Cellular automata with random initial conditions

The previous post looked at a particular cellular automaton, the so-called Rule 90. When started with a single pixel turned on, it draws a Sierpinski triangle. With random starting pixels, it draws a semi-random pattern that retains features like the Sierpinski triangle.

There are only 256 possible elementary cellular automata, so it’s practical to plot them all. I won’t list all the images here—you can find them all here—but I will give a few examples to show the variety of patterns they produce. As in the previous post, we imagine our grid rolled up into a cylinder, i.e. we’ll wrap around if necessary to find pixels diagonally up to the left and right.

rule 8 with random initial conditions
rule 18 with random initial conditions
rule 29 with random initial conditions
rule 30 with random initial conditions
rule 108 with random initial conditions
rule 129 with random initial conditions

As we discussed in the previous post, the number of a rule comes from what value it assigns to each of eight possible cellular states, turned into a binary number. So it’s plausible that binary numbers with more 1’s correspond to more black pixels. This is roughly true, though the graph below shows that the situation is more complex than that.

automata pixel density as a function of 1 bits in rule

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