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Lagniappe #3

Snowflakes and Spirals on the Hexagonal Lattice

Images Ramiro Perez

Ramiro Perez, from Universidad Tecnologica de Panama, has a nice collection of software for producing color graphics of cellular automata, fractals and the like. Particular appealing to us are his images of Cyclic Cellular Automata and related rules on the hexagonal lattice, since virtually all our PSK recipes use rectangular grids. Here we present some of Ramiro's creations, together with his captions explaining the respective rules. For instance, he describes the title snowflake above thus:

If the value of a cell is 0, then count the cells in its hexagonal neighborhood which have state value greater than 0. If this sum is 1, 3 or 6, then the new state at the center is center value mod 12 + 1; otherwise the new state is 0. If the value of a cell is positive, then its new state value is center value mod 12 + 1 in any case.

Click on his snowflake to see the same rule started from a random smattering of seeds with low density.

Perez' programs use an embedding trick, familiar to CA cognoscenti, to carry out hex neighborhood updating within the confines of a conventional two-dimensional Cartesian array. This is achieved by alternately using the neighborhoods

,
for the cells in each column.

Next, let's look at two representative Cyclic Cellular Automaton (CCA) still frames showing dynamic formation of spirals, the first on the hex lattice, the second on the triangle lattice. Perez notes that nucleation takes considerably longer in the triangle case.


Four more hex snowflakes are shown and described here. The first uses a simple two-color rule: If the number of 1's in the neighborhood (center included) is 1 or 2, then the new state is 1; otherwise it is 0.


The second is a simpler variant of the title example at the top of the page: Count the number of cells in the hex neighborhood (center included) with value greater than 0. If this number is positive but not 2, then the new state at the center is center value mod 12 + 1; otherwise the new state is 0.


I am particularly fond of Ramiro's next creation, which uses the same rule as the title example, but is based on a 12-cell Star of David neighbor set.


The final snowflake applies the simpler update rule of the example before last, but on the Star of David neighborhood.

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