>>Message text written by The thrill of minimalism >>> >>Joth Tupper wrote: >>> the Bolzano-Tarski >>> theorem proved (what, back in the 1920's?) that you could cut a solid 3D >>> sphere into finitely many chunks, then rearrange >>> the chunks to make another solid (no holes or gaps) 3D sphere with _twice_ >>> the volume. Pretty spooky, I always felt. >>> Robinson first proved that 9 chunks would do it. Then he found a way with >>> 5 (maybe it is 4) and I think he showed that this >>> was the minimum. >>Thanks so much to whoever posted the link to the reference on the >>choice axiom, which allows the BTT result. >>All the talk of "cutting >>and rearranging" can be so misleading as it makes it sound as >>if the BTT is a magic trick you can do to a common household orange >>with a common household steak knife, which simply is not the case. >>The mathematical results deal with mapping sets. For instance in >>the BTT the fifth "piece" is the single center point of the sphere, >>which isn't much of a "chunk" in any real sense. >>Equivalenting 1-1 mapping and "volume" is confusing. "Volume" must >>have become technical math jargon at some point. >>When you set up a mapping of a line segments to two other line >>segments of the same length, such as f(x)=2x, you call that length >>length, and calling the points in the BTT spheres "volume" is >>an extension of that metaphor. >>the "chunks" in question is about what is the simplest way to >>arrange the mapping, it's not a wooden puzzle. >>Still spooky? Or am I just being a raving idiot. < Volume is a technical term in every setting. In 3D analysis, volume starts out as the simple idea that a cube of side 1 contains 1 unit of volume. Trouble is that 3D has enough room to work that non-measurable sets pop up in many contexts, such as BTT. Non-measurable sets in 1D are tricky to construct but do exist. As I understand it, the sphere doubling notion is not as trivial as using f(x) = 2x to map [0,1] to [0,2]. You "cut" (with a "knife" that can separate points) the sphere into finitely many pieces, and using translations and rotations only (i.e., length preserving motions) reassemble the finitely many pieces into a bigger whole with no holes. R.Robinson was not a trivial mathematician and this is one of his better known results. ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
