3 ms·
Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. Pr
by ummonk 5y ago
Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. Presumably, if it were easy, Tarski himself would have shown it, given that he's the one who posed the question.
- thaumasiotes 5y ago> Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. As the other responses point out, the Banach-Tarski construction uses properties of 3D space that do not occur in 2D space. But I don't think the "spherical shape" of the original volume or the final volume is relevant; the point of the construction is that the intermediate pieces can't really be said to have a shape.