The parity rule on a hexagonal lattice

The MIT group of E. Fredkin, N. Margolus, T. Toffoli and G.Y. Vichniac has studied many cellular automata on the square lattice. We can use their ideas as recipes to run on the hexagonal lattice. Particularly simple is the parity rule. The cells have either the number 1 or 0 as state. We use the von Neumann neighborhood and calculate around each cell the sum of its six nearest neighbors. In the next generation a cell has state 1 if this sum is an odd number and 0 if its even. Starting with a cell of state 1 at the center of a hexagonal region and all other cells with state 0 we get very complicated patterns. Here the cells of state 1 are shown in black and the others in white.

fractale hierarchieFirst the pattern grows and goes through complicated cycles. Sometimes we have a periodic structure of cells of state 1. Then we get a hierarchical fractal structure. Six cells of state 1 form the corners of regular hexagons. The centers of six of these hexagons themselves correspond to the corners of much larger hexagons, which again are at hexagonal positions. And so on, see the figure at left.

concentric hexagonsAt other times we see concentric hexagons. In the next generation they disappear, leaving only single cells of state 1 at their corners.

von eckenAfter the growing structure has reached the border we get a growth back towards the center. The details then depend strongly on the size of the region. It determines how the pattern looks like when it hits the border.

nach innen labyrinthNow we see more complicated and less regular snowflake like patterns. Sometimes long lines dominate. This corresponds to the concentric hexagons.

nach innen punkteAt other times we see mostly isolated dots. This corresponds to the periodic or fractal arrangement of cells of state 1 in the initial growth.

nachinnenstrenNew structures can arise, such as larger regions filled with cells of state 1.

endzeitFinally, the inwards growing structure reaches the center and begins to interfere with itself. Then we get often seemingly random patterns, except for the imposed symmetries. Suddenly, interesting motifs appear for just one generation.

endzeitxOne might think endzeitlabyriththat an artist or a sophisticated computer program has created these images. It is really surprising that they result from the nearly trivial parity rule.

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