A Hyperbolic Interpretation of the Banach-Tarski Paradox

Volume 3, Issue 4
Fall 1993

The Banach-Tarski Paradox asserts that a solid ball in 3-space may be decomposed into five disjoint sets that can be rearranged to form two solid balls, each the same size as the original ball. The sets are nonmeasurable, so it is impossible to visualize the paradox. However, the algebraic idea underlying the paradox can be given a constructive interpretation in the hyperbolic plane. We show how to combine the Hausdorff paradox in a certain free group with the Klein-Fricke tessellation of the hyperbolic plane. This yields a hyperbolic paradox that uses triangles and hence can be visualized via Mathematica-generated color images.