Sequentially Divisible Dissections of Simple Polygons

A k-dissection D of a polygon P , is a partition of P into a set of subpolygons {Q1, . . . ,Qm} with disjoint interiors such that these can be reassembled to form k polygons P1, . . . ,Pk all similar to P . D is called non-trivial if none of {Q1, . . . ,Qm} is similar to P . In this paper we show that any convex n-gon has a k-dissection (resp. sequential dissection) with (k − 1)n + 1 pieces, n ≤ 5. Let k ≥ 2 and n ≥ 3 be integers and let P be an n-gon. We show that if P is a convex polygon and n ≤ 5, then there exists a dissection of P consisting of at most (m−1)n+1 polygons which combine to form sequentially 2, 3, · · · , m unequal polygons similar to P . If P is a convex polygon and n ≥ 6, then there exists a dissection of P consisting of at most b 5m−4 3 nc−2(m−1) polygons which can be assembled similarly as stated above. We also show that for m ≡ 1 ( mod .3) and general n-gon P , we can dissect P into at most 2n − 2 + m−4 3 (b 7 3 nc − 4) polygons which combine to form sequentially 4, 7, · · · , m polygons similar to P .