Preface to the Paperback Edition

Summary

The early 1980s seemed like an ideal time for a book on the Banach-Tarski Paradox, in part because of several fascinating unsolved problems, some of which had resisted attacks for more than fifty years. The attacks continued, with amazingly fruitful results in the 1980s and 1990s. Who would have predicted that, within a few years, the two central unsolved problems in the area – the Marczewski problem on whether the pieces in the paradox can have the Property of Baire and Tarski's notorious question on whether a disk and a square can be equidecomposable, would both be solved? Moreover each of the questions was solved in a surprising manner: Dougherty and Foreman showed that a Banach-Tarski Paradox using only Borel and meager sets is possible; and Laczkovich showed that a disk can be squared using translations alone.

The present volume is little changed from the first version. An Addendum summarizes some of the developments of the 1980s. The reader who is interested in pursuing up-to-date results, and finding out about current questions of interest is urged to consult the excellent survey paper by Laczkovich cited below along with some other important papers in the area.