Multifold Tiles of Polyominoes and Convex Lattice Polygons

A planar shape S is a k -fold tile if there is an indexed family T of planar shapes congruent to S that is a k -fold tiling: any point in R-2 that is not on the boundary of any shape in T is covered by exactly k shapes in T. Since a 1 -fold tile is clearly a k -fold tile for any positive integer k, the subjects of our research are nontrivial k -fold tiles, that is, plane shapes with property "not a 1 -fold tile, but a k(>= 2) -fold tile." In this paper, we prove some interesting properties about nontrivial k -fold tiles. First, we show that, for any integer k >= 2, there exists a polyomino with property "not an h -fold tile for any positive integer h < k, but a k -fold tile." We also find, for any integer k >= 2, polyominoes with the minimum number of cells among ones that are nontrivial k -fold tiles. Next, we prove that, for any integer k = 5 or k >= 7, there exists a convex unit -lattice polygon that is a nontrivial k -fold tile whose area is k, and for k = 2 and k = 3, no such convex unit -lattice polygon exists.