Can we slice an object into two pieces similar to the original? I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original.  How can I prove this?

It seems that Puppe and others proved that this is impossible for any strictly convex solid. See [B. L van den Waerden, Aufgabe Nr 51, Elem. Math. 4 (1949) 18, 140]
The reference comes from Unsolved problems in geometry by Hallard T. Croft, K. J. Falconer and Richard K. Guy.
