Defining Superstring's Additional Dimensions

[OB 50]

(Just checking:) A (tiny) tube is two dimensions, one compactified, one not.

Simply explain how the following scenario is possible, and I'll shut up:

There is a 2D observer who exists in 2D space (a plane, surface of a balloon, what have you). There is also a 3D observer looking on from a surrounding 3D space.

How does the observer within the 2D space directly observe anything in the 3D space that is not intersecting the 2D space?

 

[Hurkyl]

Simply explain how the following scenario is possible,

There is a 2D observer who exists in 2D space (a plane, surface of a balloon, what have you). There is also a 3D observer looking on from a surrounding 3D space.

I don't see how it's relevant to anything I've been talking about... in particular, it has nothing to do with my correction that the tube is two-dimensional, not one-dimensional.

I was assuming you meant the surface of the tube -- i.e. a cylinder. If you were referring to the interior of the tube, then that's three-dimensional, not one-dimensional, and none of them are compactified (just incomplete).


Anyways, the first thing to pay attention to is that we're making up laws of physics as we go along -- I'm pretty sure this scenario cannot be the product of, e.g., general relativity.


If we assume that everybody's vision works via photons, then the 2D observer really gets to see all of 3D space -- a photon emitted from any point in 3D space can reach the eye of the 2D observer. The only difference is that field of vision is a curve, rather than the surface we get to view. If there was a rectangle with red and white stripes, then our 2D observer would just see a pink interval -- unless the rectangle had certain orientations which would let our 2D observer see alternating red and white intervals, possibly with pink blotches at the boundaries.


However, the fact the 2D observer interacts with photons means that he absorbs their momentum. But since we've assumed that he is constrained to live in some 2D surface, this means that interaction is limited to deflecting photons, not absorbing them -- any photon that passes through the body of our 2D observer retains all of its momentum in the normal direction.


If we assume 3D conservation of momentum, that means either the 2D surface is perfectly flat, or it means that whenever our 2D observer passes through a curve, he emits some sort of energy into 3D space to conserve momentum.

If his 2D surface is curved, then he could take advantage of them to change the angle at which he is oriented in 3D space, and use them (at least to some extent) to resolve the position of things perpendicular to his surface.


Okay, I've had enough stream of consciousness speculation for the moment, so I'll stop here.

 

[OB 50]

The process you just described has the 2D observer inferring the properties of 3D space through interactions and measurements exclusively within his own 2D space. Never does he directly observe any element of 3D space.

It is still impossible for the observer to orient himself within the 2D space such that he can look in the direction of travel of the photons as they are traveling through the 3D space. He can never look directly "outward" into the surrounding 3D space.

This seems self-evident and basic, not stream of consciousness speculation.

I'll leave it at that.

 

[apeiron]

This is not to say we cannot detect the presence of additional or compact dimensions, but we cannot directly observe them. The compact dimension which appears to be a line (but is actually a "tube") will always appear to be a line, no matter how close or magnified we get.

.

OK, so the question comes down to whether we can or cannot in practice observe the compact dimensions. Planck-scale limits would seem to constrain us - we cannot expend the energy that would be required to "magnify" our observations to the point the dimensions are directly visible.

Probably correct. Though there may be indirect effects that are observable at more achievable temperatures. So the particle zoo and its symmetries may indeed be the outward expression of these hidden from us dimensions.

 

[Haelfix]

"Never does he directly observe any element of 3D space."

And we've been trying to tell you that's incorrect for 2 pages worth of completely elementary physics. If I'm confined to live on the 2d plane of my computer screen, I still absorb photons from the full 3d geometry around me. I not only indirectly deduce that the world is 3dimensional, I also observe it (whatever that means) and I further can infer that I am simply living on a constrained subspace. In fact, your very eyes (the surface of which is more or less 2 dimensional) does the same exact thing, and its your brain that models and extrapolates the information it receives as 3dimensional.

In the compactified dimension case (say a 2d cylinder or tube), one possible sublety is that sometimes physics as we know it isn't necessarily allowed to propagate along the compactified dimension. So for instance EM/Strong/Weak force might only be allowed to live on the uncompactified dimension, whereas say gravity might be allowed to propagate along both dimensions. The observation of the extra dimension is to simply conduct an inverse square law experiment (or whatever the r dependancy is), to probe the radius of the hidden dimension. So, if we had eyes that could interpret gravitational information (rather than electromagnetic information), we would see 2 dimensions if we looked closely enough, but if we had normal eyes that are sensitive to photons, we would only see 1.

 

[OB 50]

"Never does he directly observe any element of 3D space."

And we've been trying to tell you that's incorrect for 2 pages worth of completely elementary physics. If I'm confined to live on the 2d plane of my computer screen, I still absorb photons from the full 3d geometry around me. I not only indirectly deduce that the world is 3dimensional, I also observe it (whatever that means) and I further can infer that I am simply living on a constrained subspace. In fact, your very eyes (the surface of which is more or less 2 dimensional) does the same exact thing, and its your brain that models and extrapolates the information it receives as 3dimensional.

At this point, I can tell that what you guys think I'm saying, and what I'm actually trying to articulate are two entirely different concepts. Short of being in the same room and drawing pictures, I don't know what else to do.

There have been a lot of examples given, none of which directly refute, or even correctly address the idea I'm actually trying to discuss. I acknowledge that this is mostly due to my lack of proper terminology preventing me from accurately expressing my ideas, but I won't concede that I'm incorrect simply because I've been misunderstood.

Your example of the human eye is an instance of 3D objects arrayed in a 2D configuration. There is a profound difference between this and the idea of a 2D observer who exists within 2D space.

The 2D observer doesn't get to look out into 3D space like General Zod in the Phantom Zone. All of his observations and movement are constrained to the plane of his 2D world. Likewise, do not think of him as a 3D observer squished down to 2D, because that's what it sounds like everyone is assuming. The construction of a 2D universe, and any subsequent observers contained within would be entirely foreign to us.

Perhaps he can construct a vast array of detectors which entirely fill an area of his 2D space in order to detect the presence of photons as they pass through. Fair enough, but what will he actually observe? Likely, he will detect a number of point-like particles popping in and out of existence as the photons pass through the plane. He will not be able to directly measure any aspect of the photon's 3D velocity, but only the deflection of the 2D elements with which they interact while passing through.

He may correctly deduce that these photons originate from a higher dimensional space. He may even deduce that there is an additional degree of freedom afforded by this higher dimension, but the observer himself is never aware of his own constrains. Within his 2D space, he is free to move in all directions that exist.

Now, he sets out to find this extra dimension. He assumes that since the particles appear to originate and terminate as points, the extra dimension must be compact. It must be really really tiny. So, he starts constructing a super powerful microscope as a start. It doesn't matter, because he can only magnify in the directions afforded to him by the constrains of his 2D space. He will never directly "see" the third dimension. What aspect of the third dimension could he possibly find by looking at smaller and smaller pieces of 2D space?

Maybe extra dimensions are compactified; maybe they aren't. I'm not even proposing one over the other. My example attempts to illustrate that even if they are not, they will appear to be to us. Either way, no matter how much we magnify, or how hard we smash particles together, if we exist in 3D space we will always be looking at 3D space.

 

[Hurkyl]

Now, he sets out to find this extra dimension. He assumes that since the particles appear to originate and terminate as points, the extra dimension must be compact. It must be really really tiny.

First off, I really don't see the logic to that argument at all.

Secondly, I hadn't noticed you were using the word 'compact' in the English sense -- its use as a mathematical term has pretty much nothing to do with size. You can look up the definition yourself, the bit relevant here is that there are only two kinds one-dimensional manifolds: open intervals and circles. Circles are compact, open intervals are not.

The difference is important: If the extra dimension is a small open interval, then there is absolutely nothing keeping matter from remaining inside that interval, and GR puts no constraints whatsoever upon what happens at the endpoints. If the extra dimension is a small circle, then we don't have any of those problems.

Interesting fact: GR is 100% consistent with the hypothesis that the universe is nothing more than a ball 3' in diameter surrounding your head, and stuff just appears on the boundary as necessary to maintain the illusion of an extensive universe.

 

[IqbalGomar]

Hello again, all. Glad to see this thread has remained alive, even if it has gone a bit off topic. I've been looking into the Standard Model a bit over the last few days, and I feel like I'm a lot closer to understanding what I was hoping to understand about all this. I've seen the notation SU(3)xSU(2)xU(1) a great many times without ever really knowing what it meant, and now it makes sense to me, so thanks for the insight. If my presumption of understanding has not misled me, the model is something like the following.

D1, D2, D3: Space
D4: Time
D5: U(1), electromagnetic force
D6, D7: SU(2), weak force
D8, D9, D10: SU(3) strong force, corresponding to red, green, blue

SU(3) and the operations of QCD are fairly intuitive to me. The behavior of the strong force seems complex, but its rules are highly logical and easy to understand if you think of the three color-anticolor designations as three mutually-orthogonal continua.

I remain deeply confused about SU(2) though. I've always had trouble forming an intuitive picture of the weak force, since its operations relate to events that are so far removed from everyday life. So far I understand that when a proton changes into a neutron, it emits a weak gauge boson, thereby turning an up quark into a down quark. But the boson quickly decays into an electron and a neutrino, so the weak force is sublimated into electromagnetic force.

It is very confusing to me that the weak force is considered a force unto itself, as it seems more like a mediator between the strong and electromagnetic realms. Why does it require its own doublet symmetry? What are the two parameters of SU(2)? If U(1) has one parameter corresponding to the strength of the electromagnetic field, and SU(3) has three corresponding to the color identities of quarks bound into triplets, what does the (2) in SU(2) signify? Does it have something to do with the chiral symmetry breaking of the force? I would appreciate it if someone could put this into intuitive terms I might understand. Many thanks!

 

[IqbalGomar]

The more I look into this topic, the more it seems that quantum spin and mass must be independent degrees of freedom corresponding to extra dimensions. Am I really incorrect about this? If I am, how are these identities represented within the 10D manifold?

I am also repeatedly finding descriptions of the extra dimensions as spatial or space-like. Is this simply a misleading description designed to make the idea intuitive to laymen?

I don't feel as if the question I'm asking should require that much technical jargon to parse. All I'm looking for is a list of the identities of the extra dimensions posited in string theory and beyond.

1) Length
2) Width
3) Height
4) Time
5) Electromagnetism
6)
7)
8)
9)
10)

Is this really that tall an order?

 

[Adrian59]

Hello again, all. Glad to see this thread has remained alive, even if it has gone a bit off topic. I've been looking into the Standard Model a bit over the last few days, and I feel like I'm a lot closer to understanding what I was hoping to understand about all this. I've seen the notation SU(3)xSU(2)xU(1) a great many times without ever really knowing what it meant, and now it makes sense to me, so thanks for the insight. If my presumption of understanding has not misled me, the model is something like the following.

D1, D2, D3: Space
D4: Time
D5: U(1), electromagnetic force
D6, D7: SU(2), weak force
D8, D9, D10: SU(3) strong force, corresponding to red, green, blue

SU(3) and the operations of QCD are fairly intuitive to me. The behavior of the strong force seems complex, but its rules are highly logical and easy to understand if you think of the three color-anticolor designations as three mutually-orthogonal continua.

I remain deeply confused about SU(2) though. I've always had trouble forming an intuitive picture of the weak force, since its operations relate to events that are so far removed from everyday life. So far I understand that when a proton changes into a neutron, it emits a weak gauge boson, thereby turning an up quark into a down quark. But the boson quickly decays into an electron and a neutrino, so the weak force is sublimated into electromagnetic force.

It is very confusing to me that the weak force is considered a force unto itself, as it seems more like a mediator between the strong and electromagnetic realms. Why does it require its own doublet symmetry? What are the two parameters of SU(2)? If U(1) has one parameter corresponding to the strength of the electromagnetic field, and SU(3) has three corresponding to the color identities of quarks bound into triplets, what does the (2) in SU(2) signify? Does it have something to do with the chiral symmetry breaking of the force? I would appreciate it if someone could put this into intuitive terms I might understand. Many thanks!

I agree with you final count of dimensions. Earlier comments that SU(3) was eight dimensional appeared to be confusing the number of generators with the dimensions of the group. For SU(n) Lie group in n dimensions has n^2-1 generators. You could envisage each member of SU(3) has a point in an eight dimensional space but then you would have to for consistency regard SU(2) as three dimensional.

However, counting dimensions as: four space-time, one for QED, two for the weak force & three for the colour force; gives ten dimensions for the standard model. As an amateur I am not aware if these suggested ten dimensions correlate with the ten dimensional supersymmetry of string theory, except of course for the ordinary space-time.

Also, I can not find any convincing mathematical derivation of this imperative for all these extra dimensions in string theory. Are they real or just a mathematical device. If they are real are they compactified or macroscopic but outside our awareness like the mythical flat-landers trying to appreciate a third dimension. Finally, if they are compactified space dimensions why doesn't gravity become super strong like the curvature of space at a singularity.

 

[tom.stoer]

I tried to understand string theory for years. I newer saw this idea before!

10 = 4-dim spacetime + dim U(1) + dim SU(2) + dim SU(3)

Are the string guys not aware of it? is it too simple for the Wittens, Vafas, Stromingers etc.? is it completely silly?

I know that J. Baez asks such questions? Anybody here who is in contact with him?

 

[apeiron]

I too was hoping for the right answer from someone more expert.

My understanding was that the three gauge symmetries would be nested. So SU3 would require 6D and decompose (via unstable SU2) to U1. So 6 rather than 3 dimensions to account for SU3. And subset for the others.

Here are a few possibly relevant bits I've clipped from elsewhere.

"Isospin doublets - Isospin doublets look like: (v, e-), (u_R, d_R), (u_G, d_G), and (u_B, d_B). Note particle on the right is always one EM charge unit more negative than the one on the right. This triplet is both colour and anticolour (so giving the nine-ness)."

So we have six quark states and then electron/neutrino would fall out of this symmetry as a lower energy state.

"The two sides of SU(3) matrix – columns give three slots for the colour orientation and then the rows give three anti-slots. So this is why there are nine doublet states (with the ninth self-cancelling to white)."

So again 6D to capture SU3?

And a quote that supports the nesting story I think.

"Symmetry breaking allows the full electroweak U(1)×SU(2) symmetry group to be hidden away at high energy, replaced with the electromagnetic subgroup U(1) at lower energies. This electromagnetic U(1) is not the obvious factor of U(1) given by U(1) × 1. It is another copy, one which **wraps around inside** U(1) × SU(2) in a manner given by the Gell-Mann–Nishijima formula."

So the answer could be thus SU3 takes up three compactified dimensions (with the others each having their own), or six (with the others nested inside). Or none of the above.

I have never seen the answer stated clearly.

 

[Haelfix]

What you guys are trying to do doesn't work for string theory.

The classical Kaluza Klein model is essentially this method, where you get the gauge group from the isometries of the compactified manifold.

Otoh, in String theor(ies), the gauge structure comes from other effects and for consistency reasons, cannot be built up in the exact KK way.

It wasn't a bad guess, but it didn't work out that way.

 

[IqbalGomar]

SU(2) and U(1) as subsets of SU(3)... compelling. If I understand the concept correctly, that does seem to be one way of thinking about it. Since any particle with an identity in SU(3), namely quarks, also necessarily have identity in SU(2) and U(1)... that is to say that quarks also interact via the weak force and possesses electromagnetic charge, but particles that interact via the weak force do not necessarily interact via the strong force, but do necessarily interact via electromagnetic force, and electrons do not have identity in either SU(2) or SU(3), it is as if the dimensions are nested. It's like there are three "tiers" to the dimensionality of the universe; spacetime, which is macroscopic; three dimensions of the electroweak realm, which opens up at a much smaller scale and higher energy; and three dimensions of the strong realm, which opens up at a yet higher energy.

BTW, after doing a bit more research, the two dimensions of SU(2) which I have failed to understand seem to correspond to weak hypercharge and isospin... neither of which I can make sense of intuitively. Are there intuitive descriptions of these properties beyond the maths that describe them, or are they purely mathematical concepts?

So is this more or less correct?

D1) Length
D2) Width
D3) Height
D4) Time
D5) Electromagnetism
D6) Weak hypercharge
D7) Isospin
D8) Red
D9) Green
D10) Blue

Do these ten parameters serve to describe the total state of any particle or system of particles in the universe? Is this a complete list? If not, please enlighten us.

 

[IqbalGomar]

What you guys are trying to do doesn't work for string theory.

The classical Kaluza Klein model is essentially this method, where you get the gauge group from the isometries of the compactified manifold.

Otoh, in String theor(ies), the gauge structure comes from other effects and for consistency reasons, cannot be built up in the exact KK way.

It wasn't a bad guess, but it didn't work out that way.

Please enlighten us then. What do the additional dimensions posited by string theory represent? What are the additional parameters, and why are they necessary?

 

[Max™]

The term Dimensions used in the case of gauge theories should be in reference to parameters available in a mathematical space.

Not specific "direction" type Dimensions in a spatial manifold, which is it's own type of mathematical space.

You could describe your desk surface in terms of dimensions, with each dimension representing some state of the objects on it.

String Theory involves a literally 10 (+1) Dimensional Space, and the amazing discovery was that folding the dimensions up in certain types of shapes, then allowing strings to propagate along them seemed to make Gauge groups "fall out" of the framework as possible ways the strings could interact.

Comparing that to something like an n-dimensional Hilbert space is misleading though, as those dimensions are quantities that can be measured and whatnot, not necessarily spatial degrees of freedom.

 

[apeiron]

String Theory involves a literally 10 (+1) Dimensional Space, and the amazing discovery was that folding the dimensions up in certain types of shapes, then allowing strings to propagate along them seemed to make Gauge groups "fall out" of the framework as possible ways the strings could interact.
.

And is this not the same as saying the three gauge symmetries are/could be nested within the six compactified dimensions?

So the maths was found first. Then a physical explanation in terms of available resonances in compact spatial dimensions followed. Although that intuition has never been cashed in. The way the symmetries could actually be the shape of a 6-space has never been agreed?

 

[IqbalGomar]

And is this not the same as saying the three gauge symmetries are/could be nested within the six compactified dimensions?

So the maths was found first. Then a physical explanation in terms of available resonances in compact spatial dimensions followed. Although that intuition has never been cashed in. The way the symmetries could actually be the shape of a 6-space has never been agreed?

I think it's often assumed that there is no way to intuitively visualize higher dimensions. I can tell anyone who thinks this is the case, from experience, it is not so. I've been thinking about and in multiple dimensions for so long that the idea has become intuitive to me. Whether one relies on the 'cheat' of simply thinking about a higher dimensional manifold in three dimensions at a time, or one is able to actually visualize a system incorporating multiple degrees of freedom simultaneously, the visualization is possible.

I'm simply asking for a straightforward and intuitive description of these dimensions. If I've been wrong in thinking of these dimensions as classically spatial, I have no problem admitting that and reworking the model I have operating in my head. But even if they're not spatial, there must be some analogy that would make these conceptual manifold spaces comprehensible to a layman who does not possesses the mathematical prowess to construct equations in them.

Any help would be appreciated. I'm not asking for the meaning of life here, just a laymanized version of something which should be fairly simple. I <i>get</i> higher dimensions. What I don't get is the identities of these dimensions according to physicists' best understanding of them.

 

[Max™]

If they were, String Theory would pretty much be THE end all of theoretical physics by now, I'd think.

The mathematics of gauge theories works in 3+1 spatial dimensions, with the additional parameters which are often mathematically described as dimensions.

The natural way to produce gauge symmetries is one of the key things keeping String Theory going, it's hard to accept that it can be so naturally "right" about things like that without it being right about everything else.

It's somewhat of a sleight of hand though, the shapes were found which rather naturally produce the mathematical relations of gauge theories, but in a sense the shapes themselves were first thought of as a way to embed gauge structures into spacetime, with Kaluza and Klein's early 5-D Relativity ideas.It's kind of like noticing that you can write out numbers from 1 to 10, and write some multiplication tables for them, then writing out higher multiplication tables which give numbers from 10 to 20.

There's still MUCH work which needs to be done to fully embed String Theory in a predictive physical theory, and it still has that loose parameters problem where you can nearly always get the results you want, by setting it up to give the results you want.

 

[Adrian59]

The term Dimensions used in the case of gauge theories should be in reference to parameters available in a mathematical space.

Not specific "direction" type Dimensions in a spatial manifold, which is it's own type of mathematical space.

You could describe your desk surface in terms of dimensions, with each dimension representing some state of the objects on it.

String Theory involves a literally 10 (+1) Dimensional Space, and the amazing discovery was that folding the dimensions up in certain types of shapes, then allowing strings to propagate along them seemed to make Gauge groups "fall out" of the framework as possible ways the strings could interact.

Comparing that to something like an n-dimensional Hilbert space is misleading though, as those dimensions are quantities that can be measured and whatnot, not necessarily spatial degrees of freedom.

Thanks for your explanation. To summarize how I interpret your answer it would seem that the standard model gauge theories are using a mathematical space not a real space whereas string theory actually predicts 10 (+1 for M theory) space-time dimensions. However, your penultimate paragraph does suggest that these two may be linked since as you say the gauge groups may "fall out" of the compactified extra dimentions.

 

[Max™]

Indeed, a mathematical space with n dimensions is not necessarily the same as a physical spacetime with n dimensions.

It isn't that it predicts 10 dimensions as much as it seems to produce a description of reality when formulated in that many dimensions.

 

[tom.stoer]

nevertheless - remarkable coincidence

 

[OB 50]

First off, I really don't see the logic to that argument at all.

Well, good luck then.

When a 3D sphere passes through a 2D plane, it manifests as a point, which expands to a circle and then recedes back to a point. Any 2D observer would conclude that the 3D space was compactified because of the circle's apparent origination from an infinitely small point in space.

I see a direct similarity to the line of reasoning regarding compactified dimensions and the prospect that they might be visible if only we could look at them closely enough.

Secondly, I hadn't noticed you were using the word 'compact' in the English sense -- its use as a mathematical term has pretty much nothing to do with size. You can look up the definition yourself, the bit relevant here is that there are only two kinds one-dimensional manifolds: open intervals and circles. Circles are compact, open intervals are not.

The difference is important: If the extra dimension is a small open interval, then there is absolutely nothing keeping matter from remaining inside that interval, and GR puts no constraints whatsoever upon what happens at the endpoints. If the extra dimension is a small circle, then we don't have any of those problems.

Both uses of the word "compact" would seem to describe this accurately. Compact dimensions, as described to me, are both closed and exceedingly tiny.

As I've said a few times, any extra dimension we encounter would necessarily have to appear compact to us, in exactly the manner you described. All I'm trying to say is that even if we are living an a 3D space that is embedded in 4D space, our interactions with that 4D space would appear to originate from a compactified dimension as well. I fail to see how one could make a distinction between this scenario and that of a literal 4D "pocket dimension" using only the information available to an observer within the 3D space.

Maybe there is literally no difference and each scenario is equivalent.

Interesting fact: GR is 100% consistent with the hypothesis that the universe is nothing more than a ball 3' in diameter surrounding your head, and stuff just appears on the boundary as necessary to maintain the illusion of an extensive universe.

Seriously? Did you just pull out the internet equivalent of jingling your keys in my face?

 

[Max™]

Ya know, we actually have a great example of an extended dimension which appears otherwise.

Time.

The past and future don't stop existing because you aren't looking at them, yet as far as we can see, Now is all that exists of time.

 

[Hurkyl]

When a 3D sphere passes through a 2D plane, it manifests as a point, which expands to a circle and then recedes back to a point. Any 2D observer would conclude that the 3D space was compactified because of the circle's apparent origination from an infinitely small point in space.

Right -- that's exactly the argument that doesn't make any sense.

I can imagine how a 2D observer might be led to speculate they were seeing 2D slices of objects in \mathbb{R}^3.

I can imagine how a 2D observer might be led to speculate they were seeing a "parallel dimension^*" -- another 2D plane completely disjoint from his own, except that wormholes may connect them at times. (Other interesting geometries are possible as well)

But I cannot imagine how a 2D observer might be led to speculate that the the geometry of his space is \mathbb{R}^2 \times S^1. (No matter what the circumference of the loop is)


I notice that you used the phrase "pocket dimension" too -- but not in a way that resembles any usage I've ever seen of that term.


*: Dimension used here in the sense that laypeople use it. e.g. the kind of parallel dimensions you see in star trek, or the alternate planes of reality you see in dungeons & dragons.

Seriously? Did you just pull out the internet equivalent of jingling your keys in my face?

I'm not familiar with that phrase. Yes, I was serious. But it's irrelevant if you weren't hypothesizing that the geometry of space is similar to \mathbb{R}^3 \times (-\epsilon, \epsilon) as I thought you were.

 

[OB 50]

Right -- that's exactly the argument that doesn't make any sense.

I can imagine how a 2D observer might be led to speculate they were seeing 2D slices of objects in \mathbb{R}^3.

I can imagine how a 2D observer might be led to speculate they were seeing a "parallel dimension^*" -- another 2D plane completely disjoint from his own, except that wormholes may connect them at times. (Other interesting geometries are possible as well)

But I cannot imagine how a 2D observer might be led to speculate that the the geometry of his space is \mathbb{R}^2 \times S^1. (No matter what the circumference of the loop is)

I notice that you used the phrase "pocket dimension" too -- but not in a way that resembles any usage I've ever seen of that term.

*: Dimension used here in the sense that laypeople use it. e.g. the kind of parallel dimensions you see in star trek, or the alternate planes of reality you see in dungeons & dragons.

This is going nowhere. I was trying to make a very basic observation which requires no math or geometry beyond a grade school level, and I've somehow ended up sounding like a complete lunatic in the process of attempting to explain it.

Just to help me understand where this disconnect lies, I'm curious as to which of the following scenarios you think more accurately describes a situation where something has been constrained to 1 dimension.

A. An ant is walking on a wire.

B. The data representing an ant is being transmitted through a wire.

 

[Max™]

An ant walking on a wire could walk around it, thus it has 2 degrees of freedom, 1 dimensional would be restricted to forwards or backwards along the wire.

 

[OB 50]

I knew that would come up.

The distinction I'm trying to get to is whether the ant is on the wire or in the wire.

 

[Max™]

I meant that in my post, when I said along the wire I should have said within the wire, though again that is a failure of language to properly represent the condition of being one dimensional. How can you be within a one dimensional object?

Nonetheless, in a hypothetical sense, if the ant were restricted to the single degree of freedom on or in the wire, the effect is the same, in is a better description, but for the conceptual difficulties which arise.

 

[OB 50]

Those conceptual difficulties are the very things I've been trying to point out.

The ant on the wire is still a 3D observer that has been artificially constrained.

The ant contained within the wire has to be reimagined and constructed from the ground up using the principles of a 1D universe.

The two scenarios are profoundly different.