squaring the circle-a shocking result

[Van Vu <t-vanvu@microsoft.com>]

Hi everybody,

The problem was posed by Tarsky 60 years ago and solved by a Hungarian
mathematician M. Laczkovics
in 1991. He proved that one can cut the circle into finite number of
pieces (about 10^50) and put it together 
(using translations; if I remember well even rotation was not needed) to
have a circle of equal area.

The word CUT here is probably not the best one. Infact the pieces used
in the partition are not 
measurable, and (of course) have no boundary. The proof had rather
combinatorial flavour. 

In this sense (when unmeasurable sets are used)  this result (at least
for me) is not so unintuitive. More 
surprising is the following (proved by Tarsky himseft or Banach):

One can CUT a ball of RADIUS 1 (at least 3-dim) in finite pieces and 
put these together to obtain a ball of RADIUS 2. 

It simply means that you can double a ball from itseft. The key point
here again is in the 
word CUT, which involves nonmeasurable sets. However, it is worth to
mention that 
similar result cannot be proved in 2 dimension, since there is an
universal measure in the plane
which is invariant under translation and rotation. 

Best, Van