Re: squaring the circle-a shocking result

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Hi everybody, (to Mr Van Vu et al, Mr Aiviet and other)
First, thanks for informations about the Hungarian author who had solved 
one of the most "famous" elementary problems.
Va^'n de^` nha^n ddo^i qua? ca^`u tha`nh 2 qua? ca^`u (do ddo' co'
the^? nha^n 1 qua? ca^`u tha`nh vo^ ha.n qua? ca^`u) duuoo.c gia?i boo?i
BANACH khoa?ng nhuu~ng na(m 30.  Ha^`u he^'t nhuu~ng ai la`m Functional
Analysis, dde^`u i't nhie^`u ve^` va^'n dde^` na`y (dda(.c bie^t la` 
co' lie^n quan voo'i Polish School of Math).  Nha^n dda^y ke^? the^m la`
khi Banach che^'t, mo^t so^' nguuoo`i co' dde^` nghi. vie^'t le^n mo^.
o^ng ta ra(`ng, Banach ngu+o+`i dda^`u tie^n bie^'t la`m nhu+~ng ddie^`u
ma` tru+o+c da^y chi? co' God bie^'t la`m.  (Jezus Christ, nha^n ba'nh my`
va` ru+o+u nho).  Tuy nhie^n dde^` nghi. na`y kho^ng ddu+o+.c cha^'p
nha^.n. (Banach = God ?)
Bo+?i vi` Tarski la.i ddu+o+.c nha('c dde^'n, to^i se~ pha?i vie^'t
few things about this great logician. 
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To Mr Aiviet,
Ne^'u no'i dde^'n "curvature" thi` ca'i ddo' chi? co' y' nghi~a 
cho smooth manifold (or smooth line in one dimen.) tho^i.
Tha^.m chi' cho du+o+ng cong lie^n tu.c (but non-differ.), curvature
losses a meaning sense.

SN
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On Fri, 27 Jun 1997, Van Vu wrote:
> 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.