How to get a cellular automaton with a hexagonal lattice: The orange cells are the neighbors of the orange cell. The green lines show the connectivity.

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.

artificial snowflake

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.

To get a nice image we only have to skew in x-direction and compress distances in y-direction. Here are more results:

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