Banach-Tarski "paradoxes" can include, e.g. cutting a ball into a finite # peices, then reassembling it into two balls (i.e. twice the volume). In order to accomplish this, the"pieces" are non-measurable sets. This is far nastier than the square-circle "paradox" Plouffe mentioned. The main "use" of Banach-Tarski and their successors seems to be in demonstrating that the foundations of mathematics are a dangerous and nasty place, and maybe you shouldn't really believe some of those ZFC axioms... One starts to feel some sympathy with Brouwer and the "constructivists"... In contrast, the phrase "scissors congruence" is usually used to mean something far tamer and also more practical -- one cuts A into a finite number of pieces using "scissors" (i.e. piecewise-smooth boundaries, everything measurable) which then are re-assembled into B. This was already studied by Bolyai who showed that any two polygons with the same area, were scissors congruent. However, Dehn showed in one of the first Hilbert problem solves, that a regular tetrahedron and cube (same volume) are NOT scissors congruent. It is fairly well understood now what is scissors congruent to what in all low dimensions (say, up to 4) and there is at least one entire book giving amazing scissors congruences. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)