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The Banach-Tarski Paradox
How To Create Two Identical Spheres From One?
Lets imagine we have a regular sphere (like a basketball). We split this into a finite number of different pieces. After moving these pieces around with no stretching, compressing, deforming etc. we discover that we now have two identical spheres, both the same size as before!
This seems impossible! How can we make two objects of the same size from just one? This seems to go against one of the most fundamental rules in mathematics and Physics — Conservation of Volume.
However, surprisingly, this is what the Banach-Tarski Paradox tells us.
The Banach-Tarski Paradox is one of the most intriguing results in mathematics today. The result was developed in 1924 by Stefan Banach and Alfred Tarski, two Polish Mathematicians. This shows us that, theoretically, you can take a solid sphere and split it into a finite number of extremely unusual pieces. With rotations and translations these pieces can be rearranged to form two identical spheres.
Whilst this seems like magic, it is actually a valid mathematical theorem! To understand how it works lets look at some unusual concepts from Geometry, Set Theory and Infinity.
