Banach-Tarski paradox

The Banach-Tarski paradox states that a solid sphere can be split into a finite number of parts, which can then be reassembled—using only rotations and translations—into two identical copies of the original sphere. This counterintuitive result relies on the mathematical properties of infinity and the nature of points in space, particularly non-measurable sets, and does not hold in physical reality due to the constraints of matter and energy. It highlights the strange and fascinating aspects of mathematical infinity and the foundations of geometry, rather than suggesting such duplication is possible in the physical world.