4 ms·
If you accept the axiom of choice, then you can cut up a sphere into finitely many pieces and reassemble them to a sphere with twice the volume. (The pieces are
by jjgreen 7y ago
If you accept the axiom of choice, then you can cut up a sphere into finitely many pieces and reassemble them to a sphere with twice the volume. (The pieces are not measurable, naturally.)
- pfortuny 7y agoAs they are not measurable, they do not possess a finite number of describing equations, so “cutting” is not a good analogy.
- klyrs 7y agoNah, just enchant a broom, cut it in two, and let them take care of the rest. (Alas, no, that will only give a countable set of cuts... but the image is pleasing unless you're a mouse)
- mikorym 7y agoAs sets of real numbers, yes. However, if you have an approapriate metric (in this case I think cuts are area preserving) then you can't. [1] [1] https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#Von_Neumann_paradox_in_the_Euclidean_plane https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#.... Look at the comment about measures changing. Edit: Oh, I see you did say this at the end.