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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)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!
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.
Don't you need some assumptions, like space being entirely homogenous in that direction, and in some sense static with respect to time?If you have a flat, closed dimension, you can tell whether or not you are moving.
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'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.
I wouldn't pay much attention to me when I am making stuff up OFF the top of my head
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?
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... ...Are you afraid to improvise in science? that's what science is supposed to be for it to progress.
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.
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.
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.
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.
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?
When "Imaginary or Complex Numbers" are applied to the space-time continuum
What do you get?
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?
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I think that's about as much as I can describe.
Fortunately, things become simpler when the dimension is very tiny (e.g. if quarks can 'span' the entire dimension) -- you simply don't (directly) detect it at all!But when you try to undertand the nature of curled-up dimensions from your own point of view, as an observer, things tend to become more complicated.
Fortunately, things become simpler when the dimension is very tiny (e.g. if quarks can 'span' the entire dimension) -- you simply don't (directly) detect it at all!
Curled-up dimensions have been discussed in papers ...and probably others... the ubiquity of helices...