i have seen the the term "geometric dissection" used to mean dissection into closed pieces, each of which is the closure of its interior and such that the only overlap between is along the boundary, and the term "set-theoretic dissection" used to mean a partition into disjoint sets (as in banach-tarsky). besides considering dissections os a square into congruent pieces, we can consider dissections of a regular k-gon into congruent pieces. the case k = 4 is obviously special! for which N is it possible to (geometrically) dissect a regular hexagon into N congruent pieces? there are some "obvious" values of N that are possible, and i have found dissections for a lot of values of N that are not "obvious". the same for a regular pentagon: for which N is it possible to (geometrically) dissect a regular pentagon into N congruent pieces? as before, there are some "obvious" cases, although fewer. i have found some dissections for non-"obvious" values, although these are also scarce. i haven't yet looked at impossibility proofs; these are likely to be prohibitively difficult. mike