8 September 2004 - Banach-Tarski

Jay Scott : Daily Whale : archive : 2004 : 8 September 2004

I went to a tough school. Math students were given a sharp knife and a pea and asked to demonstrate the Banach-Tarski Paradox. The new peas were, of course, donated to the cafeteria. But, y’know, my school was no more rigorous than the school of hard knocks. Our leaders should remember that the parts of the solution can be counted even if they can’t be measured—go ahead and make the hard choices.

clue:

The Banach-Tarski theorem is called a paradox because it is slightly counterintuitive: It states that a sphere can be dissected into five pieces which can be reassembled into two spheres, each identical to the original. Math is fun! There is one trivial difficulty: the pieces are infinitely complicated (they are not Lebesgue-measurable, which means that they do not have any meaningful volume). The proof of the theorem depends on the Axiom of Choice.

give me a clue so sweet and true