I have three questions that relate to _Flatland_ in some way or other (as a whole, not just the beginning). Here goes:
1) Is it truly possible for a square with a 2-D visual system (i.e., a 1-D retina) to comprehend the third dimension even if someone WERE to move him "upwards, but not northwards" out of his plane? I believe the question is one of environment vs. equipment--in other words, is it faulty sensory equipment that prevents him from comprehending the third dimension? Or is it merely his lack of exposure to higher dimensions? I would tend to think that a square with a visual system designed to perceive two dimensions will still only perceive those two dimensions, no matter what way you orient him. Therefore, above the plane, what might we expect him to see? Extrapolating, is OUR difficulty with visualizing 3+n dimensions a limitation of our own neural equipment or our environment? The implication in _Flatland_ is that Flatland itself exists as a plane in (surrounded by) three-space; the only reason the flat creatures didn't see "up" was because they just didn't understand in what direction to look. I would tend to go with the limitations of our sensory equipment, though a good argument might be made that we might have developed (3+n)-D sensory structures were we exposed to those dimensions. What do you think?
2) A shorter question (whew): It makes perfect logical, if not visual, sense to me how one goes about making a hypercube (move a point through one dimension and get a line, move the line and get a square, move the square and trace out a cube, and move the cube along a "new" direction to get a hypercube). What really confuses me is the idea of a hypersphere. How would one construct such a beast? How would one explain or describe it? The rotation of a sphere through the next dimension? What would some characteristics of this figure be, given the unique properties of the 3-D sphere?
3) This may or may not be tangential, but it is sort of an interesting question anyhow. A. Square kept referring to his "intestines" as being visible to those in higher dimensions. I once read something (I don't remember where) called the Two-Dimensional Dog Paradigm, or some such thing. There was a picture of a two-dimensional (flat, line drawing) dog, with a simplified "food tube" leading in from its mouth and out the other end. The point of the picture was to demonstrate that a long tube through the dog would effectively bisect the poor creature, dividing its top half from its bottom half quite completely. Does this apply to the two-dimensional square, or might it have some other kind of intestine (i.e., a one-aperture tube like a jellyfish which serves both as mouth and anus)? Just an odd biology-of-the- second-dimension question. Could the square manage to avoid this problem in any way, topologically?