The simplest spatial cellular automaton I can think of is produced by computing a one-dimensional automaton and showing each succeeding generation on a different line. The value of a cell in a given row depends on the value of its "parents," the three cells directly and diagonally above it. For example, we might use the following rule:
Number of "parents" "offspring" produced
1 1 (Yes)
2 0 (No)
3 0 (no)
Or, in shorthand, (1,0,0) - so there is an "offspring" if and only if there was exactly one "parent." Then if we start with a single cell on the top row, we get:
and so on. You may recognize this as a finite version of the Sierpinski gasket.
If instead, the rule is (1,1,0), then we get:
another Sierpinski gasket. Or if the rule is (1,0,1)...
Not quite a Sierpinski gasket, but a Sierpinski-like pattern. Like Sierpinski, it's self-similar on any larger scale.
The other possible rules produce even more boring results. (1,1,1) produces a solid triangle; the rest never get started from a single cell.
But what if we allow each "live" cell to have a value of 1, 2, or 3? We can represent the different values by different colors - say, red for 1, green for 2, and blue for 3. Our rule will have to accommodate sums from 1 to 9, and map them to results from 0 to 3 (zero meaning no offspring, and offspring valued 1 to 3).
If the rule is, say, (3,2,0,1,1,1,2,2,1), we get:
Where it goes from here is left as an exercise for the reader. I have no idea what develops; I picked the rule more or less at random, and just calculated the first few rows by hand. Anyway, it's richer than the possibilities with binary cells. We can get more variety if we increase the number of possible values, and more variety still if instead of starting from a single cell, we seed multiple starting values across the top row. But the possibilities really open up when we add a dimension: start with the seeded cells in a plane, not a row, and stack the succeeding generations on top of each other to form a three-dimensional pattern.
The images in the first post of this series were produced by just that process.Each cell has nine possible parents on the layer below it, and there are nine possible values for each cell. The rule is 81 digits long. The seeds, the rules, and the colors assigned to each value are all picked at random, and the results vary widely - but are usually uninteresting. The ones I showed were hand-picked from among thousands of attempts. I programmed them in POV-Ray's scene description language and let the software do the heavy lifting; I'd come back later and pick out the few I liked.
Next time, the fun stuff: how random rules applied to random seeds can produce things that look like they were consciously designed.
What shall I leave you with? How about another flower?