Tiling with Polyominoes, Polycubes, and Rectangles

In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes

Tiling With Notched Cubes

Golomb [1] showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a "notched cube") that tiles a larger copy of itself, but does not tile any box, and obtain several related results about tiling with this figure. We also obtain analogous results in all dimensions d 3