Here's a problem that's been
bothering me recently. It seems that in most respects, a higher spatial dimension is "superior" to a smaller one. For example, a
fourth dimensional being could travel anywhere in our three dimensions as well as traveling in the
fourth, well beyond our reach. This is illustrated very beautifully in
Flatland, where two dimensional beings can only perceive slices of
3D objects.
Here's the interesting thing though: it is possible in 2D space to create something that cannot exist in 3D, namely the impossible triangle. Other examples include M.C. Escher's impossible staircase and waterfall. In both of these cases, the two-dimensional figure describes an object that cannot exist in three-space. Is this phenomenon also true for any n- and n+1-dimensional space? Are there 3D objects that describe something that can't exist in 4-space, a sort of 3D paradox? Is this just a 2D phenomenon? Is it possible to draw things in 2D that are impossible in higher dimensions, too?
The thing that also fascinates me about this is that it is not based purely on physical reality. The illusion has a significant psychological component because the whole reason the impossible triangle seems wrong is because one's mind is interpreting into 3-space. It is giving it its non-existant volume. The triangle is not really an object--it is a representation. It only becomes "impossible" when my mind imbues it with properties it doesn't have. On the other hand, it definately is not a purely psychological phenomenon because there seems to be some objective, physical truth to it, otherwise why would the majority of the population agree that the impossible triangle is impossible?
The interesting thing is that while figures like these are constrained by physical laws, they are also constructed to a certain degree. We are the ones who create and interpret them. The only reason we know how to make a figure such as the impossible triangle is because we have experience with the properties of a three-dimensional world. I would suppose that in order to construct a representation of an impossible 4D object, we would have to have some knowledge of what 4-space is like.