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 less than or equal to g(4, n) less than or equal to n/2 + o(n) and [n/4] less than or equal to g(n, 4) less than or equal to 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] less than or equal to p(4, n) less than or equal to (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).