Efficient Regular Polygon Dissections

We study the minimum number g(m, n) (respectively, p(m, n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that ⌈n/2⌉ - 2 ≤ g(4, n) ≤ (n/2) + o(n) and ⌈n/4⌉ ≤ g(n, 4) ≤ (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4, n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that ⌈n/4⌉ ≤ p(4, n) ≤ (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m, n) and p(m, n).