Sequentially Divisible Dissections of Simple Polygons

Abstract

A k-dissection D of a polygon P, is a partition of P into a set pf subpolygons {Q ₁,...,Q _m } with disjoint interiors such that these can be reassembled to form k polygons P ₁,...,P _k all similar to P. If for every j, 1 ≤ j ≤ k, the pieces of D can be assembled into j polygons, all similar to P, then D is called a sequentially k-divisible dissection. In this paper we show that any convex n-gon, n ≤ 5, has a sequentially k-divisible dissection with (k - 1)n+1 pieces. We give sequentially k-divisible dissections for some regular polygons with n ≥ 6 vertices. Furthermore, we show that any simple polygon P with n vertices has a (3k+4)-dissection with (2n - 2) +k(2n + ⌊ n/3 ⌋ - 4) pieces, k ≤ 0, that can be reassembled to form 4,7,..., or 3k + 4 polygons similar to P. We give similar results for star shaped polygons.