# Curled up dimensions

What do they mean by a dimension that's curled up? I'm having a hard time with this notion. What would such a dimension be like?

Hurkyl
Staff Emeritus
Gold Member
What do they mean by a dimension that's curled up? I'm having a hard time with this notion. What would such a dimension be like?
Let's consider "curling up" one of our existing spatial dimensions. (just geometrically -- without worrying about how it might affect physics)

Do you recall old games like asteroids or pac-man where, if you go off of one edge of the screen, you reappear on the other side? Imagine that if you walked a mile East, you would up right back where you started. (Of course, this would mean the Earth is some weird sort of shape, but don't think about that)

Now, imagine how it would look if you only had to go a few hundred feet East before you wound up where you started. What about just 10 feet? 1 inch? 1 millimeter? 1 angstrom? With these small sizes, it would be more like you were a two-dimensional object than a three-dimensional one!

String Theory is not scritctly a science yet - Its more of a mathematical philosophy that hasn't got any supporting evidence or observational data to conform any of the conclusions or predictions.

The mathematics in ST is logically consistent but that does not mean that it reflects reality in any way.

When String Theorists make statements such as "curled up dimensions" or even an infinite number of dimensions they need to support these theoretical constructs with reproducible evidence of some kind - and they cant - so far!

Let's consider "curling up" one of our existing spatial dimensions. (just geometrically -- without worrying about how it might affect physics)

Do you recall old games like asteroids or pac-man where, if you go off of one edge of the screen, you reappear on the other side? Imagine that if you walked a mile East, you would up right back where you started. (Of course, this would mean the Earth is some weird sort of shape, but don't think about that)

Now, imagine how it would look if you only had to go a few hundred feet East before you wound up where you started. What about just 10 feet? 1 inch? 1 millimeter? 1 angstrom? With these small sizes, it would be more like you were a two-dimensional object than a three-dimensional one!

Yes, I remember asteroids. Used to play it on the original bar tables. Used to stand in line to feed quarters into Pong. I told you I was almost old.

Tell me if I understand you right in these terms. I live inside this curled up dimension. Locally everything looks normal. I can survey a local map, and as I move around, overlapping at the edges of the pages I draw, I collect these maps into an atlas. After collecting, say nine maps, I notice the landmarks are familiar. I arbitrarily pick one map to be the center, and connect four maps, one to a side. That was pretty easy. It wasn't hard placing the last four corner sheets into a 3x3 grid either. As I'm making a list of distances to cities, I notice that cities on the opposite corners of this big map are actually close together. Everything on the extreme left is close to everything on the extreme right, but the top left is close to the top right, and bottom left is close to the bottom right. I devise a way to index to every point on the grid, and figure out a modulo function on the edges that let me calculate the distance between any two cities. I get up to write down this function and wonder what it means, when a breeze blows my maps apart. I reassemble them using a different map for the center, but now the mapping function I get is a little different for the corners. Any point on any sheet could be used as the origin, and I have a family of mapping functions at every point to every other point. the domain of the maps and the class of distance functions defines the topological space, right? I can generalize the family of functions into a single function in one added dimension, in the above case an oblate spheroid. It might have been a torus, but I suspect it'd take more than nine maps to figure that out. As long as that function is differentiable and conformal in my space, locally everything is guaranteed to seem normal to me. This also works even if my space is infinite, I just have to map it onto the extended Rieman sphere. Am I on track so far?

In the example you used, when you shrunk the curled up dimension from 100 feet to 10, that 10 feet was measured in your dimension. In mine, the maps I drew might be smaller, but on my maps the distance from Ank-Morpork to Quirm is still awfully far. I don't even think the feet that I use are the same kinds of feet as yours.

So you might see me as becoming more two dimensional as you curl me up, but I'd still feel three dee. The functions that define how things work in my space are still locally valid and presumably unchanged and possibly as I can measure it infinite.

It seems to me then that there isn't much difference between a dimension that's curled up and one that isn't. So unless I'm wrong in how I understood the above, I still don't understand what a curled up dimension means.

If we were living inside a curled up dimension, could we tell? How would we detect it? What properties of it could we measure?

First, it's obvious that a curled dimension is different from a non-curled one. For one, you can send a message in one direction and it could come back and hit you. Second, using just that method, you can show that it's possible to determine whether or not you are moving in a curled dimension (i.e., there is a difference between moving and not moving in a curled dimension) Consider a person sending 2 light beams in opposite directions and asking where they collide. A person standing still will give you one answer. However, if there were really no difference between standing still and moving, then a person who is moving would see them collide at a different point. However, this is a contradiction (since they could communicate to each other where they saw the light beams collide and they'd disagree. If you allowed this, there could be a lot of weird causality breaking things). This is of course assuming that the curled dimension is also flat, but I'm pretty sure similar things happen for non-flat curled dimensions.

Other odd things happen when you have multiple dimensions, some of which are curled and some of which are not curled, but I don't know any of the details.

mathematical theories produce all sorts of weird and wonderful concepts - doesnt make them REAL though - even if they are logically consistent

Or a dimension that does not interact with any other dimensions?

Or a oscillating or wave like dimension?

All mathematically consistent - BUT do they exist?

What's a semi-curled dimension? Would that be one where the maps on the sides overlap but the ones on top and bottom go off to infinity?

What's an oscillating or wave like dimension? I've never heard of that.

As for a dimension that doesn't interact with any other dimension, I guess that depends on what your definition of "is" is. If it doesn't interact with any other dimension, then I don't care if it exists or not. Makes no difference to me or to any theory.

First, it's obvious that a curled dimension is different from a non-curled one. For one, you can send a message in one direction and it could come back and hit you. Second, using just that method, you can show that it's possible to determine whether or not you are moving in a curled dimension (i.e., there is a difference between moving and not moving in a curled dimension) Consider a person sending 2 light beams in opposite directions and asking where they collide. A person standing still will give you one answer. However, if there were really no difference between standing still and moving, then a person who is moving would see them collide at a different point. However, this is a contradiction (since they could communicate to each other where they saw the light beams collide and they'd disagree. If you allowed this, there could be a lot of weird causality breaking things). This is of course assuming that the curled dimension is also flat, but I'm pretty sure similar things happen for non-flat curled dimensions.

Other odd things happen when you have multiple dimensions, some of which are curled and some of which are not curled, but I don't know any of the details.

Sounds to me like you've described a relativistic space. In that case, why are there any big problems? Not long ago, wasn't it a question in cosmology whether our universe was open, closed, or flat? Locally it looks flat, but on larger scales all the evidence was for open, until they introduced dark matter. Then there was the supernova data and dark energy, and now the universe is way open and accelerating. All this always sounded speculative and I take it with a big grain of salt. But in any case, how would a curled up dimension be any different from such a closed universe, just on a seemingly larger scale?

You also said the curled dimension would have causality breaking if it was also flat. I don't know what that means. What is a flat curled dimension? What is a non-flat curled dimension?

BTW, I've always found the notion of looking at the back of my head rather disturbing. I avoid that unpleasantness by presuming the curvature of space is an irrational number. That way my line of sight will never return to it's starting point. I take comfort that I'm probably 100% right about that.

No, no, there's no causality breaking in assuming that the universe is flat and curled. There's causality breaking in assuming that there is no preferred reference frame (i.e., that you can't tell whether you or not you are moving). What I meant by flat was that this is a simple special relativity calculation (i.e., that that you are ignoring any of the effects of mass and energy on space-time and that the closedness of the dimension is just an intrinsic property of the dimension). If you have a flat, closed dimension, you can tell whether or not you are moving. I believe that something similar is true in general relativity as well unless certain constraints are put on the way dimensions curl and on the way they interact with mas, but I do not believe that this is done in string theory.

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Hurkyl
Staff Emeritus
Gold Member
If you have a flat, closed dimension, you can tell whether or not you are moving.
Don't you need some assumptions, like space being entirely homogenous in that direction, and in some sense static with respect to time?

Anyways, I'm of the (possibly wrong!) understanding that gauge theories can be thought of exactly as that -- e.g. you can attach a U(1) dimension to classical space-time, and electric charge manifests itself as something like velocity in that direction. (This is something I don't understand well)

There's causality breaking in assuming that there is no preferred reference frame (i.e., that you can't tell whether you or not you are moving).

I don't see how this follows. I thought it was correct to say that there is no preferred reference frame, at least in the dimensions I occupy (meaning, not in some higher dimension contained the curved one), and I didn't think the curvature of space qualified that statement.

Besides that, the argument about seeing the back of your head only applies with positive curvature and doesn't happen with negative curvature. I still don't understand why there should be causality problems in either case.

What's a semi-curled dimension? Would that be one where the maps on the sides overlap but the ones on top and bottom go off to infinity?

What's an oscillating or wave like dimension? I've never heard of that.

Well I just made them up to make a point.

Not sure if they exist in mathematical theory either

I wouldnt pay much attention to me when I am making stuff up OFF the top of my head

(altough thinking about it now, I dont see why an OSCIALLTING dimension could not be a feasible mathematical construct. I would imagine it would be a dimesion that begns to curl one way and then curl the other way producing a wave like pattern - non-linear but period if you like???)

arivero
Gold Member
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I wouldnt pay much attention to me when I am making stuff up OFF the top of my head

My problem is that I don't know enough to be sure when you're making stuff up and when you aren't. Without such an indication, I'd have no choice but to simply not pay attention to you at all, since I'm really trying to learn something here.

What do they mean by a dimension that's curled up? I'm having a hard time with this notion. What would such a dimension be like?

I wish I could help you here, but I haven't even found a satisfactory description of what a dimension is, or is like, yet. This might develop into an interesting thread if you could start at the beginning and describe an ordinary dimension, say one of the ones we call "space dimensions" (the time one is too difficult). Then one might be able to get a handle on how to describe a curled-up dimension.

Come to think of it, I don't even know what "space" is.

I'm almost too old, it seems.

My problem is that I don't know enough to be sure when you're making stuff up and when you aren't. Without such an indication, I'd have no choice but to simply not pay attention to you at all, since I'm really trying to learn something here.

You learn BEST by making stuff up off the top of your head.

If you wish to regurgitate accepted standard models of nature then just read dozens of text books - that a different sort of learning - although its important, its not critical to me.

Free thinking is far more challenging than PLAIN LEARNING and it opens up many horizons and to enter this realm ONE must ponder the wrong or neglected questions

Are you afraid to improvise in science? thats what science is supposed to be for it to progress.

You learn BEST by making stuff up off the top of your head... ...Are you afraid to improvise in science? thats what science is supposed to be for it to progress.

Not at all! But you clearly don't understand science at all. Science requires that you provide experimental evidence for your theories. Or at the very least you must show that experimental evidence could be provided, in priciple, given an appropriate state of the art.

Making things up off the top of your head is, at best, philosophy and, more likely, nothing more than nonsense.

I wish I could help you here, but I haven't even found a satisfactory description of what a dimension is, or is like, yet.

What is a dimension? The flippant answer is "anything you want it to be." It's almost a religious question to some. Poetically it's a measure of a fundamental reality about the universe. In other words, it's one necessary measure of an observable. Mathematically (I think, help me out here) it is an orthogonal parameter in a mapping from the universal set onto the universal set. You just have to define your universal set.

The trick is in knowing everything about your universal set. If you know everything, you can map one point in reality (an element of the universal set, call it e) to any other point of reality within a neighborhood of x. If you know everything, the radius of that neighborhood might be unbounded. Does that seem right?

Anyway, if you don't know everything and don't have enough parameters, you can only map to subspaces of the universal set, and it won't be sensibly ordered, at least in some ways (in exactly the number of dimensions you're short ways. I think). If you have too many dimensions, then n of them measure some combination of other dimensions, and not what you think it does. In this case some combinations of parameters will be able to predict some things but won't apply to others. This makes it very hard to decide which parameters to get rid of.

You don't even have to pick just the exact right dimensions, either. As long as the n dimensions you pick contain all of the n independent orthogonal vectors, then everything will work out the same. I doubt the equations which defined the mappings would even be more cumbersome.

So while it's not exactly "anything you want it to be" it's still sort of close to that.

You don't even have to pick just the exact right dimensions, either. As long as the n dimensions you pick contain all of the n independent orthogonal vectors, then everything will work out the same.

Hurykl mentioned something about charge being a velocity in the direction of another U(1). If that's so, then it seems something like the above is going on.

If the n vectors needed to do the mapping in U were all orthogonal, they would be simply 1D. Would it then make sense to say that they were curled? In that case specifically, would it make any difference if it were curved or not?

But if instead our vectors were some combination of vectors, then the idea of curled seems more palatable, and I can see how it'd make a difference. Is this a reasonable by topology?

If you know everything, you can map one point in reality (an element of the universal set, call it e) to any other point of reality within a neighborhood of x.

Oops, I meant to say within a neighborhood of e. Sorry.

Thanks for these descriptions of a dimension.
What is a dimension? ......"anything you want it to be." .... it's a measure .....one necessary measure of an observable. Mathematically ..... it is an orthogonal parameter in a mapping from the universal set onto the universal set.

Your first would be have been admired by Lewis Carroll. I like it. The second has an operational flavour and would have suited Percy Bridgman. The last, involving concepts like "orthogonal" "parameter","set" , "mapping", "vectors" etc are also somewhat operational, but disguised with that amazingly useful invented language, namely mathematics. It's the kind of answer that might later lead to further inventions, like differentiable manifolds and fiber bundles.

Have you any further alternatives?

As someone who knows the majority of the undergraduate mathematics curriculum and wants to learn some physics (as well as more mathematics), I'd just like to say:

What??????

I'm sorry, but the last 5 posts had virtually no content even by pseudoscience standards.

The definition of dimension depends on the exact structure one is studying (i.e., Fractals have dimension in a sense, and this is completely different from the sense in which vector spaces have dimension, which is related to, but different from the sense in which manifolds/surfaces have dimension)

I don't know anything about fractals, so I can't tell you anything about them.

A vector space is a set of "vectors" along with a field (in the mathematical sense, a field is essentially a set of numbers that can be added, subtracted, multiplied, and divided (except for 0, which cannot be divided). Familiar examples of fields are the real numbers and complex numbers. This is a very loose explanation, to learn more, check wikipedia or a book on the subject).
Elements of the field are often called scalars. Any vector can be multiplied by a scalar to get another vector, and any two vectors can be added together to get another vector. One example of a vector space is $$\mathbb{R}^2$$, which is the set of all points (a,b) where a and b are real numbers (with addition and multiplication by scalars defined how you would think it should be).
Now if you think about it, you only need 2 vectors to describe this vector space. One example of 2 vectors that will work are (1,0) and (0,1), but in fact, any 2 that are not multiples of each other work (for instance, (1,0) and (1,1) work). Any other vector in R^2 can be made by adding multiples of these two together. A set of vectors that can generate the entire vector space like this is called a basis if additionally, there is no vector in the set that can be generated by other vectors in the set.
It turns out that every basis of a specific vector space always has the same number of vectors. (e.g., every basis of R^2 has exactly 2 vectors). This number is called the dimension of the vector space.

Note that the dimension of a vector space refers only to the number of objects needed to describe the entire space. It has nothing to do with orthogonality at all. In fact, although the vector spaces that physicists usually use have a notion of orthogonality, in general, vector spaces do not need one at all (and many times, orthogonality does not work quite how one would expect from the typical usage you encounter in physics and engineering)

A manifold on the other hand is essentially a set where each neighborhood of a point in the set (i.e., the point and points in some non-zero radius around that point) can locally be described by a map from the vector space R^n to the manifold. n need not be the same for the entire manifold (e.g., one part of the manifold could be like R^1 and another could be like R^2. An example of this would be a plane in R^3 along with a line that does not intersect the plane). If n however is the same for the entire manifold, then this is called the dimension of the manifold. The definition of manifold that I used is not the most general definition, but it's general enough to encompass what most physicists use and more.

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When people talk about a "curled" dimension, they usually mean that space can be modeled as an n-dimensional manifold, and that restricting space to a specific dimension, gives a circle, which is to say that for some constant C, that the point (x+C) is exactly the same as the point (x) (for all x). This probably is not what every physicist means, when ey says "curled", but it is the most common usage. Foolosophy was right in saying that although this is the easiest way for a dimension to "curl", it is not the only way. More complicated mappings can be constructed that don't necessarily map a point to a constant amount away. However, physicist almost always work with what are called "smooth" manifolds, which restrict the mappings that can be used to certain functions called diffeomorphism, which have nice properties.

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When "Imaginary or Complex Numbers" are applied to the space-time continuum

What do you get?

As someone who knows the majority of the undergraduate mathematics curriculum and wants to learn some physics (as well as more mathematics), I'd just like to say:

What?????? .................

Your familiarity with undergraduate mathematics and your clear technical exposition of what theorists mean by the "dimension" of a vector space, and how they model space as a manifold that includes in one of its dimensions a circular character is helpful.

But I don't think it goes very far in answering the second part of the original question, namely "what would a (curled-up) dimension be like?" This needs a simpler explanation, first, perhaps, of what an ordinary dimension is, or is "like", and how this would be changed were this dimension to be curled-up. I can't supply such an explanation. Perhaps you could take a less blinkered technical approach, and attempt a non-technical discussion, maybe by considering the possible topologies of manifolds?

Your familiarity with undergraduate mathematics and your clear technical exposition of what theorists mean by the "dimension" of a vector space, and how they model space as a manifold that includes in one of its dimensions a circular character is helpful.

But I don't think it goes very far in answering the second part of the original question, namely "what would a (curled-up) dimension be like?" This needs a simpler explanation, first, perhaps, of what an ordinary dimension is, or is "like", and how this would be changed were this dimension to be curled-up. I can't supply such an explanation. Perhaps you could take a less blinkered technical approach, and attempt a non-technical discussion, maybe by considering the possible topologies of manifolds?

Sorry, I haven't learned much about manifolds yet. I've just learned a definition, defined how to do integration on a manifold, a proved a few simple things about it. It was just as an end of the year thing so that we could see some more advanced topics. I probably won't actually do anything with manifolds for another 2 years at least. I'll try to explain things a bit, but I can't promise it'll be that great because what you are asking is a question about both differential geometry and topology, neither of which I've been more than just barely introduced to.

One important thing about manifolds is that any manifold can be embedded in R^n (n-dimensional Euclidean space) for some large enough n. That is to say, that you can view it as a surface in a higher dimensional space. So when someone talks about an 11-dimensional manifold, it's often good to think of it as lying in a 12 or higher dimensional space. This is similar to how when someone talks about the surface of a sphere, you imagine it being in 3-dimensional space even though the surface of a sphere is only a 2-dimensional manifold.

Since we have a difficult time thinking about higher than 3 dimensional spaces, for now just think about the case of a 2-dimensional manifold embedded in 3-dimensional space. Since physicists usually deal with smooth manifolds, what we're talking about are 2-dimensional surfaces in 3-dimensional space. If you think about all of the different 2-dimensional surfaces you've heard of, you know that these can get fairly complicated. You can have things twisting, things branching; you can have any number of sides on the surface. These can be as simple as the plane or as strange as a moebius strip. 2-dimensional surfaces embedded in higher than 3-dimensional space can be even more complicated. For instance, the klein bottle is a 2-dimensional surface embedded in 4-dimensional space. Like the moebius strip, it only has one side and appears to "turn inside out" at some point.

The manifolds that physicists tend to deal with however, are much simpler than some of the examples you're probably thinking about. The manifolds that quantum mechanics are modeled on are fairly regular. At every point, you tend to have the same sort of features. For instance, if there is a "curled dimension", this means that at every point on the manifold, there is some circle that can be traveled around to get back to where you started. Not just any circle will do though. Consider the surface of an infinite cylinder (i.e., it has a finite radius but it's length is infinite): From any point, you can draw small circle right next to the point, but this circle can be made infinitely small until you get back to a single point. This circle that you drew is considered equivalent to a point (since we can continuously deform it to a point), so that type of circle is not the type that we are talking about. However, from any point you can also draw a large circle that goes around the entire cylinder and ends up back at the point. This circle cannot be deformed back down to a point. Also note that from any point, any circle that can be drawn is either equivalent to a single point or equivalent to the circle that goes around the entire cylinder.

I think that's about as much as I can describe.

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