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The trick is to cut it up to an extent that you can't meaningfully assign area (or rather volume) any more. The only real trick here is the low number of piece
by shiandow 2y ago
The trick is to cut it up to an extent that you can't meaningfully assign area (or rather volume) any more.
The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres, that's just because infinity is weird that way (see Hilbert's hotel if you're not sure).
The question then becomes can you meaningfully talk about volume if you cut a sphere into a finite number of pieces? Turns out you can't.
- afiori 2y agoNot really, Banach-Tarski is very similar to the hilbert hotel with odd-even rooms, the genius is in finding a way to recompose these parts using only rotations.
- shiandow 2y agoNot quite sure what part you disagree with, but indeed that's more or less what's happening. Though also the fact that it's a finite number of pieces is important.
- afiori 2y ago> The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres I took this as saying that it is easy to split the sphere into an infinte number of (I implicitly assumed) non null sets and recombine them in two spheres, to which I disagreed that it is hard to do so using only rotations. But I guess what you meant is that you can build a bijective mapping S^2 -> {0,1} x S^2 just like you can build one for R -> {0,1} x R. I did not consider the extreme case of every element of your partition being allowed to be a single point.