BANACH-TARSKI PARADOX
step 5 of 5
5
Two Spheres from One
You now have two spheres, each identical in every way to the original. Same size, same volume (insofar as volume applies), same shape. Nothing was added. Nothing was created. The original sphere contained enough points — uncountably many — that when rearranged by the Axiom of Choice into non-measurable pieces, the concept of 'size' simply ceases to apply. Volume is not preserved because volume was never defined for the pieces.
VISUALIZATION
Two perfect oranges sit where one sat before. No magic — just a demonstration that infinity, the Axiom of Choice, and non-measurability can shatter our intuition about space.
The paradox is not physical — it is a theorem about the geometry of infinite sets and the limits of measure theory.