Here's an insight I just had (seriously).
Perhaps higher dimensions are 'curled up' after all. I had contradicted statements made by some string theorists that higher dimensions are in fact curled up on their vertices somehow.
But now I think I understand the concept. It involves a higher dimension of perception, but not necessarily a higher dimension of matter.
What is involved is that a line may become a circle, which is perceived as a line in its center, but becomes circular and narrower, and fades away on both sides.
In this way, a triangle might have a bottom as well as a top made of three parts. You think that's the third dimension, but it's also the fourth-dimensional view of the second dimension!
Similarly, a square might be a sphere if the lines have no where to go (e.g. in Cartesian Space), or it might become fragmented. The rational answer is that it, too must be a three-d figure when projected into the four-d realm.
And the same for every other shape!
The perceptions we observe, if they are 2-d projections in the 4-d realm, must be mere pictures if we ever live in 4-d.
Planes on the other hand will be like what we know of 3-d, except they will have properties instead of depth. I think that is the conclusive answer.