From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: comp.theory,rec.games.abstract,rec.puzzles,sci.math
Subject: Re: Chopping rectangles (PROBLEM)
Date: 9 Dec 1997 16:18:50 GMT

In article <348cb8d9.1588932@news.igs.net>,  <nobody@nowhere.on.ca> wrote:
>On 7 Dec 1997 21:35:16 GMT, shamos@ix.netcom.com (Michael Ian Shamos)
>wrote:
>
>> It is also known that a square and a
>>circle of equal area are finitely equidecomposable, but the latter
>>construction requires about 10^50 pieces. 
>
>Here, I am trying to picture the "shapes" of the pieces that would be
>taken from a circle to make a square, or visa-versa, and I can't see
>them for the life of me.  Would you please be so kind as to indicate

See Croft, Falconer, and Guy, "Unsolved Problems in Geometry", section C20.
"Laczkovich...showed that a square can be decomposed into a finite number
of pieces that can be rearranged using translations only to form a circle...
These constructions depend heavily on the axiom of choice." 
This is the construction with an estimated 10^50 pieces.

So this is not something one can "see", nor does curvature have anything
to do with it (or can curvature be defined in the absence of measurability?).

They list two papers by Miklos Laczkovich:

   Equidecomposability and discrepancy: a solution to Tarski's circle squaring
   problem, Crelle's 404(1990)77-117

   Decomposition of sets with small boundary, JLondonMS, 46 (1992) 58--64

See also his recent survey article 

   96j:28004 28A75 03E15 28A05 28E15
   Laczkovich, Miklós(H-EOTVO-AN)
   Paradoxical decompositions: a survey of recent results. (English)
   First European Congress of Mathematics, Vol. II (Paris, 1992), 159--184,
   Progr. Math., 120, Birkhäuser, Basel, 1994.

dave