spatial dimension could link general relativity and electromagnetic theory. An idea later improved by a Swedish mathematician Oscar Klein, was that space consisted of both “expanded” dimensions and “collapsed” dimensions. The extended dimensions are the three spatial dimensions we are familiar with, and the collapsed dimension is found deep within the extended dimensions. Experiments later showed that Kaluza and Klein's curled-up dimension did not unify general relativity and electromagnetic theory as originally thought, but decades later string theorists found the idea useful, even necessary.
The mathematics used in superstring theory requires at least 10 dimensions. That is, for the equations describing superstring theory and in order to connect the general theory of relativity with quantum mechanics, to explain the nature of particles, to unify forces, etc., it is necessary to use additional dimensions. These dimensions, according to string theorists, are wrapped in the folded space originally described by Kaluza and Klein.
Circles represent an additional spatial dimension folded into every point of our familiar three-dimensional space. │ WGBH/NOVA
To expand the twisted space to include these added dimensions, imagine replacing the Kaluza-Klein circles with spheres. Instead of one added dimension, we have two if we consider only the surfaces of the spheres and three if we consider the space inside the sphere. This resulted in only six measurements. So where are the others that superstring theory requires?
It turns out that before superstring theory appeared, two mathematicians Eugenio Calabi from the University of Pennsylvania and Shin-Tung Yau from Harvard University described six-dimensional geometric shapes. If we replace the spheres in twisted space with these Calabi-Yau shapes, we get 10 dimensions: three spatial, as well as six-dimensional Calabi-Yau figures.
Six-dimensional Calabi-Yau forms may account for the additional dimensions required by superstring theory. │ WGBH / NOVA
String theorists are betting that extra dimensions do exist. In fact, the equations that describe superstring theory assume a universe with at least 10 dimensions. But even physicists who think about extra spatial dimensions all the time have a hard time describing what they might look like, or how people might come closer to understanding them.
If superstring theory is proven and the idea of a world of 10 or more dimensions is confirmed, will there ever be an explanation or visual representation of higher dimensions that the human mind can comprehend? The answer to this question may forever be negative, unless some 4D life form “pulls” us out of our 3D world and allows us to see the world from its point of view.
The space of the universe is truly multidimensional. Just as sunlight coexists with pure water in the same space, freely passing through the water and at the same time interacting with it little, just as radio waves of different frequencies freely exist in the depths of space outside and inside our bodies - like this Everywhere in multidimensional depth, inside and outside any solid, liquid or gaseous objects, there are other worlds - the abodes of spirits and God.
The scale of multidimensionality is a special scale of energy states that differ as fundamental ranges. When studying this scale, the vector of attention should be directed not up, down, or in any other direction, but deep down. The layers of multidimensional space (in Greek they are called eons, in Sanskrit - lokas) differ from one another in their degree subtleties-rudeness.
The layer of the most subtle energies is God in the aspect of the Creator. It looks like the purest infinite in extent Light, similar to the light of the morning sun - gentle and warm. There are no forms in Him. Once in Him, all forms immediately dissolve.
In different earthly languages, people call Him differently: God the Father, Jehovah, Allah, Ishvara, Primordial Consciousness, Tao, etc. He is the God of the Jewish prophets, and Jesus Christ, and Muhammad, and the faithful of China, India and other countries where correct ideas about Him exist.
And only human ignorance and intellectual primitivism lead to the opinion that since the “names” are different, then the Gods are different...
It is from the Abode of the Creator, from this first, primordial eon, that each new “island” of multidimensional Creation is created. The building material for the formation of solid matter is, first of all, protomatter (protoprakriti, bhutakasha).
This layer is seen from the inside - when penetrating into it - as an endless space filled with Tender Peace and lacking bright luminosity. It is like the state of a warm and quiet gentle southern night with many stars.
It is extremely important that the Creator and the eons of Akasha are located, relative to the entire Creation, as if on the other side of the “mirror,” in “Through the Looking Glass.” Yes, just as our ordinary mirror has a light and dark side, so it is there, in the multidimensional depths of the universal Ocean.
It is this phenomenon that physicists guess about, trying to look through their theoretical calculations into the “Through the Looking Glass” from the world of matter; they call the energy of the Akashic eons... “anti-energy”, “anti-matter”...
... In order to create another material “island” in the endless Ocean of the universe, the Creator first forms a local zone of increased gravity (attraction) in it. This phenomenon is known in astronomy as “black holes”. This is how protoprakriti is drawn into the eon and transformed into elementary particles, various material cosmic “garbage” - dead planets, meteorites, cosmic dust.
Then the Holy Spirits form a compaction from this material. The gradually increasing superpressure and superheating in this clot provoke nuclear fusion reactions; This is how all the elements of the periodic table are formed, molecules are formed, including organic ones. Clots of the protopurusha begin to incarnate into the latter. This is how the parallel evolution of organic bodies - and the souls embodied in them - begins. Biologists have studied the evolution of organic bodies quite well; we just need to take into account the leading role of God in this process.
Our - human - task here is to, having developed ourselves - as a soul, consciousness - to a sufficient extent, to pass the path from Creation to the Creator, refining ourselves as consciousness - in order to merge into Him, enriching Him with ourselves.
This was God’s “plan” when He created our Earth. This is the meaning of our lives.
It is important for us to understand that we are not self-existent, we do not have the right or any grounds to claim our own egocentrism, or a sense of our own special “significance.” For only the Creator is self-existent. And He started this whole Creation with us not at all for our sake, but for His own sake, for the sake of His own Evolution.
Hence the quality of our destinies: if we develop correctly, everything goes well in our lives, if wrong, He shows us this through our pain and failures.
... After a huge amount of time, by our earthly standards, billions of human bodies and even more souls of different ages and different qualities appeared on our planet. Of these, those who achieve Perfection merge into the Creator and no longer incarnate (except as Messiahs, Avatars). The rest are incarnated again and again - until the time of existence of this material “island” ends. When it is destroyed, matter and those souls that have not become close to the Creator are destroyed to the state of akash, forming building material for future “islands” and life on them.
...At the opposite end of the scale from the Creator subtleties - rudeness there is a devilish eon - a world of rough black energies, terrible in emotional state and “sticky” like oil. How to get there - we’ll talk about this separately.
But there is also an abode for the righteous - paradise.
Each person, having disincarnated, finds himself in the eon that he deserved during his life in the body on Earth. But we must strive for higher eons.
It is difficult but necessary for us, brought up in an environment of atheism and dominant religious ignorance, to learn that God the Father does not live high in the sky, not on other planets, not on some mountain, etc. He is everywhere in the entire universe: in depth under our bodies and the entire world of matter, under all of Creation.
And the “ladder” to Him does not lead up, but deep down. Its steps are the steps of refining oneself as consciousness. And that ladder begins... in our spiritual hearts.
... Everything said was actually researched by the author of this book, and was not at all copied by him from somewhere or retold from someone else’s words. And everyone should try to walk this Path. At the same time, it is important to know that you should move along it “from step to step”, and not by jumping over “flights of stairs”.
… So, the Abode of the Creator exists everywhere, under every molecule of matter. The distance to it, as Jesus said, is no thicker than a sheet of thin paper...
God the Father is not in heaven, He is everywhere: in and around our bodies, under every particle of them. His abode is extremely close! But... - try, step into it!
You can step into it only with His blessing. And only those who have developed themselves to the proper extent according to the parameters of Love, Wisdom and Strength can receive a blessing for this.
The path to the Abode of the Creator is the Path of gradual refinement of oneself as consciousness. First, in the words of the Apostle Paul, one must “turn away from evil and cleave to goodness” [, ], that is, get out of drunken companies, from among rude and cruel people, find beauty in nature, in true art, let companions on the spiritual Path become friends .
The next stage of strengthening in subtlety will be the initial realization of the potential of the spiritual heart. Then - cleansing the chakras and the most important meridians, including chitrini (Brahmanadi). Now, leaving the body through chitrini, we will go straight into the Holy Spirit, and meditation Pranava will give the first mergers with Him... Thus, following from step to step of the multidimensional Universe, sometimes stopping to rest and get comfortable, we reach the Abode of the Creator, which now becomes our Home.
This is the true Path to God. And not evil rallies with calls for reprisals against “infidels”, not anathemas (curses) addressed to individual “dissidents” or neighboring sects or even entire nations! That is the path of devilization, the path to hell.
What's happened spatial And time coordinates? There are four explicit ones: three spatial and one temporal. Are additional, unknown to us, hidden spatial and temporal dimensions possible in our world?
Physicists say yes. In 1921, an article by Theodor Kaluza entitled “On the Problem of the Unity of Physics” appeared in the journal “Sitzungsberichte der Berliner Akademie” (the article was recommended by A. Einstein). In it, the researcher proposed to supplement the four dimensions of space-time with a fifth, spatial dimension. The introduction of the fifth dimension made it possible to describe all fundamental dimensions known at that time (gravitational and electromagnetic) through spatial categories.
A few years later, Swedish physicist Oskar Klein expanded this theory by considering other multidimensional versions of the Universe and checking their compatibility with already known fundamental physical laws. In modern physics, the Kaluza-Klein theory is any quantum theory that attempts to unify the fundamental interactions in space-time having more than four dimensions. Currently, there are a large number of theories that consider our World as 5-, 6- and even 12-dimensional, and additional coordinates may be both spatial and temporal.
However, there are a number of strong arguments against multidimensionality. First of all, it is not observable. And no matter how many theories physicists invent, not a single fact has been discovered in our world that confirms the theory of multidimensionality. Except, of course, the human mind.
Moreover, it turned out that if there is a surrounding us world additional dimensions, some existing natural phenomena would be impossible (in particular, the existence planets, stars, atoms and molecules).
Visually, although not entirely true, this can be represented as follows: if in our world there were additional spatial dimensions, then something would definitely fall through there, fall out, bend (atoms, planetary orbits, waves or particles). But this is not happening!
Naturally, multidimensional theories took into account the limitations imposed by reality. There are several ways to smooth out the contradiction between the harsh demands of our world and the dream of multidimensional realities.
First way.
Was proposed in the work A. Einstein And P. Bergman“Generalization of Kaluza’s theory of electricity,” it assumed “that the fifth coordinate can vary only within certain limited limits: from 0 up to some value T, i.e. The 5-dimensional world is enclosed, as it were, in a certain layer of thickness T.” This value is so small that even an elementary particle (electron, for example) exceeds it as much as the globe exceeds a pea. And it is impossible to place anything into this more than narrow layer of additional dimension.
If we imagine our entire visible world with its 4 dimensions as a plane, for example, a piece of paper, then the fifth dimension will appear in the form of a thin layer of space applied to this piece of paper. In all directions the sheet is infinite, and upward (into the 5th dimension) its extent is limited by the microscopic size of the layer. It is impossible for a person, even an elementary particle, to fall into such a dimension. And you can't see him. Even the most powerful microscopes will not help.
Method two.
The extent of space in the fourth dimension can be as large as desired (in principle, comparable to almost infinite length, width and height). However, this space is “collapsed into an exceptionally small circle.” And this folded 5th direction (coordinate axis) is connected to the 4-dimensional world we see only by a narrow neck, the diameter of which is comparable to the size of the 5-dimensional layer described above. “To detect this circle, the energy of the particles illuminating it must be sufficiently high. Particles of lower energies will be distributed evenly around the circle and will not be detectable. The most powerful accelerators create particle beams that provide a resolution of 10–16 cm. If the circle in the fifth dimension is smaller, then it is not yet possible to detect it.”
Acceptance of one of these provisions explains the unobservability of additional dimensions (by the way, this is why they are called hidden) and why they do not affect our world.
But in addition to physicists, representatives of other natural sciences also turned to multidimensional theories of space, in particular V.I.Vernadsky, which assumed that “ physical space is not a geometric space three dimensions”
How, in general, could these multidimensional spaces come into a person’s head if they are not in the surrounding reality? And can we come up with, imagine something that has no analogue in the outside world (until now only a wheel has been proposed as such, and even then it had analogues - moving round disks - the moon and the sun).
If the psyche is a reflection of the macrocosm, then it reflects all space-time properties of the universe, including those that we are not yet aware of. This applies to any idea of space. The more complex the World around us is, the more complex the display is. Any mirror is two-dimensional, but is capable of reflecting three-dimensional objects, just as there is a three-dimensional world behind an absolutely flat TV screen; and with a little effort, the landscapes shown can acquire depth. And if the psyche is still not a reflection, but an elusive higher substance, then, in this case, a person created “in the image of likeness” initially carries within himself the Highest plan of the structure of the Universe. And, of course, if this plan provides for higher dimensions for space-time, a person carries them within himself.
Are the hidden dimensions of the external world reflected (can they be reflected) in sensory images, if of course there are any in it?
A person can perceive and visualize only three-dimensional objects; images of higher dimensions are fundamentally inaccessible to either perception or imagination, i.e. we cannot not only see them, but also imagine them.
We have developed “organs” only for those aspects of Being-in-itself that were important to take into account for the preservation of the species.
Yes, but... What if the perception or imagination of multidimensional structures has meaning and significance for the survival of the species? If we do not realize it, does the multidimensionality of space-time play an important role in the organization of our mental life? Then it is possible that within us we somehow reflect the multidimensional structure of the Universe, although we are not aware of it, because the fish, which reflects the hydrodynamic properties of water with the structure of its body, also does not suspect this and, even more so, is not familiar with the laws of thermodynamics.
Research has established empirically that images of altered states of consciousness can be multidimensional.
In LSD sessions, subjects “familiar with mathematics and physics sometimes report that many of the concepts in these disciplines that elude rational understanding can become more comprehensible and can even be experienced in altered states of consciousness. Insights that contribute to comprehension include theoretical systems such as non-Euclidean geometry, n-space geometry, space-time, special and general Einstein's theories of relativity...
If hidden dimensions of space-time exist in any form, then their presence should be reflected in the structure of internal space, i.e. under certain conditions (possibly in altered states of consciousness), a person, to one degree or another, can visualize visual images with dimension greater than three. If this happens, then we can talk about the multidimensionality of a person’s internal space. If a person cannot do this, then his internal space is, at best, 3-dimensional.
Hypnosis. The subjects were put into a state of deep hypnosis.
1st experiment - after waking up for a while (before receiving the final signal), they will stop seeing everything that is to the right of them, and it does not matter where they look and with which eye (right-sided agnosia);
Experience 2 - after waking up for a while (before receiving the final signal), they will stop seeing everything that is to the left of them, and it does not matter where they look and with which eye (left-sided agnosia).
In the second series of experiments, visualization of images was caused 4th spatial dimension. Before the experiment, the subjects did not know what exactly they were going to visualize. Before the experiment, the subjects were reminded of some provisions of the school geometry course. A straight line, a right angle, and coordinate axes were drawn; from matches and plasticine they made up: a straight line, an angle, two straight lines at an angle of 90 degrees, three straight lines intersecting at angles of 90 degrees - Cartesian coordinate axes, an example of a volumetric right angle was demonstrated - the corner of a room in which three walls intersect at a right angle. It was unobtrusively mentioned that the 4th line could not be drawn in this way (“how could one draw another line at right angles to all the others - it doesn’t work, but oh well”).
1. Visualization in a state of hypnosis. The subjects were introduced into a similar state of deep hypnosis. Next they were asked to imagine:
1) straight line,
2) two lines intersecting at an angle of 90 degrees,
3) three lines intersecting at an angle of 90 degrees.
After which we moved on to visualizing the 4th spatial dimension. The subjects were asked to mentally draw another line (the fourth) at an angle of 90 degrees to all the others. Another option was to imagine a corner of the room and try to imagine a fourth wall, at right angles to the rest. Next, the subjects were asked to mentally “look” in the direction of this line and verbally describe everything they saw.
2. Post-hypnotic visualization. In a state of deep hypnosis, subjects were told that after waking up for a while (before receiving the final signal), they would retain their ability to visualize the 4th straight line and would be able to look in its direction from anywhere in the room. Next, they were taken out of the state of hypnosis, and the safety of the suggestion was checked. The subjects described the features of their vision of the world. At the end, a final signal was given.
7 people took part in the experiments.
Results of the second series. The phenomenon of visualization of the 4th spatial dimension was very easy to cause. All seven people completed the task.
When visualizing under hypnosis, most subjects in the direction of the 4th axis “saw” either abstract geometric figures or found it difficult to describe what they saw
In the next group of experiments, an attempt was made to physically penetrate the 4th dimensions in a situation of post-hypnotic suggestion. In this group of experiments, the well-known fact was confirmed that, alas, it is physically impossible to penetrate into the 4th dimension. Even if a person sees images of this dimension, the physical body still imposes restrictions on his freedom of movement. Thus, almost all of our subjects, “looking” into the 4th dimensions, visualized abstract geometric figures. And only in one case, the subject imagined real pictures. By the way, this was the only left-handed subject in this series of experiments.
The question that arises. Or maybe all this - game of imagination? Maybe the subjects didn't really imagine Fourth dimension, but only imagined that they were imagining? But it was precisely the space of imagination that was studied; Not physical world how it works (after all, the study of the physical world is a matter of another science - physicists), A dimensionality our space of imagination. And if Human only imagines that he imagines the fourth change, perhaps this means that he can imagine higher dimensions in his inner space.
A fact that attracts attention. Ease of completing the “imagine the fourth dimension” task by subjects. It can be assumed that the multidimensionality of the imagination space is a natural state of the human psyche, which has a completely material basis - a brain substrate.
Indeed, if multidimensionality is not alien to our world, then shouldn’t the psyche that has arisen in its image and likeness reflect it in the depths of its existence? It should be noted that this definition of internal space does not violate any of the laws of physics.
Let us now turn to the verbal sphere. The ideas embodied in the word are brought to consciousness and thereby become conscious of us. The display of multidimensional aspects of the universe occurs through the embodiment of relevant ideas in cultural achievements (from myths and fairy tales to formulas and theories). And it is in such forms that these ideas are recognized by humanity - as myths and legends, as fantasies and works of art; embodiment in the form of formulas and theories.
At first, of course, the multidimensional structure of the universe was depicted in myths. The idea that our universe consists of several worlds, communicating or almost not communicating, is quite common in the mythology of different peoples. For example, in the myths of the ancient Slavs there was an idea of three main substances of the world. The idea of the multidimensional structure of a person’s inner world is found in Egyptian mythology. This is a fairly common division of the universe into three worlds (earthly, heavenly and underworld).
Man displayed multidimensionality of our world and hidden space-like dimensions since time immemorial. But the question of how to penetrate the higher dimensions of space in our Universe remains an eternal one. The answers to this, of course, exist, it’s just not entirely clear how to use them.
Most often, to transition to higher dimensions, it is recommended to imagine your internal space as external, and the external space of multidimensional reality as internal. In terms of the topology of multidimensional spaces, this is a really nice way to imagine the fourth spatial dimension while being in the third.
Even in the apocryphal Gospel of Thomas, it is in these words that man’s path to the kingdom of God is described. “When you make the two one, and when you make the inside as the outside, and the outside as the inside, and the top as the bottom, /.../ when you make the eyes instead of the eye and the hand instead of the hand, and the leg instead feet, an image instead of an image - then you will enter [the kingdom]. Usually these words are interpreted in a figurative sense: a person must completely change, understand himself, realize the complex nature of his inner world, change it for the better, etc. But perhaps these words can also be understood in their literal sense, as another description of the transition to higher dimensions. Well, the “kingdom of heaven” is a classic representation of other realities in the mythology of many peoples.
Our psyche has additional dimensions, like some kind of reality of a higher (in the spatio-temporal sense) order that cannot be reduced to everyday life.
Or it may be otherwise, only thanks to the presence of additional dimensions in our Universe, the very possibility of mental reflection appeared, the psyche arose and the mind developed.
