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Is there any where to watch a video of a ball being decomposed into 5 pieces that are then translated and rotated to create two balls?

How is this even possible without stretching?

Is it possible to give a simple proof (along the lines of why it is impossible to cube the cube because the smallest cube will have no space above it to put a cube any larger than itself)

In other words -- how would I convince a 6 year old of this? Further questions : 1) If the sets are not measurable, why do there even need to be 5 of them, why not just 2?

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In R. French: The Banach-Tarski Theorem, The Mathematical Intelligencer 10 (1988), pp. 21-28, there are some pictures of the author performing the construction on an orange... – Per Manne Feb 16 '13 at 20:43
Which is here, cool : – Cris Stringfellow Feb 16 '13 at 20:46
I wonder though why everyone does it with oranges.. A hundred dollar bill would be my first choice... – N. S. Feb 17 '13 at 4:06
up vote 6 down vote accepted

I'm fairly sure that such a video has not been made. However, there is a paper by Randall Dougherty and Matthew Foreman called "Banach-Tarski Decompositions Using Sets with the Property of Baire" that may be relevant. It shows that that the pieces, although they cannot be measurable, can have another nice property (the property of Baire) which gives some hope for a visual representation. It would not be exact, but you could approximate the pieces by open sets and this might give some insight into what is going on. I remember Alekos Kechris mentioning something to me about this possibility. It you know some programming maybe you should ask him about it.

As for why there have to be five sets, I think this is proved by showing that a paradoxical decomposition of $S^2$ (the surface of the unit ball) requires four pieces. You can then do the same thing with the interior of the ball, except that you need a fifth piece for the center. I believe this is proved in Stan Wagon's book "The Banach-Tarski Paradox", although I do not have my copy with me.

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Cool, paper is here : – Cris Stringfellow Feb 16 '13 at 20:34
@CrisStringfellow the link is broken. – Ruslan Jun 25 '14 at 8:15

That demo of mine cited by amWhy shows that a paradox in the whole hyperbolic plane is possible using pieces that are simple (hyperbolic) triangles. These are indeed measurable sets. That is ok, since the total measure of H^2 is infinite, so it just shows that infinity = 2 infinity. More precisely, it shows that a certain type of measure in H^2 (a type of total measure 1, which DOES exist in R^2) cannot exist. Additionally, it is a nice pictorial representation of the paradox in the abstract group.

The material is discussed in detail in my book The Banach-Tarski Paradox (almost 30 years in print: I am now starting work on a second edition with G. Tomkowicz!) and also in Chapter 19 of my Mathematica in Action (3rd ed.)

And I admit I was very very terse in that demo on the WRI site (from several years ago) not even explaining, as I did above, what the consequences of such a constructive paradox in the hyperbolic plane are. More on the demo: It is really showing the Hausdorff paradox: Each of the three colored sets, orange, blue, and green, are one third of the space, since they are all congruent. But moving our viewpoint a little, one sees that two of the sets are congruent to the third, so each set is also a half of the plane. So 1/2 = 1/3 (if a congruent measure of total measure 1 existed; so such a measure does not exist).

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The last line should probably be a comment on Trevor's answer. – Asaf Karagila Feb 17 '13 at 3:30
Good for the explanation. – Cris Stringfellow Feb 17 '13 at 5:19

Here is a Wolfram demonstration of The Banach Tarski Paradox.

You might be interested in a couple of threads on the site: [References for the Banach Tarkski Paradox] (Reference about the Banach-Tarski paradox). E.g., you might want to look into Stan Wagon's text on the matter. You'll find a link to it on the linked "References" thread. (Also on that references link is a post (at MSE) by Stan Wagon himself.

See also the thread Banach Tarski Paradox for a link to a proof.

Terence Tao has written a bit on the Banach Tarski Paradox, in a fairly accessible way.

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Do you have a further reference for this? It is not clear to me what that animation has to do with the Banach-Tarski paradox. – Trevor Wilson Feb 16 '13 at 20:33
It was authored in part by Stan Wagon, who is well known for his book on the Banach-Tarski Paradox. I suspect understanding it presumes a wee bit of a background in topology, but not all that much. – amWhy Feb 16 '13 at 20:39
I don't doubt that Wagon knows what the animation in the link means, but I was hoping that we might find out as well :) – Trevor Wilson Feb 16 '13 at 20:40
Cool links @amWhy – Cris Stringfellow Feb 16 '13 at 20:43

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