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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.

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    Could you explain the asymmetry you seem to be seeing? Under the only meaning of "combine" that I can think of, two congruent balls being combinable into one ball that's congruent to them is *synonymous* with one ball being decomposable into two balls congruent to the one ball.2012-03-20
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    @joriki I was asking whether the axiom of choice can be used to do the process in reverse. So, is it the same? I know that I can use the axiom of choice make one ball decompose into two. I am asking whether I can use the axiom of choice to combine two into one. that's all. nothing special into the meaning of "combine".2012-03-20
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    joriki's point is that you should convince yourself that if you can decompose $A$ into pieces and rearrange the pieces to get $B$ then you can do this process in the reverse direction and transform $B$ into $A$ in this way. As Dejan points out in the answer below, a much better results holds.2012-03-20
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    @user25148: If you want to decompose two balls into one from scratch, you basically need to do the same construction as if you were decomposing one ball into two. So the axiom of choice is needed in this case. But if you already have a decomposition of one ball into two, you need no further axiom of choice, since as I describe below, you can just invert the isometries the axiom of choice has given you before.2012-03-20
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    I'm not sure that the tag [axiom-of-choice] fits here...2012-03-20
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    @wythagoras: I highly doubt that the BT paradox needs a separate tag.2015-05-23
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    @AsafKaragila Hm, there are about 200 questions for it. (Yeah, I first wanted to add it to all those questions, then I realized I wasn't being helpful.)2015-05-23
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    @wythagoras: It's not just a question of volume. It's a question of classification. All questions about the BT paradox contain the keywords "Banach-Tarski" making them easily searchable with the crummy internal search engine of the site. I suggest that you bring this up on meta, where an actual discussion on the merits of this tag can take place. (Also, if you search for "Banach Tarski" is:question you will see that the number is closer to 64 questions, not 200.)2015-05-23

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