[math-fun] Paradoxical decomposition of the plane or sphere?

Does there exist a decomposition of the plane into three disjoint sets A, B, C such that all three are isometric to each other but also A is isometric to the union B + C ??? Ideally each set would be connected. Also measurable, but I would be surprised if measurable is possible. And ideally, without needing the Axiom of Choice. How about the same thing but with some points omitted: A + B + C = R^2 - Z where Z is a set of measure 0 ??? And if not the plane, how about the sphere S^2 ??? —Dan