Bubblecar said:
An article for those wanting to understand the joke btm posted.
Mathematician 1: “What an anagram of Banach-Tarski?”
Mathematician 2: “Banach-Tarski, Banach-Tarski.”
This is one of the classic paradoxes in modern mathematics – if we assume that, from an infinite set of sets, we can choose one element from each, then we can slice up a solid ball into finitely many pieces and reassemble it as two balls of the same size!
Banach-Tarski Paradox
I understand that joke. Excellent one !
My best references (in books) on the topic are:
References:
Robinson, R. M., On the decomposition of spheres, Fundamenta Mathematicae 34 (1947) 246-260.
Banach, S., and Tarski, A., Sur le decomposition des ensembles de points en parties respectivement congruents, Fundamenta Mathematicae 6 (1924), 244-277.
Wagon, S. (1985) The Banach-Tarski paradox, Cambridge University Press.
Richard J. Gardner & Stan Wagon (1989) At long last the circle has been squared, Notices of the American Mathematical Society, Vol. 36, No. 10, pp. 1338-1343.
Hugo Hadwiger & Han Debrunner (1964) Combinatorial Geometry in the Plane (trans. Victor Klee) Holt, Rinehart & Winston, New York
Miklós Laczkovich (1992) Paradoxical Decompositions: A Survey of Recent Results,
First European Congress of Mathemarics Volume II, Paris July 6-10, 1992 Birkhäuser pp. 159-184.
Proof complete
