TITLE: reverse Banach-Tarski paradox
QUESTION [8 upvotes]: Is it possible to reverse the operation of the Banach-Tarski paradox? That is, I have two three-dimensional balls, and is it possible to combine them into one ball that is identical to one of the balls (using the axiom of choice)?
Thank you very much.

REPLY [5 votes]: The short answer is, of course.
The Banach-Tarski paradox tells you how to take a ball and cut it in such way that you can move the pieces around and create two balls. The process is completely symmetric. 
What does the Banach-Tarski paradox say? It says that given a ball $B$ we can write it as $B=A_1\cup\ldots\cup A_n$ which are all disjoint, and using very nice maps $f_1,\ldots,f_n$ (rotation and moving then pieces without stretching them) we get that $f_1(A_1)\cup\ldots\cup f_n(A_n)$ is actually two balls! These $f_i$ are so nice that we can reverse them, that is rotate back and shift around the other direction - again without stretching.
Given two disjoint balls we can consider a very canonical map $g$ the two balls of the Banach-Tarski recomposition two our new ones. Now we have that for $i=1,\ldots,n$ we have that $B_i=g(f_i(A_i))$ is the part which essentially decomposed the unit ball. Reverse the process that $f_i$ did to each $B_i$ and the result will recompose a single ball.
The trivial and obvious example is when your two balls are the ones you get from the Banach-Tarski paradox itself, in which case $g$ is the identity.