Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures: 1. Introduction; 2. The Hausdorff paradox; 3. The Banach–Tarski paradox: duplicating spheres and balls; 4. Hyperbolic paradoxes; 5. Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition; 6. Higher dimensions; 7. Free groups of large rank: getting a continuum of spheres from one; 8. Paradoxes in low dimensions; 9. Squaring the circle; 10. The semigroup of equidecomposability types; Part II: Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions: 11. Transition; 12. Measures in groups; 13. Applications of amenability; 14. Growth conditions in groups and supramenability; 15. The role of the axiom of choice.
The Banach–Tarski Paradox seems patently false. The authors explain it and its implications in terms appropriate for an undergraduate.
Grzegorz Tomkowicz is a self-educated Polish mathematician who has made several important contributions to the theory of paradoxical decompositions and invariant measures. Stan Wagon is a Professor of Mathematics at Macalester College, Minnesota. He is a winner of the Wolfram Research Innovator Award, as well as numerous writing awards including the Ford, Evans, and Allendoerfer Awards. His previous work includes A Course in Computational Number Theory (2000), The SIAM 100-Digit Challenge (2004), and Mathematica® in Action, 3rd edition (2010).
'The new edition of The Banach–Tarski Paradox, by Grzegorz
Tomkowicz and Stan Wagon, is a welcome revisiting and extensive
reworking of the first edition of the book. Whether you are new to
the topic of paradoxical decompositions, or have studied the
phenomenon for years, this book has a lot to offer. I recommend
buying two copies of the book, one for the office and one for the
home, because studying the book carefully (perhaps in a series of
working seminars) will be worthwhile, and casually browsing through
the book in your spare time will be simply a lot of fun.' Joseph
Rosenblatt, Department Chair, Department of Mathematical Sciences,
Indiana University-Purdue University Indianapolis
'This is the second edition of this classic and comprehensive
monograph on paradoxical decompositions. What adds to the special
appeal of this topic is the diversity of methods and the connection
to several fields including set theory, group theory, measure
theory, geometry, algebra, and discrete mathematics. The previous
edition of this book stimulated a large amount of research. The
present volume also includes these developments and furthermore
discusses the solutions to some of the problems that were solved in
the past thirty years, including the realization of the
Banach–Tarski paradox with pieces having the Baire property and
Tarski's circle squaring problem.' Miklos Laczkovich, University
College London
'Wagon's classic book on the Banach–Tarski paradox has been updated
with Tomkowicz to include major advances over the last thirty
years. It remains the definitive source for both newcomers to the
subject and experts who want to broaden their knowledge. The book
provides a basic introduction to the field with clear exposition
and important historical background. It includes complete proofs of
the Banach–Tarski paradox and related results. It continues with an
extensive survey of more advanced topics. This is far and away the
best resource for beginners and experts on the strangest result in
all of mathematics.' Matthew Foreman, University of California,
Irvine
'Several spectacular results have been proved since the first
edition of this book … All these results and problems are presented
in a penetrating and lucid way in this new edition.' Jan Mycielski,
University of Colorado, Boulder, from the Foreword
Review of previous edition: '… a readable and stimulating book.'
Ward Henson, American Scientist
'In 1985 Stan Wagon wrote The Banach-Tarski Paradox, which not only
became the classic text on paradoxical mathematics, but also
provided vast new areas for research. The new second edition,
co-written with Grzegorz Tomkowicz, a Polish mathematician who
specializes in paradoxical decompositions, exceeds any possible
expectation I might have had for expanding a book I already deeply
treasured. The meticulous research of the original volume is still
there, but much new research has also been included … I should also
mention that this book is beautifully illustrated.' John J.
Watkins, MAA Reviews
'For some people the book will be over by page 36, because by then
one has seen full treatments of the results of Hausdorff and of
Banach and Tarski. These people are short-sighted; there is much
fascinating mathematics to be learned from the further
developments. As the recent result of Marks and Unger shows, there
is probably still much to discover. Indeed, the book contains some
very interesting questions that still await solution.' Klaas Pieter
Hart, Mathematical Reviews
 |
Ask a Question About this Product More... |
|
