| I came across the Banach-Tarski Theorem many years ago.
The instructor called it the Banach-Tarski Paradox.
It isn't magic; the reassembly process allows one to
do wondrous things with infinite sets of points. For
example, consider this collection of points on the unit
circle in the x-y plane: {(cos n, sin n) | n = 0, 1,2, 3, ...}
where n is in radians. Since the integer n is a multiple
of pi only for n=0, the points all are distinct. Now
suppose that during the reassembly you rotate the set
of points by, say, 5 radians counterclockwise. Then the
rotated set is {(cos n, sin n) | n = 5, 6, 7, ...}.
The "reassembly" process, i.e., the rotation, looks as
if it just subtracted five points from the set. If you
had rotated it 5 radians in the other direction, it would
look like five new points had been added to the set.
This doesn't happen when you rotate a triangle, which
is why you may not be used to it.
The proof of the Banach-Tarski Theorem "cuts" one sphere
into a finite number of point sets which can be rotated
and translated etc. so that they now take up all of the
points of two spheres the size of the original.
I wish that I could remember the method. It would be
interesting to know why/how those two came up with this
result. "Why" if they set out to prove this particular
result, "how" if it was a by-product of other
investigations.
Dan
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| The Intelligencer article does not prove the main theorem but indicates
the direction. More important, the applications section is
interesting. To quote, "All you need is a sharp knife, a small
loaf of bread, a few fish and a large audience. ... who knows where
it might lead."
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