See Greg Frederickson's "Dissections: Plane and Fancy" (pp. 217--218) for a discussion. All you have below is a bit of computer recreation, complete with a (perhaps obvious) kludge that I have no intention of improving. Meanwhile, it is rather hypnotic.
Coded and animated (with Mathematica 7.00) on a listless morning between semesters (Aug. 4, 2009) by Rick Mabry when he should have been being productive.