Robert McNees
Mathematician Alfred Tarski was born #OTD in 1901.
The Banach-Tarski paradox asserts that a ball in three dimensions can be decomposed into a finite number of pieces which, after translation and rotation, can be reassembled into *two* distinct copies of the original ball.
The paradox also works in more than three dimensions, but not in one or two without significant technical modifications.
Robert McNees
The trick is that the ball has to be broken up into pieces that have a well-defined mathematical meaning, but are pathological when you try to think about them in terms of concepts like volume.
The decomposition relies on the axiom of choice. There are other approaches to set theory without the axiom of choice that do not lead to the Banach-Tarski paradox.
Robert McNees
The pathological nature of the subsets of the ball used in the construction make it impossible to produce a truly faithful visualization of the decomposition and assembly process. But this video still manages to do a pretty nice job!
https://www.youtube.com/watch?v=s86-Z-CbaHA&t=675s&pp=2AGjBZACAQ%3D%3D
Chirp! is a social network. It runs on GNU social, version 2.0.1-beta0, available under the GNU Affero General Public License.
All Chirp! content and data are available under the Creative Commons Attribution 3.0 license.