Proof Pertaining to Possible Paths Through 3 Dimensions

[Lynch101]

TL;DR Summary

Are the possible paths from one spatially separated region to another, through 3 dimensions, subject to mathematical proof?

I'll try to phrase this as clearly as possible but my use of terminology might need to be refined. That may be what ultimately comes of this thread, but hopefully the question as I phrase it will make enough sense. I'm not necessarily asking that a proof be provided, rather, I am interested to hear if people believe that such a proof is possible.

I'm not sure if it makes a difference but I'm thinking in terms of 3D modelling of the physical world.

The Scenario
If we create a 3D model using the 'usual' orthogonal XYZ axes and accompanying co-ordinate system. Using these axes we define two regions within the model A and B, such that A and B are spatially separated i.e. there is a non-zero distance between them. The remaining space within the model we can label C, where C is simply defined as not A and B.

Would demonstrating the following be the subject of a mathematical proofs, where 'path' means a line or combination of lines:

1. any path between A and B must necessarily contain a point in C i.e. pass through C;
2. where objects can only travel at a finite speed e.g. </= the speed of light, any line/path from A to B must pass through C;
3. any object which travels from A to B must take a unique path through C;
4. any object which travels from A to B without taking a unique path through C either:

A) Travels instantaneously form A to B​

B) Does not travel within 3 dimensions​

C) Cannot be modeled using 3 dimensions.​

​

​

​

My thinking is that 2 & 3 should be provable while 4 should probably follow by way of necessity. Obviously, I can't say for sure, so I was hoping someone might know/have an better informed opinion.

 

[Mark44]

Summary:: Are the possible paths from one spatially separated region to another, through 3 dimensions, subject to mathematical proof?

I'll try to phrase this as clearly as possible but my use of terminology might need to be refined. That may be what ultimately comes of this thread, but hopefully the question as I phrase it will make enough sense. I'm not necessarily asking that a proof be provided, rather, I am interested to hear if people believe that such a proof is possible.

I'm not sure if it makes a difference but I'm thinking in terms of 3D modelling of the physical world.

The Scenario
If we create a 3D model using the 'usual' orthogonal XYX axes and accompanying co-ordinate system. Using these axes we define two regions within the model A and B, such that A and B are spatially separated i.e. there is a non-zero distance between them. The remaining space within the model we can label C, where C is simply defined as not A and B.

Minor correction: The axes would be X, Y, and Z, not X, Y, and X.
BTW, most of the discussion could be limited to two dimensions -- the arguments would work just as well.

Would demonstrating the following be the subject of a mathematical proofs, where 'path' means a line or combination of lines:

1. any path between A and B must necessarily contain a point in C i.e. pass through C;

Yes, this could be proved mathematically.

[*]where objects can only travel at a finite speed e.g. </= the speed of light, any line/path from A to B must pass through C;

Since we're talking about paths from A to B, the speed of travel of an object along any path is not relevant.

[*]any object which travels from A to B must take a unique path through C;

Not true at all. There are an infinite number of paths from A to B, unless you're talking about the path of minimum length.

[*]any object which travels from A to B without taking a unique path through C either:

[/LIST]

A) Travels instantaneously form A to B​

B) Does not travel within 3 dimensions​

C) Cannot be modeled using 3 dimensions.​

The first point above isn't relevant.
If the "universe" is three-dimensional, as in your setup, then invoking higher dimensions is unrealistic, IMO. As already mentioned, there is no unique path from A to B. Here's a drawing in two dimensions to help explain what I am saying.

My thinking is that 2 & 3 should be provable while 4 should probably follow by way of necessity. Obviously, I can't say for sure, so I was hoping someone might know/have an better informed opinion.

 

[mfb]

You are working with a three-dimensional space, so every path is in three dimensions by definition.
The path can be (but doesn't have to be) within a two-dimensional subspace of that three-dimensional space: You might be able to find a plane in that volume that contains the whole path.

 

[Lynch101]

Minor correction: The axes would be X, Y, and Z, not X, Y, and X.

Thanks for pointing out the typo. I've changed it above.

BTW, most of the discussion could be limited to two dimensions -- the arguments would work just as well.

Ah I see. Just by making motion along one of the axes = 0?

Yes, this could be proved mathematically.
Since we're talking about paths from A to B, the speed of travel of an object along any path is not relevant.

I was thinking, if something traveled from A to B instantaneously then it might not necessarily pass through C.

Not true at all. There are an infinite number of paths from A to B, unless you're talking about the path of minimum length.

Thanks Mark. Just to clarify, by 'taking a unique path' I meant that something which travels from A to B must take one of the infinite number of paths, or a unique combination of the possible paths.

[/LIST]The first point above isn't relevant.
If the "universe" is three-dimensional, as in your setup, then invoking higher dimensions is unrealistic, IMO. As already mentioned, there is no unique path from A to B. Here's a drawing in two dimensions to help explain what I am saying.
View attachment 289054

Thanks Mark, that all makes sense. The idea of an object/system taking a 'unique path' might just be a trivial statement. What I was trying to suggest was that anything traveling between A and B must take one of the infinite possible paths or some unique combination of them.

With regard to invoking higher dimensions, that was more just a case of one possible inference from a model which says that something moves from A to B but doesn't specify the path taken, or specifies that no unique path is taken.

 

[Lynch101]

You are working with a three-dimensional space, so every path is in three dimensions by definition.
The path can be (but doesn't have to be) within a two-dimensional subspace of that three-dimensional space: You might be able to find a plane in that volume that contains the whole path.

Thanks mfb. This would this be the case where motion along one of the axes = 0?

 

[mfb]

It can also move in the plane y=2x or the plane x+3y-7z=0 or whatever else.

 

[Mark44]

I was thinking, if something traveled from A to B instantaneously then it might not necessarily pass through C.

Objects can't travel from one point to another simultaneously. In any case, as long as A and B are separated at some positive distance, the path from A to B must go through C.

 

[Lynch101]

@Mark44 Could such mathematical proofs be applied to the question of 'completeness' in quantum mechanics?

I was thinking that, for a 3D model of a physical experimental set-up to be considered complete, everything within the experiment should be represented in the model at any given time or at all times.

Where we have a preparation device (represented by the region A, above) and a measurement apparatus (represented by region B) the quantum system would have to travel some path through C.

A single line path would suggest the quantum system had a well defined value for position at all times, while a combination of paths would mean that it didn't have a single, well-defined value at all times. But what would it tell us if we had a model which did not represent the path taken by the system from A to B? Would the following be reasonable conclusions:
1) The 3D model is incomplete
2) The system traveled from A to B instantaneously*
3) The system travels from A to B via another dimension

*I know this isn't a realistic possibility, I've just included it for the sake of providing a more complete set of conclusions.

 

[Lynch101]

Objects can't travel from one point to another simultaneously. In any case, as long as A and B are separated at some positive distance, the path from A to B must go through C.

Thanks Mark. I personally don't think it is a realistic possibility that objects could travel from A to B instantaneously. I was including it just in case it might be a possible conclusion.

Would you be familiar with any formal proof of that proposition? It's not that I would doubt it, since I can't imagine how it could be any other way, but I would be interested in reading one just for the sake of seeing how something like that would be proved.

 

[Mark44]

I personally don't think it is a realistic possibility that objects could travel from A to B instantaneously. I was including it just in case it might be a possible conclusion.

Would you be familiar with any formal proof of that proposition?

Let d > 0 be the distance from A to B. If t = 0, the well-known formula $d = r \cdot t$ implies that d = 0 no matter how large r (the rate, or speed) happens to be.

 

[Lynch101]

Let d > 0 be the distance from A to B. If t = 0, the well-known formula $d = r \cdot t$ implies that d = 0 no matter how large r (the rate, or speed) happens to be.

Ah cool. Very straight forward. Thank you!

Would you have any thoughts on the post about re: applicability to the question of 'completeness' in quantum mechanics?

 

[Mark44]

Would you have any thoughts on the post about re: applicability to the question of 'completeness' in quantum mechanics?

Nope, not my area at all.
Going back to your original question, suppose you have a balloon, and mark the north and south poles with N and S using a marker pen. The distance between them is $\pi r$ (half the circumference) along any longitudinal path on the balloon's surface. However, if you squeeze the balloon by pressing the points N and S together, you can shorten the distance between N and S to essentially zero (in three dimensions), while their distance in the two-dimensional balloon surface is still $\pi r$.

 

[Lynch101]

Nope, not my area at all.

Would you have any thoughts on the completeness of 3D models in general? As in, their completeness as models of the physical world/experimental set-up?

Going back to your original question, suppose you have a balloon, and mark the north and south poles with N and S using a marker pen. The distance between them is $\pi r$ (half the circumference) along any longitudinal path on the balloon's surface. However, if you squeeze the balloon by pressing the points N and S together, you can shorten the distance between N and S to essentially zero (in three dimensions), while their distance in the two-dimensional balloon surface is still $\pi r$.

Would the distance in 3D not remain $\pi r$ also?

If the pen has to stick to the surface of the balloon then, unless the two points (A and B) are in contact, the pen would still have to follow the surface of the balloon, which would be $\pi r$, wouldn't it? If the two points are in contact, would that not reduce the distance in 2D to essentially zero, also?

 

[Mark44]

Would you have any thoughts on the completeness of 3D models in general? As in, their completeness as models of the physical world/experimental set-up?

I'm not sure what you mean by "completeness of 3D models." If you model some physical world object, the model will almost always be an approximation to the object, so won't be complete.

Would the distance in 3D not remain πr also?

No, it would be essentially zero (or two thicknesses of the balloon).

If the pen has to stick to the surface of the balloon then, unless the two points (A and B) are in contact, the pen would still have to follow the surface of the balloon, which would be πr, wouldn't it? If the two points are in contact, then would the distance in 2D not also be reduced to essentially zero?

The 2D distance measure would be $\pi r$, since we're forced to measure along the 2D surface of the balloon. In 3D, we have another dimension along which to measure, so with that in mind, the two points are in contact (or adjacent, two balloon thicknesses apart).

 

[Lynch101]

I'm not sure what you mean by "completeness of 3D models." If you model some physical world object, the model will almost always be an approximation to the object, so won't be complete.

I was thinking more along the lines of being able to represent objects/systems, even approximately, in a model.

For example, we could represent the preparation and measurement devices in an experiment. If we know that an object/system travels from one to the other (and therefore follows some path between the two) but we don't/can't represent the path it takes, can we consider the model a complete representation of the physical experiment? Or, put another way, at a series of individual times if we cannot represent the position of the system even though it must be following some path between A and B, can we consider the model complete at those values for t?

No, it would be essentially zero (or two thicknesses of the balloon).
The 2D distance measure would be $\pi r$, since we're forced to measure along the 2D surface of the balloon. In 3D, we have another dimension along which to measure, so with that in mind, the two points are in contact (or adjacent, two balloon thicknesses apart).

I can see what you're saying. I was probably focusing too much on the line drawn along the surface of the balloon.

 

[Mark44]

For example, we could represent the preparation and measurement devices in an experiment. If we know that an object/system travels from one to the other (and therefore follows some path between the two) but we don't/can't represent the path it takes, can we consider the model a complete representation of the physical experiment?

I'm thinking in terms of a block diagram, with the prep device being one block, and the measurement device another block. If we can't determine what path, if any, something (a signal?), takes between the two blocks, the model is obviously incomplete.

Or, put another way, at a series of individual times if we cannot represent the position of the system even though it must be following some path between A and B, can we consider the model complete at those values for t?

I'm still having a hard time trying to understand what you're asking here. To me, the term "system" implies the whole apparatus - prep device A, measurement device B, and any pathways between the two. You can have some object or collection of objects (bunch of electrons or photons, or whatever) traveling from A to B. If you don't know what path they take or where they are at any given time, your understanding of the model is incomplete.

 

[Lynch101]

I'm thinking in terms of a block diagram, with the prep device being one block, and the measurement device another block. If we can't determine what path, if any, something (a signal?), takes between the two blocks, the model is obviously incomplete.

Thanks Mark. That's how I was thinking of it as well and that is the conclusion I came to also. I was told that was simply a matter of opinion but I suspected it would be a matter of mathematical proof.

I'm still having a hard time trying to understand what you're asking here. To me, the term "system" implies the whole apparatus - prep device A, measurement device B, and any pathways between the two. You can have some object or collection of objects (bunch of electrons or photons, or whatever) traveling from A to B. If you don't know what path they take or where they are at any given time, your understanding of the model is incomplete.

This would be my reasoning as well. I was also thinking it would have implications for the question of 'completeness' in quantum mechanics because there are interpretations of QM which remain silent on the path taken by the quantum system from preparation device to measurement apparatus. My reasoning is that this would render any such 3D model incomplete.

Some would argue that we cannot talk about the path the quantum system takes because such reasoning does not apply to quantum systems. My thinking is that such reasoning does apply to 3D models and so, if quantum systems operate in 3 dimensions, they must 'play by the rules' of 3D models. If they don't then my reasoning would be, either the 3D model is incomplete or quantum systems travel in other dimensions.

 

[Mark44]

If they don't then my reasoning would be, either the 3D model is incomplete or quantum systems travel in other dimensions.

"... or quantum systems travel in other dimensions" -- sounds more like science-fiction than science to me.

 

[Lynch101]

"... or quantum systems travel in other dimensions" -- sounds more like science-fiction than science to me.

I wouldn't disagree. But if someone were to maintain the above, it would seem that would be one of the only possible conclusions left.