Series: Penn State Logic Seminar

 Date: Tuesday, September 20, 2005

 Time: 2:30 - 3:45 PM

 Place: 106 McAllister Building

 Speaker: John D. Clemens, Penn State, Mathematics

 Title: Continuous Banach-Tarski Paradoxes 


   In 1924, Banach and Tarski showed (using the Axiom of Choice) that
   a sphere is paradoxical, i.e. it can be partitioned into finitely
   many pieces and rearranged using isometries to form two spheres,
   each of equal volume to the first. De Groot asked whether this
   could be done continuously, that is, if the isometries could be
   extended to paths (continuous motions) so that the pieces remain
   disjoint at each instant. Recently, Trevor Wilson settled this
   question by proving the following general result: Any two bounded
   subsets of R^n (for n at least 2) which are equidecomposable using
   proper isometries are in fact continuously equidecomposable. I will
   explain these notions and present Wilson's result and some of its