Mathematical Curiosities

In this page, i will put up simple but rather curious mathematics objects. As of now, i am not exactly sure what these are going to be, but i know one which i want to put here, so hopefully the rest will follow suit.

The Banach-Tarski Paradox

This states that given any two bounded 3-dimensional bodies, which have non-empty interiors, we can always "cut" up one of them into FINITELY many pieces, move these pieces (translate and/or rotate them), and rejoin them to get the second body.
Often its stated as you can cut up a pea into finitely many pieces and rejoin them them to get the sun.

As it is stated it seem extremely counter intuitive. But there is something here which the layman assumes that makes it even more counter-intuitive for him. I was making the assumption till i actually saw the proof, which by the way is rather simple, contrary to what you might think. And the assumption is of course continuity. The finitely many objects that we get need not be continuous. With this assumption gone, i guess the theorem, is not really as paradoxical as it initially seemed. Here is a link to the proof. Its very simple and requires very basic prerequisites. And very nicely written. Have a look.
Exposition on the Banach-Tarski Paradox

Mangesh Gupte <mangesh[at]csa.iisc.ernet.in>

Last modified: Tue Apr 26 00:57:14 2005