Tiling with Notched Cubes

Tiling with Notched Cubes, by Robert Hochberg and Michael Reid
Discrete Mathematics 214 (2000), no. 1-3, pp. 255-261.
[DOI] [Math Reviews] [Zentralblatt]
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It is a simple observation that any polyomino that tiles a rectangle can also tile a larger copy of itself, and thus is a "self-replicating tile", or "rep-tile" for short. Although one does not expect the converse of this statement to be true, it is still an open question. That is, all known examples of polyomino reptiles have the property that each also tiles a rectangle.
In this paper, we consider the analogous question in higher dimensions. We exhibit a simple shape, the "notched cube", which has a simple rep-tiling. We show that, in all dimensions d ≥ 3 , the notched cube cannot tile any box. Moreover, we show that the only rep-tilings it has arise from composition of the obvious rep-tiling with itself. We deduce these from another interesting result, that in d ≥ 3 dimensions, the notched cube has a unique tiling of an orthant.
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Updated January 8, 2008.