The Klein bottle, though unorientable, is nevertheless the boundary of a three manifold. It is easy to see how to construct this 3 manifold. The projective plane is not a boundary of any manifold. How can we visualize this impossibility? Is there a nice embedding of the projective plane in R^4? How about an embedding of the projective plane in Euclidean space so that it has constant curvature? Same embedding question for the Klein bottle.