paradoxical decomposition: the banach-tarski paradox

Question

is there a simple and vivid example of paradoxical decomposition specifically on the banach-tarski paradox? is there a math software that can be used as an application of the paradox?

I don't get what you mean... do you want to see the actual pieces in which you can decompose a ball and form two balls? — Sergio Enrique Yarza Acuña, Jul 10 '17 at 02:59

Arkady · Answer 1 · 2017-07-10T08:55:08.647

0

The Banach-Tarski paradox consists of breaking a sphere into finitely many pieces and reassembling them into two disjoint using only rotations. The key point here is that the pieces are non-measurable. So there can never be an animation that depicts the decomposition and reassembly. (Thanks to user Hurkyl for the comment!)

edited Jul 10 '17 at 08:55

answered Jul 10 '17 at 04:06

Arkady

9,315

Aside: one can write down a putative paradoxical composition without invoking choice at all; the problem is proving that the decomposition doesn't leave any points out. IMO, the implausibility of an animation follows simply from the fact some pieces are non-measurable. – Jul 10 '17 at 06:04
I've edited the post accordingly, thank you :) – Arkady Jul 10 '17 at 08:55

score 0 · Answer 2 · answered Jul 10 '17 at 08:31

If you have not found so, you might want to have a look at the Youtube video made by Vsauce of about 2 years ago The Banach-Tarsi Paradox. At 11:15 he starts talking explicitly about the breaking up of a sphere and reassembling it into two spheres and adds an animation of the process. It gives a nice visual illustration of the paradox and might help you to understand it a bit better what it is all about.

paradoxical decomposition: the banach-tarski paradox

2 Answers2