rwg> Are curves essential to any known minimal dissection? I think (no proof) that you can force a required curve: Consider a big, asymmetric object, perhaps a scalene triangle. Punch a smallish asymmetric hole in it, perhaps a second scalene triangle. This is the source polygon. The target is the same object, with the internal hole rotated. The rotated hole must overlap the original; the rotation amount must be an irrational multiple of pi. This has a two piece dissection, based on cutting a circle around the hole, and rotating the internal piece. I believe I've specified enough restrictions to make polygonal solutions suboptimal. The spoiled ideas include simply cutting a new hole and filling the old hole; flipping either inner or outer triangle; and cutting portions of either triangle and refitting around the edge to do an ersatz rotation. Rich