Maths proves you can cut one ball into pieces and build two identical balls.
In 1924 two mathematicians proved that a solid ball can be split into just a handful of pieces and reassembled, by moving and rotating alone, into two balls exactly the same size as the first. It is not a trick. It is a theorem, and it says something unsettling about infinity.
The Banach–Tarski paradox, published by Stefan Banach and Alfred Tarski in 1924, states that a solid ball in three dimensions can be divided into a finite number of pieces that, rearranged using only rotations and shifts, form two copies of the original ball. No stretching, no bending, no new material. Raphael Robinson later showed that five pieces are enough, and fewer will not do. A stronger version says any two 'reasonable' solids can be converted into each other, which is why it is sometimes called the pea and the Sun paradox.
The catch is in the pieces. They are not chunks you could carve with a knife but infinitely intricate scatterings of points, so strange that they have no volume at all in the ordinary sense. Our intuition that cutting and rearranging must preserve volume is sound, and is even built into the formal definition of volume. It just cannot be applied to sets that have no definable volume to begin with. When the pieces come back together, the result has a volume again, and it happens to be double.
The construction relies on the axiom of choice, a basic rule of set theory that lets mathematicians select one element from each of infinitely many sets, even when there is no recipe for doing so. Without something like it, the paradox cannot be proved. Banach and Tarski themselves pointed out that the axiom also underpins proofs of perfectly intuitive results, so rejecting it just because it yields a strange consequence would throw those out too.
The effect only works in three or more dimensions; in one or two it fails, because the possible rotations are too simple to support it. Mathematicians call such results veridical paradoxes: they offend intuition but are not contradictions. The pieces exist only as abstract sets of points, so nobody can carve them out of a real object. The lesson is that infinity does not follow the rules of everyday stuff.
Source: Wikipedia — Banach–Tarski paradox · Text summarised from Wikipedia (CC BY-SA 4.0)
