How Many Dimensions Are There?

Factual Questions
21

On the “wrapped up small” thing I once saw it explained like this, which made sense to me. I hope it’s right!

First imagine that the geometry of the universe is “closed”, such that if you were to fly off in a given direction long enough, you’d loop back on yourself and be back where you started. Similar to walking on the surface of the earth.

Now imagine that there are dimensions like this but instead of being billions of light years in extent, they are instead a minute fraction of the radius of a proton.
These dimensions are so small in fact, that the strings of string theory are long enough to entirely wrap around some of these dimensions. In fact it’s this wrapping that’s crucial to giving the strings many of their properties.

It’s like how paper is 3D but the third dimension may be very very small compared to the size in the other dimensions, and the extra dimensions posited by string theory are like the paper’s thickness dimension.

IANA theoretical physicist and the above might be horseshit.

22

I finally found a question on SD I’m smart enough to contribute to! I’ve been looking for SIX YEARS. This will begin the process of dampening the accumulating guilt I feel I’m stealing consulting services one would otherwise pay very good money for. I went to a confession once, I said “I only ask questions on Straight Dope”

OK, so. How do I explain this so it sounds clever and not crazy.
I’m a “lite” schizophrenic. (Footnote: technically, I’m schizoaffective which is lite-schizophrenia meets lite-bipolar; I also have potent ADHD, but don’t worry about any of that! And FYI for any future employer searching the net for dirt about me, I’m also well medicated.)

Schizos often perceive reality and interpret information differently from others. These experiences can be good or bad. Once I was in a psych ward (not many stories end well that start with this phrase; this is an exception) and lying around the place for public reading material was the book Hyperspace, by a physicist Machio Kaku who has a knack for explaining things in exciting terms to layman; in the book, he explains higher dimensions in an enticing way. Another patient in the ward told me to read it.

Now in Hyperspace Kaku describes a childhood memory of watching carp in a pond, and thinking that his world was beyond their comprehension, and likens this to what higher dimensions are like. Now a NORMAL reader of the book can feel uplifted, as if Kaku’s explanation of the physics is almost lifting them up into a slightly higher awareness. Me being less sane and more easy to jump to otherworldly explanations, my personal feeling/interpretation/experience reading the book was a bit more lofty: I had the perception that some being(s) in a higher dimension, that I couldn’t understand, dropped the book into my life as a way of communicating. They saw that I was creative/etc and wanted to inject the book into my lifepath to see what I would do with it and how it would affect me, like watching the carp like Kaku did. That was a pleasant experience. (It kind of wore off… I don’t look at the book now and think “this was delivered to me through a higher dimension” but it was still a good memory).

My point here may be that “alternate dimensions” in the spirital/religious/crazy sense, can be directly related to higher physical dimensions in math/physics. Again even a normal reader of the book can feel uplifted in a way, like they are (metaphorically) being lifted up into a higher awareness. So maybe understanding higher dimensions can actually help lift you up into them.

Honestly, I think numbering them is kind of silly. Kaku goes through all the math, that there may be 10 or 20 or whatnot by XYZ theories, but some of the beauty of higher dimensions is lost in the precise numbers. I think the obvious, natural, normal answer is that there are infinite dimensions.

23

It’s just a hyperbola. As in, for instance, this diagram. The curved lines represent a fixed space-time distance.

Of course the true 4D shape is rather more difficult to visualize than 2D. But regardless, it’s a different shape than the Euclidean hypersphere with the ordinary Pythagorean distance metric.

To correct the OP slightly, a hypersphere is a shape in any dimension defined as the surface a fixed distance away by any metric. For instance, a 3D cube is a hypersphere with distance metric max(|x|, |y|, |z|).

24

A hypersphere is just the n-sphere in R^{n+1}, namely: {(x_0,…,x_n)= x_0^2+ … + x_n^2 = 1}. That’s the standard one, anyway; you could also take one with an arbitrary radius and center in the obvious way. Although the relativistic metric is ds^2 = dx^2 + dy^2 + dz^2 - dt^2 (taking c = 1), the hypersurface {(x_0, …, x_3): x_0^2 + x_1^2 + x_2^2 - x_3^2 - 1} is not a sphere; it’s not even compact.

Nope. The dimension of a vector space is the size of its basis. The dimension of a manifold (e.g. spacetime) is the dimension of its tangent space. That’s it. There’s nothing mystical, fuzzy, or ill-defined about it. There’s nothing to suggest that the universe has infinite dimensions and a bunch of reasons to suggest it doesn’t (e.g., pretty much integral ever would diverge).

25

Is that common? At least on the mathematical side of things, we (or at least I) usually just use ‘hypersphere’ as a less common synonym for ‘sphere’ or ‘n-sphere’, even in a geometric setting like a metric neighborhood.

26

“Loathe” isn’t really the right word. I’d be more inclined to say that we’re vehemently ambivalent about it.

27

Well, there are at least five.

:smiley:

28

Also remember that the Fifth Dimension hit #7 with Up, Up and Away in 1967.

29

I understand how meaningful this is for you, but it’s a nearly perfect demonstration of why trying to merge spiritual and mathematical definitions can never work. They can never be related, never be one to one, never cross over the line except as personal and fuzzy feelings.

That’s fine for an individual, but imposing personal and fuzzy feelings on science by force is what the woo crowd attempts. It’s demeaning to science, because that’s its purpose. And that’s why scientists fight woo so fiercely.

30

I defer to your understanding. I may have falsely generalized from other terminology, and cannot find a reliable cite. The best I can find is the Wikipedia article on taxicab geometry, in which it declares a circle to be a diamond shape using that metric. Saying that “hypersphere” extends that concept to additional dimensions makes sense to me but I don’t know if that’s really the common usage.

31

Although it may lurk as a technical distinction, I’ve never seen it in popular books on math or remember hypersphere being defined as anything but an n-sphere. It certainly confused me when you used it in a larger context but l’m just an amateur.

32

Thought I’d link to this (YouTube video). It’s not a very scientific site, nor is this guy a scientist, but it’s a reasonably good explanation of dimensions, at least as far as my own (now 30 years out of date) math goes. The only thing I’ll add is that, from my own Science Channel understanding of String Theory, there is a debate between 10 and 11 possible dimensions…I THINK that 11 is what you need to make some of the math work, but it’s frankly way beyond me.

33

I don’t understand how a dimension can be “small”. Isn’t a dimension infinite? The location/value of a point in a dimension can be small, but how can the dimension itself be small?

Same question about the idea of a dimension being curled up. “Curl” is described by three dimensions , how can a dimension be curled?

34

Dimensions aren’t always infinite. If you’re considering the surface of the Earth (and only the surface), it has two dimensions (latitude and longitude (or any equivalent set of parameters - you can make New York the north pole of your coordinate frame if you’d like)) but neither of them is infinite - a point at latitude 71 degrees and longitude -80 is the same point as one described as latitude 71 degrees and longitude 280 degrees (or an infinite number of other choices).

You can describe curvature of a two-dimensional surface as taking place in three dimensions - but it’s not necessary to do so - there’s “intrinsic curvature” that is detectable locally inside the manifold that is curved (like the curvature of the Earth’s surface - you could detect the distortion on the shape of triangles made by lines that were straight (great circles)), but there’s also extrinsic curvature that’s not detectable locally on the inside (triangles aren’t distorted, but if you traveled far enough on a straight line you’d still get back to your point of origin). A good example of a surface that’s extrinsically curved but not intrinsically curved is the world of the video games Asteroids - it’s flat and triangles look normal, but because the screen “wraps around” the whole universe is equivalent to a torus, even though it doesn’t look like one. Chronos can correct me (and probably will), but I think the extra dimensions in string theory (etc.) are treated as extrinsically curved.

35

Picture a really long, but thin, cylindrical surface. Now there are two dimensions, but one is long or even infinite, while the other direction is small as well as curled up (I think just compactness is meant by the word here; perhaps the word “curl” is misleading, especially since in string-type theories there appear spaces like Calabi-Yau manifolds which have flatness properties.)

36

I’ve always assumed that the small dimensions described in String Theory mean that those dimensions aren’t accessible unless you’re something the size of a quark. Or something.

I’d love if someone knowledgeable could elucidate on that.

37

Wow, thanks. The sphere and long thin cylinder examples really help me to grasp the concept.

38

I have to admit, I do not understand the intricacies, but the background of string theory is the product manifold of a Calabai-Yau manifold* and flat Minkowski spacetime. Calabai-Yau manifolds have the property of being Ricci-flat, which is not the same as lacking intrinsic curvature, as, for Rimenanian manifolds of dimension greater than 3, Ricci flatness does not imply conformal flatness.

*or an ‘almost’ Calabai-Yau manifold that is ‘almost’ Ricci-flat.

39

Glad to help.

40

Please don’t link to that, as it’s a terrible explanation of dimensions, from a guy who has no clue of what he’s talking about. I wish that stupid video would just die already; it’s a major impediment to understanding.

Andy L, an Asteroids screen is neither intrinsically nor extrinsically curved. It’s flat, but with a nontrivial topology. You can embed a surface into 3-dimensional flat space that has the same topology and which is both intrinsically and extrinsically curved (it’s the surface of a torus), or you can embed such a surface into 4-d flat space with extrinsic curvature but no intrinsic curvature, or you can have the space on its own with no sort of curvature. In any event, the extrinsic curvature of space is never relevant to physics; if the extra dimensions of string theory are said to be curved (they might or might not be), that means intrinsic curvature.

As an example of intrinsic curvature, by the way: One of the standard relativity textbooks has a map of Middle Earth in it, with four locations marked, and distances between each pair of locations, with the question “Is Middle Earth flat?”.