The Pea and the Sun: A Mathematical Paradox [Hardcover]

Leonard M. Wapner (Author)

Book Description

ISBN-10: 1568812132 | ISBN-13: 978-1568812137 | Publication Date: April 29, 2005

Take an apple and cut it into five pieces. Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

Editorial Reviews

Review

" … engaging, thorough, and fascinating explanation of one of mathematics' most perplexing paradoxes. Wapner's book is a skillful blend of history, mathematics, and philosophy that will please both mathematicians and the merely math curious."" -Keith Devlin, Stanford University, April 2005
""The author gives a fascinating account, in a journalistic style, of the history of the Banach-Tarski Theorem, devoting a chapter to the cast of characters, including Georg Cantor, Kurt Gödel, Paul Cohen, and of course Stefan Banach and Alfred Tarski. . . .What is presented in this book is maths for its own sake: beautiful, elegant, artistic, astonishing. . . . it would surely make a great present for a budding pure mathematician - and what a present it would be, to give someone their first inkling of the wonders that lie at the heart of pure mathematics."" -Helen Joyce, Plus Magazine, September 2005
""Written in a fresh, captivating, friendly style, The Pea And The Sun is remarkably engaging and will appeal to any reader with a discerning, inquisitive mind into the nature of the so-called impossible, regardless of their particular mathematical background."" -Bookwatch, October 2005
""Leonard Wapner, Professor of Mathematics at El Camino College in Torrance, California, makes the BT paradox the centerpiece of a marvelous book (his first, by the way)."" - Martin Gardner -Martin Gardner, The New Criterion, October 2005
""...the book has a little something for everyone."" -J.M. Plotkin, American Mathematical Society, November 2005
The Pea and the Sun makes Librarian's Choice in Non-Fiction for 2005 -Librarian's CHOICE Magazine , January 2005
""Wapner reaches out and tries to explain a truly amazing result to an audience without serious mathematical training. He keeps the reader interested and he explains some very interesting things about set theory in particular and mathematics in general."" -Kenneth R. Davidson, CMS Notes, February 2006
""This book is sure to intrigue, fascinate, and challenge the mathematically inclined reader."" -Mathematics Teacher, May 2006
Wer Einblick in die Schönheit und Eleganz der theoretischen Mathematik erlangen möchte, aber nur ein einziges Buch lesen will, dem sei Wapners Werk wärmstens empfohlen. -George Szpiro, NZZ am Sonntag, January 2006
""One does not need to have a degree in mathematics in order to follow the lively and readable, highly intriguing story of the paradox. Yet the exposition is serious, correct and comprehensive, and it presents a detailed proof of the result. The presentation is light-hearted, highly entertaining and illustrated with many examples, puzzles, etc. This book is (already) a classic in an area that needed one."" -Newsletter of the European Mathematical Society , June 2006
""The book is written in a fashion that is easily understood by non-mathematicians, providing a popular discussion of this bizarre paradox, little known outside the mathematical community."" -Efstratios Rappos, Zentralblatt MATH 1080, June 2006
""The book is well written and entertaining."" -Peter Komjath, Notices of the AMS, October 2006
""In his immensely engaging book... Wapner has chosen as the basis for his book the Banach-Tarski Paradox: quite simply, the finest paradox in all of mathematics."" -John J. Watkins, The Mathematical Intelligencer , July 2006
""With some real substance as well as plenty of entertainment, all elegantly packaged and lucidly presented, The Pea and the Sun is a cut above most popular mathematics books."" -Danny Yee, Danny Yee's Book Reviews, October 2006
""Very readable and aimed at non-technical readers ... an ideal book for undergraduates or sixth form students."" -Anthony C. Robin, The Mathematical Gazette, November 2006
""The Banach-Tarski theorem states, in everyday words, that it is possible to take a sphere, partition it into a finite number of pieces, and reassemble those pieces into two spheres each the same size as the original. What?! Doesn't that violate several laws, or something? Well, yes it would, in the physical world, but these are mathematical spheres, which can behave in apparently paradoxical ways. Wapner, in this lovely little book, builds up to the statement and proof of the theorem in a very gentle manner. By the end, you understand how the theorem could possibly be true, and why not to bother trying to use it to get rich by doubling golden balls. "" -Susan Stepney, Susan Stepney, August 2008
""Where Wapner diverges from the canonical popular maths book types is in his decision to present, in detail, the proof of the Banach-Tarski theorem . . . Wapner's goal is a very ambitions one! The proof presented is complete and is certainly accessible to any later-year undergraduate student . . . ."" -The Australian Mathematical Society Gazette, January 2006
""... a very well-written introduction to the Banach-Tarski Theorem. The mathematical exposition is sound, but simple. The prose is lively and entertaining. It is a pleasurable read that is accessible to all students of mathematics."" -James V. Rauff, Mathematics and Computer Education, March 2009"

About the Author

Leonard Wapner is a Professor of Mathematics at El Camino College in Torrance, CA. He received his BA and MAT degrees in mathematics from the University of California, Los Angeles. During his thirty-year tenure at El Camino, his writings on mathematics education have appeared in The Mathematics Teacher and The AMATYC Review. This is his first book. Len lives in Seal Beach, CA.

Product Details

• Hardcover: 232 pages
• Publisher: A K Peters/CRC Press (April 29, 2005)
• Language: English
• ISBN-10: 1568812132
• ISBN-13: 978-1568812137
• Product Dimensions: 9.1 x 6.1 x 0.9 inches
• Shipping Weight: 1.1 pounds
• Average Customer Review: 4.6 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #1,549,585 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

30 of 31 people found the following review helpful:

5.0 out of 5 stars A fascinating introduction to the Banach-Tarski Paradox, October 11, 2005

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Midwest Book Review (Oregon, WI USA) - See all my reviews

This review is from: The Pea and the Sun: A Mathematical Paradox (Hardcover)

The Pea And The Sun: A Mathematical Paradox is a fascinating introduction to the Banach-Tarski Paradox, a mathematical riddle that asserts it could be possible to create something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again. Written to be accessible to lay readers and non-mathematicians, The Pea And The Sun outlines the history of the paradox, introduces readers to the basics of such matters as set theory, isometrics, scissors congruence and equidecomposability, and walks the reader through the theorem and proof that object duplication is indeed mathematically possible. But just because it is mathematically possible, is it physically possible? The highly counterintuitive nature of the mathematical theorem demands a critical response. A final chapter dwells on speculation as to how the Banach-Tarski Paradox may apply to the modern and future world. Written in a fresh, captivating, friendly style, The Pea And The Sun is remarkably engaging and will appeal to any reader with a discerning, inquisitive mind into the nature of the so-called impossible, regardless of their particular mathematical background.

26 of 27 people found the following review helpful:

5.0 out of 5 stars It made my top ten list of best popular mathematics books, January 13, 2007

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: The Pea and the Sun: A Mathematical Paradox (Hardcover)

The Banach-Tarski paradox is a candidate for the most counter-intuitive mathematical result ever published. Basically, the conclusion of the theorem is that it is possible to decompose a small object into a finite number of pieces and then reconstruct the pieces a certain way to make two objects identical to the original. Phrased another way, it means that an object the size of a pea can be deconstructed into a finite set of objects that can be reconstructed in a manner to make an object the size of the sun. As bizarre as this sounds, the proof has stood up against all manner of assaults until there is no doubt that it is in fact true.
Wapner does an excellent job in setting the mathematical, historical and philosophical groundwork for explaining the theorem. The book starts with a brief explanation of the lives of Alfred Tarski and Stefan Banach as well as others such as Georg Canto and Kurt Godel who helped create the mathematical framework. This is followed by a lengthy and thorough discussion of the mathematical background needed to understand the theorem and the proof. It begins at the very basic level, so very little mathematical knowledge is needed before you begin.
The next step is the proof of the theorem, which by this time is very easy to understand. It is done step-by-step with not even the slightest "leap of faith." The final chapters deal with the consequences of the theorem. I found these chapters to be the most interesting in the book. In "Resolution", Wapner discussions the possible reactions to the theorem. They are:

*) Declare the result fallacious.
*) Accept the theorem at face value.
*) Reinterpret the result.

The first is not realistic as there is no longer any doubt that the theorem is true and the second is self-evident. Performing the mental gyrations necessary to accept the third option is the most interesting. Wapner resolves it by saying, "Yes, the theorem is true, but the actions needed to do something like duplicating a gold bar are not possible." Chapter 7, called "Real world" mentions some of the principles of quantum mechanics and how they can be related to the Banach-Tarski paradox.
This book is one that will fascinate you, it proves in the mathematical sense what you "know" cannot be true in the real sense. It also demonstrates a fundamental problem of philosophy, which is to consider to what extent a mathematical result can be applied in the real world. I loved this book, it made my top ten list of best popular mathematics books.

Published in Journal of Recreational Mathematics, reprinted with permission.

21 of 25 people found the following review helpful:

5.0 out of 5 stars Actually proves the theorem, October 20, 2005

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Mark Weitzman (USA) - See all my reviews
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This review is from: The Pea and the Sun: A Mathematical Paradox (Hardcover)

The book is wonderful because it actually proves the theorem in a way that a non-expert in mathematical foundations can actually understand. I wish all popular mathematics books were written at this level where the goal is to educate and entertain. Now as I suggested to the author all we need is a book like this one that will explain Godel's and Cohen's results on the independence of the continuum hypothesis.