This is part of a letter by Morris W. Hirsch (who was my thesis advisor) to the Notices of the AMS (2014): ----- The term "scissor congruent," introduced in [1], refers to a pair of planar convex bodies A = A_1 u ··· u A_n, B = B_1 u ··· u B_n, where A_i and B_i are congruent compact 2-cells, andfor i,j the interiors of A_i and A_j (respectively, B_i and B_j) are disjoint. The main result is that A and B are scissor congruent iff they have the same area and their boundaries have partitions ∂A = I_1 u ··· u I_m, ∂B = J_1 u ··· u J_m, where I_k and J_k are congruent 1-cells. It follows that a square and a circle are not scissor congruent, andscissor congruent ellipses are congruent. No analog in higher dimensions is known. The term "scissor congruent" was introduced by Dubins et al. in [1] to describe a pair of convex planar bodies A, B having cell decompositions such that there is a bijection between the two-cells of A and those of B with corresponding two-cells congruent through rigid motions. The main result in [1] is that A and B are scissor congruent iff they have the same area and their boundaries have partitions ∂A = I_1 u ... u I_m, ∂B = J_1 u ... u J_m with I_k and J_k congruent arcs. It follows that a square is not scissor congruent to any strictly convex body, and if two ellipses are scissor congruent, then they are congruent. References [1] L. Dubins, M. Hirsch, and J. Karush, Scissor congruence, Israel J. Math. 1 (1963), 239–247. [2] S. Gold, The sets that are scissor congruent to an unbounded con- vex subset of the plane Trans. Amer. Math. Soc. 215 (1976), 99–117. [3] R. Gardner, A problem of Sallee on equidecomposable convex bod- ies, Proc. Amer. Math. Soc. 94 (1985), 329–332. [4] C. Richter, Affine congruence by dissection of discs—appropriate groups and optimal dissections, J. Geom. 84 (2005), 117–132. [5] ________, Squaring the circle via af- fine congruence by dissection with smooth pieces, Beiträge Algebra Geom. 48 (2007), no. 2, 423–434. ----- —Dan
On Apr 14, 2016, at 5:18 AM, Bill Gosper <billgosper@gmail.com> wrote:
http://home.btconnect.com/GavinTheobald/HTML/Octagon.html#Hexadecagon I don't recall curved cuts in a polygonal dissection. Can't the circular segments be replaced by segments of hexadecagons?
Are curves essential to any known minimal dissection?