How can one imagine curled up dimensions?
Solution 1
Annav gave the correct answer, but here's some help on visualisation.
First thing: We cannot imagine more than 3 space dimensions. You can try, and get tantalizingly close, but it's extremely hard to wrap one's brain around it Due to this, I shall explain this in lower dimensions,and you can try to generalise it. Try.
Alright. Let's imagine a thin hose. You're standing nearby, and to you, the hose looks like a line. Onedimensional. On the other hand, an ant on the hose notices that it can move along two dimensionsalong the hose and around the hose. In this way, the 2nd dimension is "curled up" and hidden except at small levels.
For a 2D example, consider an extremely thin sheet. This sheet is made of layers of mesh. For an external observer, the system looks two dimensional. But, an ant can move along the two dimensions of the sheet, and also perpendicular to it (up/down sheets). You can also look at this as a thin film of water and a tiny fish inside it.
We can't really extend this further than that. We can look at projections (like the CalabiYau manifold linked in Anna V's answer)but these are like a crosssection. Not the whole thing..
We go out of the universe to see a 4th dimension
Well, this is sort of correct. Let's take a 2D ant living on a 2D scrap of paper. It perceives that there are two dimensions, and it is restricted to moving along these two only. If, by some unknown force(your hand), the ant is picked up, it will have moved through the third dimension. But, being a 2D ant, the ant will still "see" only two dimensions. The important part here is that something happened that could not have happened if the universe was restricted to 2Da smart ant would reason thus "I can only see two dimensions, but something picked me up; thus there are more than two dimensions"
Similarly with humans. If we "exited" the 3D universe in that sideways manner, we would be moving through another dimensionbut we would still see only three dimensions. But we can reason that more dimensions exist from this.
Solution 2
Here is a nice illustration of the Calabi Yau manifold. One can visualize at each point of our 3 dimensional space as a tiny manifold like that which encloses the extra dimensions.
Alternatively: if our third dimension were curled up we would be living in Flatland , without knowledge of the third dimension. One can rotate a two dimensional figure into the third dimension in our world, since the third dimension is not curled up. If it were as in Flatland, then the figure would not fit into the third, curled, dimension, because of dimensional incompatibility. Thus the flatlander cannot interact/see the third curled dimension.
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funtime
Updated on April 28, 2020Comments

funtime over 3 years
Actually I'm learning String Theory, and one of its proposals is that there are actually 25+1 dimensions of which only 3+1 are visible to us and the remaining are curled up. However, superstring theory says that there are 9+1 dimensions; and M theory says 10+1 dimensions.
I'm having trouble imagining "curled up dimensions"
Is it that we go out of the universe to see a 4th dimension? (This was suggested by a friend of mine)

Ron Maimon over 11 yearsYou can start by considering a 5 dimensional asteroids game, where the spaceship goes around the asteroids board and comes back to where it started after a very short distance in 2 of the dimensions, but never comes back in the remaining 3. This is a 2d square torus compactification.

Chris Marie over 8 yearsYou said you had trouble VISUALIZING ... check out this video: youtube.com/watch?v=XjsgoXvnStY


Jabavu Adams over 10 yearsMy understanding of popularscience descriptions of these curled up extra dimensions seems to imply that we can't see them because they're curled up. This has always confused me. Wouldn't we be unable to see them regardless of whether they are curled up, or not? For example, the 2D Flatlander can't see into the third dimension, regardless of whether the third D is infinite in extent, or curled up.

Jabavu Adams over 10 yearsThe ant on a string analogy has always bothered me. We're watching the ant while we're embedded in 3D. The ant isn't moving in any extra dimension from our POV. Its motion is constrained, but still embedded in 3D. Its motion isn't higherdimensional, so how does the analogy explain anything about extra dimensions. Basically, I don't see how a curved path embedded in the same number of dimensions or lower as us explains anything about extra spatial dimensions beyond those we can perceive. What am I missing?

anna v over 10 years@JabavuAdams The curl is very tiny too. "see" means if there are interactions . If there were measurable interactions into the third dimension of the flatlander, then he would be able to "see" that another dimension existed. When string theory speculatively started extra dimensions were not curled, there were the so called "cosmic strings" . The curl was introduced "because" in our experiments we do not "see" interactions into extra dimensions. Some theorists proposed that some of these dimensions might be as big as a milimeter, experiments are still looking for them, we would "see" them.

Manishearth over 10 years@jab because any sufficiently large entity living on the hose/mesh will not perceive 3 dimensions. When I say "large" I'm talking about resolutionthe entity sees larger objects better than smaller ones.

Manishearth over 10 yearsCurrently, humans don't have the experimental resolution necessary (or we do have it, just that we haven't devised the right experiment yet.) to verify the existence of any more spatial dimensions.

brightstar almost 10 yearsJust to be clear, none of this curling means dimensions have any "whereness", correct? A dimension might be compactified making it impossible to interact with, but for all intents it is present everywhere as part of the description of the universe, right?

anna v almost 10 years@TrevorAlexander Yes. The same holds true for (x,y,z) normal three dimensions. Except that we draw the axis to + and  infinity . At each point in the ten dimensional space one will be drawing the 3 space axis, the 4th (imaginary numbers)for the time and the rest curled up.