That reminds me of one of the oldest unsolved problems in geometry: show that 4 pieces are minimal for a scissors dissection of a square to an equilateral triangle. The classic dissection (believed to be optimal) is here: https://oeis.org/A110312/a110312_3v.jpg and this "4" is the conjectured value of a(4) in https://oeis.org/A110000 (one of my favorite sequences, but regrettably only one term, a(3)=1, is known to be correct) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Apr 14, 2016 at 11:43 AM, <rcs@xmission.com> wrote:
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
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