The Banach–Tarski Paradox (Encyclopedia of Mathematics and its Applications Book 163) 2nd Edition by Grzegorz Tomkowicz (PDF)

Description

The Banach–Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher’s famous ‘Angel and Devils’ woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Stan Wagon and Grzegorz Tomkowicz have given mathematicians a great gift with this Second Edition. Long ago I waded through Waclaw Sierpinski’s “Congruence of Sets,” Chelsea, 1953, which covered this result, but the ideas did not stand out clearly and sharply. Here you can see the idea beautifully displayed on the cover, and it is made completely clear in the first few pages. Moreover, the authors cover relatively recent and surprising material such as Miklós Laczkovich’s solution to Tarski’s squaring the circle problem and the proof that the pieces in the paradoxical decomposition of the Banach-Tarski paradox canoe taken so that the disassembly and reassembly can be done my Euclidean motions without the pieces colliding in the process.On a deeper level, the authors make clear how this whole exploration springs from the effort to see what quantities, such a s number, length, area, volume may be assumed to be conserved under decomposition, rearrangement, and reassembly.Buy it. Browse it, Read it. Learn.

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