Stefan banach alfred tarski biography
Closed range theorem (proved by banach in 1932: théorie des opérations linéaires)!
Banach–Tarski paradox
Geometric theorem
For the book about the paradox, see The Banach–Tarski Paradox (book).
The Banach–Tarski paradox is a theorem in set-theoreticgeometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjointsubsets, which can then be put back together in a different way to yield two identical copies of the original ball.
Stefan banach alfred tarski biography
Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their original shape. However, the pieces themselves are not "solids" in the traditional sense, but infinite scatterings of points.
The reconstruction can work with as few as five pieces.[1]
An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other.
This is often stated informally as "a pea can be