The Golden Hexagon Hat Tiling Substitution Rules

Inspired by the new substitution rules that Shigeki Akiyama and Yoshiaki Araki propose (along with a new proof of aperiodicity!), here are some drawings from the last few days. These aren’t quite the same as rendered in their paper, but fundamentally amount to the same thing:

In green are all of the tiles that are adjacent to another with a 180 degree turn (a 2-fold rotation) like this pair, we’ll call an S. These S’s, in turn only appear in the two clusters further right: four S’s in a patch with 2-fold symmetry, and three S’s in a patch with 3-fold symmetry. Together these form infinite trees of S’s like these:

Meantime, other tiles meet in ribbons, all oriented thew same way, with the occasional reversed one, shown in red. These ribbons collide, and form infinite trees themselves, complimentary to those of the S’s. The ribbons are made out of two pieces, at right.

A simple recursive rule generates these ribbons and these trees:

From further back, some interesting structure emerges. Trace for yourself what is connected to what, in the green tree below, and in the missing brown tree in its complement:

Pulling back to an enormous triangle, that is, one of these, we’ve been looking at the center of this:

And all of this from the very simple rules proposed by Shigeki and Yoshi, with beautiful new versions by Erhard Künzel and others. Here are some drawings, which are pretty much the same as Erhard’s, just drawn for the hat rather than the turtle and not fully complete. I’m withholding an opinion on the 1-d rules — an L-system dendrite type production would seem promising.

Now, the thing is, there are other, seemingly completely different hat substitution rules that produce the hat tilings! Here is the H7/H8 system overlaid on the coloration of the drawings above.

The coloring refers to the golden hex system, but is drawn to be consistent on some fixed level of supertile: Here in these pictures, the coloring level is consistent on level 3 supertiles, but the color some of the hats cannot be determined. In the picture below, these white hats are now inside of a higher level supertile and the colors can be determined. On the boundary of this much larger supertitle, there are still white hats for which the color can not yet be determined. The sequence of addresses of these remaining white hats is an example of a “vertex wire” in the manner of [GS98].

And here is the H7/H8 system overlaid on top of the original metatiles that are used in the first proof that the hat tile forms only hierarchical tilings — It does, and for different overlays of hierarchies, that at first all seemed really very different from each other!

Are these overlays of hierarchies actually a new phenomenon, or something we’ve long seen before?

I’m going to start posting this stuff on a more regular basis. Enjoy!

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