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[Unfortunately, I made some mistakes in my original question. I tacitly corrected them wherever I found them.] Wolfram's rule 18 gives rise to fractal patterns, but when started with two black cells ...

I investigated the long term behaviour of a pair of black cells ■■ on a circle of $N$ cells under the action of each of Wolfram's rules $R$. For each combination $(R,N)$ I determined the first ...