Golomb defines a plane figure to be rep-k if it can be cut into k similar but smaller copies. It is known that the only convex rep-2 regions in 2D are the isosceles right triangle and the rectangle with sides in ratio 1: sqrt(2).
Questions: Consider the analogous 3D situation. It appears there are no convex rep-2 3D regions. The closest to rep-2 that one could get is only a 'loft' of one of the above two 2D rep-2 regions in the z directions. This gives a prism which of which only a fraction of 1/sqrt(2) is covered by 2 similar scaled down prisms. Is this the best one could get in 3D (have no proof)?
If so, does the same setup generalize to higher dimensions? ie. can one say: the 4d convex region closest to being rep-2 is got by just lofting the above 3d prism (generated from lofting the 2d region) into the 4th dimension to have a 4D region with coverage only 1/2 by two self-similar 4d regions.
Moreover, we don't know if one discards convexity, there are 3d regions which are rep-2.
Update (July 17th 2005)
From Dirk Frettloeh, I got a nice example of a 3D 2-reptile - a rectangular box of dimensions 1, 2^(1/3), 2^(2/3). Two of them piled together gives a 2^(1/3), 2^(2/3),2 box. This is generalizable to n-reptiles in 3D.
No idea yet about rep-3 convex
Shapes in 3d ... and rep-4 and so on...
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