Hi Paul and Ed,

When I say "Eisenstein Integers" I just mean hexagonal lattice + group action. Divisibility properties are not necessary in any of my calculations so far. It's possible that Eisenstein integers have some interesting relative structure, but it's unlikely to be more than just a curiosity. The reason is that the tiling (or one of its MLD color patterns) is a Delone set, while Eisenstein primes are known not to be a Delone set. Thus it would be kind of an "apples to oranges" comparison, but you might be able to get some trivia out of it.

In more recent calculations I've done, I actually used complex numbers in the data model. Unfortunately, the calculation design was constrained by my own ignorance, and didn't even bother to limit the number of complex multiplications + expand. It turned out slow, but the results are still amazing:

I'm giving away too many of my secrets here, but the reason is that they don't make good secrets anyways. If we keep following leads from the color patterns, I think we're going to get a more intuitive proof of the tiling and all its properties sooner rather than later (and in WL). Students might be interested to participate during summer school. It would be a great exciting project, with no relation to LLM's whatsoever.

Now that HatTrialityTree is available through WFR, we have a new way to generate hat tilings from the "triality forest", depicted here:

Module[{treeData}, 
 treeData = TreeFold[Function[Flatten[Append[#2, #1]]],
   ResourceFunction["HatTrialityTree"][1, 4, "Seeds" -> True]];
 Graphics[{EdgeForm[Gray], Map[If[#["Class"] == 0, Nothing,
      {If[#["Symmetry"], Lighter[Red, .6], Lighter[Blue, .6]],
       Polygon[CirclePoints[#["Coordinates"], {2/Sqrt[3], Pi/6}, 6]
        ]}] &, treeData]}]]

which uniquely determines the following patch of the hat tiling:

As we can see in the first picture, each tree has a finite number of vertices. These integer numbers can be calculated using TreeCount:

TreeCount[  ResourceFunction["HatTrialityTree"][1, #, ImageSize -> 500],
        _] & /@ Range[0, 5]
Out[]= {1, 9, 53, 295, 1623, 8903}

TreeCount[ ResourceFunction["HatTrialityTree"][2, #, ImageSize -> 500], 
        _] & /@ Range[0, 4]
Out[]= {2, 13, 72, 392, 2137} 

For more terms, closed forms for the linear recurrences can be read out of case tables in the source code.

These are also related to the parity pattern from Ed:

More speculation about MLD relations will have to wait. It's getting late around here.

Very nice! I decided to take a look at the component hexagons. They are either part of a Y division of a hexagon, or an upside down-Y division of a hexagon. So we can color each hexagon.

Alternately, we can just color the triples.