As mentioned earlier in “The benefit of programming mistakes” I wanted to make a computer program that creates snowflake images. Something that resembles Bentley’s photos. Thus I made up a modified cellular automaton, that designs ornaments of four-fold symmetry, see the last posts.
Now, snowflakes have six-fold rotational symmetry and that requires a hexagonal lattice. Fortunately, a square array of cells can also play the role of a hexagonal lattice. For each cell we simply consider only six of the eight nearest neighbors, see the figure at left. The connections between nearest neighbors form a lattice with triangular meshes. This is topologically equal to a hexagonal lattice and has the same symmetry, if all connections have the same properties. This makes programming about as simple as for four-fold rotational symmetry.