Rich, maybe I didn't make it clear. The requirement is that the sets be equi-tilable BY RECTANGLES. The example of the 1 by 2 rectangle and the square with side sqrt(2) shows that equal area is not sufficient for equi-tilability. (if you don't restrict to rectangular tiles but allow say triangles then equal area is necessary and sufficient in R2, -but in even dimensions? I never heard of that) Dan, are you perhaps thinking of a different Boltyansii book "Equivalent and Equidecomposable Figures"? This is the book which I used for my website exhibit. (I still have hopes someone will look at it). http://mathsite.math.berkeley.edu/polygon/polygonGalleryIntro.html The book "Hilbert's Third Problem" is non-elementary and it's devoted mostly to proving Sydler's converse. David At 03:34 PM 10/25/2004 -0600, you wrote:
My memory from a long-ago popular article is that in even dimensions, all that's required is equal measure (area, hypervolume, ...) for hyper-polyhedra to be interdissectable. It works for D=2.
Rich
-----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com]On Behalf Of Daniel Asimov Sent: Monday, October 25, 2004 1:52 PM To: math-fun Subject: Re: [math-fun] tiling rectangles with rectangles.
David Gale wrote:
<< Dehn only proved the necessity. The sufficiency was proved (?) seventy years later by a Danish mathematician named Sydler and a clear proof was given by Borge Jessen shortly afterwards. The proof is difficult and uses homological algebra (!). The book "Hilbert's Third Problem" by Vladimir Boltyanskii 1978 is the definitive reference. ...
I have that book and it's very nice, written at a level most upper division math undergrads could understand, though I don't think he tries to prove Sydler's sufficiency proof.
QUESTION: I've never heard of any generalization of Dehn-Sydler to higher dimensions than 3. Does anyone out there know of progress in higher dimensions?
--Dan Asimov
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