The Banach-Tarski Paradox

The Banach-Tarski Paradox is a mathematical theorem stating that a solid ball in space can be divided into a finite number of pieces, which can then be reassembled—using only rotations and translations—into two identical copies of the original ball. This counterintuitive result hinges on the properties of infinite sets and the concept of "non-measurable" sets in pure mathematics, rather than physical reality. While it defies everyday intuition, it illustrates the strange and counterintuitive aspects of infinity and abstract mathematical structures, and does not imply such division is possible in the real world.