Get the latest tech news
Banach–Tarski Paradox
paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape.
"Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. 2000: Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2, R) (Wagon, Question 7.4). Analogous results were obtained by John Frank Adams[19] and Jan Mycielski[20] who showed that the unit sphere S 2 contains a set E that is a half, a third, a fourth and ... and a 2 ℵ 0{\displaystyle 2^{\aleph _{0}}}-th part of S 2.
Or read this on Hacker News