Comment by akoboldfrying
2 days ago
Interesting article! I'm stuck on the following claim about tiling Hats though:
> In each center of an hexagon you can have any of the 12 possibilities:
> Any of the 6 rotations of the Hat
> Any of the 6 rotations of the anti-Hat
For this claim to hold, it must be the case that a Hat (or anti-Hat) occupies the same area as a hexagon. But they don't: a hexagon is made of 6 kites, while a Hat is made of 8. So, some hexagons must contain no corresponding (anti-)Hat -- specifically, for every 8 hexagons, there must be 6 (anti-)Hats.
This seems to complicate the SAT formulation. But could the fix be as simple as adding a 13th possibility, "No hat at this hexagon centre occupies more than half of its kites"? Or are additional constraints needed?
Oof, that is poorly written. I'll update the text.
Notice how every hat has a special "marked" vertex. It is the red dot in this image:
https://www.nhatcher.com/images/hats/hat-marked.png
This is what I mean by "at the center ofthe hexagon you have the hat". What should say is "the center of the hexagon coincides with the marked vertex of a hat". Hope that makes more sense.
Thanks for responding, but this doesn't address the issue. It still cannot be the case that some hat's marked vertex coincides with each hexagon centre -- since that would imply that (in a region large enough that boundary effects can be ignored) there are at least as many hats as there are hexagons, and since hats and hexagons each tile the plane, this would imply that a hat has size less than or equal to the size of a hexagon, which is not true. (As an aside, observing that the marked vertex has an internal angle of over 180 degrees for the hat lets us additionally conclude that at most one hat's marked vertex coincides with each hexagon centre, meaning that if your claim were true then any large enough region would be covered by exactly the same number of hats as hexagons (and they would be forced to have exactly the same size), but this doesn't help make the original claim hold.)
It must instead be the case that some hexagon centres do not coincide with any hat's marked vertex, and indeed this is the case -- I've marked out a few with red circles in the first image here: https://imgur.com/a/Kat6oXf
In the second image at that link I've marked two other vertices of a hat in green. It looks to me like the hexagon centres I circled in the first image are incident on one of the green vertices of each of 3 different hats, each contributing "120 degrees of coverage".
Oh, I don't think I understood you at all the first time around.
But you are absolutely right. The center of an hexagon may or may not coincide with the vetex of a marked hat.
In particular in the set of statements, for a given (m, n) all 12 these might be false:
There is Hat (or an anti-Hat) in vertex (m, n) with rotation 60×i (with i=0,..,5)
I'll update the text when I get a chance