The Banach-Tarski Paradox

About: The Banach-Tarski Paradox in the Wikipedia

RE:
.
.
.
.
.
This was mentioned in the What is Mathematics paper.







The Banach"Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3'dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e. subsets), which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun".
...
Reading the paper, however, reignited my interest in mathematics.

Comments


Mark de LA says
The Wolfram description:
http://mathworld.wolfram.com/Banach-TarskiParadox.html

Mark de LA says
OTOH, most people abandon proofs that end up with absurdity.  Somewhere their idea &/or definition of a solid must differ substantially from the real World.


Mark de LA says
M 2012-09-21 09:35:36 16197
And, maybe that's why physicists had to pack solids in the real world with atoms, protons, neutrons, electrons & quarks & bozons etc...