Talk:Banach-Tarski paradox

Can this construction (perhaps the one using five pieces) be shown in a picture? Can it be animated on a computer screen (in simulated 3D)?

I find pictures much more intuitive than symbolic manipulation. David 16:10 Sep 17, 2002 (UTC)

All Banach-Tarski constructions involve non-measurable pieces, so I don't think a useful picture (or animation) would be possible. --Zundark 16:46 Sep 17, 2002 (UTC)

What about the series of pictures in Scientific American magazine some years ago that showed how a ball can be sliced up and rearranged to become something else? (I forget what it was.) I believe that example involved non-measurable pieces as well. I think non-measurable pieces can be visualized, just not realized in nature because they may be infinitely thin or whatever. In any case, the pictures were interesting. David 17:05 Sep 17, 2002 (UTC)

A piece which is infinitely thin has Lebesgue measure 0. Non-measurable pieces are much worse than this, and cannot even be explicitly described. I still maintain that a useful picture of a Banach-Tarski dissection is not possible, especially as it's impossible to even specify such a dissection (rather than merely prove one exists). I can't comment on the Scientific American pictures, as I haven't seen them. --Zundark 18:27 Sep 17, 2002 (UTC)

The paradoxical decomposition of the free group in two generators, which underlies the proof, could maybe be visualized by depicting its (infinite) Cayley graph[?] and showing how it consists of four pieces that look just like the whole graph. AxelBoldt 18:41 Sep 17, 2002 (UTC)

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The following text has been copied from Talk:Banach Tarski Paradoxical Decomposition:

How about a full name for Hausdorff[?] so it can be linked?

Is this "doubling the interval" thingie related to the fact that on the Real line there are the same number of points in any interval of any length? Or am I simply showing my ignorance? Seems we need an article on infinity. --Buz Cory

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I've put in Felix Hausdorff's first name, but there's no article for him yet.

Doubling the unit interval would be impossible if the doubled interval didn't have the same number of points as the original. But there's more to it than that, because only countably many pieces are used, whereas breaking it up into individual points would involve uncountably many pieces.

Zundark, 2001-08-09