Sorry, I should have been more clear at some point. The usual way to formalize this kind of cutting-up problem sweeps the points on the cut under the rug: your goal is to find disjoint *open* sets -- in our case, the interiors of the three obvious 1-by-1/3 rectangels -- whose *closures* cover the square. The points added in taking the closure are those pesky cutting edges, and they're allowed to overlap. --Michael Kleber On 4/20/06, Torgerson, Mark D <mdtorge@sandia.gov> wrote:
I'm still stuck on the trivial way of thirding the square.
If you use pencil and paper, sure two horizontal lines divide the thing into three pieces that are the same. But y'all have been talking, open sets, closed sets, point sets, etc. Down to that level of detail I don't see how to trivially make equal thirds. Actually, I don't see how to do it at all. If the trivial way is to make two equally spaced cuts, the middle third shares two sides, the other two share one. How are the points on the cuts assigned?
Mark -----Original Message----- From: math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com [mailto:math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com] On Behalf Of Fred lunnon Sent: Wednesday, April 19, 2006 2:38 PM To: dasimov@earthlink.net; math-fun Subject: Re: [math-fun] Square thirds
On 4/19/06, dasimov@earthlink.net <dasimov@earthlink.net> wrote:
A RELATED PROBLEM that some people could've predicted I'd ask is:
In what ways can the square torus be decomposed into n congruent pieces for any positive n (other than the obvious n parallel annuli decomposition [whose slopes, btw, can be any fixed rational number]) ?
--Dan
Well, hexagonal and triangular tilings as well, obviously: including the famous dissection into 7 hexagons, each adjacent at an edge to all the others. Fred Lunnon
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