Max G. Levy in Quanta:
Imagine two friends hiking in the woods. They grow hungry and decide to split an apple, but half an apple feels meager. Then one of them remembers one of the strangest ideas she’s ever encountered. It’s a mathematical theorem involving infinity that makes it possible, at least in principle, to turn one apple into two.
That argument is called the Banach-Tarski paradox, after the mathematicians Stefan Banach and Alfred Tarski, who devised it in 1924. It proves that according to the fundamental rules of mathematics, it’s possible to split a solid three-dimensional ball into pieces that recombine to form two identical copies of the original. Two apples out of one.
“Right away, one sees that it’s completely counterintuitive,” said Dima Sinapova of the University of Illinois, Chicago.