Dear Ioan,

IH: I suppose this is the result of our sense of vision that is based on both of our eyes.

No, it is based on the fact that we can only arrange three rulers such that each is perpendicular to the others, you cannot add a fourth that is perpendicular to all the other three.

Dear George,

Your answer imply that independent space variables should be represented by perpendicular axes. This is true IF we accept that the orthogonal system (Cartesian Coordinate System) is the right one for examining spacetime. Is it?

What is the meaning of time in a 3D (2+1) spacetime represented by an orthogonal system with origin before Big Bang (negative time)? If we are forced by the reference system to consider only part of it (time>=0) is it the proper reference system for our case?

Orthogonality is the result of Cartesian Coordinate System (CCS) and not the opposite. In other words, the condition for the rulers to be perpendicular by two, and not any other reasonable angle (e.g. 109.5 degrees), is a matter of conviction (the reference system we are using).

Hello Ioannis Hadjidakis : Thank you, as you say, for taking the risk of asking this question. We live in, at least, 4D. This is the dimension of spacetime, not 3+1. Time and space can interchange, they are not isolated.

To repeat the famous words of Minkowsky, "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." [1], 120 years ago.

Cheers, Ed Gerck

[1] http://rgs.vniims.ru/books/spacetime.pdf

Dear Ed, I would agree with you that the three space dimensions and the fourth dimension of "time" are interchangeable and furthermore there is no separation of space and "time" in spacetime. However, if we assume the unit of space dimensions to be real we should have the unit of "time" to be imaginary (or vice versa).[1] This is the only reason of (3+1)D and not just 4D. It does not mean any kind of isolation.

Even more, the four dimensions of the (3+1)D spacetime are not independent as there is an equation that relates these (c 2 t 2 − x 2 − y 2 − z 2 = 0, if consider BB as the origin of spacetime reference frame). This equation reduce the four dimensions to three independent dimensions (3D or more strictly (2+1)D). In other words, for spacetime we should have a system with 4 dimensions with three(3) of these independent and one(1) of these (3 independent) representing "time". This certainly DOES NOT refer/correspond to any orthogonal coordinate system that is inappropriate for spacetime examination.

[1] according to Minkowski in the ref. you kindly mentioned (p48).

Dear Ioan,

IH: Your answer imply that independent space variables should be represented by perpendicular axes.

That was the simplest way to explain that it has nothing to do with vision.

IH: This is true IF we accept that the orthogonal system (Cartesian Coordinate System) is the right one for examining spacetime.

You don't need to do that, you could for example use polar coordinates, latitude, longitude and altitude, but the point is that whatever system you choose, you cannot define all locations in space with less than three independent variables, and if you try to use four you find they are degenerate. Theories with more than three spatial dimensions like Kalusa-Klein have to propose they are "compact" to get round that problem.

Dear George,

GD: You don't need to do that, you could for example use polar coordinates,

Polar coordinates are of different nature (length, angle,angle).

GD: you cannot define all locations in space with less than three independent variables,

But these three (independent) space variables define the forth dimension of "time" of spacetime according to the equation mentioned by my last answer.

Ioan, I don't know that I would worry about how stereoscopic vision can perceive a 3D scene. That's merely one technique for perceiving 3D objects. For example, you can also perceive such objects with touch, in principle even smell. I'd make the discussion more generic.

In order to actually perceive that 3D scene, we need a passage of time to occur. If for no other reason, because it takes time for the light reflecting off the scene to reach our optical sensors. Or, it takes time to feel the object with your hands. So, time is needed. Without a passage of time, we are nothing and we perceive nothing.

My contention is, to correctly perceive space-time, the observer would have to exist at a higher dimension than our 4D existence. Space-time is a blob that we can't imagine properly. Just as we can only properly perceive 3D objects because we exist in 4D space. My hypothesis is that you cannot correctly perceive the highest dimension in which you exist.

Lucky for us, we have mathematics. Math can model any multi-dimensional space you ask it to. Only our imaginations limit what very well could be out there, unperceived by mere humans. These nD dimensions we talk about are simply orthogonal axes. There's nothing to limit the existence of orthogonal axes beyond just 3 of them. Not in math, anyway.

Hello Ioannis Hadjidakis

Maybe some of you know the French physicist Henri Poincaré (he is best known for the famous controversy the paternity of the special relativity). Henri Poincaré wrote three remarkable masterpieces (he wrote others):

- Science and the Hypothesis

- The value of science

- Science and method

I quote several passages "the value of science" regarding your question about dimensions. I will only cite the conclusion, and I invite you to read what precedes it, to understand (Chapter 4: Space and its three dimensions). Here is an English source:

https://www3.nd.edu/~powers/ame.60611/poincare.pdf

"I seem to be coming to conclusions that are consistent with empiricist ideas, because I have sought to highlight the role of experience and to analyze the experimental facts involved in the genesis of three-dimensional space. But whatever may be the importance of these facts, there is one thing that we must not forget and on which I have more than once called attention.These experimental facts are often verified, but not always.

This does not mean that space often has three dimensions, but not always.

I know that it is easy to get away with it and that, if the facts are not verified, it will be easily explained by saying that the external objects have moved. If the experiment succeeds, it is said that it tells us about the space; if it does not succeed, one attacks the external objects which one accuses of having moved; in other words, if she does not succeed, she is given a helping hand.

These pushes are legitimate; I do not disagree; but they suffice to warn us that the properties of space are not experimental truths proper. If we had wanted to check other laws, we could have done it too, by giving other similar helpers? Would not we always have been able to justify these pokes for the same reasons? At most we could have been told: "Your help is probably legitimate, but you abuse it; What is the use of moving external objects so often? "

In short, experience does not prove that space has three dimensions; it proves to us that it is convenient to allocate three to him, because that is how the number of the strokes is reduced to the minimum.

Will I add that experience would never touch us except the representative space which is a physical continuum, and not the geometric space which is a mathematical continuum. At most, it could teach us that it is convenient to give geometric space three dimensions so that it has as much as representative space.

The empirical question may arise in another form. Is it impossible to conceive of physical phenomena, mechanical phenomena, for example, other than in three-dimensional space? We would thus have an objective experimental proof, so to speak, independent of our physiology, of our modes of representation. "

[...]

"The experiment played only one role, it was used as an opportunity, but that role was nonetheless very important, and I thought it necessary to bring it out. if there existed a form a priori imposing on our sensitivity and which would be three-dimensional space.

Does this form exist, or, if you will, can we imagine space as more than three dimensions? And first of all, what does this question mean? In the true sense of the word, it is clear that we can not represent four-space or three-dimensional space; we can not first represent them as empty, nor can we represent an object in four-space or three-dimensional space: (1) because these spaces are one and the same; another infinite and that we could not represent a figure in space, that is to say the part in the whole, without representing the whole, and that is impossible, since this all is infinite; (2) because these spaces are both mathematical continuums and we can only represent the physical continuum; 3 ° because these spaces are both homogeneous, and the frames where we enclose our sensations, being limited, can not be homogeneous. "

[...]

"We must therefore return to the Mach-Delage theory.What can we know about the nerves of the channels is the difference of pressure on the two ends of the same channel, and thus:

1 ° The direction of the vertical with respect to three axes invariably linked to the head;

2 ° The three components of the translational acceleration of the center of gravity of the head;

3. The centrifugal forces developed by the rotation of the head;

4. The acceleration of the rotational movement of the head.

It follows from the experiments of M. Delage that it is this last indication which is by far the most important; no doubt because the nerves are less sensitive to the pressure difference itself than to the sudden variations of this difference. The first three indications can thus be neglected. "

[...]

"In these series S and Σ, these sensations of semicircular canals obviously have a very important place, but they alone would not be enough, for they can only inform us of the movements of the head. learn nothing about the relative movements of the trunk or limbs in relation to the head, and it seems that they only tell us about the rotations of the head and not about the translations it may undergo. "

Obviously, these are just excerpts, but I urge you to read it! I give the sources:

In English: https://www3.nd.edu/~powers/ame.60611/poincare.pdf

In French: https://en.wikisource.org/wiki/The_Value_of_Science/Text

Hope this helped you

Regards,

BAUDRIMONT Roman

I would admit that we could not discuss about space without taking "time" into consideration. The interconnection between space and "time" does not allow us to consider any of them separately. Starting from the equation (ict)2+x2+y2+z2=0, we should choose which variables we are going to consider as independent. Even if we choose the three space variables as independent we are dealing with spacetime (and NOT space alone) as we could determine time and a point within spacetime is explicitly defined. In the case we presume that time is kept constant, the space independent variables reduce to two.

When we measure the convictional "length","width"and "height" of an object we presuppose that our position (as observers) within spacetime is well defined each "time" and we measure the appropriate lengths (in different "times"), that are related to our position the appropriate "time". It is obvious that we consider the three measurements as if they would represent the difference of the values of the three space variables xobject-xobserver (or yobject-yobserver or zobject-zobserver). Then, we transfer the origin of the reference frame from BB(Big Bang) to the observer(xobserver,yobserver,zobserver) and we mean that these values refer to the independent x,y,z space variables of the object measured. This is far from a valuable conclusion because observer's position differs to each measurement and hence the transfer of the reference frame to an uncertain origin is meaningless.

@ Albert Manfredi : I would be extremely cautious to the practice of addition a new dimension for solving an unresolved problem. The respond to your 3D perception issue is very well documented in Poincaré 's book that Roman Baudrimont mentioned.

@ Roman Baudrimont : The final conclusion of Poincaré 's is " In short, experience does not prove that space has three dimensions; it proves to us that it is convenient to allocate three to him, because that is how the number of the strokes is reduced to the minimum.", and "The empirical question may arise in another form. Is it impossible to conceive of physical phenomena, mechanical phenomena, for example, other than in three-dimensional space? We would thus have an objective experimental proof, so to speak, independent of our physiology, of our modes of representation." (Is it impossible ...? This is what this contribution/thread is about) Furthermore at the end Poincaré mentioned time as an influencing factor in p71 and this, in a sense, weaken most of his excellent arguments.

Hello all: People can view nature as 3+1 D, but nothing is gained by this image. Nature is, according to what experiments tell us, to be viewed at least as 4D. The 4D view does not imply that rotations and boosts are the same, but that they operate in 4D. We live in at least a 4D spacetime, where space can become time and vice-versa.

Physically and mathematically, 3 + 1 D were not the words of Minkowsky [1], and not the difficulty Einstein [1] momentarily had with them. The original reference by Minkowsky on spacetime, began with, "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." [1], a 4D view already 120 years ago.

The difficulty was that the Lorentz transformation on 3+1 D became second-fiddle. They were introduced already in 4D spacetime to keep the Minkowsky metric valid, not the other way around. These false controversies need to stop at RG, and stop being used to create traffic.

The transformations of special relativity, thus, may be defined as those which do not change the 4D Minkowski metric, since all such observers will agree on physical distances and so also agree on physical laws that are expressed in terms of them.

So, we do need to abandon the Lorentz transformations or 3+1 D idea in the basis of "old" special relativity. It is finally time everyone does that. For more than one century, the Lorentz transformation and 3+1 D are NOT an axiom, NOR the basis of special relativity, NOR "abandoned", though, but used as a CONSEQUENT of the 4D spacetime approach, and so validated. This was said and done long ago, it is not a matter of research anymore, nor questions in RG, not even in history.

The answer is, the dimensionality of spacetime is >=4D.

Cheers, Ed Gerck

[1] http://rgs.vniims.ru/books/spacetime.pdf

Dear Ioan,

The critical state of a dynamical system for which there are still cyclic orbits in the phase space, but also an active evolution with a complication of the structure, is characterized by a set of degrees of freedom somewhere in the neighborhood of percolation constant. This is a necessary and sufficient condition for the existence of self-organized criticality; an obligatory condition for the existence of us as part of reality. Hence, the conclusion about the infinite dimensionality of space is inevitable, and, of course, this is a characteristic of the Multiverse. The critical state is characterized by scale invariance, traces of which are easily detected in all areas, for example, all sorts of gauge theories rely on this, processes in biological systems, etc. So, reasoning about integer finite dimensionality are meaningless, even computers in such a world would not be possible.

Hello Christian Baumgarten , You can view that way, the end result will be the same, as it should. Electromagnetic interactions are relativistic, the Maxwell equations comply with Mikowski spacetime in tensor notation. When one tries to use vector notation, as in 3 + 1 D Gibbs notation, the transformation uses vectorial products, which result is not a vector, and one gets pseudo-escalars, polar and axial vectors, strange things.

When we are stationary and looking at a stationary scene we are looking at a planar surface spread out before our eyes. We do not see the third dimension until we move our head and, thanks to parallax, three dimensionality springs forth. By experience we learn to perceive a three dimensionality in the two-dimensional images on our retinas of otherwise stationary scenes.

Time enters as a given because without it nothing would ever happen; the Universe would be static; we would not exist and the dimensionality of spacetime would be a moot point.

The three dimensionality of the space we occupy together with the addition of time appears to be a necessary and suficient condition for there to be a Universe like our own. Abbott's "Flatland" does not provide a viable living space.

We do not pass through the petty-pace of our lives in "spacetime" but rather in a three-dimensional space under ever advancing time. Spacetime is the realm of relativistic speeds and phenomena that are beyond our usual experience. We don't need spacetime and our immediate world is 3D space plus 1D time.

We have to take extra measures to create the spacetime of four dimensions that serves to underlie the relativistic realm of the Universe. This spacetime may be a reality but it is not our reality. We endow spacetime with its properties to create relativistic physics whose applicability in Nature we test against observation and experiment.

We are given a world of three-dimensional space in which we can move about with the passage of one-dimensional time. Space and time are a reality inherent to the Universe in which we live and we have evolved our optical senses to perceive the reality of space. We have gone beyond this initial offering to contemplate other more complex realities to represent those realms within the Universe that are beyond our direct experience. Three-dimensional space and one-dimensional time are our given reality; four-dimensional spacetime is a hypothetical reality of our own making.

Hello Christian Baumgarten , Of course, you can use Clifford Algebra (CA). But then, you are not working with Gibbs vectors anymore, but with multivectors. Better yet, try to check with tensors. There is a physics fundamental reason why tensors are better than CA, when you transform coordinates.

Also, every element of a geometric algebra, such as CA and spacetime algebra by Hestenes, can be identified with a tensor, but not every tensor can be identified with an element of a geometric algebra. In that sense, tensors are more general than pure grade multivectors. The rank of a tensor is not restricted by the dimension of the base vector space like the grade of a multivector. This means that there is no way to represent any 2nd-rank tensor by a bivector, vector, and scalar. Or, in particular, no way that any 3×3 matrix could be represented by a 2-vector in 3D!

So, it is not a 3D here and 1D there, but a union of the two, in 4D. One can transform length into time, and vice-versa. Tensors allow that, CA does NOT do that. And nature may be at least 4D, maybe 6D or more, which tensors can represent, but Cl(3,1) would not.

Similar question: what's the relationship of tensor and multivector . The short answer is that all multivectors are tensors, but not all tensors are multivectors, so geometric algebra is not and cannot be isomorphic to tensor algebra. See https://math.stackexchange.com/questions/806671/whats-the-relationship-of-tensor-and-multivector/1846516#1846516

Hello Dwight Hoxie , You are ignoring the 3D gimbal lock effect, which requires quaternions to solve, not Euler angles in 3D. This applies to stationary 3D, one does not need movement. We are given a world of four-dimensional spacetime, at least. Saying otherwise is a false controversy, not physics, not math, not history.

Dear Ed,

I choose to disagree. We are given a world that we perceive and we invent methods, such as physics, by which to describe and represent it. Anything beyond our perception entails (hopefully, testable) invention, such as the Hamiltonian formulation of mechanics in the variables p and q, quantum mechanics, complex numbers and quaternions. We perceive a three-dimensional space and a ticking clock from which we invent a four-dimensional spacetime to address issues beyond our perception but not beyond our comprehension. "Quaternion" is a noun; have you ever seen one? Nor have I.

Dear Ioan, as for the topic header, I think it will be interesting for you to read about Cayley and Klein ideas:

https://www.researchgate.net/publication/278017020_An_eye_as_the_tool_of_a_choice_of_the_Euclidean_metric_for_the_description_of_real_physical_space

It is even not about dimensionality (that we perceive through a receivier of same nature), it is rather about the kind of space.

Comprehension of the problem together with problems of cyclic orbits of dynamic systems raise even more serious and reasonable consequences for understanding - awareness of the illusory nature of the abstract concept of a point of space. Cyclic orbits (which are equivalent to the appropriate attractions in the system) are enough; then, just dynamic of pure hypergeometry (not curved space with placed objects) is the case. Perceived linearity of structures associated with "discreteness" (cycles) is a separate question.

NB: The continuum, apparently, is similar to the scenery in the theater, which presentable from the visible side only: only what interacts is exist, wherever we look, feel it, or think about it, it always gives the impression of something solid - this forms our natural habitual feeling of macroworld illusion. Interaction is the essence of connectivity/continuity.

Hello Dwight Hoxie , You are free to disagree, and it pays in RG. But software developers use quaternions all the time to represent 3D movement without gimbal lock, and Hamilton explained why quaternions (a noun) are needed, more than one century ago. Maybe video-gamers can show you one :-), even in a video game next to you. Physics also uses them, but there too we find that quaternions have mathematical limitations to represent reality, and octonions too.

Hello Christian Baumgarten , As I wrote, there is no way to represent any 2nd-rank tensor by a bivector, vector, and scalar. Or, in particular, no way that any 3×3 matrix could be represented by a 2-vector in 3D! There is missing structure in geometric algebra. The question is, does it matter in electromagnetism? Not enough to answer by citing results using geometric algebra.

Dear all,

As we should have realized the issue is still open, though nobody feels comfortable to talk about it and tries to transfer the discussion to advanced incomprehensible areas of modern science (e.g. mathematics).

Please put forward to this thread (again if necessary) whether a strong argument against (2+1)D spacetime, compared to (3+1)D spacetime, has expressed here or elsewhere.

@ Ed Gerck : “This was said and done long ago, it is not a matter of research anymore, nor questions in RG, not even in history.”, “The answer is, the dimensionality of spacetime is >=4D.”, or historical evidences do not sound as strong arguments for spacetime 4D dimentionality. I am not a specialist (and do not want either) to be able to deepen into this kind of fields (quaternions, Clifford algebras, ...).

@ Vasyl Fedorovych Komarov : (as a non mathematician, I would try to express my reservations to some points of your reply, to the extend I am able to understand of what you are saying) The different phases within phase space correspond to different entities that could be in cyclic orbit according to the conditions applied (e.g. gas-liquid). Each of these entities exist within spacetime of a certain dimensionality. If we try to represent all of them together, at the same time, within the same reference system we are not dealing with spacetime anymore but an arbitrary reference system of properties, that may include as many properties/dimensions we like.

“The critical state is characterized by scale invariance” This is not so, e.g. liquid needs a certain number of gas(molecules) in order to be characterized as liquid, the same with solid in liquid-solid phase system.

Why: Is it “an obligatory condition for the existence of us as part of reality”, or “the infinite dimensionality of space is inevitable”, or “even computers in such a world would not be possible.”?

@ Christian Baumgarten : Are “interactions” (“dominant” or not) entities by their own, or they relate entities (the second)? Are these entities, interactions are dealing with, have space and time properties as dominant ones (yes)? Where these properties are referred to (spacetime, fictitious or NOT)?

@ Dwight Hoxie : “The three dimensionality of the space we occupy together with the addition of time appears to be a necessary and sufficient condition for there to be a Universe like our own. Abbott's "Flatland" does not provide a viable living space.” Time is included to the three dimensions of space (and not “... with the addition of time ...”). We occupy a 2D space that is changing with time and becomes a 3D. Is there any sound argument against it? It is sometime useful to remember the pre-Einstein point of view but the notion of physical reality tries to change since then.

PS: I think, advanced mathematical theoretical structures obscure the issue, even more if they do not serve to a solution to an existed problem. I may be wrong and science (as a try to explain Nature) is a matter of specialists that could discuss only among themselves.

Hello Ioannis Hadjidakis and all, Thanks for the summary that you see. No controversy exists when 4D Mikowsky spacetime is used. However, as an alternative to Minkowsky spacetime, one replaces the unit imaginary i with the Clifford bivector I for the plane, that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis, trying for 3 + 1 D. That 3D does not work is known already, from Maxwell's equations.

All seems to go very well, but there is no isomorphism between tensors and multivectors, medium with a varying dielectric constant cannot be used, collective effects are not represented, the electron is just a point particle, and one misses the important fusion of space and time, to 4D. This is strong evidence that we do live in at least 4D, as it must be used.

As for static position in what seems to be 3D, where the speed is immaterial, the gimbal lock problem shows that quaternions are necessary to perform unlimited rotations In the space we live. It is well-used in games, and is supported mathematically by hardware video processors. We do NOT occupy a space that is changing with time and is 3D. Even static positioning is affected.

Cheers, Ed Gerck.

".... We occupy a 2D space..."

Earth is flat?

Cheers

I wrote " We occupy a 2D space that is changing with time and becomes a 3D. " but I meant: "We occupy a 2D space the contents of witch are changing with time and seems as if we are living in a 3D environment."

@ Sergey Shevchenko : I do not like to be provocative but Universe is flat so why not an entity within it (e.g.Earth) ... . It is easy to mix things for making a distasteful soup.

What we "see/detect" consist our detected physical reality (2+1)D. All the rest exist but in a "virtual" form, that e.g. means that their wave-function has not "collapsed" ...

Yes, these threads in RG tend to acquire lives of their own and often depart from the original question and sometimes produce angry charges and countercharges. As scientists we should be above such petty antics but we also are human beings with all of our egos and failings. We should realize that the fate of world likely does not depend on the outcome of our debates and deliberations here in RG's realm. Hopefully, however, we learn from others' points of view and hone our own arguments or, possibly, abandon them. We are here to learn, to meet and share with others and, above all, enjoy ourselves in the pursuit of knowledge, beauty and truth.

Now about the issue at hand. There seems to be two questions under consideration. The one in bold print concerns the dimensionality of spacetime and the second deals with the rise of the human perception of three-dimensional space.

Perhaps there were others before him but I beieve it was Minkowski who invented the four-dimensional spacetime of three space dimensions and one time dimension that, thanks to relativity, we have come to love or hate depending on our persuasion. This is how the spacetime we use today arose.

I don't quite understand the argument regarding the two-dimensional surface of a sphere. If we observe the surface from the center of the sphere, we see before us a three-dimensional curved spherical surface in three-dimensional space. The surface attains two dimensions only if we the observers are in the surface, e.g., like us on the surface of the Earth with the two-dimensions of latitude and longitude. Even so the two-dimensional surface of the Earth is embedded in three-dimensional space. Our two eyes give us depth perception as well as the flat composite image of that that is in front of us.

Dear Dwight Hoxie,

You said:"I don't quite understand the argument regarding the two-dimensional surface of a sphere. If we observe the surface from the center of the sphere, we see before us a three-dimensional curved spherical surface in three-dimensional space."

From Wikipedia: "Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point)."

For our case (surface of sphere) in order to determine the position of a point on it two values are required (x,y), or latitude and longitude if you prefer. The same for the plane: in order to determine a point on it two(2) values are required (x,y). If we stay outside of the structure (the surface of sphere or the plane) three values are required in order to determine the position of a point on the structure, because the position of the structure (the surface of sphere or the plane) has to be determined in relation to the new hosting super-structure. This could be continued endlessly. After all, when we say that a spherical surface is 2D we assume that we are on it and when we say that spacetime is (2+1)D we assume that we are within it (spacetime).

PS: as a guide, line/length denotes 1D, surface/area denotes 2D and geometric solid/volume denotes 3D. (e.g. the periphery of a circle is a line, hence 1D. The point is 0D because we assume we are on the point itself and hence no value is required for determination of its position, although three values are required to determine its position in the 3D space(time) super-structure.)

Dear Dwight Hoxie,

If you refer to the spherical surface on witch the photons are - time t - before they are detected by our eye, yes our eye is not on the spherical surface and the required parameters should be three. However, these parameters obey to the equation of the sphere (x2+y2+y2=R2=(ct)2). So any two(2) parameters are enough for any point on the spherical surface to be determined (the third one is found by the equation). So, the spherical surface is again 2D. I hope this is more clear.

Hello Ioannis Hadjidakis , Dwight Hoxie , and all, There is another meaning to "two dimensions" in physics and electromagnetism, based on the fact that special relativity only requires two axes, the orthogonal and parallel directions to the relative velocity vector.

P.S. To Dwight: no comment on the 3D gimbal lock problem in my posting?

Dear Ed,

Thank you for all of your comments! I must confess that , no, I am not familiar with the "3D gimbal lock problem" . Physics (and mathematics) are daunting to grasp in their foundations and keeping up with their advances nigh unto impossible. My intended point is that there is the raality that we can perceive directly through our senses that we hope and trust is congruent with the "reality" by which the Universe is constructed and operates. We can go beyond this given reality to invent through our intellectual abilities new constructions like quaternions, Hamiltonian p's and q's, and the imaginary unit i that we apply to problems that have basis in our given reality but whose solutions are framed in our intellectual constructions. Thus we can construct a quark that we cannot see, touch, smell, hear or taste. the same is true for quaternions, methinks. The real number pi = 3.1415926 . . . is part of our inherent reality since it is just the ratio of the circumference a circle to its diameter but i such that i2 = -1 is of our invention.

Hello Dwight Hoxie , Thanks. A good reference, with context, is at https://youtu.be/mHVwd8gYLnI (long) and https://youtu.be/N5PDboNJwks (short).

@ Ed Gerck : SR uses min 2 dim. (x-distance vs time) but it uses also 3 dim. x,y distances vs time, e.g. light cone), and more than 3 dimensions (as physicists like to use), x,y,z,... vs time. "3D Gimbal lock problem" (and quaternions as well) is another indication that orthogonal systems are not the proper one for explaining Nature. These extra subjects that certainly are related to the dimensionality of space but in a very indirect way could be discussed to separate threads. (https://www.youtube.com/watch?v=oj7v3MXJL3M)

@ Dwight Hoxie : In our attempt to explain physical phenomena we invent new structures and relations that we suppose they could be part of our reality. Are numbers (e.g. 1,2,6,...) part of reality or of our invention? ( title for an interesting thread)

Orthogonality is the relation that connects the two forms/frames of reality (wave/particle, energy/matter, R-numbers/I-numbers, electricity/magnetism, "symmetry"/"super-symmetry", Black Hole (BH)-interior/BH-exterior, ...).

But all these are out of the scope of this thread: What is the dimensionality of spacetime?

Hello Ioannis Hadjidakis , and all: My first posting may be hidden above, in the RG system, but the answer is well-known, for 120 years. No controversy exists when 4D Minkowsky spacetime is used. So, the dimensionality of spacetime is, at least, 4D.

(6-9-2018) What is the dimensionality of spacetime?

Please add (your) rational argument(s) about the issue and not your believes. We all know what is considered as the answer but no any strong evidence appears to support it. (Apart from Poincaré's treatise ( in BAUDRIMONT Roman's answer) who - with honesty+objectivity - states also the weakness of his conclusion).

Dear Ioan, under phase space I used the common meaning from theory of dynamic systems - that is, space of states of the system. For an unclosed system, it evolves along with the system, in fact, it is an extended phase space.

Any phase transitions, including (thermodynamic), of course, are directly related to changes in the degrees of freedom of the system and, accordingly, to the rearrangement of topology, but I meant common mechanism.

For example, if the rails is bent at point - this will lead to the catastrophe, the transition between rails with jump of the second derivative (from the straight line to the circle) can be easily felt with the body being in the train by sudden occurrence of acceleration, the third derivative in kinematics is called a jerk, what on good railroad track must be also taken into account - the question of the kind of manifestation of a phase transition for mechanical system is only a matter of how we, as an observers, are related to the changing degrees of freedom: something is noticeable, on something the existence of the structure of our order of complexity seriously depends (the constancy of blood temperature, etc.), something is very dangerous and leads to dissipation of the most higher levels of order (transition to the ionized state of the plasma, for example). The mankind is an extremely fine-tuned ordered superstructure on a huge (or may be it is better to say, complete) list of critical parameters.

As for the space-time - it is a matter of specific parameters undergoing a change. Obviously, this should affect all possible order parameters that are hierarchically dependent on the degrees of freedom undergoing drastic change. It is obvious that for an ordered structure of a human it is better not to experience this particular transition (at least in one of the ways).

It is difficult to discuss this if you have narrowed attention to a very limited ontology localized within the demarcation of the physics section.

Hello Ioannis Hadjidakis , The answer is >=4D. This is not a matter of personal belief, it is NOT a mathematical choice, anyone can empirically test, it is well-well-known for more than 120 years, and the cause of "corrections," not solution, by using spurious things when ignored.

Hello Mohamed Hassani

For usual life 3D is sufficient with consideration of time as a separate quantity. Relativity needs 4D where time multiplied by light speed is treated as a fourth special dimension, but one that is imaginary, the negative of a square.

At the highest energy where QM can not be ignored, 4D is no longer sufficient. Researchers require 10D in a locality, 11D in a region, and super symmetry of 12D on a cosmic scale.

In other threads I have shown how all of the higher dimensions can be constructed and measured in a reference frame composed of velocity base units instead of distance base units. These larger frames are needed in deep space at high speed.

Hello Jerry Decker and all: For about 120 years, physics knows that, for usual life, 3D is NOT sufficient, and consideration of time as a separate quantity is false.

Dear Jerry Decker,

“Relativity needs 4D where time multiplied by light speed is treated as a fourth special dimension, but one that is imaginary, the negative of a square.” Is this fourth special dimension independent of the other three? If yes, How the equation 0=x2+y2+z2+(ict)2 (for the reference, frame of an “observer”, with origin the singularity (BB)) could be valid? If no, Why the dimensionality of spacetime is said to be (3+1)D and not (2+1)D?

“At the highest energy where QM can not be ignored, 4D is no longer sufficient.” Why? Is there any physical phenomenon that is not possible to be explained by QM by using 4D (or 3D) and is explained by using >=10D?

Thank heavens, a bottle with a good cognac has 3D+1 dimensionality…

Cheers

(11/9/2018) I wonder whether Physics can go further with such a basic issue open and such weak arguments for this.

Good luck to everyone

Dear all,

If we accept that positive numbers (and specially the natural ones (N), for our quantized Universe) are the only ones needed for Nature to be described, then the dimensionality of spacetime is concluded to be (2+1) and it is explicable why: a) the observable Universe is the one forth of the Whole, b) Black Hole horizon (2D) includes all its information, c) future is related to the past through present (and vice versa), d) we can not observe backward in time deeper than half of Universe's life, e) the inverse-square law is applicable (and not the inverse-cubic one), f) imaginary realm (e.g. I-numbers) is half of our whole “reality”, g) ...

1a. Let us suppose that the space of our present Universe is the flat equilateral triangle that is the base of a triangular pyramid whose edges from its apex are the axes of a coordinate system (with origin pyramid's apex) and any observer is always on the centroid of this flat equilateral triangle. According to this set-up, observer identifies a point within his Universe by three coordinates (x,y,z) that seem to be independent of each other. So, he may be mistaken that his Universe is a 3D-space (although it is 2D).

Is this the cause of our misconception about our (3D)-space?

(24-2-2019)

1b. Let us consider the above set-up not for a pyramid but for one of pyramid's face. Our coordinate system has reduced to two axes (let us call them R and I) and our present Universe is getting linear. Also, let us suppose that the two axes are perpendicular to each other (see fig). Now, observer is at point O, by definition (following the shown red world-line as time pass) , and his present Universe is extended on the red line., from +1 (arbitrary) to -1. Observer (O) may define his present Universe by two coordinates (R,I) or by one “coordinate” (X) on his present Universe, due to the special symmetry that these observer-axes have at this case (on the same line). “Coordinate” X is related to observer and not to the Universe (as a whole, not only the present observer's Universe) where coordinates R and I are referred to.

It is interesting to examine this observer's X “coordinate” further. Let us suppose that the range of possible values on X axis are from -1 to +1 (arbitrary). In order to connect this model to QM, let us suppose that the horizontal axis R is the Real one and the vertical axis I is the Imaginary one. It is evidence that there is a normalization factor between observer's X-unit and R-unit (or I-unit) that is equal to 1/2.5. Note that a) point Λ might be defined as (Λr,Λi), or as Λ= Λr+iΛi , b) Λr+Λi=1 and c) observer's present universe is 1D-space with two(2) coordinates.

PS: It is not the right place to extend this model further but keep in mind that 1/2.5 is a normalization-factor to transform observer's X-unit to Universe's R(I)-unit. This factor would be different if R and I axes were not perpendicular (see fig).

(27/2/2019)

The answer on this thread’s question see the last SS post in the thread

https://www.researchgate.net/post/What_is_Spacetime_how_does_space_time_work ,

and in https://www.researchgate.net/post/Is_time_particulate

Cheers

The SS post in

https://www.researchgate.net/post/Does_dimension_relate_to_cosmogenesis

is relevant/useful for this thread.

Cheers

How the axial Doppler shift should add dimensions to observed time at relativistic velocities:

Independent of the MM experiment: The Fourier time series proves all Doppler shifts change observed time and adds dimensions to time.  It does this by breaking any signal into sine and cosines if the frequency of each are multiplied a Doppler shift frequency ratio,  each value now happens at t/K instead of t. Therefore by superposition it is true for the original function.   That adds dimensions to time because the axial shift is directional. That impacts a lot of relativity. Also, we sense all things by forces traveling at the speed of light and have Doppler shifts. For the full proof and another different one and some impacts open this link:

https://drive.google.com/file/d/16oJTvHAaV6TYPXMtGdz-NBqc1RW3yY-n/view?usp=sharing

The importance of that is for most angles the axial shift is larger than the transverse; the axial shift is directional giving direction to all things it affect and we measure all things by forces whose transmitted energy have Doppler shifts.The source of Einstein and Lorentz NOT giving the axial shift that power is Lorentz wrote about aether winds which cancel each other out in the axial direction. Which enabled isolation of the transverse shift. But after the light in the Michelson experiment hit the first stationary mirror, the Doppler shift puts it in the stationary reference plane. No more shifts to cancel the first. Also in most situations the axial shift exists.

Link to a paper describing an omission in the application of relativity to beams in most books: https://drive.google.com/file/d/17wia8LJNxjRAnDqK_XwAOABK5yqs8qM/view?usp=sharing

This link is to a paper on: One important constant is NOT invariant during change of reference frame: https://drive.google.com/file/d/1ljTtjvx5HiFi3vguznj8p-8XRtXa85NV/view?usp=sharing

Thank you for your time and I hope the above gave you some new ideas. Samuel Lewis Reich [email protected]

The SS post in

https://www.researchgate.net/post/What_is_the_nature_of_time

- is relevant /useful to/for this thread.

Cheers