From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: newbie question: Banach-Tarski
Date: 17 Sep 1999 13:00:08 -0500
Newsgroups: sci.math
In article ,
Barnaby Finch wrote:
>Newbie questions:
>The seemingly counter-intuitive Banach-Tarski theorem, in which a solid
>sphere may be decomposed and then reassembled into a sphere with twice the
>volume (and no spaces), seems less paradoxical if one is comfortable with
>the notion that the infinitude of reals between 0 and 1 is no less than
>the infinitude of reals betweeen 0 and, say, 1000. Am I on the right track
>conceptually, or are these notions unrelated?
Not really. Banach-Tarski cannot be done in two dimensions. It
is related to the proposition that there does not exist an average
with respect to rotations of 3-space of all bounded functions on
the sphere. The functions for which this cannot be done have to
be quite nasty. There does in 2-space, although it cannot be
constructed for arbitrary functions. The average does exist for
continuous, or even measurable, functions.
>Is is possible to prove Banach-Tarski without the Axiom of Choice?
It is possible to prove it with less than the full Axiom of Choice.
It has even been recently done with the Hahn-Banach Theorem.
>I've heard that the decomposition/reassembly is complicated. Has anyone
>produced a video of it? You know, like the famous "Inside Out", a clip of
>the hard-to-visualize sphere eversion.
It is not "visualizable". The decomposition must be into sets
which are not approximable by those we can see, or even really
describe. Such sets have to be measurable.
>Thanx, Barnaby
>(I am aware that the actual phrasing of the theorem goes something like:
>If A and B are bounded subsets of Euclidian space of three or more
>dimensions and both sets have interior points, then A can be finitely
>decomposed and reassembled using rigid motions to form a set congruent to
>B.)
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
==============================================================================
From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Banach-Tarski
Date: 21 Oct 1999 17:10:26 -0500
Newsgroups: sci.math
Keywords: no such paradox in R^2
In article <8sszewruc584@forum.swarthmore.edu>,
Ricky Der wrote:
>It says basically what you are saying. For instance, you can take up
>a circle, divide it up into no more than 5 regions, and then take
>those regions and subjecting them to nothing more than the Euclidean
>motions, create a circle 20 times the area of the original. It is
>merely a theorem in mathematics - no paradox at all. The mathematical
>trick used in this theorem is to make sure that those sets (the
>regions you split it up into) are not measurable. Thus, they cannot
>be assigned the concept of "an area". Thus, you can no longer say
>that, OK, pi*r^2 = A1 + A2 + A3 + A4 + A5, because all the areas
>cannot be defined.
You can do it for a sphere, but not for a circle.
The reason for this is that the group of rigid motions on
the plane, as a discrete group, is solvable, hence amenable;
this means that one can find a group average for any bounded
function. This allows the existence of a finitely additive
area function, invariant under all rigid motions, for ALL
bounded subsets of the plane.
When one goes to 3-space, the group of rotations, as a
discrete group, is not amenable. This means that volumes
of nasty sets need not be preserved by rotations.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558