Get a compass and a straightedge, and draw some figures. I'll bet you haven't used a compass since high school geometry - I know I haven't. Construct polygons, bisect angles, go wild!
While you're constructing polygons, think about their duality. Why is it that all regular polygons are self-dual, while the majority of higher-dimensional regular shapes are not self-dual? What properties must an object have in order to be self-dual?
Is there any explanation for the existence of 6 regular 4-dimensional polytopes when there are only 5 regular polyhedra? I would expect the number of regular figures to decline continuously with higher dimensions, not increase and then decrease again. Why does such a counter-intuitive phenomenon occur?