The Banach Tarski Paradox

The Banach Tarski Paradox. visualizingmath The BanachTarski Paradox Visualizing Math 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 Instead, it is a highly unintuitive theorem: brie y, it states that one can cut a solid ball into a small nite number of pieces, and reassemble those

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 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

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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 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 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."

BanachTarski Paradox. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated For example, the set I, J, K are congruent, and seemingly should have the same volume.

Figure 2 from The BanachTarski Paradox Semantic Scholar. Instead, it is a highly unintuitive theorem: brie y, it states that one can cut a solid ball into a small nite number of pieces, and reassemble those 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