paradox · logic
Banach-Tarski paradox
One ball, then two of the same size.
How can one solid ball become two balls of the same size? Yellow names start with A. Red names start with B. S is the start, before either turn. The bottom chip stands for every longer name.
Each point of the ball is named by a sequence of two turns, A and B. S is the start, before any turn.
One piece is every point whose name starts with A. The other piece here is every point whose name starts with B.
Turn the A piece by the turn that undoes A. The first letter drops, so A, AA, and AB become S, A, and B.
Those names are now every name, so that one piece fills a whole ball. The B piece does the same for a second ball of the same size. No chip is stretched.
A knife cut cannot do this. A cut has a volume, and a turn keeps that volume. These pieces have no volume, because the points are an infinite scatter with holes at every size.
The axiom of choice is what assigns the names. No formula can list the points, which is why a drawing can show the idea and cannot show the real pieces.