2005*The Pea and the Sun*.

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.

It all hinges on the weird properties of infinities, which is (one of
the reasons) why it doesn't work in physical reality: matter is not
infinitely divisible. Wapner explains these weirdnesses, like the fact
that there are just as many even numbers, or prime numbers, as there are
whole numbers. This is beautifully exemplified by the
Hilbert
Hotel, and Wapner includes Lesser's wonderful song "Hotel
Infinity" as a musical version of how this works. [And
Kevin Wald's equally fine "Banned
from Aleph" goes infinitely further!] He also covers some
different kinds of infinities -- that there are more (infinitely more!)
real numbers than whole numbers -- and the undecidability of the
continuum hypothesis. Using these
counter-intuitive but true ideas, he builds up to, not only a proof of the
theorem, but an *intuition* of how it works.

After the theorem itself, Wapner provides some musings on its
applicability (or lack thereof!), and the differences between physics and
(pure) mathematics. Once the theorem has been understood, it is clear why
it can't be used to duplicate physical objects. However, Wapner goes on to
muse about duplicating regions of *phase space*, and speculates on
the application to chaos and strange attractors. Now, phase space *is*
a mathematical object (mapping the velocities and momenta of particles
moving in real space). But I remain unconvinced: positions and momenta are
quantised in physical space; nevertheless there certainly appears to be
more wiggle room here.

Recommended. And for the more adventurous reader, there is always more technical detail available.