The above figures are two of the most interesting artifacts from my career in mathematics. Mainly they're just nice to look at. But they share a special property as well. Don't know what it is? Well, here's a hint: The center square in the right-hand figure should really be a sixth color, instead of simply red. Okay, here's another hint. Think about the discriminantal arrangement of hyperplanes determined by the generating vectors (including multiplicities) of the zonotopes, and the natural connection between chambers of the discriminantal arrangement and the set of tilings of those zonotopes. It's easy to see it now, right?

Okay, so by now you're probably sorry you asked. The answer is that each of these geometric doohickeys is an example of an incoherent tiling of a 2-zonotope. In other words, there is no convex (ball-like) three-dimensional object which looks like either of these things when you take its picture. An "easy" way to see this for the figure on the right is to imagine that each set of borders shaped like >- is an arrow. Then going around the outside, there's a collection of arrows going counterclockwise around the figure, with the head of one arrow pointing at the tail of the next. If there really were a ball-like object which looked like this when you took it's picture, it would have to have infinite depth. In short, it's an optical illusion, like M.C. Escher's print Ascending and Descending. The two-dimensional representation makes it look like the object can exist in three dimensions, but it really can't.