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Banach–Tarski paradox - Wikipedia, the free encyclopedia - http://en.wikipedia.org/wiki/Banach–Tarski_paradox#Obtaining_...

a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. · ati