I have been reading these, along with slowly chipping away at the two books Margenstern has produced:
- A universal cellular automaton in the hyperbolic plane
- A Universal Cellular Automaton on the Ternary Heptagrid
- A new universal cellular automaton on the ternary heptagrid
The first one gives the explanation of the Railway model. I understand it, up to a point. I can imagine how it works building more complex circuits, but I am currently having a hard time following the written natural language describing how the circuit works, too much going on, too much motion for my little mind.
But I get the gist.
However, to satisfy my curiosity, what I would really like to know is how this model of computation actually works on the heptagonal tiling.
Without studying (yet) all of hyperbolic geometry and all of cellular automata stuff (I read about Poincaré's disk model and know the basics of how simple cellular automata work), I am wondering how the railway model is actually implemented in the tiling. Can you outline how it works? There is a section, The implementation of the railway circuit, but it doesn't go into enough detail. The detail is referred back to the other two papers, such as this in section The implementation of the railway circuit. Etc.
How does it relate to the fibonacci tree/numberings? How do the 7 corners and several lines and paths relate to the railway circuit? How does the cellular automaton part fit in? I don't get where it's going with this yet.