UC Irvine

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 R2 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.