A Hyperbolic Interpretation of the Banach-Tarski Paradox
Volume 3, Issue 4
Stan Wagon, Macalester College
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.