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I think the title says it all. I am planning on giving a talk in a few weeks about the Banach-Tarski paradox and I have some pdfs found online which describe the paradox a little but I am looking for a solid reference which covers the construction from A to Z and on which I can extract the main ideas for my talk from (I understand the ideas beneath the paradox, I am just looking for a formal proof with no details excluded,i.e. a well-structured document). Anyone has a reference in mind?

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Doesn't Wikipedia give more than enough references? – t.b. Aug 11 '11 at 2:29
If you could only read one book about it, go with Stan Wagon's. – J. M. Aug 11 '11 at 2:30
I said references. I didn't say Wikipedia is any good for the proof (this is only rarely the case except for utter trivialities). But you have Wagon (including the Wolfram demo), Stromberg, Banach-Tarski, von Neumann among many other things, even the circle-squaring problem is mentioned. Only the pointer to the Ruziewicz problem is somewhat disguised. – t.b. Aug 11 '11 at 5:06
By the way Terry Tao's blog contains two rather nice blog entries on amenability and the BTP. David Feldman's thread on MO contains some nice ways of getting free subgroups of SO(3) which in my view is the heart of the matter. – t.b. Aug 11 '11 at 22:41
Sorry I copied the wrong link from Tao's blog (it's getting late here...): that's the one – t.b. Aug 11 '11 at 23:01
up vote 12 down vote accepted

I gave a similar presentation at MathFest 2011 in Kentucky last week, using Stan Wagon's book as a guide. Here is a list of definitions/theorems/etc that are on the direct thread to reaching the Banach-Tarski Paradox (stated as a corollary in the book).

  • Def 1.1: G-Paradoxical
  • Thm 1.2: Free group of Rank 2 F-paradoxical
  • Prop 1.10: Group acting without nontrivial fixed points
  • Thm 2.1: $SO_3$ has free subgroup of Rank 2
  • Thm 2.3: Hausdorff Paradox
  • Def 3.3: G-Equidecomposable
  • Prop 3.4: Equidecomposability preserves Paradoxes
  • Thm 3.9: $S^2$ and $S^2$ minus a countable set are equidecomposable
  • Cor 3.10: The Banach-Tarski Paradox

As others have stated, there are MANY interesting results along the way in this book, and the development is superb. Here is a link to a modified version of the presentation I gave at MathFest. It is an attempt at illustrating exactly what is presented in the material of the text, rather than providing an alternative interpretation (baby steps, right?). For the web version, I've added some annotations so that it's better suited for reading as the original slide presentation didn't have a lot of textual development, though unfortunately I was not able to add detailed descriptions of the animations without significant re-work.

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Yes, that would be of a great help. Thanks =) – Patrick Da Silva Aug 11 '11 at 13:24
Rachel, these are really nice slides. I hope your presentation went well! – t.b. Aug 14 '11 at 22:09
@Theo - Thank you!! The presentation went unexpectedly well, actually! The room was absolutely packed--probably 80 students came to listen. The majority of students had heard of the paradox, but only one kept their hand up when I asked if they had an idea of the shape the pieces resembled. In the end I was awarded a student speaker award by the AMS. It was a great experience for my first mathematics presentation!! – Rachel Aug 14 '11 at 23:14
That is a wonderful story to hear, thanks for sharing it! Congratulations, and as far as I can tell from the slides this must have been deserved. The challenge of speaking in front of 80 people for a first time serious math presentation must have been quite exciting, no? I for one am glad that my first presentation was on an about ten times smaller scale... – t.b. Aug 14 '11 at 23:25
I've switched my check ; this answer is awesome. Thanks for everything, it'll definitively help! I'll give you news on my presentation. =) – Patrick Da Silva Aug 16 '11 at 16:56

In my view the images that show the constructive version of the BTP in hyperbolic space are the best to motivate what is going on. Sure, there are some details that are different in $R^3$ compared to $H^2$, but they are just details really. The underlying group theory -- the way a free group leads to a paradox -- is so clear in the hyperbolic plane, disk model. Of course, this is discussed in my BTP book, but this demo/movie (which anyone can look at with the free software) shows the story nicely.

Stan Wagon

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Hello Dr. Wagon, thanks for posting that demo! Hopefully you can register here... – J. M. Aug 11 '11 at 3:31
Thanks for your answer, I looked at the movie and it is very interesting. – Patrick Da Silva Aug 11 '11 at 4:56

The Banach-Tarski Paradox, a great book by Stan Wagon, quite detailed. Most university libraries would have it. The book also discusses a lot of interesting ancillary material, very useful for a lecture!

Comment: The result does not extend to $\mathbb{R}^2$. Roughly speaking, this is because there is a (finitely additive) translation invariant "measure" on all subsets of $\mathbb{R}^2$ that extends Lebesgue measure.

The following is an old question of Tarski: Given a disc and a square of equal area, can the disc be decomposed into a finite number of regions, which can be reassembled to form the square? About $20$ years ago, Laczkovich proved, to everyone's surprise, that the answer is yes.

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While I was taking issue with the first version of your answer (because it was the canonical one) the circle-squaring problem is an excellent suggestion! Re your comment: here's Banach's original article on the Ruziewicz-Problem Sur le problème de la mesure. By the way, many of the relevant original references on Banach-Tarski are available on the site of the Polish virtual library. – t.b. Aug 11 '11 at 3:51
@Theo: Thank you, I had not read the Banach paper, but now will. – André Nicolas Aug 11 '11 at 4:08
If your library doesn't have Stan Wagon's book, you should buy it. One of the most clearly written math books I have seen. – Ross Millikan Aug 11 '11 at 4:50
I am definitely going to take a look at this book. Thank you! – Patrick Da Silva Aug 11 '11 at 4:56

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