For a change I am looking again at cellular automatons. The results using a square lattice were not really satisfying, see “cellular automaton with color on a square lattice“. There, horizontal and vertical lines dominated too much. To get different and may be more interesting patterns of four-fold rotational symmetry I was thinking of the semiregular tessellation of squares and octagons, see the figure at the left. The neighborhood of a cell is simply given by all cells having a common edge, as shown by the blue lines in the figure at the left. Such a cellular automaton is easy to program. I simply shrink the common edges of the octagons and get a square lattice, see the figure at the right. Half of the cells are actually octagons and have the Moore neighborhood. The other half are really squares and have the von Neumann neighborhood.
Further using the methods of “cellular automaton with color on a square lattice” I get results, that typically reflect the tessellation of squares and octagons:
Note that the vertical and horizontal lines resulting from the square grid have disappeared.