# The Banach-Tarski Paradox

*22 Mar 2018**07 Mar 2023*

Given a solid ball in 3â€‘dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.The reassembly process involves only moving the pieces around and rotating them without changing their shape.

This ‘paradox’ is a firm result in set theory. It’s not up for debate.

Now, as you might have guessed, there *is* a wrinkle. That is, the individual pieces - the ‘disjoint subsets’ - aren’t ordinary solid pieces: each piece is an infinite scattering of points. This is where the usual concept of volume being preserved runs into a difficulty.

Nevertheless I’m continuing my researches into this using a small sphere made of gold. A stronger form of the theorem, you see, is that any sphere can be cut into pieces and reassembled into any other sphere of *any volume*. It’s often stated as, *‘a pea can be chopped up and reassembled into the Sun’*. Or…a gold ball can be chooped up and reassembled into another gold ball of any desired size…