----- Banach-Tarski "paradoxes" can include, e.g. cutting a ball into a finite # pieces, 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"... ----- After learning that the Axiom of Choice is equivalent to "The cartesian product of nonempty sets is nonempty", I decided that any counterintuitive consequences of AC were problems with my intuition, not with AC. ----- 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. ----- I thought scissors congruence normally refers just to decomposition of two spaces into (finitely many) topological closed disks with disjoint interiors, where the collection of disks in one case is isometric to the collection of disks in the other case. ----- This was already studied by Bolyai who showed that any two polygons with the same area, were scissors congruent. ----- In fact, congruent by dissection into finitely many convex polygons. --- However, Dehn showed in one of the first Hilbert problem solves, that a regular tetrahedron and cube (same volume) are NOT scissors congruent. ----- I thought Dehn proved only that they could not each be dissected into the same finite set of polyhedra (up to isometries), analogous to the Bolyai dissections. Incidentally, Sydler found an exact algebraic invariant for when two polyhedra can be dissected into each other. ----- 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. ----- Is there a proof that a 4-dimensional cube and 4-dimensional ball (sphere + interior) of the same volume are not scissors congruent? --Dan