### Some articles on *paradoxical decomposition, decomposition, paradoxical decompositions*:

Banach–Tarski Paradox - A Sketch of The Proof

... Essentially, the

... Essentially, the

**paradoxical decomposition**of the ball is achieved in four steps Find a**paradoxical decomposition**of the free group in two generators ... Use the**paradoxical decomposition**of that group and the axiom of choice to produce a**paradoxical decomposition**of the hollow unit sphere ... Extend this**decomposition**of the sphere to a**decomposition**of the solid unit ball ...Banach–Tarski Paradox - The Von Neumann Paradox in The Euclidean Plane

... motions are necessarily of the same area, therefore, a

... motions are necessarily of the same area, therefore, a

**paradoxical decomposition**of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible ... and rotations, and rules out**paradoxical decompositions**of non-negligible sets ... Von Neumann then posed the following question can such a**paradoxical decomposition**be constructed if one allowed a larger group of equivalences? It is clear that if one permits similarities, any two squares in the ...### Famous quotes containing the word paradoxical:

“So far as we are human, what we do must be either evil or good: so far as we do evil or good, we are human: and it is better, in a *paradoxical* way, to do evil than to do nothing: at least we exist.”

—T.S. (Thomas Stearns)

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