La Paradoja de Banach-Tarski en el espacio euclídeo y en el plano hiperbólico

Abstract

This work deals with paradoxical descompositions in diferent spaces. First of all we deal with the Hausdorff paradox, which states that the three dimensional Euclidean sphere except for a countable subset is paradoxical. After this, we improve this result building a paradoxical descomposition of the whole sphere and we treat the Banach-Tarski paradox that proves that the three-dimensional euclidean ball is also paradoxical. We also include a section about equidecomposability and the strong form of the Banach-Tarski paradox, which states that any two bounded subset of ℝ3 having nonempty interior are equidecomposable. In addition, we will study the minimum number of pieces necessary reach out the paradoxical decompositions of the sphere and the ball. Finally we move to the hyperbolic plane. We start by building the basic elements of this. Then, we prove the Hausdorff paradox in the hyperbolic plane which provides a paradoxical decomposition of the upper half plane except for a set of null measure. Finally we provide a decomposition of the complete upper half plane by the Banach Tarski paradox.