In the last post I did not do a good description how the cellular automaton with parity rule evolves on a hexagonal lattice. Thus I made a video using the von Neumann neighborhood. You can see reappearing inflating generations with only six cells of state 1, periodic patterns and fractal, self-similar structures:
On a square lattice I get similar results. Some generations have only four active cells instead of six because of the different lattice. Then there are fractal structures too. Checkerboard patterns replace the concentric rings of hexagons:
I was first surprised that fractal structures appear simply repeating the parity rule and that we need no recursive procedure as for quasiperiodic tilings. But noting that Stephen Wolfram has found a cellular automaton that generates the fractal Sierpinski triangle, see http://mathworld.wolfram.com/ElementaryCellularAutomaton.html, this seems not to be so exceptional.