[m00n]

Actually, a Banach-Tarski-like result is impossible in 2D space, since there is a Banach measure (= volume definition to all subsets of the plane) that extends the usual volume definition (e.g. for circles).

The crucial idea that makes Banach-Tarski work in 3D is the insight that the set of rotations around an axis through the origin in 3-space has a free subgroup F on 2 generators (finite strings of A's, B's and their inverses). From this fact the proof is quite easy, but this comment is too small for it.