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I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five pieces. My question is :

"Is it possible to construct (mathematically) these five sets or is the proof more of an existence result and not a construction?" If it is possible to construct, I would really appreciate if someone can show the construction here.

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Hardly a definitive treatise, but it's readable: – Blue Nov 11 '10 at 2:28
[ Very related.]( – J. M. Nov 11 '10 at 3:28
I recommend Stan Wagon's book on the subject very highly. – Ross Millikan Nov 11 '10 at 6:25
up vote 10 down vote accepted

No, it is not possible to explicitly construct the pieces. The pieces are, by necessity, non-measurable. This means that they cannot be Borel sets, which covers essentially every "explicit construction" technique you might think of.

Moreover, there are models of ZF set theory (without the axiom of choice) in which the Banach-Tarski paradox fails. So the construction must, necessarily, make use of some form of the axiom of choice. This means that an even wider range of construction techniques - those that can be carried out in ZF - are insufficient to form the decomposition.


After clarification, it seems that one part of the question is to find an explicit proof that (using the axiom of choice) it is possible to get a decomposition using exactly 5 pieces. A proof of this is given by Francis Su's thesis on the paradox (PDF). Theorem 20 gives the proof of the five-piece decomposition, and by tracing back the previous results you can work out exactly what the pieces are. That thesis is, by the way, a wonderful reference for many other aspects of the paradox as well.

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I am not sure what "mathematically construct" could possibly mean. Why would a "mathematical construction" rule out the use of choice? – Andrés Caicedo Nov 11 '10 at 2:25
Obviously we can mathematically construct the decomposition in some sense - it's a theorem. I was reading to question to ask whether there is some sort of explicit construction, rather than just an existence result. The difficulty with that sort of question is that "explicit" can mean a lot of things, which is why I phrased the answer to point out that if "explicit" means either "Borel" or "doable without the axiom of choice" then there is no explicit construction in that sense. – Carl Mummert Nov 11 '10 at 2:28
A crude first approach at formalizing what I mean: Proving the existence of an object by arguing that its non-existence leads to contradiction would not be a "mathematical construction". But I would be fine with a construction that uses a choice function or a well-ordering. I'm sure there are many levels of subtlety I'm ignoring here. – Andrés Caicedo Nov 11 '10 at 2:42
I added a reference that gives complete proofs for the 5 piece decomposition. – Carl Mummert Nov 11 '10 at 3:24
There is a nice appendix to the story. A few years ago, Trevor Wilson (then an undergraduate at Caltech) showed that the 5 pieces can be moved continuously without overlapping from their original position in the sphere to their new positions in the two spheres. (So in a concrete sense the construction is not that abstract, since it admits such nice analysis). The paper is Wilson, Trevor M. "A continuous movement version of the Banach-Tarski paradox: a solution to de Groot's problem." J. Symbolic Logic 70 (2005), no. 3, 946–952. – Andrés Caicedo Nov 11 '10 at 3:42

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