4 ms·
> I don't understand Banach-Tarski, but it seems like the common version which you just explaing must be wrong. It is not possible to double the volume of an ob
by mcherm 6y ago
> I don't understand Banach-Tarski, but it seems like the common version which you just explaing must be wrong. It is not possible to double the volume of an object by cutting and reassembling.
Well, it seems that you DO understand why it is considered to be very surprising and unintuitive.
The definitions of "ball" and "rotate" are perfectly normal. The definition of "piece" is slightly odd in that the "pieces" don't have smooth surfaces -- they are "jagged" or "fuzzy" down to an infinite level. (This is why common notions like "volume is conserved" don't apply -- such notions don't apply normally even to simpler objects like a fractal Serpinski Sponge.)
What is especially weird about the pieces (axiom-of-choice weird) is that specifying exactly what the boundaries of the pieces are is so hard that no clear description or algorithm can be given that specifies them. It's sort of like if you said "cut a ball so the prime-numbered points are in piece 1 and the composite points are in piece 2", except MORE strange because the definition of "prime" is easy to understand.
My favorite explanation is this one: https://www.irregularwebcomic.net/2339.html https://www.irregularwebcomic.net/2339.html
- xyzzyz 6y ago> This is why common notions like "volume is conserved" don't apply -- such notions don't apply normally even to simpler objects like a fractal Serpinski Sponge. No, volume always is conserved just fine, as long as you deal with measurable pieces. Sierpinski’s sponge has a well defined volume, and this volume behaves in a sensible manner. Normal sets have well defined volume. The thing with Banach-Tarski pieces is that they cannot be assigned any volume in any sensible way. It has nothing to do with them being “jagged” or “fuzzy”, but rather with their weird behavior when it comes to self-overlapping translations.