# A circle cut into smaller pieces can become a smaller circle?

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). – André Nicolas Nov 28 '11 at 20:38
It sounds similar to the Banach-Tarski paradox (en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). I do not think this is possible in two dimensions using finitely many pieces; see terrytao.wordpress.com/2009/01/08/… . – Qiaochu Yuan Nov 28 '11 at 20:38
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." – Thomas Andrews Nov 28 '11 at 20:43
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! :) – Qqwy Nov 28 '11 at 20:45

## 1 Answer

There are two meanings to the word circle. Among non-mathematicians, it often means the curve together with its interior. Among mathematicians, circle refers to the curve, and disk refers to the curve together with its interior.

Using the standard axiomatization of set theory (ZFC), it can be proved that no paradoxical decomposition of the circle exists, nor of the disk.

However, there is a famous related result, called the Banach-Tarski paradox, that says that the $3$-dimensional ball of radius $1$ can be decomposed into a finite number of sets, which can then be reassembled to make two complete $3$-dimensional balls of radius $1$. There is a large family of related results. The one that is closest to your question is that for any $r$ and $R$, a ball of radius $r$ can be decomposed into a finite number of sets which can be reassembled to make a ball of radius $R$.

Unrelated to your question, but interesting, is the following beautiful theorem of Laczkovich. The disk of area $1$ can be decomposed into a finite number of sets which can be reassembled to make a square of area $1$.

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