The search for regular polytopes in four-space answers some questions for myself about the mystification of the fourth dimension-- our world is a spatially 3-dimensionsal one, one can think of it as a spatio-temporal world of four-ish dimensions, but this really isnšt the case. Higher dimensionality is easily explored in algebra of mutiple variables and modern computing in the case that dimension refers to degree of freedom. But why does the fourth dimension specifically draw in so much study and speculation, when in this analogous manner any number of dimensions is possible. I frequently thought that it was simply due to it being the next one up. Each successively higher dimension being increasingly difficult to visualize and understand, so that at the present time we were just getting to the fourth dimension and wešre still stuck on it, until we can truly understand it--maybe we never will. Anyway, the search for regular polytopes and the finding that there are six in four-space (more than three and all successively higher dimensions) actually does make the fourth dimension distinct and special. The fourth dimension is one up fom the reality of three perpendicular dimensions, how does this distinct and special dimension relate back to our three dimensions in which it is possible to model and appreciate the higher dim.
Can we see more of the four-dimesional art? Art seems to be the most viable and real-world (excepting several variable sytems) scenario for the usage of higher dimensions.