PPT A Top Down Look at the BanachTarski Paradox PowerPoint Presentation ID26879

The Banach Tarski Paradox. PPT A Top Down Look at the BanachTarski Paradox PowerPoint Presentation ID26879 THE BANACH-TARSKI PARADOX Second Edition The Banach-Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large 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

The Banach-Tarski paradox is a theorem which states that the solid unit ball can be partitioned into a nite number of pieces, which can then be reassembled into two copies of the same ball The Banach-Tarski paradox May 3, 2012 The Banach-Tarski paradox is that a unit ball in Euclidean 3-space can be decomposed into finitely many parts which can then be reassembled to form two unit balls in Euclidean 3-space (maybe some parts are not used in these reassemblings)

PPT A Top Down Look at the BanachTarski Paradox PowerPoint Presentation ID26879

That argument is called the Banach-Tarski paradox, after the mathematicians Stefan Banach and Alfred Tarski, who devised it in 1924 For example, the set I, J, K are congruent, and seemingly should have the same volume. It proves that according to the fundamental rules of mathematics, it's possible to split a solid three-dimensional ball into pieces that recombine to form two identical copies of the original

A Geometric Oddity The BanachTarski Paradox YouTube. It is not a paradox in the same sense as Russell's Paradox, which was a formal contradiction|a proof of an absolute falsehood The Banach-Tarski paradox May 3, 2012 The Banach-Tarski paradox is that a unit ball in Euclidean 3-space can be decomposed into finitely many parts which can then be reassembled to form two unit balls in Euclidean 3-space (maybe some parts are not used in these reassemblings)

(PDF) The BanachTarski Paradox. (Five pieces are minimal, although four pieces are sufficient as long. This result at rst appears to be impossible due to an intuition that says volume should be preserved for rigid motions, hence the name \paradox."