To find the future state of a cell we choose some cells around the cell. This may include the cell itself. Then we calculate the sum of the states of these cells. If it is odd, the cell will have state 1 and state 0 if the sum is even. Starting with a single cell of state 1 we get a growing structure. At some generations images of the chosen cells appear at increasing scales. This seems to be quite universal.

In the last posts we have already seen this for the von Neumann neighborhood in the square and the hexagonal lattices. This also happens if the cell itself is counted too. I have not yet found a case with an different evolution.

Neighborhood (orange) used for getting Sierpinski triangles with the parity rule.

Another example. We look only at three cells of the von Neumann neighborhood of the hexagonal lattice, see the figure at left. The parity rule gives images with repeated Sierpinski triangles. They increase in size and sometimes we have only one very great Sierpinski triangle:

Clearly this figure has exact look-alikes if you search the internet. But the figure here is a snap-shot of one particular generation of the two-dimensional cellular automaton discussed above. In the next generation there will be only three cells of state one, outside the big triangle at its corners. Next there will grow again Sierpinski triangles out of these three cells.

If we count the cell itself too in the parity rule we get a similar Sierpinski triangle, but now some triangles are black:

In the next generation there will be four cells of state 1, three at the corners and one at the center of the triangle.

These are probably the most simple cellular automaton that generate Sierpinski triangles, but thats probably all they can do.

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