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.