Efficient Regular Polygon Dissections

Abstract

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 $\lceil n/2\rceil -- 2 \leq g(4,n)\leq\frac{n}{2} + o(n)$ and $\lceil n/4\rceil\leq g(n,4)\leq\frac{n}{2} + o(n)$, 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 $\lceil n/4\rceil\leq p(n,4)\leq\frac{n}{2} + o(n)$, holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m.n).