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While I was on a plane, a Math professor once told me that it was possible to divide a circle in multiple smaller pieces in such a way that, when those smaller pieces are assembled in another way, they create another circle, but this circle somehow is smaller than the first one.

It sounds kind of paradoxical to me, and that's why I want to ask: Is this true? If so, where can I find more information about it?

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    Not true. But true of a $3$-dimensional ball, if you interpret "pieces" as subsets. See the Banach-Tarski paradox (Wikipedia).2011-11-28
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    It sounds similar to the Banach-Tarski paradox (http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). I do not think this is possible in two dimensions using finitely many pieces; see http://terrytao.wordpress.com/2009/01/08/245b-notes-2-amenability-the-ping-pong-lemma-and-the-banach-tarski-paradox-optional/ .2011-11-28
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    Banach-Tarski is what you want, although the word "cut" is a misnomer, because the division of the points on the sphere doesn't match any intuitive notion of "cut."2011-11-28
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    It could very well be that he was talking about a ball instead of a circle, and that I misremembered. Thanks very much, all three of you! :)2011-11-28

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