The Banach-Tarski Paradox

The Banach-Tarski Paradox is a mathematical theorem that states that it is possible to take a solid ball and decompose it into a finite number of non-overlapping pieces, and then reassemble those pieces in a different way to create two identical copies of the original ball.

This result appears to violate basic principles of geometry and common sense, since it suggests that it is possible to create something out of nothing, or to duplicate a finite object without adding or subtracting anything.

The paradox arises from the fact that the pieces into which the ball is decomposed are not simple geometrical objects like cubes or cylinders, but rather more complex shapes that have a fractal-like structure. These shapes have the property of being "paradoxical".

This is a purely theoretical result and has no practical applications in the real world. Nevertheless, the paradox has important implications for the philosophy of mathematics and the nature of infinity, and has led to new developments in the field of measure theory.