From: ags@seaman.cc.purdue.edu (Dave Seaman)
Newsgroups: sci.math
Subject: Re: This joke wasn't mine!
Date: 7 Oct 1998 09:08:29 -0500
In article ,
Virgil Hancher wrote:
>In article <361a9097.7667145@news.tct.net>, daveva#earthling.net wrote:
>>I don't want to sound thick or anything, but what *is* the
>>Banach-Tarski paradox?
>
>In one form, The Banach-Tarski theorem says that you can disassemble
>a solid sphere int a finite number of subsets (IIRC, 5 subsets, one of
>which is a single point) that can be reassembled into 2 spheres each
>congruent to the original. All motions of the sets being rigid motions,
>no stretching or compressing is required.
That's a corollary. The actual theorem says that the surface of a
sphere can be partitioned into regions A1, A2 and B such that:
(1) A1, A2, B are pairwise disjoint,
(2) The union A1 U A2 U B covers the entire sphere,
(3) Any two of A1, A2, and B are congruent (there is a rotation
that carries one into the other), and
(4) The union of A1 and A2 is congruent to B.
The construction depends on the Axiom of Choice, and therefore
demonstrates that AC implies the existence of nonmeasurable sets.
The result about solid balls follows by replacing each individual point
on the surface by a radius ending in that point, but excluding the
center. If you ignore the center point, the solid ball can be cut into
just four pieces that can be reassembled to form two balls, also
without the center points. There is supposed to be a way to account
for the center point by using just one additional cut.
--
Dave Seaman dseaman@purdue.edu
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