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
There's no way to do the dissection if you play the game that way (and if the pieces have polygonal interiors and boundaries composed of open, half-open, and closed intervals). Each region of this sort has an "Euler measure" (aka combinatorial Euler characteristic), which you can compute by decomposing it into points, open intervals, and open 2-cells, and then computing V-E+F, where V counts the points, E counts the open intervals, and F counts the open 2-cells. This quantity is always a whole number; it doesn't depend on how you decompose a region, and it's invariant under congruence. For the closed square, the most natural decomposition gives 4-4+1, or 1. And since this is not divisible by 3, there's no way to "exactly" decompose the square into 3 congruent pieces. Klain and Rota's book "Introduction to Geometric Probability" talks about Euler measure. The people who have explored this notion are (roughly in chronological order) Hadwiger, Lenz, McMullen, Rota, Schanuel, Morelli, Chen, and myself. Jim Propp