The Banach–Tarski paradox is a theorem in mathematics that says that any solid shape can be reassembled into any other solid shape. It was made by mathematicians Stefan Banach and Alfred Tarski. For example, a sphere can be split up into a limited number of parts which can then be put back together again to make two identical copies of the original sphere. The Banach–Tarski paradox can also be used to turn a small sphere into a huge sphere. It is important when thinking about this to realize that the parts that the solid shape is being split into are not solid shapes themselves; they are infinite sets of points spread around the solid shape. For the Banach–Tarski paradox to work, you need to use the axiom of choice.
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