Tiling with notched cubes

In 1966, Golomb 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. Golomb [1] shows that any polyomino that tiles a rectangle also tiles a larger copy of itself. There is no reason to expect that the converse holds; however, every polyomino that is known to tile a larger copy of itself also tiles a rectangle. This is considered, for example, in [2, Problem 6.10]. We examine here the corresponding question in higher dimensions. Definitions. A cell in d-dimensional space R is a region C(n1, n2, . . . , nd) = {(x1, x2, . . . , xd) ∈ R d | ni ≤ xi ≤ ni + 1 for i = 1, 2, . . . , d} where n1, n2, . . . , nd are integers. A (d-dimensional) polycube is a finite union of cells, whose interior is connected. A (d-dimensional) box is a subset of R which is congruent to {(x1, x2, . . . , xd) ∈ R d | 0 ≤ xi ≤ ai for i = 1, 2, . . . , d} for some positive a1, a2, . . . , ad. A (d-dimensional) orthant is a subset of R d which is congruent to the positive orthant {(x1, x2, . . . , xd) ∈ R d | xi ≥ 0 for i = 1, 2, . . . , d}. A reptile is a figure that tiles a figure similar to itself, with ratio of similitude greater than 1. An N -reptiling by a figure is a tiling of a larger figure similar to the original, which uses N tiles. A (d-dimensional) doublecell is a region Q(2n1, 2n2, . . . , 2nd) = {(x1, x2, . . . , xd) ∈ R d | 2ni ≤ xi ≤ 2ni + 2 for i = 1, 2, . . . , d} for some integers n1, n2, . . . , nd. Note that different doublecells do not share any cells. Typeset by AMS-TEX 1 2 ROBERT HOCHBERG AND MICHAEL REID Definition. A (d-dimensional) notched cube is a polycube congruent to the closure of Q(0, 0, . . . , 0) r C(1, 1, . . . , 1), i.e. the polycube which is the union of the 2 − 1 cells {C(n1, n2, . . . , nd) | each ni = 0 or 1, and some ni = 0}.