It could be a compact[0] dimension, i. e. of finite length. In the simplest case you might imagine it as a circle attached to every point in our 3-dimensional Euclidean space. The aforementioned length scale would be the circumference of that circle.
Trying to wrap my head around this explanation and I’m picturing a looping gif. You have your normal x and y dimensions and then time through the gif. If the loop length is very short then distance between any two pixels will mostly only depend on x and y. Is that right?
Interesting case if we are the “ants” and it is our 3 dims happen to be compact looping somewhere beyond our event horizon. Multitude of Universes in that garden hose in which gravity can be falling as cube or more while at small scale if our compact Universe we’ll see square, and only very precise measurements may notice a bit larger than square.
Another possibility is if our brane has a lot of folds coming close/touching - that would make gravity there stronger like say that dark matter idea inducing rotation speed curve of the disk stars.
> Interesting case if we are the “ants” and it is our 3 dims happen to be compact looping somewhere beyond our event horizon. Multitude of Universes […]
I think you're mixing up two different cases here: 1) Our established 3 dimensions are actually compact, i.e. loop around or hit a boundary somewhere. No multiverse here. 2) There are extra dimensions, meaning that for every point in that extra dimension there's another 3-dimensional universe as we know it.
In the simplest case, yes. Though, once curvature (gravity) enters the picture, it could (in theory) become more complicated, as the additional dimension could get stretched or compressed.
Imagine if Flatland were a very long string in a big circle. In one direction you go around the big circle and it's a long distance. At a right angle to that, you go around a tiny little circle.
[0]: https://en.m.wikipedia.org/wiki/Compact_space