A Book to Read...Think About...and Believe!
Summer is coming...and you need a good mathematics book to read. My suggestion is to start with Leonard Wapner's The Pea and the Sun (A.K. Peters, 2005).
The "root" of the book is the BanachTarski Paradox, a subject I think everyone should know about.....both because of its strangeness and its mathematical implications. First, consider this summary from the publisher's website: "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 BanachTarski paradox, which asserts exactly such a notion."
Doesn't that description grab you....entice you to order a copy and read it....to find out how can this be!
Also, this description sounds so much more interesting that the actual statement of the paradox: "It is possible to dissect a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five...although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected.)..." or its generalization: "Any two bodies in R^{3} that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are equidecomposable)." [Both from Wolfram's MathWorld].
Len Wapner does an outstanding job of meandering gently through other geometrical paradoxes, plus the historical events and mathematicians/logicians connected with the difficult mathematics that underlies the outlandish BanachTarski Paradox. In contrast to other resources that try to describe the paradox, this book is the best...and should be your next step into the weird world called mathematics.
