The term "equidecomposable" has two related but distinct meanings: "equidecomposable" and "equidecomposable by dissection." The difference is that the pieces in an equidecomposition by dissection have overlapping boundaries, as in the case of the four small squares into which a large square can be cut.
The pieces in the Banach-Tarski paradox, on the other hand, do not overlap even over the boundary. Their intersections are empty, and where one is closed, the adjacent one is open.
