From: Jeff Erickson
Newsgroups: comp.theory,rec.games.abstract,rec.puzzles,sci.math
Subject: Re: Chopping rectangles (PROBLEM)
Date: Sun, 07 Dec 1997 18:19:14 -0500
Yann DAVID wrote:
> Given two rectangles A and B of equal area, is it always possible
> to cut A in finitely many pieces and to use them to tile B ?
Yes. This has been known since the 1700's. Think "torus".
Moreover, for any two polygons X abd Y of equal area, there is a finite
set of pieces that can be assembled into either X or Y. Hint: every
polygon can be borken into triangles.
See Greg Frederickson's marvelous new book "Dissections: Plane and
Fancy" for (MANY) further details.
--
Jeff Erickson Center for Geometric Computing
jeffe@cs.duke.edu Department of Computer Science
http://www.cs.duke.edu/~jeffe Duke University
==============================================================================
From: shamos@ix.netcom.com (Michael Ian Shamos)
Newsgroups: comp.theory,rec.games.abstract,rec.puzzles,sci.math
Subject: Re: Chopping rectangles (PROBLEM)
Date: 7 Dec 1997 21:35:16 GMT
In Yann DAVID
<100552.1400@CompuServe.COM> writes:
>Given two rectangles A and B of equal area, is it always possible to
>cut A in finitely many pieces and to use them to tile B ?
Answer: yes. In 1924, Tarksi proved that any two plane polygons are
finitely equidecomposable iff they have equal areas. They need not
have the same number of sides. 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. For polygons it's much
simpler. Are you interested in minimizing the number of pieces or the
time required to perform the decomposition (or something else)?
Mike Shamos
School of Computer Science
Carnegie Mellon University
==============================================================================
[The "pieces" in the circle decomposition have to be nonmeasurable. See
http://www.math.niu.edu/~rusin/known-math/97/51ddecomp
--djr]
==============================================================================