There are two similar but distinct concepts related to equidecomposability: "equidecomposable" and "equidecomposable by dissection." The difference is in that the pieces involved in the "equidecomposition by dissection" have overlapping boundaries, such as in the case of four small squares into which a big 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.
