[ben_w]

If a compactified spatial dimension exists in our universe, and was big enough to fit an atom, why couldn't we see two atoms that seem like they're in the same 3-dimensional coordinates?

Sometimes compactified dimensions are analogised to a straw: seen from a distance it seems one dimensional, up close (an ant's perspective) it's got one long dimension and one short dimension.

I don't know how far to take the analogy. It sounds like surely photons with wavelengths smaller than the compactified dimension would be likely to take a spiral path, looping around compact dimension n times for every m units of 3-space travelled, which would seem like they were mysteriously slow if you weren't expecting the compact dimension to exist.

I vaguely remember the idea of wavelength-dependent speed of light is a thing that's been ruled out by tests with supernova data, but not to what wavelength or sigma.

[MichaelZuo]

The same reason why flatlanders don’t see two circles in the same 2D coordinates, even if a 3D tube was penetrating through their world.

Because they can’t see above or below to the rest of the tube. They can only see a single infinitely thin slice of the tube.

[ben_w]

I think you're describing a completely different geometry than I'm describing.

An ℝ²-brane such as flatland existing in a ℝ³ bulk is different to an ℝ²⨯S¹.

If the S¹ part* is present in our universe to the degree that it can explain anything about gravity, it should also have an impact on everything else in the universe larger than the radius of the S¹ dimension's circumference.

* well, S^n ⨯ T^m, the version of string theory I hear most about has n+m = 6, but there are others, and this thread is a toy model where n=1, m=0

Edit: Apparently the U+1D54A character is stripped, so put a plain ASCII "S" back in.

[MichaelZuo]

I’m describing why the flatlanders wouldn’t see multiple circles even though a 3D tube is composed of infinitely many 2D circles.

[ben_w]

I noticed you were doing so, yes.

The "tube" (compactified dimension) isn't a higher dimensional object going through our space, in string theory it is an actual part of our space.

To put it another way: for compactified dimensions, we're not in flatland.

(For brane theory, we are in flatland, but they're two different ideas about how stuff might work).