In 1953 Solomon W. Golomb defined a polyomino as a rook-wise, connected subset of squares of the infinite checkerboard. The first polyomino puzzles were tiling problems. Most of the time in tiling problems one restricts to simply-connected polyominoes (i.e., polyominoes without holes). But almost all polyominoes have holes—the number of polyominoes with holes grows exponentially faster than polyominoes without holes. In this talk, we are interested in studying the asymptotic behavior of polyominoes with holes by exploring: What is the maximum number of holes that a polyomino with n tiles can enclose? In answering this question, we have found a sequence of polyominoes that reaches the maximum possible number of holes that approaches some limiting fractal shape. This sequence generates a non-periodic tessellation of the plane. We have also proved that the maximum number of holes that a polyomino can have, divided by the number of tiles, converges to 1/2.
