Paradoxical Decomposition

Some articles on paradoxical decomposition, decomposition, paradoxical decompositions:

Banach–Tarski Paradox - A Sketch of The Proof
... 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 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)