Math Says You Can Cut One Ball Into Two Identical Balls
A famous theorem proves a solid ball can be split into five pieces and rebuilt into two balls, each the size of the first.
In 1924, mathematicians Stefan Banach and Alfred Tarski proved something that sounds impossible: a solid ball can be cut into just five pieces and, using nothing but rotations and slides, reassembled into two solid balls, each identical in size and shape to the original.
This is the Banach-Tarski paradox, and it is a genuine theorem, not a trick. There is no stretching, no hidden gaps, no extra matter conjured up. The catch is that the five pieces are not ordinary chunks you could carve with a knife. They are infinitely intricate clouds of points so jagged that no consistent notion of volume can be assigned to them at all.
The pieces are non-measurable, so the usual rule that volume is preserved simply does not apply.
The proof relies on the axiom of choice, a foundational rule of set theory that lets mathematicians pick points from infinitely many sets at once. It works only in idealized mathematical space, never with real atoms, which is why you cannot duplicate a real ball this way.
Sources & references
2 referencesWell-established. Corroborated by 2 independent sources.
