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Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?

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what do you mean by 'similar disjoint pieces'? do you maybe just mean 'disjoint pieces'? or do you mean disjoint pieces that also have the same shape? If you cut the circle (disk) up into pieces by a straight lines for example - i think there is only one possiblity to make those pieces the same shape (namely the lines all pass through the center of the disk and always halve the remaining pieces). – Peter Sheldrick Oct 12 '11 at 10:55
I mean "disjoint pieces that also have the same shape". But not necessarily the same size or rotation. I suspect its not possible but I'd like a proof. – user17497 Oct 12 '11 at 11:37

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A version of this question was asked on MO a while back: "Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?." It was determined that that particular problem (which I now think is identical to yours as currently posted) is open, and appears in Unsolved Problems in Geometry.

There Anton Geraschenko posted this interesting dissection:
          
which answers another version (perhaps the first version?) of your question. In the comments another dissection was given by 'sobe86':
          

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