The Banach-Tarski theorem

It is theoretically possible, believe it or not, to cut an orange into a finite number of pieces that can then be reassembled to produce two oranges, each having exactly the same size and volume as the first one. That’s right: with sufficient diligence and dexterity, from any three-dimensional solid we can produce two new objects exactly the same as the first one! Mathematicians, upon first hearing of this result (otherwise known as the Banach-Tarski Theorem), are generally somewhat blasé; they know that funny counterintuitive things crop up all the time whenever infinity is involved. Most mathematicians encounter the result for the first time in graduate school and file it away in their strange results category (along with space-filling curves, Cantor functions, and non-measurable sets). But in spite of the relative simplicity of the proof, discovered by Stefan Banach and Alfred Tarski in 1924 and hinging on the Axiom of Choice, many mathematicians go no further than the lay scientist who comes across the result.