Given the possibility for automata like Langton's Ant to lead to complex, intricate structures, I'm curious whether a Penrose tiling can be generated via this sort of local exploration. More formally:
Suppose we have an "ant" which can take steps of $e^{k\cdot\pi i/5}$ for any $0\le k <10$, leaving an edge behind it. Does there exist an algorithm for the ant which takes as input only the edges it has drawn within some radius $R$ and outputs a direction to take the next step in, such that the ant draws all and only those edges present in some Penrose tiling?
We might give the ant some more flexibility by permitting it a finite number of internal states to adjust as it moves around, turning it into a sort of 2D Turing machine.
It seems plausible that an argument from the aperiodicity of the Penrose tilings could show this impossible, but it's not clear to me quite how this would go through.