The problem of infinite thinness in dimensional interaction

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

The problem of infinite thinness in dimensional interaction

Probably starting with Flatland it has been very popular to explain higher dimensional concepts by analogy to an interaction between beings in 2 space and 3 space.
To me a true 2D universe has zero extension in the third dimension and for all practical purposes (even if such a world did exist inside our 3D space) it would be impossible for either world to interact with the other except in the most hypothetical thought experiment manner.
For instance, the atoms (and everything else) in the 2D world must be 2D and somehow confined to the plane of their universe.
The idea that we could take a 2D object that is infinitely thin and turn it upside down to reverse it in 2D space so that it became its mirror image is absurd.
We are talking INFINITELY thin, not just very thin. ZERO thickness
Supposing we could even detect an infinitely thin being... and with god-like powers we succeed in flipping it out and back into its own dimensional space by breaking and then restoring whatever forces were confining it to that space... the idea that with infinitely thin eyes it could see anything of its own world from above is again ridiculous.
I know some people have thought about his as I have seen it suggested that a two dimensional world should be allowed some degree of 'thickness' in the third dimension.

My point here is not really about the use of the metaphor in order to explain the concept of dimensions -
it does seem to be a useful way for beginner to understand the concept, and some geometrical progressions help to understand higher dimensional geometries.
My real point is the question of whether our third dimensional space has any thickness in any higher dimensions.
I am not referring to string theory, although I suppose that is the first thing that comes to mind.
I am questioning whether anything that has ZERO extension in a particular dimension can be considered 'real' from the perspective of that dimension.

Holographic theories that propose that the universe is a holographic projection of information contained in the 2D boundary of the universe seem to me to be the opposite of common sense.
As I see it, it is higher dimensions that give reality to lower dimensions not vica versa.
Please excuse the inadequacy of my wording for this next sentence, I hope you will get the gist of what Im saying .
Without the existence of infinite dimensional space in which lower dimensional space can exist, it seems to me that there will always be this problem of thinness to deal with.
Eleven or thirteen dimensional string theory still would mean that in the 14th dimension the universe is infinitely thin.

Okay Ive made a fool of myself now, I think I'll shut up before I sound even more confused

Sphericality
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Re: The problem of infinite thinness in dimensional interact

Well, it's an interesting concept to think about. It does seem a bit circular, like the chicken or the egg.

What you're proposing is that in order to have interaction with hyperbeings, we must have some measurable thickness in n+1 (or whatever) dimensional space. 2D beings can't interact with 3D beings, unless 2D beings are ever so slightly 3D.

I've thought the same thoughts, too. What if Mr. Square wants to stab Mr. Sphere with a 2D knife? Would it even do any damage? Would 2D electron orbitals slide right past 3D orbitals, rendering the stabbing useless?

We've created 2D environments using electrons and graphene, among other such materials. Only, it's the rules of interaction that's confined to 2D. It's still an electron, with unknown number of dimensions.

The best way I like to deal with this apparent issue, is to assume atoms are infinite dimensional. They can stack into arrays of however many dimensions as you want. Where, perhaps it is our space that's truly 3D, giving us 3D slices of matter (stacked into unknown number of dimensions) that we perceive in real life.
in search of combinatorial objects of finite extent
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Re: The problem of infinite thinness in dimensional interact

I like the idea that the basic building blocks (call them atoms for the moment) should be infinite dimensional.

I am in a sense proposing that infinity is the prerequisite of quantity.
That nothing is 'real' unless everything is real.
That it is the relationship to Infinity that allows us to borrow our reality from it.

And therefore finite things (such as finite geometries) should be considered to be cross sections, projections, or surfaces of infinite things.
If I imagine Flatland as being the surface of a 3d ball it makes both the problem of confinement to its two dimensions,
and the problem of its infinite thinness, seem more intuitively approachable.

Sphericality
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Re: The problem of infinite thinness in dimensional interact

Sphericality wrote:Probably starting with Flatland it has been very popular to explain higher dimensional concepts by analogy to an interaction between beings in 2 space and 3 space.
To me a true 2D universe has zero extension in the third dimension and for all practical purposes (even if such a world did exist inside our 3D space) it would be impossible for either world to interact with the other except in the most hypothetical thought experiment manner.
For instance, the atoms (and everything else) in the 2D world must be 2D and somehow confined to the plane of their universe.
The idea that we could take a 2D object that is infinitely thin and turn it upside down to reverse it in 2D space so that it became its mirror image is absurd.
We are talking INFINITELY thin, not just very thin. ZERO thickness
Supposing we could even detect an infinitely thin being... and with god-like powers we succeed in flipping it out and back into its own dimensional space by breaking and then restoring whatever forces were confining it to that space... the idea that with infinitely thin eyes it could see anything of its own world from above is again ridiculous.
I know some people have thought about his as I have seen it suggested that a two dimensional world should be allowed some degree of 'thickness' in the third dimension.

My point here is not really about the use of the metaphor in order to explain the concept of dimensions -
it does seem to be a useful way for beginner to understand the concept, and some geometrical progressions help to understand higher dimensional geometries.
My real point is the question of whether our third dimensional space has any thickness in any higher dimensions.
I am not referring to string theory, although I suppose that is the first thing that comes to mind.
I am questioning whether anything that has ZERO extension in a particular dimension can be considered 'real' from the perspective of that dimension.

Holographic theories that propose that the universe is a holographic projection of information contained in the 2D boundary of the universe seem to me to be the opposite of common sense.
As I see it, it is higher dimensions that give reality to lower dimensions not vica versa.
Please excuse the inadequacy of my wording for this next sentence, I hope you will get the gist of what Im saying .
Without the existence of infinite dimensional space in which lower dimensional space can exist, it seems to me that there will always be this problem of thinness to deal with.
Eleven or thirteen dimensional string theory still would mean that in the 14th dimension the universe is infinitely thin.

Okay Ive made a fool of myself now, I think I'll shut up before I sound even more confused

Yes, we have always been cheating when we look at a 2D Flat World from our 3D perspective. The hard part is conceptualizing a 2D Space without that extra 3rd dimension. How is it possible that there is no Space out in that 3rd direction? But Space is a thing after all and it can be 2D, 3D, 4D etc. or even no Space at all. Try to conceive of no Space? We think of Space as being the ultimate noting, but there is Absolute Nothing which implies no Matter, no Energy, and no Space.
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Re: The problem of infinite thinness in dimensional interact

Yes, this has been thought of before.

The assumption of microscopic (but non-zero) thickness is really is just a mental crutch to help us understand dimensional analogy by positing higher-dimensional beings interacting with lower-dimensional ones. But an actual interaction of this sort poses all sorts of difficult questions. For example, suppose Bob, a 2D being, interacts with Fred, a 3D being. This immediately raises the following problems:

1) Since Fred exists in 3D space, and one would presume that he wouldn't fit inside a 2D space, how does he even get to the space where Bob exists, which is a 2D space? If the 2D space is a completely separate thing from the 3D space, then where does Bob even go in order to meet Fred? If the two are completely separate, no amount of travel within the respective space will ever get anywhere near the other space. So interaction is impossible.

2) Maybe you'll say that Bob's 2D space is really just a subspace of Fred's 3D space. This in turn raises the question: Is Bob's 2D space then really a 2D space? Or is it really just some 3D atoms confined to a 2D-like slice of 3D space, for example, confined between two glass panes, so that macroscopically-speaking, it feels like 2D space, but in reality the atoms are 3D atoms?

3) If Bob's 2D space is actually 2D, i.e., with zero 3D thickness, then how can it even exist inside of Fred's 3D space? Because in order for this 2D space to have zero thickness, it means any atoms that Bob is made of must also have zero thickness. The question then is, if this 2D space actually sits inside of Fred's 3D space, then what holds it together? If the surrounding space is actually 3D, wouldn't any 2D atoms simply fly off out of the 2D plane into the surrounding 3D space? What prevents them from doing so? Furthermore, supposing the 2D atoms are made of the 2D analogs of electrons and protons, what stops the electrons from occupying a 3D-shaped cloud around the protons -- since the surrounding space is actually 3D? So the so-called 2D atoms would actually become 3D atoms, unless they are somehow constricted into 2D by something.

4) So something must be there to keep the contents of the 2D space within the 2D space. Assuming that it sits inside a 3D space, of course. And assuming such a scenario actually exists, of a 2D space that sits inside a 3D space. Then there is the question of how something with zero thickness can even interact meaningfully with anything in the surrounding 3D space.

5) To start with, it wouldn't even be visible, even if such a thing exists, because 3D light can't reflect from it. Why not? Because light carries momentum, and in order for this light to be visible to a 3D being, it must bounce off the 2D subspace in a direction outside the 2D space. Meaning that whatever it bounces off of inside the 2D space must recoil in the opposite direction with the same momentum. But that happens at all, that means the 2D atom or whatever the object is would recoil away from the source of light. This is a big problem, because if the 2D subspace has actual, zero thickness, that means its 3D mass is zero. So the only possibility is that it must recoil from the light at light speed. And all it takes is for one photon to strike this 2D universe, and either the 2D atom would instantly disappear from the 2D subspace, flying off in the opposite direction at light speed, or if something is forcing the 2D atoms to stick inside the 2D subspace, then the entire 2D universe would fly off in the opposite direction at light speed. Not a very promising prospect for Bob and Fred to have a meaningful interaction!

6) Then there is the question of what is the something that holds the 2D subspace in one piece, as opposed to instantly evaporating into the surrounding 3D space. First of all, it must be something that exists in 3D, in order for it to shield the 2D world from, you know, stray 3D photons that would send it flying or cause 2D atoms to randomly vanish (which would be a rather concerning state of affairs for poor Bob!). So Fred wouldn't even get to interact with Bob directly, because this barrier prevents him from doing so, and breaching the barrier would also remove the constraints that hold the 2D subspace in one piece. So as soon as Fried tries to break the barrier, the 2D universe would begin to evaporate away. Again, not very promising for meaningful interaction between Fred and Bob.

7) But how would such a barrier prevent 2D things from passing right through it? If the 2D atoms in the 2D subspace have actual zero thickness, then they have zero volume, so they can't possibly have any meaningful interaction with the 3D atoms that, presumably, is what the barrier is made of. So the barrier wouldn't be a barrier at all: the 2D atoms wouldn't even "know" the barrier is there, and the barrier therefore wouldn't even be able to stop the 2D atoms from flying off the 2D plane, causing the 2D universe to basically disintegrate.

8 ) But if the barrier does interact with the 2D subspace, how can any such interaction not be detrimental to the 2D universe? Because remember, the barrier itself must be 3D, so it would be able to transfer momentum from 3D directions to the 2D world. Which means the 2D atoms will recoil at light speed off the 2D plane. So the only way it could hold together is if the barrier exists on both sides of the 2D subspace, so that any recoiling 2D atoms are stopped by the barrier on the other side, which, presumably, would absorb the excess momentum and pass it off to the 3D world on the other side. But there is another problem with this: such a transfer of momentum would cause the 2D atom to vibrate in a 3D direction. So it wouldn't be strictly 2D anymore, and we're basically back to the case of the 2D world being actually 3D, but just squished sufficiently in the 3rd direction so that it only appears to be 2D macroscopically, but microscopically it's actually 3D.

9) But even if we ignore the previous problem, there is another problem: since the 2D atoms in the 2D universe have zero thickness, what prevents them from just "slipping between the cracks" of the barrier? Because the 3D barrier is presumably made of atoms, and as we know atoms contain lots of empty spaces both inside and between atoms, a 2D atom of zero thickness would easily "leak out" through these empty spaces and evaporate off the 2D subspace.

10) So the only possible solution is for the barrier to be some kind of force field that isn't made of 3D particles that 2D atoms can "leak" through. But even this has problems. The force field must be 100% uniform, otherwise it would introduce a force gradient in the 3rd direction in the 2D subspace it's holding together, and so it is no longer a 2D subspace, but will exhibit 3D effects. Furthermore, this force field must be so strong that it can keep zero-thickness 2D atoms strictly within the 2D plane: imagine if there are two 2D atoms, A and B, flying towards each other, and due to a slight variation in the force, A has a 3D displacement of 0.000001 units (whatever unit you choose, it can be arbitrarily small). Then A and B won't collide and interact in a 2D way, since they have zero thickness. It doesn't matter how small the displacement is, as long as it's not exactly zero, A and B won't collide as they ought to. So then the 2D subspace won't actually behave like a 2D universe at all.

So basically, it is impossible for a strictly 2D universe to exist as a subspace inside a 3D universe. Either the 2D universe exists as a completely separate entity that has no interaction with the 3D universe at all, or the only other possibility is that it's actually a 3D universe, but just confined by something to have 2D macroscopic behaviour, but microscopically it actually has non-zero thickness.

//

And now, the interesting part: we think of our own universe as 3D, because that's how it appears macroscopically. But many of the things I hinted at above are actually observed in our supposedly-3D universe! For example, ever heard of quantum tunnelling? That's when a subatomic particle that really doesn't have enough energy to pass through a barrier, somehow appears on the other side of the barrier anyway, in spite of it all. Does this sound like 2D atoms A and B having non-zero displacement in an external dimension so that they slide past each other instead of interacting with each other as they "ought" to?

Also, we know that all matter is ultimately made of waves, and we have the Schrodinger Equation to describe that. But here's the problem: what is "waving"?! When we're talking about photons, we know that the wave is made of changes in the electromagnetic field. An ocean wave is the fluctuation of water levels -- the water molecules are bobbing up and down. A photon has the intensity of magnetic poles / electric charges fluctuating. But what on earth is "waving" in a matter wave like an electron?! We don't know. As far as we can tell, it has no physical measurement. We know something is waving, but what it is, is anybody's guess. But it does have physical consequences: the squared amplitude of this abstract matter wave is directly proportional to the probability the particle will be found in that location. Now, doesn't this sound like what I described about the 2D subspace sandwiched between two barriers, such that transfer of momentum causes vibrations outside of the 2D universe?

There is plenty more to say here, but it suffices to make my point: there are many signs that seem to hint that our universe is actually not a 3D universe after all, but a higher-dimensional one, except that all but 3 of these dimensions are confined (for whatever reason) to the subatomic scale. If we assume that the universe is actually higher-dimensional, then many of the "strange" things we observe at the subatomic level begin to make sense, they are just natural consequences of higher dimensional space that got compressed into a very small scale. This is basically the underlying thrust of string theory.
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Re: The problem of infinite thinness in dimensional interact

Being infinitely thin means zero length and it belongs in a lower dimension. You are running into a potential 0/0 paradox when you try to apply physics. For this reason space, and everything within it, is essentially 3D (extra dimensions are very thin but nonzero in length).

To follow, consider an analogy:

Consider the rational expression (x^2+2x-3)/(x-1). Suppose we're finding its vertical asymptote, it becomes apparent that there are no such lines. This is because both the numerator and denominator assume a value of 0/0, which is undefined. Simply put, it is no other than the line y = x+3 which has no denominator.

Another is to consider the function x/x, which is defined to be 1 for all x not equal to 0. If we try to put in 0, we get 0/0, a known contradictory value. This shows how dangerous is when applying division by zero. Physics is generally intolerant when it comes to infinity or dividing by zero.

To confirm, consider a 2D being. The 3D volume of a 2D being is zero, because a 2D being can take on the form of a square, which is a cube with zero height. Put density as mass divided by volume. You get a division by zero. In that sense, it is not possible to conceptualize perfect thinness. I think that quantum foam is actually an effect of microscopic extra dimensions and that we are actually in a higher-dimensional space.

In short, an object in a lower dimension will have zero volume in a higher-dimensional space, causing interactions to process division by zero. We can't even challenge it, it's a mathematical fact.
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Re: The problem of infinite thinness in dimensional interact

Division by zero is an artifact of the mathematical model chosen to represent physical phenomenon, not a property of the phenomenon itself. There are ways of dealing with subdimensional objects without running into that kind of problem.

Even in the realm of non-subdimensional physics, there is such a thing as renormalization in various quantum theories, where we "cancel out" infinite terms or terms with potential zero division by mathematical trickery. It doesn't really reflect an actual, physical process called "renormalization"; it's just a device to work around limitations in the mathematical models we use to describe how physics works.
quickfur
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Re: The problem of infinite thinness in dimensional interact

P.S. There exist mathematical devices that can work even in the face of division by zero, such as wheel theory. Just because these things are less well-known doesn't mean they don't exist.

Furthermore, there exist various systems that can deal with infinite or transfinite arithmetic, in cases where you might need to work with infinite thinness. True, these systems are harder to work with than traditional arithmetic, but it is possible to do this in a non-contradictory way.
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