What is equidecomposable?

Some articles on equidecomposable:

Banach–Tarski Paradox - Formal Treatment
... by the group of Euclidean motions and introducing the notions of equidecomposable sets and paradoxical set ... Two subsets A and B of X are called G-equidecomposable, or equidecomposable with respect to G, if A and B can be partitioned into the same finite number of respectively G-congruent ... gi (Ai ) = Bi , then we will say that A and B are G-equidecomposable using k pieces ...
Banach–Tarski Paradox - Obtaining Infinitely Many Balls From One
... a ball can be cut into k pieces so that each of them is equidecomposable to a ball of the same size as the original ... partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−1 using rotations ... into as many pieces as there are real numbers (that is, pieces), so that each piece is equidecomposable with two pieces to Sn−1 using rotations ...