I was thinking about if we can find out how these (4,12,12) vertex configurations distribute, it is easy
to replace them by corresponding super-tiles, which requires only few information of cluster.
As for the connection between these substitution rule and TralityTree, maybe more likely there is a
connection between TralityTree and H-supertile, which are both in a "triangle" shape. We will find out
if there is a further connection.
Notice that the
$H$ cluster is in 1-to-1 correspondence with reflected tiles only occurring at location
$H_1$. The
$A+$ edge of
$H$ is (from what we've found) the strongest starting place for growing
$H_7$ and
$H_8$. Counting the two clusters separately, {
$H_7$,
$H_8$} are in a 2-to-1 correspondence with reflected hats (but the ratio of
$H_7$ :
$H_8$ is not
$1:1$).
Do not be deluded by boundary shape, which is really just an accident of how well the rules have been chosen on edges. My first choice was not so nice, so I kept going through subsequent choices until arriving at something aesthetically pleasing.
I've repeated the code from the WFR site, with extra highlighting to show how two of the symmetric trivalent vertices are selected from the set of eight, and then used to project supertiles to cover the Cartesian space:
Your highlighted
$(4,12,12)$ are seen in this image as green and yellow plus one extra gray tile on the outside of the
$A+$ edge. Green always falls on an
$H_7$
supertile, but in my convention, yellow does not. Yellow tiles are found on both
$H_7$
and
$H_8$
in an extended position outside of the central
$H$ meta-tile.
From what I've heard, Craig made the choice to have both
$H_7$ and
$H_8$ with complete surrounds of the central tile and thus complete
$H$ meta-tiles directly inlaid. This choice is inconsistent with the map from vertex figures, because it would result in overlap of the
$H_2$ occurring in every
$H_7$. The solution was to parallel-translate the
$H_2$ in
$H_7$ to the extended location, which gives (imo) a more intrinsically-motivated definition for replacement rules.
Back to my question above: If we suspend belief in the aesthetic choice of completing a surround for every reflected tile, and pursue the tree-motivated definitions more directly, does the resulting substitution system on two-dimensional tiles have any additional nice properties from being defined more naturally?
