The mathematical point is voluminous. Critical point (mathematics)

This term has other meanings, see point. A set of points on a plane

Dot- an abstract object in space that does not have any measurable characteristics (zero-dimensional object). The point is one of the fundamental concepts in mathematics.

Point in Euclidean geometry

Euclid defined a point as "an object without parts". In the modern axiomatics of Euclidean geometry, a point is a primary concept, given only by a list of its properties - axioms.

In the chosen coordinate system, any point of the two-dimensional Euclidean space can be represented as an ordered pair ( x; y) real numbers. Likewise, point n-dimensional Euclidean space (as well as vector or affine space) can be represented as a tuple ( a 1 , a 2 , … , a n) from n numbers.

point is:

dot dot noun, well., use Often Morphology: (no) what? dots, what? dot, (see) what? dot, how? dot, about what? about the point; pl. what? dots, (no) what? points, what? points, (see) what? dots, how? dots, about what? about points 1. Dot- this is a small round speck, a trace from a touch with something sharp or writing.

Dot pattern. | Puncture point. | The city on the map is indicated by a small dot and one can only guess about the presence of a bypass road.

2. Dot- this is something very small, poorly visible due to remoteness or for other reasons.

Point on the horizon. | As the ball approached the horizon in the western part of the sky, it began to slowly decrease in size until it turned into a dot.

3. Dot- a punctuation mark that is placed at the end of a sentence or when abbreviating words.

Put a point. | Don't forget to put a dot at the end of the sentence

4. In mathematics, geometry and physics dot is a unit having a position in space, the boundary of a line segment.

Math point.

5. dot name a certain place in space, on the ground or on the surface of something.

placement point. | Pain point.

6. dot name the place where something is located or carried out, a certain node in the system or network of any points.

Each outlet must have its own sign.

7. dot they call the limit of development of something, a certain level or moment in development.

The highest point. | point in development. | The state of affairs has reached a critical point. | This is the highest point of manifestation of the spiritual power of man.

8. dot called the temperature limit at which the transformation of a substance from one state of aggregation to another occurs.

Boiling point. | Freezing point. | Melting point. | The higher the altitude, the lower the boiling point of water.

9. Semicolon (;) called a punctuation mark used to separate common, more independent parts of a compound sentence.

In English, almost the same punctuation marks are used as in Russian: dot, comma, semicolon, dash, apostrophe, brackets, ellipsis, question and exclamation marks, hyphen.

10. When they talk about point of view, mean someone's opinion about a certain problem, a look at things.

Less popular now is another point of view, previously almost universally recognized. | Nobody shares this point of view today.

11. If people are said to have points of contact so they have common interests.

We may be able to find common ground.

12. If something is said dot to dot, meaning an absolutely exact match.

Dot to dot in the place where it was indicated, there was a coffee-colored car.

13. If a person is said to be reached the point, which means that he has reached the extreme limit in the manifestation of some negative qualities.

We've reached the point! You can't live like this anymore! | You can't tell him that the secret services have reached the point under his wise leadership.

14. If someone puts an end in some business, it means that he stops it.

Then he returned from emigration to his homeland, to Russia, to the Soviet Union, and this put an end to all his searches and thoughts.

15. If someone dot the "and"(or over i), which means that he brings the matter to its logical conclusion, leaves nothing unsaid.

Let's dot the i's. I didn't know anything about your initiative.

16. If someone hits one point, which means that he concentrated all his forces on achieving one goal.

That is why his images are so distinct; he always hits one point, never getting carried away by secondary details. | He understands very well what the task of his business is, and purposefully hits one point.

17. If someone hit the spot, which means that he said or did exactly what was needed, guessed it.

The very first letter that came to the next round of the competition pleasantly surprised the editors - in one of the listed options, our reader immediately hit the mark!

point adj.

Acupressure.

Explanatory dictionary of the Russian language Dmitriev. D.V. Dmitriev. 2003.

Dot

Dot Can mean:

Wiktionary has an article "dot"

A point is an abstract object in space that does not have any measurable characteristics other than coordinates.
A period is a diacritical mark that can be placed above, below, or in the middle of a letter.
Point - a unit of distance measurement in Russian and English systems of measures.
The dot is one of the representations of the decimal separator.
Dot (network technologies) - designation of the root domain in the hierarchy of global network domains.
Tochka - chain of electronics and entertainment stores
Tochka - album of the group "Leningrad"
Point - Russian film of 2006 based on the story of the same name by Grigory Ryazhsky
Dot is the second studio album by rapper Sten.
Tochka is a divisional missile system.
Tochka - Krasnoyarsk Youth and Subcultural Journal.
Tochka is a club and concert venue in Moscow.
The dot is one of the characters in Morse code.
The point is the place of combat duty.
Point (processing) - the process of machining, turning, sharpening.
POINT - Information and analytical program on NTV.
Tochka is a rock band from the city of Norilsk, founded in 2012.

Toponym

Kazakhstan

Dot- until 1992, the name of the village Bayash Utepov in the Ulan district of the East Kazakhstan region.

Russia

Tochka is a village in the Sheksninsky district of the Vologda region.
Tochka is a village in the Volotovsky district of the Novgorod region.
Tochka is a village in the Lopatinsky district of the Penza region.

Can you give a definition of such concepts as a point and a line?

Our schools and universities did not have these definitions, although they are key in my opinion (I don’t know how this is in other countries). We can define these concepts as "successful and unsuccessful" and consider whether this is useful for the development of thinking.

Wrestler

Strange, but we were given the definition of a point. This is an abstract object (convention) located in space, which has no dimensions. This is the first thing that was hammered into our heads at school - a point has no dimensions, it is a "zero-dimensional" object. A conditional concept, like everything else in geometry.

Straight lines are even more difficult. First of all, it's a line. Secondly, it is a set of points located in space in a certain way. In its simplest definition, it is a line defined by the two points through which it passes.

Medivh

A point is some kind of abstract object. A point has coordinates but no mass or dimensions. In geometry, everything begins precisely from a point, this is the beginning of all other figures. (In writing, by the way, too, without a point there will be no beginning of a word). A straight line is the distance between two points.

Leonid Kutny

You can define anything and anything. But there is a question: will this definition "work" in a particular science? Based on what we have, it makes no sense to define a point, a line and a plane. I really liked Arthur's remarks. I would like to add that a point has many properties: it has no length, width, height, no mass and weight, etc. But the main property of a point is that it clearly indicates the location of an object, an object on plane, in space. That's why we need a point! But, a smart reader will say that then a book, a chair, a watch and other things can be taken as a point. Absolutely right! Therefore, it makes no sense to define a point. Sincerely, L.A. Kutniy

A straight line is one of the basic concepts of geometry.

The period is a punctuation mark in writing in many languages.

Also, the dot is one of the symbols of Morse code

So many definitions :D

The definitions of a point, a line, a plane were given by me back in the late 80s and early 90s of the 20th century. I give a link:

https://yadi.sk/d/bn5Cr4iirZwDP

In a 328-page volume, the cognitive essence of these concepts is described in a completely new aspect, which are explained on the basis of a real physical worldview and a sense of I exist, which means "I" exist, just as the Universe itself to which I belong exists.

Everything written in this work is confirmed by the knowledge of mankind about nature and its properties that have long been discovered and are still being researched at this point in time. Mathematics has become so complex to understand and comprehend in order to apply its abstract images to the practice of technological breakthroughs. Having revealed the Foundations, which are the fundamental principles, it is possible to explain even to an elementary school student the reasons underlying the existence of the Universe. Read and come closer to the Truth. Dare, the world in which we exist opens before you in a new light.

Is there a definition of the concept of "point" in mathematics, geometry.

Mikhail Levin

"indefinable concept" is a definition?

In fact, it is the uncertainty of concepts that makes it possible to apply mathematics to different objects.

A mathematician can even say "by point I will mean the Euclidean plane, by plane - the Euclidean point" - check all the axioms and get a new geometry or new theorems.

The point is that to define term A, you need to use term B. To define B, you need term C. And so on ad infinitum. And in order to be saved from this infinity, one has to accept some terms without definitions and build definitions of others on them. ©

Grigory Piven

In mathematics, Piven Grigory A point is a part of space that is abstractly (mirrored) taken as the minimum length segment equal to 1, which is used to measure other parts of space. Therefore, a person chooses the scale of a point for convenience, for a productive measurement process: 1mm, 1cm, 1m, 1km, 1a. e., 1 St. year. etc.

MKOOST SANATORIUM SCHOOL - BOARDING SCHOOL

Point and geometric shapes.

Research work in mathematics.

Completed by: Anatoly Vasiliev, 3rd grade student

Work manager:

Dubovaya Natalya Leonidovna,

Primary school teacher.

Tommot, 2013

Brief annotation. ................................................. ....................2
Annotation. ................................................. .................................3
Research Article. ................................................. ......................6
Conclusion................................................. ...............................................7

Bibliography.

Brief annotation.

The paper discusses the point and geometric shapes: line, ray, segment, angle, triangle, quadrilateral, circle and circle, as well as the role of the point in the composition and construction of these figures.

Annotation.

Purpose of the study:find out what is meant by the concepts of a point and what geometric shapes consist of: a straight line, a ray, an angle, a quadrilateral, a triangle, a circle.

Object of study:point and definitions of geometric shapes: line, ray, angle, quadrilateral, triangle, circle.

Subject of study:point and geometric shapes: straight line, ray, angle, quadrilateral, triangle, circle.

Research hypothesis:point - the only geometric figure, and all the rest consisting of many points.

Research objectives:

study materials on the topic: “Point and geometric shapes: straight line, ray, angle, quadrilateral, triangle, circle.”;
find the definitions of a point, a straight line, a quadrilateral, a triangle, an angle, a ray, a circle;
present their analysis and reflections on the topic;
present a presentation based on this research paper.

Research methods:study of literature, work with dictionaries, analysis of the study, conclusion.

Research Article.

Mathematics arose in ancient times from the practical needs of people. No one will argue about the antiquity of mathematics, but there is another opinion about what prompted people to do it. According to him, mathematics, as well as poetry, painting, music, theater and art in general, was brought to life by the spiritual needs of man, his, perhaps not yet fully realized, desire for knowledge and beauty.

Have you ever thought about what a point is and what geometric shapes consist of?

At first glance, everything is clear here: a point is a point, a straight line is a straight line, what could be incomprehensible here? Well, all the same, how to explain this to someone who does not know this at all and, moreover, understands everything very literally? Is it that simple? It turns out not at all!

In labor lessons, when we studied the isothread technique, I had an assumption that all geometric shapes consist of points. It is to this topic that I decided to dedicate my research work.

“I know that I don’t know anything,” Socrates said, and tried to find out through dialogue with the interlocutor what exactly he knows. Therefore, I decided to first find out what I know about geometric shapes.

So, let's look at the definitions of geometric shapes indicated by the topic of my research work.

Dot - this is a mark, a trace from a touch, an injection with something sharp; small round speck, speck; something very small, barely visible. A point is a basic geometric figure

Line- it's a lot of points. If the basis for constructing geometry is the concept of distance between points in space, then a straight line can be defined as a line along which the distance between two points is the shortest. Direct - there is a line that is equally located with respect to all its points. The term "line" originated from the Latin linum - "linen, linen thread".

_________________________________________________

Ray is a part of a line that consists of all points of this line that lie on one side of its given point.

Line segment is the part of a line that consists of all points of this line that lie between two given points on it.

Injection- this is a figure that consists of a vertex point of an angle and two different half-lines descending from this point, the sides of the angle.

Quadrilateral- this is a figure that consists of four points and four segments connecting them in series.

Triangle - a figure composed of three points that do not lie on one straight line, connected by segments.

A circle -

Circle is a figure that consists of all points of the plane equidistant from a given point. A closed line around a circle.

CONCLUSION.

The concepts of a point and a straight line are found in our life everywhere and everywhere. For example, if you look into the Russian language, then a period is a punctuation mark (.) that separates a complete sentence. Also in Russian there are such punctuation marks as a semicolon, colon, ellipsis.

In physics, a point is a specific value of a quantity.

In geography, a point is considered as a specific place in space.

In biology, this is the growing point of plants.

In chemistry - freezing point, boiling point, melting point.

In music, a dot is a sign that is one of the basic elements of musical notation.

In mathematics, a point is a basic geometric figure; the intersection of two lines, the boundary of a line segment, the beginning of a ray, etc.

To build any figure, we need a point. Based on the definition of a straight line,A LINE IS A LOT OF POINTS, and from the definitions, we know that any figure is built using a point and a line, therefore all figures consist of points.

In our life, a dot is an injection badge, a small speck.

My research work leads to the conclusion that the point is the only geometric figure. Everything begins with a point and ends with it, and it is not yet known what opening it will serve as the beginning.

Literature:

1 .Aksenova M.D. Encyclopedia for children. T.11. - Mathematics, M .: Avanta +, 1999. P. 575.

2 .Atanasyan L.S., geometry, 7-9: textbook for educational institutions / 12th ed. - M.: Enlightenment, 2002. Pp. 5, 146, 177,178.

3. Atanasyan L.S., geometry, 10-11: a textbook for educational institutions / 15th ed., add. - M.: Education, 2006. Pp.5-7.

4 .Vinogradov I.M., mathematical encyclopedia / M.: Soviet encyclopedia. pp. 410, 722.

5 .Evgenyeva A.P. Dictionary of the Russian language. - M.: Enlightenment, 1984.

6 .Kabardin O.F. Physics: reference materials. - M.: Education, 1991.

7 .Kramer G. Mathematical methods of statistics, translated from English, 2nd ed., M., 1975.

8 .Lapatukhin M.S. School explanatory dictionary of the Russian language. - M.: Education, 1981.

9 .Prokhorov A.M. Big encyclopedic dictionary. - M.: Education, 1998.

10. Prokhorov Yu.V. Mathematical Encyclopedic Dictionary. - M.: Education, 1998.

11 .Savin A.P. Encyclopedic Dictionary of a Young Mathematician. - M.: Pedagogy, 1985, p.69.

12 .Sharygin I.F. visual geometry. - M.: Education, 1995.

The concept of a critical point can be generalized to the case of differentiable mappings , and to the case of differentiable mappings of arbitrary manifolds f: N n → M m (\displaystyle f:N^(n)\to M^(m)). In this case, the definition of a critical point is that the rank of the Jacobian matrix of the mapping f (\displaystyle f) it is less than the maximum possible value equal to .

Critical points of functions and mappings play an important role in areas of mathematics such as differential equations, calculus of variations, stability theory, as well as in mechanics and physics. The study of critical points of smooth mappings is one of the main questions in catastrophe theory. The notion of a critical point is also generalized to the case of functionals defined on infinite-dimensional function spaces. Finding critical points of such functionals is an important part of the calculus of variations. Critical points of functionals (which, in turn, are functions) are called extremals.

Formal definition

critical(or special or stationary) a point of a continuously differentiable mapping f: R n → R m (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(m)) a point is called at which the differential of this mapping f ∗ = ∂ f ∂ x (\displaystyle f_(*)=(\frac (\partial f)(\partial x))) is an degenerate linear transformation of the corresponding tangent spaces T x 0 R n (\displaystyle T_(x_(0))\mathbb (R) ^(n)) and T f (x 0) R m (\displaystyle T_(f(x_(0)))\mathbb (R) ^(m)), that is, the dimension of the transformation image f ∗ (x 0) (\displaystyle f_(*)(x_(0))) smaller min ( n , m ) (\displaystyle \min\(n,m\)). In the coordinate notation for n = m (\displaystyle n=m) this means that the jacobian is the determinant of the jacobi matrix of the mapping f (\displaystyle f), composed of all partial derivatives ∂ f j ∂ x i (\displaystyle (\frac (\partial f_(j))(\partial x_(i))))- vanishes at a point. Spaces and R m (\displaystyle \mathbb (R) ^(m)) in this definition can be replaced by varieties N n (\displaystyle N^(n)) and M m (\displaystyle M^(m)) the same dimensions.

Sard's theorem

The display value at the critical point is called its critical. According to Sard's theorem, the set of critical values of any sufficiently smooth mapping f: R n → R m (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(m)) has zero Lebesgue measure (although there can be any number of critical points, for example, for the identical mapping, any point is critical).

Constant rank mappings

If in the vicinity of the point x 0 ∈ R n (\displaystyle x_(0)\in \mathbb (R) ^(n)) rank of a continuously differentiable mapping f: R n → R m (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(m)) is equal to the same number r (\displaystyle r), then in the vicinity of this point x 0 (\displaystyle x_(0)) there are local coordinates centered at x 0 (\displaystyle x_(0)), and in the neighborhood of its image - points y 0 = f (x 0) (\displaystyle y_(0)=f(x_(0)))- there are local coordinates (y 1 , … , y m) (\displaystyle (y_(1),\ldots ,y_(m))) centered on f (\displaystyle f) is given by the relations:

Y 1 = x 1 , … , y r = x r , y r + 1 = 0 , … , y m = 0. (\displaystyle y_(1)=x_(1),\ \ldots ,\ y_(r)=x_(r ),\ y_(r+1)=0,\ \ldots ,\ y_(m)=0.)

In particular, if r = n = m (\displaystyle r=n=m), then there are local coordinates (x 1 , … , x n) (\displaystyle (x_(1),\ldots ,x_(n))) centered on x 0 (\displaystyle x_(0)) and local coordinates (y 1 , … , y n) (\displaystyle (y_(1),\ldots ,y_(n))) centered on y 0 (\displaystyle y_(0)), such that they display f (\displaystyle f) is identical.

Happening m = 1

In case, this definition means that the gradient ∇ f = (f x 1 ′ , … , f x n ′) (\displaystyle \nabla f=(f"_(x_(1)),\ldots ,f"_(x_(n)))) vanishes at this point.

Let's assume that the function f: R n → R (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ) has a smoothness class of at least C 3 (\displaystyle C^(3)). Critical point of a function f called non-degenerate, if it contains a Hessian | ∂ 2 f ∂ x 2 | (\displaystyle (\Bigl |)(\frac (\partial ^(2)f)(\partial x^(2)))(\Bigr |)) different from zero. In a neighborhood of a nondegenerate critical point, there are coordinates in which the function f has a quadratic normal form (Morse's lemma).

A natural generalization of the Morse lemma for degenerate critical points is Toujron's theorem: in a neighborhood of a degenerate critical point of the function f, differentiable an infinite number of times () of finite multiplicity µ (\displaystyle \mu ) there exists a coordinate system in which a smooth function has the form of a polynomial of degree μ + 1 (\displaystyle \mu +1)(as P μ + 1 (x) (\displaystyle P_(\mu +1)(x)) one can take the Taylor polynomial of the function f (x) (\displaystyle f(x)) at a point in the original coordinates) .

At m = 1 (\displaystyle m=1) it makes sense to ask about the maximum and minimum of a function. According to the well-known statement of mathematical analysis, a continuously differentiable function f (\displaystyle f), defined in the whole space R n (\displaystyle \mathbb (R) ^(n)) or in its open subset, can reach a local maximum (minimum) only at critical points, and if the point is nondegenerate, then the matrix (∂ 2 f ∂ x 2) = (∂ 2 f ∂ x i ∂ x j) , (\displaystyle (\Bigl ()(\frac (\partial ^(2)f)(\partial x^(2)))( \Bigr))=(\Bigl ()(\frac (\partial ^(2)f)(\partial x_(i)\partial x_(j)))(\Bigr)),) i , j = 1 , … , n , (\displaystyle i,j=1,\ldots ,n,) must be negatively (positively) definite in it. The latter is also a sufficient condition for a local maximum (respectively, minimum) .

Happening n = m = 2

When n=m=2 we have a mapping f plane onto a plane (or two-dimensional manifold onto another two-dimensional manifold). Let's assume that the display f differentiable an infinite number of times ( C ∞ (\displaystyle C^(\infty ))). In this case, the typical critical points of the mapping f are those in which the determinant of the Jacobian matrix is equal to zero, but its rank is equal to 1, and hence the differential of the mapping f has a one-dimensional kernel at such points. The second condition of typicality is that in a neighborhood of the considered point on the inverse-image plane, the set of critical points forms a regular curve S, and at almost all points of the curve S core ker f ∗ (\displaystyle \ker \,f_(*)) does not concern S, while the points where this is not the case are isolated and the tangency at them is of the first order. Critical points of the first type are called crease points, and the second type assemblage points. Folds and folds are the only types of singularities of plane-to-plane mappings that are stable with respect to small perturbations: under a small perturbation, the fold and fold points move only slightly along with the deformation of the curve S, but do not disappear, do not degenerate, and do not fall apart into other singularities.

See also: http://akotlin.com/index.php?sec=1&lnk=2_07

For two and a half millennia, mathematics has been using the abstraction of a dimensionless point, which contradicts not only common sense, but also knowledge about the world around us, obtained by such sciences as physics, chemistry, quantum mechanics and computer science.

Unlike other abstractions, the abstraction of a dimensionless mathematical point does not idealize reality, simplifying its cognition, but deliberately distorts it, giving it the opposite meaning, which, in particular, makes it fundamentally impossible to understand and study spaces of higher dimensions!

The use of the abstraction of a dimensionless point in mathematics can be compared to the use of a base currency with zero value in economic calculations. Fortunately, the economy did not think of this.

Let us prove the absurdity of the abstraction of a dimensionless point.

Theorem. The mathematical point is voluminous.

Proof.

Since in mathematics

Point_size = 0,

For a segment of finite (nonzero) length, we have

Segment_size = 0 + 0 + ... + 0 = 0.

The obtained zero size of the segment, as a sequence of its constituent points, contradicts the condition of finite length of the segment. In addition, the zero point size is absurd in that the sum of zeros does not depend on the number of terms, that is, the number of "zero" points in the segment does not affect the size of the segment.

Therefore, the original assumption about the zero size of a mathematical point is WRONG.

Thus, it can be argued that a mathematical point has a non-zero (finite) size. Since the point belongs not only to the segment, but also to the space in which the segment is located, it has the dimension of space, that is, the mathematical point is volumetric. Q.E.D.

Consequence.

The above proof, performed with the involvement of the mathematical apparatus of the younger group of the kindergarten, inspires pride in the boundless wisdom of the priests and adherents of the “queen of all sciences”, who managed to carry through the millennia and preserve for posterity in its original form the archancient delusion of mankind.

Reviews

Dear Alexander! I'm not strong in mathematics, but maybe YOU can tell me where and by whom it is stated that the point is equal to zero? Another thing is that it has an infinitely small value, up to a convention, but not zero at all. Thus, any segment can be considered zero, since there is another segment that contains an infinite number of initial segments, roughly speaking. Maybe we should not confuse mathematics and physics. Mathematics is the science of being, physics is about the existing. Sincerely.

I mentioned Achilles twice in detail and many times in passing:
"Why won't Achilles catch up with the tortoise"
"Achilles and the tortoise - a paradox in a cube"

Maybe one solution to Zeno's paradox is that space is discrete and time is continuous. He considered, as it is possible for you, that both are discrete. The body can remain at some point in space for some time. But it cannot be in different places at the same time at the same time. This is all, of course, amateurishness, like our entire dialogue. Sincerely.
By the way, if a point is 3D, what are its dimensions?

The discreteness of time follows, for example, from the aporia "Arrow". “Simultaneously stay in different places” can only be an electron for physicists who, in principle, do not understand and do not accept either the structure of the ether or the structure of 4-dimensional space. I don't know of any other examples of this phenomenon. I see no "amateurism" in our conversation. On the contrary, everything is extremely simple: a point is either dimensionless or has a size; continuity and infinity either exist or they do not. The third is not given - either TRUE or FALSE! The fundamental foundations of mathematics, unfortunately, are built on false dogmas, accepted out of ignorance 2500 years ago.

The point size depends on the condition of the problem being solved and on the required accuracy. For example, if a gear is designed for a watch, then the accuracy can be limited by the size of the atom, that is, eight decimal places. The atom itself here will be the physical analogue of the mathematical point. You may need 16-character precision somewhere; then the role of a point will be played by a particle of ether. Note that talk about allegedly "infinite" accuracy in practice turns into wild nonsense, or, to put it mildly, absurdity.

I still don't understand: does the point exist? If it exists objectively, therefore it has a certain physical value, if it exists subjectively, in the form of an abstraction of our mind, then it has a mathematical value. Zero has NOTHING, it does not exist, this is the abstract definition of Non-existence in mathematics or emptiness in physics. The point does not exist by itself outside of the relationship. As soon as the second point appears, a segment appears - Something, etc. This topic can be developed endlessly. With uv.

It seemed to me that I gave a good example, but probably not detailed enough. Objectively, there is a World that science cognizes, and at present it cognizes mainly by mathematical methods. Mathematics cognizes the world by building mathematical models. To build these models, basic mathematical abstractions are involved, in particular, such as: point, line, continuity, infinity. These abstractions are basic because it is no longer possible to further subdivide and simplify them. Each of the basic abstractions can either be adequate to objective reality (true) or not (false). All of the above abstractions are initially false, because they contradict the latest knowledge about the real world. Hence, these abstractions hinder the correct understanding of the real world. One could somehow put up with this while science was studying the 3-dimensional world. However, the abstractions of a dimensionless point and continuity make all worlds of higher dimension unknowable in principle!

The brick of the universe - a point - cannot be a void. Everyone knows that nothing comes from emptiness. Physicists, declaring the ether non-existent, filled the world with emptiness. I believe that mathematics with its empty point pushed them to this stupidity. I'm not talking about atoms-points of worlds of higher dimension than 4D. So, for each dimension the role of an indivisible (conditionally) mathematical point is played by the (conditionally) indivisible atom of this world (space, matter). For 3D - a physical atom, for 4D - an ether particle, for 5D - an astral atom, for 6D - a mental atom, and so on. Sincerely,

So, does the brick of the universe have any absolute value? And what does it represent, in your opinion, in the ethereal or mental world. I'm afraid to ask about the worlds themselves. With interest...

Ether particles (these are not atoms!) are electron-positron pairs, in which the particles themselves rotate relative to each other at the speed of light. This fully explains the structure of all nucleons, the propagation of electromagnetic oscillations and all the effects of the so-called physical vacuum. The structure of the atom of thought is unknown to anyone. There is only evidence that ALL the highest worlds are material, that is, they have their own atoms. Up to the matter of the Absolute. You're being ironic, though. Do wormholes and big explosions seem more plausible to you?

What is the irony here, just a little taken aback after such an avalanche of information. I, unlike you, am not a professional and I find it difficult to say anything about the five- or six-dimensionality of spaces. I'm all about our long-suffering point ... As far as I understand, you are against material continuity, and the point is that you have a really existing "democratic" atom. "Brick of the Universe". Maybe I was inattentive, but still, do not hesitate to repeat what its structure, physical parameters, dimensions, etc. are.
And also answer, does the unit exist in itself, as such, outside of any relations? Thank you.