"William E. Emba"|
The Banach-Tarski paradox is invoked repeatedly as the underlying
explanation for shapeshifting. And higher-dimensional generalizations
prove crucial to the plot. The author goes so far as to cite (with
no actual relevance) G T Sallee `Are Equidecomposable Plane Convex
Sets Convex Equidecomposable?' American Mathematical Monthly (Oct
This story appeared in the February 2003 issue of Asimov's Science
Fiction Magazine and parts of this story are (at least at present) available on-line for
free by following the link above or below. Note also that this is a
sequel to the story Milo and Sylvie published
in the same magazine a few years earlier.
For those who may not know, the Banach-Tarski Theorem is a real,
surprising, and somewhat disturbing theorem of geometry. What it says,
essentially, is that any sphere can be broken apart into a finite number of
pieces and then reassembled into another sphere of any desired volume.
Certainly this is disturbing: one is inclined either to be impressed that mathematics has shown us that volume
is not what we think it is, or perhaps one will conclude that mathematics
doesn't make sense after all! [See Division by Zero].
When I learned about it as an undergraduate (back in the 20th century) we
were told that this was an indication of possible problems with the Axiom
of Choice (an axiom of set theory that is not universally popular), but
that viewpoint seems to be out of date. This is now seen as just one of
many indications that volume is a slipperier topic than one might expect.
In particular, as this theorem and others like it show, the notion of
volume is not "finitely additive"...and there is no alternative measure for
arbitrary sets in dimensions 3 or higher which are! In other words, when
it comes to volume, the whole may NOT be equal to the sum of its parts.
For more information about this funky theorem, check out this
link or even this one.
Now, please forgive me for being too serious, but it annoyed me that the
story misuses the theorem. If one were to ignore the atomist
view of matter, and if one had a way to break matter (even your own body)
up into pieces of arbitrary shape, then the Banach-Tarski theorem WOULD
give you a way to reassemble those pieces into something of a different
volume. However, this story makes it sound as if it is the theorem that
gives them the power to break their body into pieces, and that's just