Klarner systems and tiling boxes with polyominoes

Let T be a protoset of d-dimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by T? A nice result of Klarner and Göbel asserts that the answer to this question can always be given in a particularly simple form, namely, by giving a finite list of “prime” boxes. All other boxes that can be tiled can be deduced from these prime boxes. We give a new, simpler proof of this fundamental result. We also show that there is no upper bound to the number of prime boxes, even when restricting attention to singleton protosets. In the last section, we determine the set of prime rectangles for several small polyominoes.