> 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

> 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.