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January 24

Connecting line between hex centers on hex grid that intersects exactly one vertex

While playing a wargame on a hex map, I thought of the following question: on a finite (regular) hexagonal tiling, is it possible to draw a straight line connecting the center of two hexagons such that the line intersects one and only one vertex? (It is trivially easy to see that you can draw a line connecting the center of one hex with that of another hex two tiles distance away so that this line intersects a vertex, but such a line will necessarily intersect two vertices; I'm asking whether it's possible to draw a line between hex centers that intersects at least but not more than one vertex.) —SeekingAnswers (reply) 21:08, 24 January 2016 (UTC)[reply]

Consider rotating the tiling by 180 degrees about the mid-point of line segment connecting the centres of hexagons A and B. This maps the hex tiling onto itself exchanging A and B. Any vertex-crossing between A and the mid-point will be mapped to a vertex-crossing between B and the mid-point. So there can be an odd number of vertex crossings only if the mid-point itself is a vertex crossing. Next consider rows of adjacent hexagonal tiles running along any one of three possible axes. If A and B are an even number of rows apart, then the mid-point of the line joining them will be on the centre-line of some middle row - along which there are no vertices. If A and B are an odd number of rows apart, then the mid-point will fall on a line passing between rows - which also misses all vertices. So the mid-point can't be a vertex. So there are an even number of vertex crossings. --catslash (talk) 12:26, 25 January 2016 (UTC)[reply]

Thanks, but I don't think the second part of that argument is quite complete: it works for tiles A and B that are a straight row of tiles apart along one of the three possible axes in a hex grid, but what if tiles A and B are offset so that they are not a straight row of hexes away from each other (do not both lie on the same axis)? —SeekingAnswers (reply) 00:37, 26 January 2016 (UTC)[reply]

Sorry, I should have said I was considering a pair of parallel rows of adjacent tiles, one row containing A and the other B (any one of the three such pairs), and then constructing a line parallel to these two rows and halfway between them. I reckon this line does not pass through any vertices, but it does bisect the line between the centres of A and B. --catslash (talk) 01:08, 26 January 2016 (UTC)[reply]