In article <email@example.com>, Robin Chapman wrote: >> I'd be suspicious of any use of well-known curve theorems without going >> over their proofs and making sure they apply to paths on the Hex board, >> because a path on the Hex board can, without intersecting itself, close >> off a region of the board. > > I don't see that this is relevant. One replaces the path of pieces on the > hex board by a curve built from line segments joining the centres of the > hexagons in question. These paths are between vertices
Yeah, you're right. For what we need here, that's going in the "safe" direction. That is, take a curve derived from the placement of hexes, and that curve has to satisfy all the general theorems about 2D curves.
The "unsafe" direction would be taking a general property of 2D curves, and trying to apply that property to paths of hexes. E.g., a simple closed curve has an inside and an outside, and the inside is connected. However, a simple closed path of hexes doesn't necessarily have an inside, and if it does, the inside is not necessarily connected--it can be pinched off into multiple disconnected regions.