[quuxplusone]

> We can visualize 3D objects, and therefore can draw 2D cross-sections of 3D objects relatively well, and relatively easily.

I don't think that's true. For example, consider a regular octahedron: take a parallel pair of its faces and bisect the octahedron between those faces. What's the resulting figure? What happens to the figure as you tip the plane?

I mean, obviously the task I just set isn't impossible; and with a little reasoning anyone can give the answer in a few seconds; but it feels too me like the answer is not simply intuited merely by the virtue of our being 3D creatures.

Sure, part of the difficulty stems from that the octahedron (to most folks) is both less familiar and slightly more complicated than the cube. But the same applies to the hypercube!