DID YOU KNOW…
Duplicating a Sphere: The Bizarre Banach-Tarski Paradox
Prepare to have your mind bent: the Banach-Tarski Paradox proves that, mathematically speaking, you can cut a solid sphere into just a few pieces and reassemble them to create two identical spheres, each the same size as the original. This is achieved through rigid motions like rotations and translations, without stretching or deforming.
Proposed in 1924 by Stefan Banach and Alfred Tarski, this paradox relies heavily on the Axiom of Choice, a foundational principle in set theory. This axiom permits the selection of an element from each set in an infinite collection, even without a specific rule for selection. The paradox demonstrates that such a decomposition and reassembly is possible for sets in three-dimensional Euclidean space, showing that 'volume' can be a tricky concept in abstract mathematics. While astounding, it applies to theoretical mathematical sets and cannot be replicated with physical objects, as real-world matter has continuous properties that this abstract decomposition does not account for.
Love discovering things like this?
A new fact every day with photos, on the free app.