Dear Pushkin,

Plotting such surfaces through MATLAB are very simple : https://www.bu.edu/tech/support/research/training-consulting/online-tutorials/visualization-with-matlab/ https://au.mathworks.com/help/matlab/ref/ndgrid.html

https://au.mathworks.com/help/matlab/ref/delaunayn.html

etc.

10D space is a manifold which can be still analyzed via imbedding into Euclidean vector space. Unfortunately, computer's vision is compactable to flat/smooth surfaces, as evident from the image as drawn via linearization and normalization techniques. For details, I suggest Riemannian Geometry http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf

Dear Muhammad Ali

Thanks for your reply. Your link to the source will help me a lot in understanding the correct solution to this question.

Regards, Pushkin Sergey Viktorovich

Dear Pushkin,

Plotting such surfaces through MATLAB are very simple : https://www.bu.edu/tech/support/research/training-consulting/online-tutorials/visualization-with-matlab/ https://au.mathworks.com/help/matlab/ref/ndgrid.html

https://au.mathworks.com/help/matlab/ref/delaunayn.html

etc.

Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.

Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed.

More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional space (Euclidean) ℝ7 that are a fixed distance from the origin. This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, and in six dimensions they have far more applications.

In the mid-1990s, Edward Witten and other theoretical physicists found strong evidence that the various superstring theories represent different limiting cases of the as yet undeveloped 11-12-dimensional M-theory. This discovery marked the second superstring revolution.

As a rule, the classical (non-quantum) relativistic dynamics of n-branes is built on the basis of the principle of least action for a manifold of dimension n + 1 (n spatial dimensions plus temporal) located in a space of higher dimension. The outer spacetime coordinates are treated as fields defined on the brane manifold. In this case, the Lorentz group becomes the internal symmetry group of these fields

What happens then with T-duality, S-duality, U-duality?

Dear Muhammad Ali

You helped me a lot. Thank you!

Dear Aseel Nadhum

Thank you for your valuable input in resolving this issue. Yours sincerely

Pushkin Sergey Viktorovich

It is well knowen that, the 10th dimension is a single point that represents all the possible branches of every possible timeline of all the potential universes. ... To recall string theory, super strings vibrating in the 10th dimension are what create the subatomic particles that make up not only our universe, but all universes.

Dear Maged Gumaan Bin-Saad

Thanks for your reply. You helped me look at this question differently. That is, from the perspective of superstring theory.

Regards, Pushkin Sergey Viktorovich

Dear Pushkin Sergey Viktorovich

I think the following link shows interesting information about dimension ten.

https://en.wikipedia.org/wiki/10-cube#:~:text=In%20geometry%2C%20a%2010%2Dcube,9%2Dcube%209%2Dfaces.

PS. I have the following notes:

(1)The attached graph represents how the cube appears in dim10.

(2) Dimension six and other dimensions show different cubes.

(3) We have nothing to do with Riemann Geometry. It tackles another issue for any dimensional manifold.

(4) To understand what is going on about the n-dimensional spaces and cubes, their vertices, and how to construct their graphs and their mathematical interpretation, I advise the reader to start with dimension four about the tesseract.

https://en.wikipedia.org/wiki/Tesseract

Best regards

Dear Pushkin Sergey Viktorovich as I mentioned earlier, I'm absolutely no specialist to give you a qualified answer to your question. However, I found the article cited below rather interesting in this respect.

https://phys.org/news/2014-12-universe-dimensions.html

Dear Frank T. Edelmann

I am glad for your response thank you. You brought a more realistic photo, in contrast to my geometric image.

Regards, Pushkin Sergey Viktorovich

DearIssam Kaddoura

Thank you for your helpful materials. For your contribution to the discussion.

Regards Pushkin Sergey Viktorovich

In its most common modification, string theory claims that the universe exists in ten dimensions, but six of them we are not able to perceive.

When

someone says "other dimensions", most often you think about things like parallel Universes — alternative realities that exist parallel to our own, in which the world is arranged somewhat or quite differently. However, the reality of dimensions and their role in the structure of the Universe are very different from this popular understanding.

In a nutshell: dimensions are different facets of what we perceive as reality. We are well aware of the three spatial dimensions that we encounter and live in every day. They determine the length, height, and depth of all objects in the Universe (and correspond to the x, y, and z coordinate axes).

However, some scientists believe that, in addition to the three visible dimensions, there may be others. According to the basics of string theory, the universe exists in ten different dimensions. Recently, we published a material about how these additional dimensions that we do not perceive can be twisted and compactified. you can read it at this link. Thus, these different aspects determine the fundamental forces of nature and all the elementary particles in the Universe.

Let's start in order. The first dimension, as we have already noted, defines the length (x-axis). It is convenient to describe a one-dimensional object as a straight line that exists only within the concept of length and has no other distinctive features. If you add a second dimension to it — the y — axis, or height-you get a two-dimensional object (for example, a square).

The third dimension characterizes the depth (z — axis) - it gives all objects the concept of area and cross-section. An ideal example would be a cube: it exists in three dimensions — it has length, height, and depth, and therefore volume.

The fourth dimension is time, and this can already be called the classical, generally accepted understanding of it. This is an inseparable part of the space-time continuum. It determines the properties of all known matter at any given time. Along with three other dimensions, to determine the position of an object in the Universe, you need to know its position in time. So these four dimensions define our reality — the Universe that we are used to and understand in one way or another.

In addition to the dimensions described above, there are seven other dimensions that are not so obvious, but can still be perceived by their direct impact on the Universe and reality as we know it. Other, additional dimensions are associated with deeper capabilities. Physicists face serious questions when trying to explain their interactions with the four "main" dimensions.

According to superstring theory, the concept of possible worlds arises in the fifth and sixth dimensions. If we could perceive the fifth dimension, we would see a world that is somewhat different from what we are used to. We would be able to measure the similarities and differences between possible worlds and our own.

In the sixth dimension, we would see the plane of possible worlds, where we could compare and determine the location of all possible universes that started under the same conditions as ours (i.e., the Big Bang). Theoretically, if we were able to master the fifth and sixth dimensions, we would be able to travel to the past or to different variations of the future.

In the seventh dimension, we would have access to possible worlds that were born under different initial conditions. Whereas in the fifth and sixth dimensions the original conditions were the same and the consequences were different, in this dimension everything is different from the beginning of time. The eighth dimension also provides access to the plane of such possible universes, each of which began under different conditions. These universes branch endlessly, which is why they are called infinities.

In the ninth dimension, we can compare the histories of all possible universes that were born under all possible laws of physics and initial conditions. Finally, in the tenth dimension, we find ourselves at a point where everything possible and imaginable is open. Beyond that, limited beings like us can't imagine anything, which makes this dimension a natural limitation of what we can comprehend on this plane.

The existence of these additional six dimensions, which we cannot perceive, is necessary for string theory: they naturally follow from mathematical calculations and models of the theory, and therefore describe the Universe within the framework of this theory. The fact that we perceive only four dimensions of space-time can be explained by one of two mechanisms: either the extra dimensions are compactified on very small scales, or we live in a three — dimensional submanifold-a kind of brane that restricts all known particles, not counting gravity (brane theory).

If the extra dimensions are indeed compactified, they must exist as so — called Calabi-Yau manifolds. Despite the fact that they are inaccessible to our senses, in this case they would determine the formation of the Universe from the very beginning. This is why scientists believe that looking back through telescopes and observing light from the early Universe will probably help them see how the existence of these extra dimensions may have affected the evolution of the cosmos.

As a candidate for the theory of everything, string theory attempts to reconcile the Standard model of particle physics with General relativity (the theory of gravity) by arguing that the universe consists of ten dimensions (or more, depending on which theory it is). In essence, this is an attempt to explain and describe how all the known forces of the Universe interact and how other possible universes can be arranged.

Dear Pushkin,

Mathematicians can define N-dimensional spaces with N variables and it is then possible to solve these problems with N dimensional analytical or numerical methods. A corresponding function can also been used in this case like in a minimization or a maximisation problem. N=ten is the specific cited case.

In the other hand, physicians can physically understand 2-dimensional or 3-dimensional problems using physical representations. The 4-dimensional problem defined with the x, y, z and t coordinates can also be seen in a 3-dimensional space when moving the studied body versus the time t.

A polytope can be a convex or a non-convex body. In 2-dimensions, a triangle and a rectangle are convex polytopes called also polygones. In 3-dimensions, a tetrahedron and a hexahedron are convex polytopes called also convex polyhedra.

A non-convex polyhedron is a general polyhedron with convexities and local concavities.

If one can define a problem with ten dimensions, the corresponding function will use these dimensions. For example, one can define a V volume function to be minimised with V=V1+V2+V3 such that V1=length1*width1*thikness1, V2=length2*width2*thikness2 and V3=heigth3*length3*width3+heigth3*width3*thikness3

You can see in this case that you have ten independent variables where the problem can be mathematically handled in ten dimensions.

Dear Mohammed Lamine Moussaoui

Thanks for your answer to my question. You helped me understand the principle of geometric construction that represents this space.

Regards, Pushkin Sergey Viktorovich

Dear Zagir Ataev

Thank you for your participation in the discussion. You helped me a lot in exploring this question.

Regards, Pushkin Sergey Viktorovich

It is a mental construction of multiple time dimensions. I do think retro-causality should be reconsidered as expressed in the first relativity formulae of A. Einstein, but initially by L.Fantappie, i.e. time as a created physical continuum that moves forward and back. M.Carmeli has also to be mentioned here, concerning the relative length of temporal intervals.

Dear Stephen I. Ternyik

I am very grateful for your input. It is time as a physical quantity that worries me a lot. This is inherent in all biological disciplines, when it is difficult to imagine that it is possible to exist in the past and in the future.

Regards, Pushkin Sergey Viktorovich

Congratulations on your construction.

http://bijectivephysics.com/

Pushkin Sergey Viktorovich The works of A. Sorli should also be cited here:

http://bijectivephysics.com/the-group/

to add some more spices on temporology, but also:

http://www.chronos.msu.ru/ru/rindex

Dear Pushkin Sergey Viktorovich

Thank you for your response.

I want also to notice that you must avoid to confuse the following cases:

* Mathematics formulations use ten axes in which each variable varies independently in a R-N space. General cases use Hyperplanes and Half-spaces to define a Polytope.

* The volume can be represented in a 3-dimensional space.

Dear Mohammed Lamine Moussaoui

Thanks for your reply. This is an important point for me.

Regards, Pushkin Sergey Viktorovich

Dear Stephen I. Ternyik

Thanks for your input on this discussion. Your suggested sources helped me a lot.

Regards, Pushkin Sergey Viktorovich

Superstring theory posits that the universe exists in 10 dimensions at once. These different aspects are what govern the universe, the fundamental forces of nature, and all the elementary particles contained within. ... The first dimension, as already noted, is that which gives it length https://phys.org/news/2014-12-universe-dimensions.amp

the 10th dimension is a single point that represents all the possible branches of every possible timeline of all the potential universes. ... To recall string theory, superstrings vibrating in the 10th dimension are what create the subatomic particles that make up not only our universe, but all universes.

https://thetartan.org/2012/11/12/scitech/ten-dimensions

10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects.

https://en.m.wikipedia.org/wiki/Dimension

Dear Chinaza Godswill Awuchi

I am very glad for your answers. They helped to look at this issue from the position that physical quantities are not limited to measurement.

Regards, Pushkin Sergey Viktorovich