TITLE: Decomposing a circle into similar pieces
QUESTION [6 upvotes]: Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?

REPLY [8 votes]: 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':